# Text as Causal Mediators: Research Design for Causal Estimates of
Differential Treatment of Social Groups via Language Aspects

Katherine A. Keith, Douglas Rice, and Brendan O’Connor
University of Massachusetts Amherst
kkeith@cs.umass.edu,drrice<EMAIL_ADDRESS>
###### Abstract

Using observed language to understand interpersonal interactions is important
in high-stakes decision making. We propose a causal research design for
observational (non-experimental) data to estimate the natural direct and
indirect effects of social group signals (e.g. race or gender) on speakers’
responses with separate aspects of language as causal mediators. We illustrate
the promises and challenges of this framework via a theoretical case study of
the effect of an advocate’s gender on interruptions from justices during U.S.
Supreme Court oral arguments. We also discuss challenges conceptualizing and
operationalizing causal variables such as gender and language that comprise of
many components, and we articulate technical open challenges such as temporal
dependence between language mediators in conversational settings.

##  1 Introduction

Interactions between individuals are key components of social structure Hinde
(1976). While we rarely have access to individuals’ internal thoughts during
these interactions, we often can observe the language they use. Using observed
language to better understand interpersonal interactions is important in high-
stakes decision making—for instance, judges’ decisions within the United
States legal system Danescu-Niculescu-Mizil et al. (2012) or police
interaction with citizens during traffic stops Voigt et al. (2017). In these
settings, analysts may be interested in understanding the behavior of decision
makers as individuals or at the subgroup or aggregate level.

Figure 1: Causal diagrams in which nodes are random variables and arrows
denote causal dependence for A. proposed general framework for _differential
treatment of social groups via language aspects_ and B. instantiation of the
framework for a theoretical case study of U.S. Supreme Court oral arguments.
In both diagrams, $T$ is the treatment variable, $Y$ is the outcome variable,
and $M$ are mediator variables. This is a simplified schema; see Fig. 2 for an
expanded diagram.

Important decision makers sometimes treat some social groups (e.g. women,
racial minorities, or ideological communities) differently than others Gleason
(2020). Yet, quantitative analyses of this problem often do not account for
all possible mechanisms that could induce this differential treatment. For
instance, one might ask, _During U.S. Supreme Court oral arguments, is a
justice interrupting female advocates more because of their gender, because of
the content of the advocates’ legal arguments, or because of the advocates’
language delivery_ (Fig. 1B)? Accounting for these language mechanisms could
help separate and estimate the remaining “gender bias” of justices.

We reformulate the previous question as a general _counterfactual_ query Pearl
(2009); Morgan and Winship (2015) about two speakers: _How would Speaker 2
respond if the signal they received of Speaker 1’s social group flipped from A
to B but Speaker 1 still used language typical of social group A?_ Here, our
question is about the direct causal effect of _treatment_ —Speaker 1’s
signaled social group—on _outcome_ —Speaker 2’s response—that is not through
the causal pathway of the _mediator_ —an aspect of language (Fig. 1A).111See
§4.2 for a discussion on when and how social groups (e.g. gender or race) can
be used as causal treatments.

The fundamental problem with this and any counterfactual question is that we
cannot go back in time and observe an individual counterfactual while holding
all other conditions the same Holland (1986). Furthermore, in many high-
stakes, real-world settings (e.g. the U.S. Supreme Court), we cannot run
experiments to randomly assign treatment and approximate these
counterfactuals. Instead, in these settings, causal estimation must rely on
_observational_ (non-experimental) data.

In this work, we focus on this observational setting and build from causal
mediation methods Pearl (2001); Imai et al. (2010); VanderWeele (2016) to
specify a research design of causal estimates of _differential treatment of
social groups via language aspects_. Other work has used causal mediation
analysis to better understand components of natural language processing (NLP)
models Vig et al. (2020); Finlayson et al. (2021). However, this work is more
closely aligned with studies that focus on causal estimation in which text is
one or more causal variables (e.g., Veitch et al., 2020; Roberts et al., 2020;
Keith et al., 2020; Zhang et al., 2020; Pryzant et al., 2021).

Our focus is on the research design, and we therefore intentionally do not
present empirical results. Instead, we discuss the potential promises and
challenges of this causal research design with both general examples and
concrete examples from a theoretical case study of U.S. Supreme Court
arguments. This aligns with Rubin (2008) who argues “design trumps analysis”
in observational studies and emphasizes the importance of conceptualizing a
study before any outcome data is analyzed.

Overall, we make the following contributions:

  * •

We propose a new causal research design to estimate the natural indirect and
direct effects of social group signal on speakers’ responses with separate
aspects of language as causal mediators (§3).

  * •

We illustrate the promises and challenges of this framework via a theoretical
case study of the effect of an advocate’s gender on interruptions by justices
during U.S. Supreme Court oral arguments. (§2).

  * •

We discuss challenges researchers might face conceptualizing and
operationalizing the causal variables in this research design (§4).

  * •

We directly address critiques of using social groups (e.g. race or gender) as
treatment and construct gender and language as _constitutive_ variables,
building from Sen and Wasow (2016); Hu and Kohler-Hausmann (2020) (§4.2 and
§4.4).

  * •

We articulate potential open challenges in this research design including
temporal dependence between mediators in conversations, causal dependence
between multiple language mediators, and dependence between social group
perception and language perception (§5).

_(A) Case: Kennedy v. Plan Administrator for DuPont Sav. and Investment Plan
(2008-07-636)_
---
Mark Irving Levy: […] The QDRO provision is an objective checklist that is
easy for – for plan administrators to follow.
Antonin Scalia: What if they had agreed to the waiver apart from […] We’d be
in the same suit that you’re - - that you say we have to avoid, wouldn’t we?
Mark Irving Levy: I don’t think so. I mean I think that would be an
alienation.
Antonin Scalia: Well, if it’s an alienation, but his point is that a waiver is
not an alienation.
_(B) Case: Lozano v. Montoya Alvarez (2013-12-820)_
Ann O’Connell Adams: Well - -
Antonin Scalia: I mean, it seems to me it just makes that article impossible
to apply consistently country to country.
Ann O’Connell Adams: \- - No, I don’t think so. And - - and, the other
signatories have - - have almost all, I mean I think the Hong Kong court does
say that it doesn’t have discretion, but it said in that case nevertheless it
would, even if it had discretion, it wouldn’t order the children returned. But
the other courts of signatory countries that have interpreted Article 12 have
all found a discretion, whether it be in Article 12 or in Article 8. And if I
- -
Antonin Scalia: Have they exercised it? Have they exercised it, that
discretion which they say is there?
Table 1:  Selected utterances from the oral arguments of two U.S. Supreme
Court cases, A Oyez (a) and B Oyez (b), with advocates Mark Irving Levy (male)
and Ann O’Connell Adams (female) respectively. Justice Antonin Scalia responds
to both advocates. Hedging language is highlighted in blue. Speech
disfluencies are highlighted in red. Gray-colored utterances directly proceed
the target utterances (non-gray colored) in the oral arguments.

##  2 Theoretical Case Study: Gender Bias in U.S. Supreme Court Interruptions

To motivate our causal research design and illustrate challenges that arise
with it, we focus on a specific theoretical case study—the effect of advocate
gender on justice interruptions via advocates’ language during United States
Supreme Court oral arguments (Fig. 1B). The substantive motivation for this
theoretical case study is built from previous work examining the role of
interruption and gender on the Court. Patton and Smith (2017) found female
lawyers are interrupted earlier in oral arguments, allowed to speak for less
time, and subjected to longer speeches by justices; Jacobi and Schweers (2017)
found female justices are interrupted at disproportionate rates by their male
colleagues; and Gleason (2020) found justices are more likely to vote for the
female advocate’s side when the female advocate uses emotional language.

Counterfactual questions. We present a novel causal approach to understanding
gender bias in Supreme Court oral arguments that corresponds to the following
counterfactual questions:

  1. 1.

_(NDE)_ : How would a justice’s interruptions of an advocate change if the
signal of the advocate’s gender the justice received flipped from male to
female, but the advocate still used language typical of a male advocate?

  2. 2.

_(NIE)_ : How would a justice’s interruptions of an advocate change if a male
advocate used language typical of a female advocate but the signal of the
advocate’s gender the justice received remained male?

which we show correspond to the _natural direct effect_ (NDE) and _natural
indirect effect_ (NIE) respectively in §3. In §4, we walk through the
theoretical conceptualization and empirical operationalization of advocate
gender (treatment), interruption (outcome), and advocate language (mediators).

Intuitive example. We describe intuitive challenges of our causal research
design by contrasting Examples A and B in Table 1. Levy—a male advocate—is not
interrupted by Justice Antonin Scalia, but Adams—a female advocate—is
interrupted (Oyez, a, b). _Why was the female advocate interrupted?_ _Was it
because of her gender or because of _what_ she said or _how_ she said it_? We
hypothesize one causal pathway between gender and interruption is through the
mediating variable hedging—expressions of deference or politeness.222Previous
work has shown hedging is used more often by women Lakoff (1973); Poos and
Simpson (2002), and we hypothesize judges might respond more positively to
more authoritative language (less hedging) from advocates. Suppose we
operationalize hedging as certain key phrases, e.g. “I don’t think so” and “I
mean I think.” An initial causal design might assign a binary hedging
indicator to utterances and then compare average interruption outcomes for
male and female advocates conditional on the hedging indicator.

However, advocate utterances matched on this hedging indicator could have a
number of latent mediators and confounders. In Table 1, Adams has speech
disfluencies (“and - - and” and “have - - have” shown in red) which might
cause Scalia to get frustrated and interrupt. The cases are from different
areas of the law,333The Supreme Court Database codes Ex. A as “economic
activity” and Ex. B as “civil rights” Spaeth et al. (2021). and Scalia may
interrupt more during cases that are in areas he has more personal interest.
The advocate utterance in Ex. B is longer (more tokens) and longer utterances
may be more likely to be interrupted. In Ex. B, Scalia interrupts Adams just
prior to the target utterance which possibly indicates a more “heated” portion
of the oral arguments during which interruptions occur more on average. With
these confounding and additional mediator challenges, a simple causal matching
approach (e.g. Stuart (2010); Roberts et al. (2020)) is unlikely to work and
we advocate for the causal estimation strategy presented in §3.4. We move from
this case study to a formalization of our causal research design in §3.

##  3 Causal Mediation Formalization, Identification, and Estimation

Many causal questions involve _mediators_ —variables on a causal path between
treatment and outcome. For example, what is the effect of gender444See §4 for
discussion of operationalizing difficult causal variables such as gender.
(treatment) on salary (outcome) with and without considering merit (a
mediator)? If one intervenes on treatment, then one would activate both the
“direct path” from gender to salary _and_ the “indirect path” from gender
through merit to salary. Thus, a major focus of causal mediation is specifying
conditions under which one can separate estimates of the _direct effect_ from
the _indirect effect_ —the former being the effect of treatment on outcome
_not_ through mediators and the later the effect through mediators.

We use this causal mediation approach to formally define our framework. For
each unit of analysis (see §4.1), $i$, let $T_{i}$ represent the treatment
variable—the social group, e.g. gender of an advocate—and $Y_{i}$ represent
the outcome variable—the second speaker’s response, e.g. a judge’s
interruption or non-interruption of an advocate. For each defined mediator
$j$, let $M^{j}_{i}$ represent the mediating variable—an aspect of language,
e.g. an advocate’s speech disfluencies or the topics of an utterance. Let
$X_{i}$ represent any other confounders between any combination of the other
variables.

We use the potential outcomes framework Rubin (1974) to define the natual
direct and indirect effects.555Pearl (2001) notes _do_ -notation cannot
represent causal mediation questions, since they concern counterfactual paths,
not interventions of variables. Let $M_{i}(t)$ represent the (counterfactual)
potential value the mediator would take if $T_{i}=t$. Then
$Y_{i}(t,M_{i}(t^{\prime}))$ is a doubly-nested counterfactual that represents
the potential outcome that results from both $T_{i}=t$ and potential value of
the mediator variable with $T_{i}=t^{\prime}$. With this formal notation, we
define the individual _natural direct effect (NDE)_ and _natural indirect
effect (NIE)_ :666Pearl et al. (2016) defines the NDE and NIE in terms of the
non-treatment condition, $T=0$. Others (e.g. Imai et al. (2010) and Van der
Laan and Rose (2011)) give alternate definitions of these quantities in terms
of $T=1$. We follow Pearl et al.’s definitions in the remainder of this work.

$\displaystyle\text{NDE}_{i}$
$\displaystyle=Y_{i}(1,M_{i}(0))-Y_{i}(0,M_{i}(0))$ (1)
$\displaystyle\text{NIE}_{i}$
$\displaystyle=Y_{i}(0,M_{i}(1))-Y_{i}(0,M_{i}(0))$ (2)

These correspond to the two counterfactual questions from §2 if $T_{i}=0$ and
$T_{i}=1$ represent the gender signal of the advocate being male and female
respectively.

###  3.1 Estimands

We second the advice of Lundberg et al. (2021) and recommend researchers
explicitly state their estimand of interest. As we briefly touch on in the
introduction, some studies may be interested in the estimand as the
_individual_ -level natural direct and indirect effects (Equations 1 and 2).
For example, a legal scholar may be interested in an individual U.S. Supreme
Court case and estimate the individual NIE and NDE for this single case in
order to evaluate how “fair” the case was with respect to the gender of an
advocate. Machine learning approaches to estimating individual-level causal
effects are promising Shalit et al. (2017) but may not be applicable to all
datasets. In contrast, more feasible—and potentially equally substantively
valid—estimands may be at the _subgroup_ level (e.g. effects of all cases
about civil rights or all cases for a particular justice) or aggregate level.
Here, the estimands are some kind of aggregation over Equations 1 and 2. Thus,
in Section 3.4, we provide estimators for general population-level (_not_
individual-level) estimands.

###  3.2 Interpretation of the NDE as “bias”

Many applications of causal mediation aim to quantify “implicit bias” or
“discrimination” via the natural direct effect. However, if all relevant
mediators are not accounted for, one cannot interpret the estimand of the
natural direct effect as the actual direct causal effect (Van der Laan and
Rose, 2011, p.135). Nevertheless, if we separate the total effect into the
proportion that is the NDE and the NIE with the mediators to which we have
access, our analysis moves _closer_ to estimating the true direct effect
between treatment and outcome. Thus, in this work we emphasize the value of
having interpretable mediators (i.e. language aspects) for which the NIE is a
meaningful quantity to analyze in itself.

###  3.3 Identification

Like any causal inference problem, we first examine the _identification
assumptions_ necessary to claim an estimate as causal. The key assumption
particular to causal mediation is that of _sequential ignorability_ Imai et
al. (2010):

  1. 1.

Potential outcomes and mediators are independent of treatment given
confounders

$\\{Y_{i}(t^{\prime},m),M_{i}(t)\\}\rotatebox[origin={c}]{90.0}{$\models$}T_{i}\mid
X_{i}=x$ (3)

  2. 2.

Potential outcomes are independent of mediators given treatment and
confounders

$Y_{i}(t^{\prime},m)\rotatebox[origin={c}]{90.0}{$\models$}M_{i}(t)\mid\\{T_{i}=t,X_{i}=x\\}$
(4)

for $t,t^{\prime}\in\\{0,1\\}$ and all values of $x$ and $m$.

_Mediator Independence Assumption:_ 777This is similar to the assumptions
Pryzant et al. (2021) make for linguistic properties of text as treatment. For
our particular framework, we make an additional assumption that for each
language aspect we study, the mediators are independent conditional on the
treatment and confounders

$\forall j,j^{\prime}:\
M^{j}_{i}(t)\rotatebox[origin={c}]{90.0}{$\models$}M^{j^{\prime}}_{i}(t)\mid\\{T_{i}=t,X_{i}=x\\}$
(5)

With this assumption, we can estimate the NIE and NDE of each mediator
successively, ignoring the existence of other mediators. Imai et al. (2010);
Tingley et al. (2014). We discuss the validity of this assumption in §5.

These assumptions correspond to the causal relationships of a graph similar to
Fig. 1, with the addition of confounder $X$ as a parent of all $T$, $M^{j}$,
and $Y$ (to be more precise may require a richer formalism; e.g. Richardson
and Robins (2013)).

###  3.4 Estimation

Given the satisfaction of sequential ignorability, mediator independence, and
other standard causal identification assumptions,888Overlap, SUTVA etc.; see
Morgan and Winship (2015). we propose using the following estimators of
population-level natural direct and indirect effects for each mediator $j$
Imai et al. (2010); Pearl et al. (2016):

$\displaystyle\begin{split}&\text{SA-NDE}^{j}=\\\
&\frac{1}{N}\sum_{i=1}^{N}\sum_{x\in\mathcal{X}}\sum_{m\in\mathcal{M}^{j}}\bigg{(}\hat{f}^{j}(Y|M_{i}^{j}=m,T_{i}=1,X_{i}=x)\\\
&-\hat{f}^{j}(Y|M_{i}^{j}=m,T_{i}=0,X_{i}=x)\bigg{)}\hat{g}^{j}(m|T_{i}=0,X_{i}=x)\\\
\end{split}$ (6) $\displaystyle\begin{split}&\text{SA-NIE}^{j}=\\\
&\frac{1}{N}\sum_{i=1}^{N}\sum_{x\in\mathcal{X}}\sum_{m\in\mathcal{M}^{j}}\hat{f}^{j}(Y|M^{j}_{i}=m,T_{i}=0,X_{i}=x)\\\
&\bigg{(}\hat{g}^{j}(m|T_{i}=1,X_{i}=x)-\hat{g}^{j}(m|T_{i}=0,X_{i}=x)\bigg{)}\end{split}$
(7)

Each is a Sample Average estimate from $N$ data points, relying on models
trained to predict mediator and outcome given confounders and treatment:
$\hat{g}^{j}$ infers mediator $j$’s probability distribution, while
$\hat{f}^{j}$ infers the expected outcome conditional on mediator $j$. The
estimators marginalize over confounders and mediators from their respective
domains ($x\in\mathcal{X}$, $m\in\mathcal{M}^{j}$), which for our discrete
variables is feasible with explicit sums (see Imai et al. for the continuous
case).

Model fitting. When fitting models $\hat{f}$ and $\hat{g}$, we recommend using
a cross-sample or cross-validation approach in which one part of the sample is
used for training/estimation ($S_{\text{train}}$) and the other is used for
testing/inference ($S_{\text{test}}$) in order to avoid overfitting
Chernozhukov et al. (2017); Egami et al. (2018). With text, one must also fit
a model for the mediators conditional on text, $h(m|\text{text})$ using
$S_{\text{train}}$. In some cases, such as measuring advocate speech
disfluencies, $h$ may be a simple deterministic function. However, when using
NLP and other probabilistic models (e.g topic models or embeddings), $h$ could
be a difficult function to fit and have a certain amount of measurement error.
A major open question is whether to jointly fit $h$ and $g$ at training time
as advocated by previous work Veitch et al. (2020); Roberts et al. (2020) or
if $h$ and $g$ should be treated as separate modules. At inference time, we do
not use the inference text from $S_{\text{test}}$ since Eqns. LABEL:eqn:nde-
estimate and LABEL:eqn:nie-estimate only rely on the mediators through
estimates from $\hat{g}$.

##  4 Conceptualization and Operationalization of Causal Variables

For any causal research design—and particularly those in the social
sciences—there are often challenges _conceptualizing_ the theoretical causal
variables of interest. Even after these theoretical concepts are made
concrete, there are often multiple ways to _operationalize_ these concepts. We
discuss conceptual and operational issues for our both our general research
design and our theoretical case study. In particular, we recommend researchers
formalize variables such as gender and language as _constitutive_ variables
made of multiple components (Fig. 2) as per Hu and Kohler-Hausmann (2020), or
Sen and Wasow (2016)’s “bundle of sticks.”

###  4.1 Unit of analysis

As with most causal research designs, one starts by conceptualizing the _unit
of analysis_ —the smallest unit about which one wants to make counterfactual
inquiries. In our framework, the _unit of analysis_ is a certain amount of
language ($L$) between speakers of two categories: the first category of
speakers, $P_{1}$, are those belonging to a group of interest (e.g. advocates)
for which treatment values (e.g. female and male) will be assigned; and the
second, $P_{2}$, is the set of decision-makers responding to the first
speakers (e.g. judges).

Operationalizations. There are several possible operationalizations of $L$:
pairs of single utterances—whenever a person from $P_{1}$ speaks and a person
from $P_{2}$ responds; a thread of several utterances between persons from
$P_{1}$ and $P_{2}$ within a conversation; or the entire conversation between
persons from $P_{1}$ and $P_{2}$. In §5, we note that selecting the unit of
language could have implications for modeling temporal dependence between
mediators.

###  4.2 Treatment

At the most basic level, _treatment_ , $T$, in our research design is _the
social group_ of persons in $P_{1}$ (Fig. 1). However, inspired by the _causal
consistency_ arguments from Hernán (2016),999 _Consistency_ is the condition
that for observed outcome $Y$ and treatment $T$, the potential outcome equals
the observed outcome, $Y(t)=Y$ for each individual with $T=t$. Hernán (2016)
presents eight versions of treatment for the causal question “Does water
kill?” to illustrate the deceptiveness of this apparently simple consistency
condition. Hernán points out that “declaring a version of treatment
sufficiently well-defined is a matter of agreement among experts based on the
available substantive knowledge” and is inherently (and frustratingly)
subjective.  we examine several competing versions of treatment for our
theoretical case study of U.S. Supreme Court oral arguments and explain the
reasons we eventually choose version #5 (in bold):

  1. 1.

Do judges interrupt at different rates based on an advocate’s _gender_?

  2. 2.

Based on an advocate’s _biological sex assigned at birth_?

  3. 3.

An advocate’s _perceived gender_?

  4. 4.

An advocate’s _gender signal_?

  5. 5.

An advocate’s _gender signal_ as defined by (hypothetical) manipulations of
the advocate’s clothes, hair, name, and voice pitch?

  6. 6.

An advocate’s _gender signal_ by (hypothetical) manipulations of their entire
physical appearance, facial features, name, and voice pitch?

  7. 7.

An advocate’s _gender signal_ by setting their physical appearance, facial
features, name, and voice pitch to specific values (e.g. all facial features
set to that of the same 40-year-old, white female and clothes set to a black
blazer and pants).

In critique of treatment version #1, most social groups (e.g. gender or race)
reflect highly contextual social constructs Sen and Wasow (2016); Kohler-
Hausmann (2018); Hanna et al. (2020). For gender in particular, researchers
have shown social, institutional, and cultural forces shape gender and gender
perceptions Deaux (1985); West and Zimmerman (1987), and thus viewing gender
as a binary “treatment” in which individuals can be randomly assigned is
methodologically flawed. In critique of version #2, _biological sex assigned
at birth_ is a characteristic that is not manipulable by researchers and the
“at birth” timing of treatment assignment means all other variables about the
individual are post-treatment. Thus, researchers have warned against
estimating the causal effects of these kinds of “immutable characteristics”
Berk et al. (2005); Holland (2008).

Greiner and Rubin (2011) propose overcoming the issues in versions #1 and #2
by shifting the unit of analysis to the _perceived gender_ of the decision-
maker (#3) and defining treatment assignment as the moment the decision-maker
first perceives the social group of the other individual. Hu and Kohler-
Hausmann (2020) critique this _perceived gender_ variable and emphasize that
we, as researchers, cannot actually change the internal, psychological state
of decision-makers, but rather we can change the _signal_ about race or gender
those decision-makers receive (#4). However, as Sen and Wasow (2016) discuss,
defining treatment as the _gender signal_ (#4) is dismissive of the many
components that make up a social construct like gender. Instead, Sen and Wasow
recommend articulating the specific variables one would potentially
manipulate. For _gender_ in our case study, this could mean hypothetical
manipulations of an advocate’s dress, name, and voice pitch (#5).

Shifting from versions #5 to #6 and #7, we define treatment in terms of more
specific manipulations. However, we also enter the realm of Hernán’s argument
that precisely defining the treatment never ends, and some aspects of #6 and
#7 are impossible to manipulate in real-world settings such as the U.S.
Supreme Court. What does it mean to manipulate an advocate’s “entire physical
appearance?”101010Would justices have to interact with advocates through a
computer-mediated system in which one could customize avatars of the
advocates? We note, using computer-mediated avatars to signal social group
identity has been used effectively in other causal studies, e.g. Munger
(2017). When we define treatment very specifically—e.g. using the same 40-year
old white woman as the treatment for “female advocate” (#7)—are we estimating
a causal effect of gender _in general_? Thus, we back-off from versions #6 and
#7, and advocate using #5 as our definition of treatment.

Figure 2: _Constitutive_ causal diagram for gendered interruption in U.S.
Supreme Court oral arguments. Latent theoretical concepts are unshaded circles
and observed operationalizations (measurements) of concepts are shaded
circles. We provide alternative operationalizations in the text. The causal
variables _gender_ and _language_ are represented as dashed lines around their
constituent parts, building from the arguments of Sen and Wasow (2016); Hu and
Kohler-Hausmann (2020). The shaded portion of _gender_ consists of the gender
variables that one could potentially manipulate in a hypothetical
intervention.

Constitutive causal diagrams. With these considerations, drawing a causal
diagram in which a _gender_ is represented as a single node seems flawed.
Instead, building from Sen and Wasow (2016) and Hu and Kohler-Hausmann (2020),
we represent treatment (the social group) as cloud of components (a
_constitutive_ variable), some of which are latent, some observable, and some
manipulable. In Fig. 2, we shade the “outward” components of _gender_ —hair,
appearance, clothes, voice pitch, and name—that are our hypothetical
manipulations and would influence the latent variable of a judge’s perceived
gender of the advocate. Other “background” components of gender—gender norms,
education, and socialization—are the components that could causally influence
language.

Case study operationalizations. Even after selecting version #5 as our
conceptualization of treatment, there are still multiple operationalizations
for our theoretical case study:

Treatment operationalization 1: Previous work operationalizes gender in
Supreme Court oral arguments by using norm that the Chief Justice introduces
an advocate as “Ms.” and “Mr." before their first speaking turn Patton and
Smith (2017); Gleason (2020). The advantage of this operationalization is that
it is simple, clean, and consistent, and occurs directly before an advocate’s
first utterance.111111The treatment assignment timing is potentially important
for the rest of the causal diagram. If we can define _gender signal_ and thus
latent _perceived gender_ as happening right before an advocate first speaks,
and it is not adapted or updated by the judge over the course of the oral
arguments, then we can eliminate the causal arrow between variables “language”
and “perceived gender.”

Treatment operationalization 2: Alternatively, one could focus on even more
specific components of gender for (hypothetical) manipulations. For instance,
Chen et al. (2016) and Chen et al. (2019) measure voice pitch when studying
gender on the U.S. Supreme Court. While being more cumbersome to measure, this
operationalizes gender as a real-valued (instead of binary) variable and thus
potentially measures more subtle gender biases.

###  4.3 Outcome

In our general framework, we define the _outcome_ , $Y$, as _the response of
the second speaker_ (Fig. 1A), and we intentionally leave this variable vague
and domain-specific. However, if making the leap from _differential treatment_
to claiming _discrimination_ or _bias_ , conceptualizing a causal outcome
requires normative commitments and a moral theory of what is harmful Kohler-
Hausmann (2018); Blodgett et al. (2020). In our case study, we conceptualize
the outcome variable as a judge interrupting an advocate. This outcome is of
substantive interest because, in general, interruptions can indicate and
reinforce status in conversation Mendelberg et al. (2014), and, specifically
to the U.S. Supreme Court, justice’s behavior in oral arguments has been
connected to case outcomes.

Outcome operationalization 1: Previous work uses the transcription norm of a
double-dash (“- -”) at the end of a advocate utterance when a justice
interrupts in the next utterance Patton and Smith (2017). However, the
validity of this operationalization relies on consistent transcription
standards.

Outcome operationalization 2: An alternative operationalization could classify
interruptions into positive (agreeing with the first speaker’s comment),
negative (disagreeing, raising an objection, or completely changing the
topic), or neutral categories Stromer-Galley (2007); Mendelberg et al. (2014).
While estimating the effects of only negative interruptions could further
refine the causal question— _Do justices negatively interrupt female advocates
more?_ —this operationalization could also introduce measurement error since
it could prove difficult difficult to design an accurate NLP classifier for
this task.

###  4.4 Language Mediators

Our framework explicit focuses on _language as a mediator_ in differential
treatment of social groups. Yet, language consists of multiple levels of
linguistic structure Bender (2013); Bender and Lascarides (2019), so as with
social groups (§4.2), it is a variable that is non-modular and we believe it
should be represented as constituent parts (Fig. 2).

Mediator Operationalizations: We focus on three potential language aspects for
our Supreme Court case study: (A) _hedging_ —expressions of deference or
politeness—with an operationalization as lexical matches from a single-word
hedging dictionary (e.g. Prokofieva and Hirschberg (2014)); (B) _speech
disfluencies_ —repetitions of syllables, words, or phrases—which we
operationalize as the transcript noting a repeated unigram with a double dash,
“ _word_ \- - _word_ ”; and (C) semantic _topics_ operationalized as a topic
model Blei et al. (2003) applied to utterances.

Recommendations. We discuss the choice of these particular language aspects,
$M^{j}$, for our case study as well as general recommendations for researchers
operationalizing language as a mediator.

  * •

Is $M^{j}$ _interpretable_? Is there a _hypothetical manipulation_ 121212To be
precise, the _controlled direct effect_ is the estimand in which the mediator
is manipulated, $do(M)$ Pearl (2001). In contrast, the _natural_ direct and
indirect effects are counterfactuals on paths. However, we still find thinking
through potential manipulations is helpful in refining the conceptualization
of a language aspect. of $M^{j}$? In contrast to prior work that treats
language as a black-box in causal mediation estimates Veitch et al. (2020), we
advocate for using interpretable aspects of language. If language mediators
are interpretable, then the NIE is both meaningful (see §3.2) and potentially
more fine-grained (we can estimate an NIE for each aspect of language that we
are studying instead of a black-box approach that lumps all text into one
effect). Furthermore, since identification is essential to claiming an
estimate is causal and identification can only be verified qualitatively and
through domain expertise, interpretable text mediators will be much easier to
evaluate.

  * •

Is there _substantive theory_ for causal pathways $T\rightarrow M^{j}$ and
from $M^{j}\rightarrow Y$? Without such theory, studying certain aspects of
language is not meaningful. For example, see §2 for our theoretical reasoning
about the causal dependence between gender, hedging, and interruption.

  * •

To what extent does one expect _measurement error_ of $M^{j}$ when using
automatic NLP tools? Our operationalizations of hedging lexicons and speech
disfluencies are deterministic; however, topic model inferences are
probabilistic and sensitive to changes in hyperparameters and pre-processing
decisions Schofield et al. (2017); Denny and Spirling (2018). These kinds of
measurement errors are still open questions although there is recent work that
examines measurement error when text is treatment Wood-Doughty et al. (2018).

  * •

Is $M^{j}$ _causally independent_ from other measured language aspects,
$M^{j^{\prime}}$? If not, our proposed estimator from §3.4 is invalid. Thus,
one must scrutinise which aspects of language are separable and thus able to
be included in the causal analysis—e.g. we could include content (topics)
versus delivery (speech disfluencies) since one could hypothetically modify
one without affecting the other. We discuss this assumption further in §5.

###  4.5 Non-language Mediators

Returning to §3.2, there is often a tendency to interpret the NDE as something
like “pure” _gender bias_ —What is the effect of gender on interruption when
all other possible causal pathways are stripped away? Conceptualizing and
operationalizing language aspects as mediators (§4.4) moves the NDE towards
the desired “gender bias.” However, there may be other mediator pathways that
explain these effects. For example, in our case-study, two additional
mediators of interest are advocate ideology (e.g. liberal or conservative) and
the level of “eliteness” of the advocate’s law firm. A major validity issue is
the _causal independence_ of these mediators from the language mediators. For
instance, ideology could influence certain aspects of language (topic), and
“eliteness” of the advocate’s law firm could be a proxy for level of training
which could influence the advocate’s delivery.

##  5 Challenges and Threats to Validity

We discuss additional challenges and threats to validity for our research
design that should be addressed before implementing the design and claiming
the estimates from the design are causal.

Temporal dependence of utterances. So far, we have assumed the “units of
analysis” of text are independent (§4.1). However, previous utterances in a
conversation often influence the target utterances. For our case study, if
Judge A interrupted Advocate B often in $t^{\prime}<t$, interruption at $t$ is
more likely (the two speakers are possibly in a “heated” part of the
conversation) and Advocate B’s speech disfluencies at $t$ are also more likely
(the advocate could be mentally fatigued). Potential avenues forward include
changing the unit of analysis to the entire conversational thread between the
two target speakers or building extensions to the multiple mediator
literature, i.e. Imai and Yamamoto (2013); VanderWeele and Vansteelandt
(2014); Daniel et al. (2015); VanderWeele (2016).

Dependence between multiple language mediators. Our framework assumes one can
computationally separate aspects of language.131313This assumption is made in
other NLP applications such as style transfer or machine translation
Prabhumoye et al. (2018); Li et al. (2018); Hovy et al. (2020). However, some
sociolinguists argue aspects of language such as “style” cannot be separated
from “content” because style originates in the content of people’s lives and
different ways of speaking signal socially meaningful differences in content
Eckert (2008); Blodgett (2021). If our mediator independence assumption (Eqn.
5) is violated, then we would have to turn to alternate estimation strategies
to deal with this dependence.

Dependence between social group perception and linguistic perception.
Separating the direct and indirect causal paths in our framework relies on
there being a _decision-maker’s latent perception of social group_ variable on
the direct path between treatment and outcome and that this variable is
independent from a _decision-maker’s latent perception of language_ variable
on the indirect path from treatment through mediators to outcome. However,
“indexical inversion” considers “how language ideologies associated with
social categories produce the perception of linguistic signs” Inoue (2006);
Rosa and Flores (2017). Suppose Judge A perceives Advocate B as female, then
Judge A might perceive Advocate B’s language as more feminine even if it is
linguistically identical to language used by male advocates. Furthermore,
latent gender perception and latent language perception might interact in
affecting the outcome through mechanisms such as rewarding “conforming to
gender norms”—an advocate who is perceived as a man might get penalized for
using feminine language whereas an advocate perceived as a woman might get
rewarded, e.g. Gleason (2020).

##  6 Conclusion

In this work, we specify a causal research design for _differential treatment
of social groups with language as a mediator_. We believe this research design
is important for studying the direct and indirect causal effects in high-
stakes decision making such as gender bias in the United States Supreme Court.
Separating the indirect effect of treatment on outcome through interpretable
language aspects allows us to estimate counterfactual queries about
differential treatment when speakers use and do not use the same language.
Despite open theoretical and technical challenges, we remain optimistic that
researchers can build upon this framework and continue to improve our
understanding of decision makers’ differential treatment of social groups.

## Acknowledgments

The authors thank Abe Handler, Su Lin Blodgett, Sam Witty, David Jensen, and
anonymous reviewers from the First Workshop on Causal Inference & NLP for
helpful comments. KK gratefully acknowledges funding from a Bloomberg Data
Science Fellowship.

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# Size Dependence in Flux-Flow Hall Effect using Time Dependent Ginzburg-
Landau Equations

Vineet Punyamoorty  Department of Electrical Engineering, Indian Institute of
Technology Bombay, Mumbai 400076, India  Aditya Malusare  Department of
Physics and Astronomy, Purdue University, West Lafayette, IN 47907  Shamashis
Sengupta  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France
Sumiran Pujari  Department of Physics, Indian Institute of Technology Bombay,
Mumbai 400076, India  Kasturi Saha  Department of Electrical Engineering,
Indian Institute of Technology Bombay, Mumbai 400076, India
<EMAIL_ADDRESS>
(2 February 2021)

###### Abstract

We study the Hall effect in square, planar type-II superconductors using
numerical simulations of time dependent Ginzburg-Landau (TDGL) equations. The
Hall field in some type-II superconductors displays sign-change behavior at
some magnetic fields due to the induced field of vortex flow, when its
contribution is strong enough to reverse the field direction. In this work, we
use modified TDGL equations which couple an externally applied current, and
also incorporate normal-state and flux-flow Hall effects. We obtain the
profile of Hall angle as a function of applied magnetic field for four
different sizes $(l\times l)$ of the superconductor:
$l/\xi\in\\{3,5,15,20\\}$. We obtain vastly different profiles for each size,
proving that size is an important parameter that determines Hall behavior. We
find that electric field dynamics provides an insight into several anomalous
features including sign-change of Hall angle, and leads us to the precise
transient behavior of order parameter responsible for them.

##  I Introduction

One of the most interesting aspects of superconducting systems is the physics
of vortices. Advances in experimental methods have made it possible to probe
type-II superconductors at small lengthscales where the behaviour of
individual vortices becomes visible. Vortices have been imaged with electron
spins in diamonds [1, 2], scanning superconducting quantum-interference
devices (SQUIDs) [3, 4] and Hall-probe magnetometry [5, 6]. The behaviour of a
superconducting system is expected to vary significantly if its dimensions are
reduced to be comparable to the coherence length. In this work, we
theoretically investigate the effect of finite size on the properties of
vortices with numerical simulations of solutions of the time dependent
Ginzburg-Landau (TDGL) equations. The flux-flow Hall effect in square planar
type-II superconductors are studied for different sample sizes, given by
$l/\xi$ ratio of 3, 5, 15 and 20 (where $l$ is the length of the square and
$\xi$ is the superconducting coherence length). It will be seen that the
electric field and hall angle profiles under a unidirectional current are
widely different depending upon the dimension of the superconductor.

Hall effect in superconductors has been observed to display anomalous
properties below the critical temperature, most significant of which is the
sign-reversal of Hall voltage at certain magnetic fields [7, 8, 9]. This could
not be explained by either the phenomenological models of vortex flow of the
time, namely the Bardeen-Stephen (BS) model [10] and the Nozières-Vinen (NV)
model [11], or by the microscopic theory [12]. It was later proposed that the
induced electric field of magnetic vortex flow could contribute to the Hall
field and cause anomalous behavior [9]. Dorsey [12] and Kopnin et al. [13]
proved using analytical approximations of a modified time dependent Ginzburg-
Landau (TDGL) system, that indeed sign reversal of Hall effect is possible
under some circumstances. These theories make use of microscopic quantities
(related to the electronic structure) to define the regimes of sign-reversal.
Alternatively, one may numerically compute the Hall effect in a
superconducting sample governed by the modified TDGL system of Dorsey [12] and
Kopnin et al. [13], and find the magnetic field regimes of sign-reversal. This
could provide insights into Hall effect behaviour of a superconductor, as a
function of macroscopic quantities alone (e.g. GL parameter $\kappa$, sample
size, etc.).

In this work, we follow the alternative numerical route mentioned above, which
is direct and does not resort to any analytical approximations. We first use
standard time dependent Ginzburg-Landau (TDGL) equations [14, 15, 16] to
numerically compute the time-varying order parameter of a planar
superconductor in the vortex state. We benchmark our simulations by comparing
the numerically obtained fluxoid value of each vortex against $\Phi_{o}$, the
superconducting flux quantum, as a rigorous test (III.1). With these
benchmarks in hand, we simulate a modified TDGL system that includes an
externally applied current to probe the Hall effect in these systems.
Importantly, we go beyond existing literature by incorporating the normal-
state Hall conductivity and flux-flow terms into the dynamical equations that
we numerically simulate. In this, we have taken inspiration from the
analytical works of Dorsey [12] and Kopnin et al. [13] (III.2). Next, in III.3
we study the resultant changes in flux-flow and resultant induced electric
fields based on the numerical simulation of our modified TDGL system. We
compute the Hall angle profile for various sizes of the superconductor, and
find vastly different profiles. We find that transient electric fields and
related order parameter behavior give us good insight into explaining the
anomalous Hall behavior.

##  II Theoretical Model

The TDGL equations are [16, 14, 15]:

$\gamma\left(\hbar\frac{\partial\psi}{\partial
t}+ie_{s}\Phi\psi\right)+\frac{1}{2m_{s}}\left(i\hbar\nabla+\frac{e_{s}}{c}\mathbf{A}\right)^{2}\psi+\alpha\psi\\\
+\beta|\psi|^{2}\psi=0$ (1a)
$\nu\left(\frac{1}{c}\frac{\partial\mathbf{A}}{\partial
t}+\nabla\Phi\right)+\frac{ie_{s}\hbar}{2m_{s}}\left(\psi^{*}\nabla\psi-\psi\nabla\psi^{*}\right)+\frac{e_{s}^{2}}{m_{s}c}|\psi|^{2}\mathbf{A}\\\
+\frac{c}{4\pi}\nabla\times\nabla\times\mathbf{A}=0$ (1b)

where, $\psi$ is the complex-valued order parameter, $\alpha$ and $\beta$ are
the phenomenological parameters of Ginzburg-Landau theory [17, 18] and $\phi$,
$\mathbf{A}$ are the electric and magnetic potentials respectively. $\nu$ is
the normal-state conductivity and $\gamma$ is a relaxation constant for the
order parameter. The charge and mass of Cooper pairs are denoted by $e_{s}=2e$
and $m_{s}=2m_{e}$ respectively.

###  II.1 Gauge invariance and normalization

For an arbitrary function $\chi(x,y,t)$ with well-defined spatial and time
derivatives, the gauge invariance can be expressed as ($\psi\rightarrow\psi
e^{i\kappa\chi}$, $\mathbf{A}\rightarrow\mathbf{A}+\nabla\chi$,
$\phi\rightarrow\phi-\frac{\partial\chi}{\partial t}$) . We choose the zero
electric potential gauge (i.e. $\phi=0$) [15, 19, 14]. The physical quantities
in (1b) are renormalized as follows:

$\displaystyle\frac{x}{\xi}\rightarrow x,\hskip
8.00003pt\frac{t}{\gamma\hbar/(-\alpha)}\rightarrow t,\hskip
8.00003pt\frac{\mathbf{A}}{\sqrt{2}H_{c}\xi}\rightarrow\mathbf{A},\hskip
8.00003pt\frac{\psi}{\sqrt{-\alpha/\beta}}\rightarrow\psi$ (2)

where $H_{c}=\sqrt{\frac{4\pi\alpha^{2}}{\beta}}$ and $\xi$ is the GL
coherence length [19]. In the chosen gauge, the resulting dimensionless
equations applicable over the superconducting domain are:

$\frac{\partial\psi}{\partial
t}+\left(i\nabla+\frac{1}{\kappa}\mathbf{A}\right)^{2}\psi-\left(1-|\psi|^{2}\right)\psi=0$
(3a) $\sigma\frac{\partial\mathbf{A}}{\partial
t}+\nabla\times\nabla\times\mathbf{A}+\frac{i}{2\kappa}\left(\psi^{*}\nabla\psi-\psi\nabla\psi^{*}\right)+\\\
\frac{1}{\kappa^{2}}|\psi|^{2}\mathbf{A}=0$ (3b)

where, $\kappa$ is the GL parameter and $\sigma$ is normalized normal-state
conductivity. When the sample is surrounded by vacuum on all sides and no
external current is passed, the following boundary conditions (BCs) apply [20,
14, 19, 21]:

$\displaystyle\nabla\times\mathbf{A}$ $\displaystyle=\mathbf{H}_{\text{ext}},$
$\displaystyle\left(i\nabla+\frac{1}{\kappa}\mathbf{A}\right)\psi\cdot\mathbf{n}$
$\displaystyle=0,$ $\displaystyle-\sigma\frac{\partial\mathbf{A}}{\partial
t}\cdot\mathbf{n}$ $\displaystyle=0$ (4)

(where $\mathbf{H}_{\text{ext}}$ is the externally applied magnetic field,
perpendicular to the sample). The first condition imposes continuity of
transverse magnetic field across the boundary, while the second and third
ensure that neither supercurrent nor normal current crosses the boundary,
respectively.

In principle, one also needs to solve Maxwell’s equation
$\nabla\times\nabla\times\mathbf{A}=0$ (considering vacuum) in the surrounding
domain [19]. However, in the case of two-dimensional samples (i.e. infinitely
long cylinders) with perpendicularly applied magnetic field, as in our system,
solving the “interior problem” (3) alone is sufficient to a good approximation
[15, 22]. For reference, the “full problem” and its boundary conditions are
discussed in Refs. [14] and [19].

###  II.2 Inclusion of externally applied current

Including an externally applied transport current in the TDGL system entails
two tasks: (a) accounting for the magnetic field induced by the transport
current and (b) modifying the boundary conditions (4) to account for flow of
normal current across boundaries. The former can be achieved by modifying the
first boundary condition to $\nabla\times\mathbf{A}=\mathbf{H}_{\text{tot}}$
where $\mathbf{H_{\text{tot}}}=\mathbf{H}_{\text{ext}}+\mathbf{H}_{c}$ is the
sum of applied ($\mathbf{H}_{\text{ext}}$) and induced ($\mathbf{H}_{c}$)
magnetic fields [20]. $\mathbf{H}_{c}$ is to be computed from the current
profile (a function of $\psi$, $\mathbf{A}$), making the system self-
consistent. However, an approximation is frequently used in literature [23,
24, 25, 26, 20] to simplify the computation of $\mathbf{H}_{c}$: the current
profile is assumed to be a uniform band, which reduces $\mathbf{H}_{c}$ to a
simple expression involving $\mathbf{J}_{a}$, the applied (uniform) current
density. In this paper, $\mathbf{J}_{a}$ is assumed to be in the
$+\mathbf{\hat{x}}$ direction (fig. 1), which gives us $\mathbf{H}_{c}$ along
all four boundaries as $\mathbf{H}_{c}^{\text{top, bottom}}=\pm
WJ_{a}/2\leavevmode\nobreak\ \mathbf{\hat{z}}$ (where $W$ is the length of the
sample along $\mathbf{\hat{y}}$) and $\mathbf{H}_{c}^{\text{left, right}}$
vary linearly between the bottom and top edges.

Secondly, to account for flow of normal current across the boundary, the
“vacuum-superconductor” BCs (4) are now replaced by “metal-superconductor” BCs
(5b) [26, 20] at the left and right edges, while retaining the former BCs (5a)
at the top and bottom edges:

$\displaystyle\nabla\times\mathbf{A}=\mathbf{H}_{\text{tot}},\quad\left(i\nabla+\frac{1}{\kappa}\mathbf{A}\right)\psi\cdot\mathbf{n}=0,\quad-\sigma\frac{\partial\mathbf{A}}{\partial
t}\cdot\mathbf{n}=0$ (5a)
$\displaystyle\nabla\times\mathbf{A}=\mathbf{H}_{\text{tot}},\qquad\psi=0,\qquad-\sigma\frac{\partial\mathbf{A}}{\partial
t}\cdot\mathbf{n}=\mathbf{J}_{a}\cdot\mathbf{n}$ (5b)

In (5b), the third BC accounts for flow of current across the boundary, and
the second condition ensures that the density of superconducting electrons is
zero at the edges. This ensures that the injected normal current transitions
to supercurrent gradually, rather than abruptly, inside the superconductor.

Figure 1: Schematic representation of our simulation domain when an external
current is applied – $l\times l$ square superconductor with metallic contacts
on the left and right, and vacuum at top and bottom. Current $\mathbf{J}_{a}$
is applied along $\hat{\mathbf{x}}$ and magnetic field
$\mathbf{H}_{\text{ext}}$ along $\hat{\mathbf{z}}$.

###  II.3 Inclusion of Hall effect

Normal state hall effect is a result of the conduction of electrons transverse
to applied electric field. This can be incorporated in the TDGL system (3),
(5) by rewriting the normal-state conductivity $\sigma$ as a tensor:

$\sigma=\begin{pmatrix}\sigma_{xx}&\sigma_{xy}\\\
\sigma_{yx}&\sigma_{yy}\end{pmatrix}$ (6)

We assume an isotropic sample ($\sigma_{xx}=\sigma_{yy}$), and by symmetry
$\sigma_{yx}=-\sigma_{xy}$. In order to determine $\sigma_{xx}$ and
$\sigma_{xy}$, we use the following as the model for the normal-state
conductivity [27, 12]:

$\displaystyle\sigma_{xx}$
$\displaystyle=\sigma_{0}\frac{1}{1+\omega_{c}^{2}\tau^{2}},$
$\displaystyle\sigma_{xy}$
$\displaystyle=\sigma_{0}\frac{\omega_{c}\tau}{1+\omega_{c}^{2}\tau^{2}}$ (7)

where $\omega_{c}$ is the cyclotron frequency
$e_{s}\left(\nabla\times\mathbf{A}\right)/m_{s}$ and $\tau$ is the electron
scattering time. Under typical conditions, we have $\omega_{c}\tau\ll 1$ (low-
field limit) [27, 12]. Thus, $\sigma_{xx}\approx\sigma_{0}$ and
$\sigma_{xy}\approx\omega_{c}\tau\sigma_{0}$. Due to the spatially varying
magnetic field, cyclotron frequency $\omega_{c}$ and consequently,
$\sigma_{xy}$ are also spatially varying. In order to enforce the low-field
limit, we take $\omega_{c}\tau=10^{-2}\left(\nabla\times\mathbf{A}\right)$,
where the pre-factor of $10^{-2}$ ensures that $\omega_{c}\tau\ll 1$.

Josephson [28] proved that macroscopically, a vortex moving at velocity
$\mathbf{v}_{L}$ gives rise to an induced electric field
$\mathbf{E}=-\frac{1}{c}\mathbf{v}_{L}\times\mathbf{H}$,where all the
quantities are spatially and temporally averaged [9, 12, 13]. In the Bardeen-
Stephen (BS) model [10], under an applied current $\mathbf{J}_{a}$, vortices
experience a Lorentz force $\sim\mathbf{J}_{a}\times\mathbf{H}$ [12, 9]. This
force gives rise to velocity $\mathbf{v}_{L}$ along
$\mathbf{J}_{a}\times\mathbf{H}$ and in a system such as ours (fig. 1), this
results in vortex motion along $-\hat{\mathbf{y}}$. This velocity
$\mathbf{v}_{L}$ along $-\hat{\mathbf{y}}$ produces a field
$\mathbf{E}=-\frac{1}{c}\mathbf{v}_{L}\times\mathbf{H}$ along
$\hat{\mathbf{x}}$, the same direction as $\mathbf{J}_{a}$, thereby causing
dissipation. However, if we were to have an additional component of
$\mathbf{v}_{L}$ along $\hat{\mathbf{x}}$ (the same direction as
$\mathbf{J}_{a}$), this would create a field contribution in the Hall
direction. Dorsey [12] and Kopnin et al. [13] proved that adding a non-zero
imaginary part to the relaxation parameter $\gamma$ in the TDGL system (1b)
produces such flux-flow contribution to the Hall field by lending the vortices
a velocity component parallel to the applied current. Thus, we write
$\gamma=\gamma_{1}+i\gamma_{2}$. In the microscopic picture, the value of
$\gamma_{2}/\gamma_{1}$ has been shown to depend upon the electronic structure
of the material [12, 13]. The sign of $\gamma_{2}/\gamma_{1}$ determines
whether vortices travel along $\mathbf{J}_{a}$ or against, and therefore
crucially affects the sign of flux-flow contribution to Hall field. Kopnin et
al. [13] proved that a sign reversal in Hall effect would be observed for
negative values of $\gamma_{2}/\gamma_{1}$.

##  III Results and discussion

We use COMSOL Multiphysics® [29], a commercial finite element tool to
numerically simulate TDGL equations (3)–(5). Throughout the paper, we take GL
parameter $\kappa=2$ and normalized normal-state conductivity $\sigma_{0}=1$.
In section III.1, we obtain a vortex state solution and propose a procedure to
rigorously verify the solution to help identify any numerical errors or
artifacts. In sections III.2–III.3, we study a system with externally applied
current and Hall effect included. Throughout these simulations, we apply an
external current $\mathbf{J}_{a}=0.04\thinspace\hat{\mathbf{x}}$. We consider
four geometries: $l\times l$ square superconductors for
$l/\xi\in\\{3,5,15,20\\}$. We apply an external magnetic field
$\mathbf{H}_{\text{ext}}$ along $\hat{\mathbf{z}}$, whose value is swept
between 0 and $H_{c2}=\kappa=2$ in steps of 0.05. For each combination of size
$l$ and $\mathbf{H}_{\text{ext}}$, we solve the modified TDGL system (3), (5)
and compute the time-varying order parameter $\psi\left(\mathbf{r},t\right)$
and vector potential $\mathbf{A}\left(\mathbf{r},t\right)$. This gives us a
complete insight into the dynamic vortex motion and electric fields
$E_{x}=-\partial{A}_{x}/\partial t$ and $E_{y}=-\partial{A}_{y}/\partial t$,
which form the basis for much of our analysis.

Since we use normalized units throughout the paper, we compute the order of
magnitude for these units to understand the typical physical values. We chose
$\kappa=2$, which broadly corresponds to Niobium [30]. Using the physical
parameters of Nb, we compute the time units as
$\sim${10}^{-12}\text{\,}\mathrm{s}$$ [31], current density units as
$\sim$1\text{\,}\mathrm{A}\text{\,}{\mathrm{m}}^{-1}$$ [32, 33] and the
applied magnetic field units as $\sim 1$ kOe [33].

###  III.1 Vortex state and verification

We first solve TDGL equations (3) on a $20\xi\times 20\xi$ planar sample (GL
parameter $\kappa=2$) with no applied current or Hall effect enabled. Thus, in
this case we use the vacuum-superconductor BCs (4) on all four sides, and
observe a vortex state solution at
$\mathbf{H}_{\text{ext}}=0.9\hat{\mathbf{z}}$ (fig. 2).

Figure 2: $|\psi|^{2}$ at various instants of time, depicting major events in
the formation of a vortex state in the $20\xi\times 20\xi$ sample. Applied
magnetic field is $|\mathbf{H}_{\text{ext}}|=0.9$.

We perform a thorough verification of our simulation as described in the
following. First, we confirm that the simulation results are stable with
respect to mesh size. We then sweep $|\mathbf{H}_{\text{ext}}|$ widely and
observe the existence of both upper and lower critical fields marked by the
vanishing of vortices, as expected from a type-II superconductor
($\kappa>{1}/{\sqrt{2}}$). We also observe that for the same $\kappa$, the
number of vortices increases (decreases) when the external field
$|\mathbf{H}_{\text{ext}}|$ is increased (decreased). This is an expected
qualitative behaviour of the superconductor to let more (less) incident flux
pass through the superconductor. One of the hallmarks of superconducting
vortices under Ginzburg-Landau theory is fluxoid quantization. It states that
the fluxoid value $\Phi^{\prime}$ associated with each vortex is quantized by
$\Phi_{0}=\frac{hc}{2e}$ [18].

$\Phi^{\prime}=\Phi+\frac{4\pi}{c}\oint\lambda^{2}\mathbf{J}_{s}\cdot dl$ (8)

where $\Phi$ is the magnetic flux $\int\mathbf{B}\cdot d\mathbf{s}$ associated
with the vortex, and the integral term involving super current
$\mathbf{J}_{s}$ and penetration depth $\lambda$ is performed on a closed
contour enclosing the vortex. We compute the fluxoid value, and obtain
$\Phi^{\prime}\approx 0.96\Phi_{0}$ for each vortex, confirming that our
simulations firmly uphold fluxoid quantization (fig. 3).

Figure 3: We verify that the fluxoid value of each of our vortices is
$\approx\Phi_{0}$. To compute the fluxoid value we take a circular area
centred at the vortex-centre defined by the local maxima in magnetic field,
with a radius $2.4\xi$ (shown with a dotted hatch on the top-left vortex).
Figure shown for $|\mathbf{H}_{\text{ext}}|=0.9$.

###  III.2 Flux flow under applied current with Hall effect incorporated

In the Bardeen-Stephen (BS) model [10], magnetic vortices experience a Lorentz
force under applied external current along $\mathbf{J}_{a}\times{\mathbf{H}}$
[9, 12]. We apply an external current
$\mathbf{J}_{a}=0.04\thinspace\hat{\mathbf{x}}$ and $\mathbf{H}_{\text{ext}}$
along $+\hat{\mathbf{z}}$, and accordingly observe vortex flow along
$-\hat{\mathbf{y}}$ with vortices entering at the top edge and leaving at the
bottom (fig. 4). Consistent with the BS model, we observe faster (slower)
motion of vortices with increased (decreased) magnitudes of $\mathbf{J}_{a}$.
Choosing a larger value of $|\mathbf{J}_{a}|$ would increase the magnitude of
current-induced magnetic field $\mathbf{H}_{c}$, driving the superconductor
into normal state (after some time) for even low values of applied field
$\mathbf{H}_{\text{ext}}$. With our chosen value of $|\mathbf{J}_{a}|=0.04$,
we find that the system is ultimately driven into normal state for all values
of $|\mathbf{H}_{\text{ext}}|\geq 0.35$ in the case of sizes
$l/\xi\in\\{15,20\\}$. For the other two smaller sizes, the system is driven
into normal state for all values of $|\mathbf{H}_{\text{ext}}|$.

For sizes $l/\xi\in\\{15,20\\}$, we observe vortices and their movement,
before the system eventually goes into normal state
($\psi\left(\mathbf{r}\right)=0$), whereas for sizes $l/\xi\in\\{3,5\\}$, we
do not observe vortices for any value of $|\mathbf{H}_{\text{ext}}|$. To
understand further, we solve the standard TDGL system (3), (4) (with no
applied current) for $l/\xi\in\\{3,5\\}$ and sweep the magnetic field
$|\mathbf{H}_{\text{ext}}|$ as earlier. We do not find a vortex state solution
for any value of $|\mathbf{H}_{\text{ext}}|$. We conclude that this is a size-
effect: the system is smaller than a critical size, forbidding the possibility
of a vortex state solution.

Figure 4: We observe vortex motion under an applied external current in the
transverse direction. Vortices enter at the top edge and leave at the bottom.
This motion is followed by the superconductor being driven into normal state.
($|\mathbf{H}_{\text{ext}}|=0.5$)

Next, we also enable normal-state and flux-flow Hall effects following the
discussion in II.3, with $\gamma_{2}/\gamma_{1}=0.4$ and $-0.4$ as separate
cases. The choice of $|\gamma_{2}/\gamma_{1}|$ was made in such a way that the
vortices gain a perceivable amount of velocity in the direction of
$\mathbf{J}_{a}$, but not significantly enough, so that vortices still
primarily move along $-\hat{\mathbf{y}}$. With Hall effect enabled, we observe
complex vortex motion (fig. 5). Vortices enter the system at the top edge and
traverse smooth but irregular trajectories through the domain, due to complex
vortex-vortex and vortex-boundary interactions. The key difference is that
they also obtain velocity in the $\pm\hat{\mathbf{x}}$ direction, unlike when
Hall effect is not enabled.

Figure 5: Complex vortex flow under an applied external current with Hall
effect enabled. The setting in of normal state from the right edge starting at
$t=1500$ induces a gradual transient field largely affecting Hall behavior
(fig. 8). ($|\mathbf{H}_{\text{ext}}|=0.5$ and $\gamma_{2}/\gamma_{1}=0.4$)

We seek to capture the effect of this complex motion on the longitudinal and
Hall electric fields and explain the observed behavior.

###  III.3 Analysis of electric fields and Hall angle

We obtain the spatially averaged electric field by averaging across the entire
domain. We find that the so obtained $\mathbf{E}$ profiles can be grouped into
three distinct types (fig. 6) of behaviour based on the $l/\xi$ ratio and
value of $|\mathbf{H}_{\text{ext}}|$. First, at a large $l/\xi$ of 20 (fig.
6a), the longitudinal field saturates to $\sigma_{0}J_{a}$ (normal state) with
the observation of a series of spikes prior to that. This spiking occurs due
to vortex entry and exit, as discussed in Ref. [23]. The Hall field $E_{y}$
also exhibits similar spiking as a result of the vortex velocity component in
the direction of $\mathbf{J}_{a}$. Such behaviour occurs for sizes
$l/\xi\in\\{15,20\\}$ and applied field $|\mathbf{H}_{\text{ext}}|\geq 0.35$.
For fields lower than $0.35$, the system rapidly evolves to the
superconducting Meissner state with $E_{x}$ reaching zero as shown in fig. 6b
for $|\mathbf{H}_{\text{ext}}|$ of 0.2. Sizes $l/\xi\in\\{3,5\\}$ are always
driven into normal state. This results in the third type, with $E_{x}$
saturating to $\sigma_{0}J_{a}$ (fig.6c). These differences in electric field
behavior produce significantly different Hall effect profiles, which we
address next.

(a)

(b)

(c)

Figure 6: The three different types of electric field profiles observed in our
simulations across all sizes and magnitudes of applied field
$|\mathbf{H}_{\text{ext}}|$. (a) Saturates to $\sigma_{0}J_{a}$, exhibits
spiking behavior corresponding to entry and exit of vortices from sample of
size $l/\xi=20$, and $|\mathbf{H}_{\text{ext}}|=0.8$. (b) Saturates to $0$, of
sample size $l/\xi=20$, and $|\mathbf{H}_{\text{ext}}|=0.2$). (c) Saturates to
$\sigma_{0}J_{a}$, for sample size $l/\xi=5$, and
$|\mathbf{H}_{\text{ext}}|=1.2$.

We characterize the Hall effect using Hall angle, the ratio of effective
transverse to longitudinal conductivity:
$\tan{\theta_{H}}=\sigma_{xy}^{\prime}/\sigma_{xx}^{\prime}$ [12, 13]. These
macroscopic effective conductivties are marked with a prime to distinguish
them from the normal-state quantities $\sigma_{xx}$ and $\sigma_{xy}$ (7).
From the macroscopic equation $\mathbf{J}_{a}=\sigma^{\prime}\cdot\mathbf{E}$,
we have:

$\displaystyle\begin{pmatrix}J_{a}\\\ 0\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\sigma_{xx}^{\prime}&\sigma_{xy}^{\prime}\\\
-\sigma_{xy}^{\prime}&\sigma_{xx}^{\prime}\end{pmatrix}\cdot\begin{pmatrix}E_{x}\\\
E_{y}\end{pmatrix}$ (9)

Thus, we get
$\tan{\theta_{H}}=\sigma_{xy}^{\prime}/\sigma_{xx}^{\prime}=E_{y}/E_{x}$,
where the fields are both spatially and temporally averaged. We obtain the
tangent of Hall angle $\tan{\theta_{H}}$ for each combination of applied
magnetic field $|\mathbf{H}_{\text{ext}}|$ and size $l/\xi\in\\{3,5,15,20\\}$
(fig. 7). We first note that the Hall angle profiles resulting from our choice
of $|\gamma_{2}/\gamma_{1}|$ closely resemble the experimental data with
respect to orders of magnitude ($\tan{\theta_{H}}\sim 10^{-2}$) [8, 34].

Figure 7: (a)Hall angle profiles for $\gamma_{2}/\gamma_{1}=+0.4$.(b) Hall
angle profiles for $\gamma_{2}/\gamma_{1}=-0.4$. Evidently, size is an
important parameter in determining Hall behavior, with the smallest size
$l/\xi=3$ behaving identical to a normal conductor. Sign-reversal is only seen
for $l/\xi=20$, in which case $\gamma_{2}/\gamma_{1}=0.4$ and $-0.4$ show
opposite behavior. Fields were averaged from $t=0$ to $11000$.

We find that as $|\mathbf{H}_{\text{ext}}|\rightarrow H_{c2}$, when the
superconductor is rapidly driven into normal state, Hall angle varies linearly
with $|\mathbf{H}_{\text{ext}}|$, as in the case of normal metals. In this
regime, flux-flow contribution is negligible and all sizes
$l/\xi\in\\{3,5,15,20\\}$ have the same profile. This is in agreement with
expected behavior because we do not have any size-effect for normal metals.
When $|\mathbf{H}_{\text{ext}}|\approx 0$, the Hall angle approaches 0. At
these low fields, sizes $l/\xi\in\\{3,5\\}$ are driven to normal-state. As
normal metals, they have negligible Hall fields at
$|\mathbf{H}_{\text{ext}}|\approx 0$, and consequently a very small Hall
angle. On the other hand, sizes $l/\xi\in\\{15,20\\}$ are in the
superconducting state at these low fields, and therefore have negligible
$E_{y}$ and $E_{x}$. The small non-zero contribution is a result of transient
behavior as seen in fig. 6b. Thus, although the Hall angle is small for both
groups of sizes at $|\mathbf{H}_{\text{ext}}|\approx 0$, the underlying states
are different.

We next note that the Hall angle profile is completely linear for size
$l/\xi=3$. This is a result of it behaving as a normal metal with no vortex
state, and electric field $E_{x}\left(t\right)$ saturating to
$\sigma_{0}J_{a}$ for all values of $|\mathbf{H}_{\text{ext}}|$ (third type
shown in fig. 6c). However, size $l/\xi=5$ also exhibits identical behavior,
but we find significant deviations from linearity. To understand further, we
look at the difference in transient fields between the two sizes. Large
transients in $E_{x}\left(t\right)$ are common for size $l/\xi=5$, such as the
one shown in fig. 6c, whereas the transient is highly suppressed in size
$l/\xi=3$. In fact, across all values of $|\mathbf{H}_{\text{ext}}|$, the
maximum percentage change in $E_{x}\left(t\right)$ (relative to saturation
value $\sigma_{0}J_{a}$) is limited to $\approx 0.02\%$ for size $l/\xi=3$,
and to $\approx 760\%$ in the case of size $l/\xi=5$. We conclude that this
enormous difference in transient levels leads to deviations from linearity in
size $l/\xi=5$, in spite of non-superconducting behavior.

We next look at sign-reversal of Hall angle, a key anomaly in the Hall
behavior of some superconductors. For size $l/\xi=20$, we observe a region of
sharp deviation from linear, “normal-conductor” behavior for intermediate
values of applied field, i.e. around $|\mathbf{H}_{\text{ext}}|\approx 0.5$
(fig. 7). This is the region where flux-flow contribution to the Hall effect
is most significant, because for higher fields, normal state sets in rapidly,
and for lower fields, there are no vortices (Meissner state). We find that
strong, negative flux-flow contribution leads to sign-reversal for
$\gamma_{2}/\gamma_{1}=-0.4$, and positive contribution leads to a peak for
$\gamma_{2}/\gamma_{1}=0.4$, in agreement with Kopnin et al. [13]. In order to
understand the precise transient behavior leading to this vast difference, we
look at the spatially averaged fields $E_{x}$ and $E_{y}$ as functions of time
(for $|\mathbf{H}_{\text{ext}}|=0.5$). First, we find that for both
$\gamma_{2}/\gamma_{1}=0.4$ and $-0.4$, $E_{x}$ has an identical profile. This
is expected because $\gamma_{2}$ only influences the motion of vortices along
$\mathbf{J}_{a}$, thus affecting only $E_{y}$ (ref. II.3). On the other hand,
we find significant difference in $E_{y}\left(t\right)$ profiles (fig. 8). It
is evident that the difference in Hall behavior between
$\gamma_{2}/\gamma_{1}=0.4$ and $-0.4$ can be attributed almost entirely to
the transients starting at $t\approx 1500$. Although spiking behavior results
in peaks in the opposite direction, they have negligible contribution to the
average. Instead, the gradual transient starting at $t\approx 1500$ leads to
different profiles for $\gamma_{2}/\gamma_{1}=0.4$ and $-0.4$, giving a strong
positive and negative contribution to Hall field, respectively. We try to
relate this transient with order parameter (fig. 5) in order to determine the
precise behavior causing such transients. We find that this transient is the
induced field resulting from variation in $\mathbf{A}$ during the onset of
normal state (fig. 5). This onset occurs from the right boundary for
$\gamma_{2}/\gamma_{1}=0.4$ (as seen in fig. 5), and from the left for
$\gamma_{2}/\gamma_{1}=-0.4$ (fig. 8). For both $\gamma_{2}/\gamma_{1}=0.4$
and $-0.4$, this transient leads ultimately to saturation of $E_{y}$ to the
same, small positive value. This is the non-flux-flow, or normal-state
contribution to the Hall field.

Figure 8: $E_{y}$ fields for size $l/\xi=20$ at
$|\mathbf{H}_{\text{ext}}|=0.5$

The gradual transients starting at $t\approx 1500$ are responsible for the
vastly different Hall angle for $\gamma_{2}/\gamma_{1}=0.4$ and $-0.4$.
Similar transients are seen for $|\mathbf{H}_{\text{ext}}|\approx 0.5$,
leading to the Hall angle profile in fig. 7.

Interestingly, we find that although size $l/\xi=15$ exhibits vortex behavior
similar to $l/\xi=20$, we do not find sign-reversal in Hall angle. This is due
to the suppressed gradual transient fields in $l/\xi=15$, leading to a much
weaker contribution of flux-flow Hall effect. Therefore, it is evident that
along with $\gamma_{2}$ (whose value depends on the electronic structure) and
several other parameters, size alters Hall behavior significantly, adding to
the reasoning behind the observation of a diverse variety of Hall angle
profiles in various materials.

In summary, we have simulated the anomalous Hall effect using the modified
TDGL equations in COMSOL Multiphysics and shown that the solutions provide
insights into the precise temporal dynamics of transient fields and vortex
behavior that scale with the sample size. We have explored theoretically how
features of the anomalous Hall effect evolve with a variation of the linear
dimensions when the lengths are only a few times the coherence length. The
Hall effect behaviour predicted by these simulations may be probed with
advanced experimental techniques that have already been applied to image
vortices [1, 2, 35, 3, 4, 5, 6]. Such studies would be important for
understanding finite-size effects in superconductors and their evolution at
microscopic lengthscales.

## Acknowledgments

K.S. acknowledges financial support from IITB-IRCC Seed grant number
17IRCCSG009, DST Inspire Faculty Fellowship - DST/INSPIRE/04/2016/002284 and
AOARD Grant No. FA2386-19-1-4042. SP acknowledges financial support from IRCC,
IIT Bombay (17IRCCSG011) and SERB, DST, India (SRG/2019/001419).

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