Spatio-temporal variation of conversational utterances on Twitter

Christian M. Alis, May T. Lim∗

National Institute of Physics, University of the Philippines, Diliman, 1101
Quezon City, Philippines

$\ast$ E-mail<EMAIL_ADDRESS>
## Abstract

Conversations reflect the existing norms of a language. Previously, we found
that utterance lengths in English fictional conversations in books and movies
have shortened over a period of 200 years. In this work, we show that this
shortening occurs even for a brief period of 3 years (September 2009-December
2012) using 229 million utterances from Twitter. Furthermore, the subset of
geographically-tagged tweets from the United States show an inverse proportion
between utterance lengths and the state-level percentage of the Black
population. We argue that shortening of utterances can be explained by the
increasing usage of jargon including coined words.

## Introduction

Utterances, the speaking turns in a conversation, relay short bits of
information. Though utterances adapt strongly to medium [1], utterance
shortening has been observed over a span of two centuries. Here we show that
utterances in the online social medium Twitter did not only significantly
shorten in a span of a few years but also varied geographically—providing
evidence of increasing usage of jargon brought about by formation of groups.
Our use of Twitter conversations provided us with a large, highly resolved and
current dataset.

Twitter (twitter.com) is an online social medium that allows its users to post
messages (tweets) of up to 140 characters in length, which are public by
default. Previous studies [2, 3] on Twitter conversations focused on modelling
the structure of conversations rather than the form of utterances. Recent
studies have ranged from characterizing the graph of the Twitter social
network [4, 5] to inferring the mood of the population [6, 7, 8, 9]. Owing to
the large number of Twitter users (about half-billion in June 2012 [10]) and
easy access via the provided application programming interfaces (API), Twitter
has become a platform for studying the usage of the English language. For
example, it has been found that longer Twitter messages (tweets) are more
likely to be credible [11] and, by determining where tweets were posted,
dialects [12] and geographical diffusion of new words [13] are observable.

Conversations in Twitter are typically performed in one of two ways:
privately, using direct messages; or publicly, using replies. Replies [14] are
tweets that begin with the username of the recipient prefixed with an at (@)
sign, for example, @bob Hello! How are you?. Since replies may be viewed by
other users aside from the recipient, replies are used for public
conversations [15] akin to having conversations while other people are
listening.

Conversation analysis usually investigates the structure of conversations [16]
by looking at the interaction of utterances instead of the individual
utterances themselves. Since we are more interested with the encoding of
information or idea into an utterance, this paper focused instead on the
construction of individual utterances and not in their interaction. More
specifically, the length of utterances are measured because the production
time and the amount of information of an utterance should be correlated with
its length.

Sentence lengths are not as widely studied as words, and conversational
utterances less so. The study of sentence lengths began with the work of Udny
Yule [17] in 1939 and eventually led to the discovery that sentence length
distributions may be approximated by a gamma distribution [18]. On the other
hand, the mean length of utterance is used to evaluate the level of language
development of children [19, 20]. The length of sentences and utterances are
usually measured in terms of words or morphemes but we used the number of
characters (orthographic length) as unit of length because Twitter imposes a
maximum tweet length in terms of characters. The use of orthographic length of
sentences has previously been shown to be a valid unit when comparing
utterance length distributions [1]. Furthermore, the orthographic length of
words is highly correlated with word length in terms of syllables [21].

In this paper, 229 million conversational utterances collected from 18
September 2009 to 14 December 2012 are first characterized. By comparing
expected (fitted) and empirical utterance length distributions, we show that
the character limit of tweets has little to no effect on the median utterance
length. We also identify some factors that significantly affect the utterance
length. The dataset is then disaggregated to reveal that utterances shortened
in a span of more than three years. Possible mechanisms of shortening are then
explored. Finally, the variation of utterance length across different US
states and its correlation with demographic and socioeconomic variables are
investigated.

## Results

### Aggregate utterance length distribution

The utterance length distribution (ULD) of the entire data set (Fig. 1A) is
bimodal and can be fitted with a gamma distribution after taking the
140-character limit into account [1]. It is bimodal due to the mixture of the
natural (unconstrained) ULD and shortened (constrained) ULD forced by the
140-character limit. To estimate the unconstrained ULD, a generalized gamma
distribution,

$\mathrm{Pr}(x)=\frac{\tilde{x}^{\alpha-1}e^{-\tilde{x}}}{{\Gamma(\alpha)}},$
(1)

where ̵̃$x$ is the utterance length, $\tilde{x}=(x-x_{0})/s$ is the scaled
utterance length, and $\alpha$, $x_{0}$ and $s$ are fitting parameters that
describe the shape, translation and ordinate scaling factor, respectively, was
fitted on the utterance length distribution from $x=1$ char. to a cut-off
length $x=x_{c}$ using least squares as was done in Ref. [1]. The estimated
natural ULD ($\alpha=1.46$, $x_{0}=1.01$ char., $s=30.0$ char.) fits the
empirical ULD with an $r^{2}=0.950$.

Both empirical and fitted (unconstrained) ULD are skewed to the right and the
quartiles (Q1=25th percentile; Q2=median=50th percentile; Q3=75th percentile)
are either the same (Q1=19 char., Q2=36 char.) or differs by 3 characters
(Q3${}_{\text{empirical}}=65$ char., Q3${}_{\text{fitted}}=62$ char.). From
here on, we used the quartiles of the empirical ULD to describe the
distributions.

Starting 10 October 2011, all URLs in tweets are automatically shortened by
Twitter [22] into a 20-character URL (http://t.co/xxxxxxxx) and this caused
the spike at $x=20$ char. in the ULD. The spike at $x=26$ char. is due to non-
English tweets while the spike at $x=3$ char. is due to the acronym LOL
(laughing out loud). Restricting utterances to English and removing URLs and
LOL result to a smoother ULD (Fig. 1B) but with the same quartiles as the
original distribution.

### Temporal dependence of utterance lengths

Utterance length distributions for tweets aggregated over a 24-hour period
that were sampled during Fridays follow the general characteristics of the
utterance length distribution for the entire dataset as shown by the
representative utterance length distributions in Fig. 2. The right peak of the
plots seems to get smaller and shifted to the left as the date becomes more
recent, suggesting shortening of utterances over time. This shortening is
clearly shown when the quartiles are plotted with respect to time (Fig. 3A).
The quartiles roughly follow their corresponding regression line except for 26
Nov 2010, which shows an unexpected spike due to spam.

As expected the linear regression line of the median (2nd quartile) is not at
the middle of Q1 and Q3 regression lines because the utterance length
distributions are skewed. The regression line of Q1 (Table 1 and Fig. 4, all)
is less steep than the regression line for the median, which, in turn, is less
steep than the regression line for Q3. The shortening of utterances is,
therefore, mostly due to the decreased occurrence of longer utterance lengths
rather than the shifting of the whole utterance length distributions to the
left.

Table 1 and Fig. 4 demonstrate the robustness of the decrease in utterance
length. The months included in the dataset differ for each year yet shortening
is still observed even if only utterances from the common included months of
September to December are considered (Table 1 and Fig. 4, Sep–Dec). Similarly,
the number of utterances per day and the percent of public data collected are
not constant throughout the entire dataset. To remove any size effects on the
results, $10^{5}$ utterances, an amount slightly smaller than the smallest
daily sample size, were sampled without replacement for each day (Fig. 3B) but
the same observations remained (Table 1 and Fig. 4, resampled). Another
possible reason for the shortening is the increased usage of link shorteners.
However, the shortening trend (Table 1 and Fig. 4, URLs removed) persisted
even if all links in the utterances were removed (Fig. 3C). Finally,
restricting the analysis to only English tweets (Fig. 3D) resulted to the same
observations (Table 1 and Fig. 4, English only).

### Possible mechanisms for shortening

A possible mechanism for utterance length shortening is the shortening of the
most frequent words either by a change in orthography (spelling) of the most
frequent words or their replacement by shorter words.

The median word length of all words is 4 characters (Fig. 5A) for all years
from 2009 to 2012. Although the median length of the 1000 most frequently used
words from 2009 to 2012 is constant at 4 characters (Fig. 5B), the peak (mode)
moved from 4 characters in 2009 to 3 characters (Fig. 5C) in the succeeding
years. Based on Kruskal-Wallis tests, the word length distributions of the
1000 most frequently used words for 2010–2012 are not significantly different
($H=1.112$, $p=0.5734$) with each other but are significantly different with
the distribution for 2009 ($H=10.31$, $p=0.0161$). However, the observed
shortening is not just due to a sudden shortening of the 1000 most frequently
occurring words from 2009 to 2010 because it was still observed in 2010–2012
(Table 1 and Fig. 4, 2010–2012)

From $60.32\pm 0.0185$% in 2009, the relative occurrence of the 1000 most
frequently used words (Fig. 5D) with respect to all words decreased to
$52.80\pm 0.0267$% in 2012. In that same timespan, the median utterance length
in words decreased from 8 words to 5 words (Fig. 5E) while the median tweet
length in words (Fig. 5F) decreased from 10 words to 8 words.

A topic is a word, usually in the form of #topic, or a phrase that is
contained in a tweet. Trending topics are the most prominent topics being
talked about in Twitter within a period of time. The shortening of trending
topics could potentially explain the observed shortening of utterances but
instead of decreasing, the median length of trending topics increased from 11
characters in 2009 to 13 characters in 2012 (Fig. 5G). Utterances about a
trending topic are shortening but the $r^{2}$-values (Table 1 and Fig. 4,
trending topics) are too small to cause the observed shortening of utterances.

The shortening of utterances is a global phenomenon and is not restricted to
the US since utterances that were geolocated outside the US also exhibited
shortening (Table 1 and Fig. 4, outside US). It was previously observed in
utterances from movies and books [1] albeit at a rate 1 and 3 orders of
magnitude smaller (-0.266 char./year in books; -0.001897 char./year in
movies), respectively. Although conversations do tend to get shorter in time,
our current findings show that it is occurring faster now on Twitter.

### Geographical variation of utterance lengths

Out of the 229 million utterances, only 795,048 utterances (0.347%) have
geographic information pointing to one of the US states (utterances-byloc.txt
in SI). The number of geolocated utterances per US state is strongly
correlated ($r^{2}=0.944$) with the 2010 census population of the US state and
ranges from 396 utterances in Wyoming to 96,120 utterances in California. The
medians are not correlated with the number of utterances ($r^{2}=0.104$)
although resampling to 300 utterances, a slightly smaller number of tweets
than the smallest sample size, resulted to changes in the quartiles, unlike in
the previous section where resampling did not change the quartiles for almost
all days after resampling.

To estimate how the quartiles change, the quartiles were bootstrapped using
$10^{4}$ repetitions but the bootstrapped values (Fig. 6A) turned out to be
the same as the empirical values. The spread in the bootstrapped medians is
very small that the interquartile range (IQR=Q3-Q1) of 40% of the bootstrapped
medians is zero. Any difference, therefore, in the median between two US
states is almost guaranteed to be significant. Both Kruskal-Wallis H-test
($H=8011$, $p<10^{-3}$) [23] and pairwise Mann-Whitney U-test [24] on the
empirical ULD of each US state conclude that not all ULD of the US states are
the same.

Plotting the medians over a US map (Fig. 6B) suggests southeastern and eastern
US states tend to have shorter utterance lengths. This clustering of
neighboring US states is very tenous, however, since pairwise Mann-Whitney
U-tests on the median utterance length of each US state yielded non-
neighboring US state pairings.

To check for possible correlates, the bootstrapped median utterance length was
regressed with demographic and socioeconomic information available in the
United States Census Bureau State and Country QuickFacts [25] (Table 2). Out
of the 51 variables (listed in SI Text S1), only the percent Black resident
population (latest data from 2011, $r^{2}=0.685$) and percent Black-owned
firms (latest data from 2007, $r^{2}=0.613$) have $r^{2}>0.5$. A detailed
description of both variables are in SI Text S1. The two variables are
strongly correlated though ($r^{2}=0.947$) so the correlation of the
bootstrapped median is really with the percentage distribution of Black
residents. The bootstrapped median is inversely proportional to the Black
resident population (Fig. 7A). Restricting utterances to English and removing
URLs improved the correlation to $r^{2}=0.707$.

For comparison, the median utterance length was also plotted against the
percent of persons 25 years and over who are high school graduates or higher
from 2007 to 2011 (Fig. 7B) and median household income from 2007 to 2011 in
thousands of dollars (Fig. 7C), which are both described in detail in SI Text
S1. Both variables are uncorrelated or only slightly correlated, at best, with
the median utterance length because the values of the coefficient of
determination are $r^{2}=0.397$ and $r^{2}=0.068$, respectively.

### Multivariate regression of median utterance length

We explored the possible dependence of the median utterance length on several
variables by considering linear combinations of the QuickFacts variables. To
ease the comparison of variable effect size and to avoid numerical problems,
the variables were standardized by subtracting the sample mean for the
variable then dividing by the sample standard deviation for the variable. That
is, the $z$-scores of the variables were considered in the multiple
regression. Aside from standardization, no other transformation e.g., power
transformation, was performed on the variables.

The parameter estimates of the linear model (Model 2) with percent Black
resident population $B$, percent high school graduates $H$, and median
household income $I$ as predictors are shown in Table
LABEL:tab:multiregression. Only the coefficient for $B$ is significantly
different from zero and its magnitude is 5 to 10 times larger than the
coefficients for $H$ and $I$. Further supported by an F-test ($F=2.87$, df =
2, $p=0.067$, $\alpha=0.05$), the three-variable model can be simplified into
the one-variable model.

Performing a stepwise regression
($\alpha_{\text{in}}=\alpha_{\text{out}}=0.05$) with race, educational
attainment and income QuickFacts variables as candidate predictors will yield
a two-variable model, Model 3. The variable $B$ is still included in the model
and the magnitude of its coefficient (-3.30) is about 4.5 times that of the
other predictor (0.73), percent of persons 25 years and over who are holders
of bachelor’s degree or higher from 2007 to 2011 (denoted as variable $C$).
The $r^{2}$ value using only $C$ as predictor is 0.093. The two-variable model
cannot be reduced to a single-variable model with either $B$ ($F=5.01$, df =
2, $p=0.030$, $\alpha=0.05$) or $C$ ($F=2.87$, df = 2, $p=0.067$,
$\alpha=0.05$) as the only predictor. The two-variable model improved $r^{2}$
by 0.03 or 4.3% from that of the single-variable model with $B$ as the only
predictor.

Expanding the set of candidate variables to all QuickFacts variables then
performing another stepwise regression
($\alpha_{\text{in}}=\alpha_{\text{out}}=0.05$) results to a five-variable
model, Model 4. Both variables $B$ and $C$ are included in the model with $B$
still having the largest coefficient magnitude. By adding three more
predictors to Model 3, $r^{2}$ increased by 0.121 or 16.9%. The adjusted
$r^{2}$ of Model 4 is larger by 0.114 or 16.2% than the two-variable model
and, since the latter is not equivalent to the former ($F=10.8$, df = 3,
$p<10^{-3}$, $\alpha=0.05$), the former can be considered as a better model
despite having more predictors. Model 4 suggests that shorter utterances are
correlated with US states having larger percentage of Blacks and lower
percentage of bachelor’s degree holders but has more owner-owned houses,
larger manufacturing output and less dense population.

The values of QuickFacts variables are regularly updated by the US Census
Bureau and only the most recent values are retained. By looking up each
variable in the source dataset, one can reconstruct the QuickFacts for
previous years (up to 2010). Repeating the regression analysis for the
different models using the data for previous years resulted to coefficient
estimates that are within the standard errors of the quoted variables above.
Thus, the coefficients remained essentially the same from 2010 to 2012.

## Discussion

The observation of geographic variability is not entirely unexpected because
of the existence of dialects. What is more surprising is that the utterance
length is (anti)correlated with the resident Black population. This factor
also dominates other predictors when combined with other demographic and
socioeconomic factors using multiple regression. A possible explanation is
that Blacks converse more distinctly and more characteristically than other
racial groups. Since utterances are only weakly correlated with median income
and educational attainment then perhaps the shorter utterance lengths is a
characteristic of their race—perhaps pointing towards the controversial
language of Ebonics [26]. The strong correlation does not imply causality, and
it is beyond the scope of this work to look for actual evidence of Ebonics in
the tweets.

Results show that people are communicating with fewer and shorter words. The
principles of least effort communications [27] provide us with two possible
implications. If the information content of each word remains the same then
the information content of each utterance is lesser and more utterances are
needed to deliver the same amount of information—a phenomenon that could be
verified by tracking the complete conversations between individuals, and not
just samples as we are doing now. On the other hand, if the amount of
information content of each utterance remains the same then encoding becomes
either more efficient (comprehension remains the same) or more ambiguous
through time. When ambiguity increases, speaker effort is minimized at the
expense of listener effort.

Based on anecdotal evidence, replies broken into several tweets are not more
frequent than before but shorter spelling and omission of words do seem to be
more prevalent. That is, encoding appears to become more efficient without
sacrificing as much precision.

The shortening, it seems, can be explained by increased usage of jargon, which
in turn provides evidence of segregation into groups. People who are engaged
in a conversation communicate using a shared context, which may utilize a more
specialized lexicon (jargon or even coined words). Although utterances are
expected to be less clear due to the use of fewer words, the use of context
prevents this from happening. The decrease in the frequency of words from
60.32% to 52.8% could mean that the use of jargon increased by about 60.32% -
52.8% = 7.52%. Furthermore, one of these groups might be composed of African
Americans hence the dependence on percent Black population can be readily
explained. Since no other demographic or socioeconomic variable is correlated
with utterance length then these groupings cannot be entirely demographic or
socioeconomic in nature.

There is no obvious remaining factor that could bias the temporal analysis of
utterance lengths after the shortening was shown to be robust. There are
several approaches in determining the proper location of users from tweets
[12, 28] but we used the simplest method of assigning the location of the user
to the location of the tweet. The geolocated tweets are relatively few and the
tweets (users) were then aggregated by US state. Statistical data from the US
Census Bureau were then used in the analysis. The inherent assumption,
therefore, is that the sample used by the Bureau can also be used to describe
the sample of Twitter users. A survey done by Smith and Brenner [29], however,
showed that among the different races, Blacks significantly use Twitter more
than other races. This could be the reason why only the dependence on Black
population was observed. More data are needed to verify if our assumption is
justified but our results are tantalizing enough to warrant a second look.

## Materials and Methods

Tweets were first retrieved using the Twitter streaming application
programming interface [30] and corresponds to 15% (before August 2010), 10%
(between August 2010 to mid-2012) or 1% (mid-2012 to present) of the total
public tweets. For ease of computation, we analyzed only tweets posted every
Friday from 18 September 2009 to 14 December 2012. Although issues in our
retrieval process prevented us from getting the entire sampled feed for the
entire data collection period, only Fridays with uninterrupted and complete
data were considered resulting to a total of 124 days analyzed. Conversational
utterances in the form of replies were selected by filtering for tweets that
begin with an at sign (@), which yielded 229 million utterances (utterances-
bydate.txt in SI).

The utterance length of a tweet was measured by first stripping off all
leading @usernames with the python regular expression ((^|\s)*@\w+?\b)+. The
utterance length is the number of remaining characters after leading and
trailing whitespace characters were removed. Utterances having lengths equal
to zero (0.483%) and greater than the maximum length of 140 char. (0.0935%)
were excluded from the dataset.

Tweet language identification was performed using langid.py [31], which claims
88.6% to 94.1% accuracy when identifying the language of a tweet over 97
languages. ldig [32] stands to be the most accurate automated language
identification system for tweets having a claim of 99% accuracy over 19
languages ($>98\%$ accuracy for English), however, it has not yet been
formally subjected to peer review. Nevertheless, we repeated the analysis
using ldig and found similar results and conclusions.

Tweets were geolocated using the geo and coordinates metadata of the tweets
and were categorized by US states using TIGER/Line shapefiles [33] prepared by
the US Census Bureau. A user must opt-in to have location information be
attached to their tweets. Previously, only exact coordinates (latitude and
longitude) are attached as location information and these become the values of
the geo and coordinates metadata of the tweet. More recently, users may opt to
select less granular location information e.g., neighborhood, city and
country, and these less precise place information are now the default. [34] A
user, though, may still choose exact location information or omit location
information for every tweet. Geolocation is possible with both mobile clients
and browsers but geolocation for the latter is not yet available for all
countries.

A parallel [35] version of the Space Saving [36] algorithm for selecting the
most frequent $k$ words was used instead of a naive histogram of word
occurrences because of the prohibitive amount of resources needed. The Space
Saving algorithm maintains a frequency count of up to $k$ words only. An
untracked word replaces the least frequent word if the maximum number of $k$
words are already being tracked. The parallel Space Saving algorithm involves
partitioning the data then running the Space Saving algorithm for each chunk.
The results of each chunk are merged using an algorithm similar to Space
Saving. The word frequency of both Space Saving and its parallel version are
approximate for near-$k$-ranked words. To have a guaranteed list of the 1000
most frequently occurring words, a much larger value of $k=10^{5}$ was used.

## Acknowledgments

Some computational resources were provided by an AWS in education grant and
the Advanced Science and Technology Institute, Department of Science and
Technology, Philippines.

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  * 33. United States Census Bureau (2012).  2012 TIGER/Line shapefiles [machine-readable data files].
  * 34. Twitter (2013).  FAQs about tweet location.  Available: https://support.twitter.com/articles/78525-about-the-tweet-location-feature.  Accessed: 24 January 2013.
  * 35. Cafaro M, Tempesta P (2011) Finding frequent items in parallel.  Concurrency and Computation: Practice and Experience 23: 1774–1788.
  * 36. Metwally A, Agrawal D, Abbadi AE (2005) Efficient computation of frequent and top-k elements in data streams.  In: Eiter T, Libkin L, editors. Database Theory - ICDT 2005. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 398–412.

## Figure Legends

Figure 1:  Utterance length distribution of the entire dataset. A. Unfiltered
utterance length distribution of the entire dataset B. Utterance length
distribution of English tweets with URLs and LOL removed. The solid line in
both plots is the best fit of Eq. (1).  Figure 2:  Representative utterance
length distributions per year. Utterance length distribution of every first
available Friday of December in the dataset.  Figure 3:  Utterance length
distribution over time. First quartile Q1 (square), median Q2 (circle) and
third quartile Q3 (triangle) of the A. original dataset, B. after resampling
into $10^{5}$ utterances per day, C. removing URLs and D. restricting to
English tweets.  Figure 4:  Slopes of utterance length quartiles temporal
regression lines. Visualization of Table 1.  Figure 5:  Exploring possible
mechanisms of shortening. Annual values of A. median word length of all words,
B. median word length of the 1000 most frequently occurring words, C. most
frequent word length of the 1000 most frequently occurring words, D. fraction
of 1000 most frequently occurring words relatively to all occurrences of
words, E. median utterance length in number of words F. median tweet length in
number of words, and G. median trending topic phrase length. Figure 6:
Utterance lengths across US states. A. Box plot of the utterance length
distribution of each US state sorted by increasing median utterance length.
The notches were estimated using 10,000 bootstrap repetitions but the
resulting bootstrapped median values are the same as the empirical median
values B. Contiguous US states colored with the bootstrapped median utterance
length.  Figure 7:  Median utterance length against demographic and
socioeconomic variables. The bootstrapped median utterance length plotted
against A. 2011 resident Black population in percent ($r^{2}=0.685$), B.
persons 25 years and over who are high school graduates or higher from 2007 to
2011, in percent ($r^{2}=0.397$) and C. Median household income from 2007 to
2011 in thousands of dollars ($r^{2}=0.068$). The linear regression line is
also shown in each plot.

## Supporting Information Legends

Text S1. Information on State and County QuickFacts variables.

Dataset S1. Utterance length frequencies by date. The rows of this comma-
separated file correspond to tweets posted on a certain UTC date. The first
column is the date in ISO format (yyyy-mm-dd) and the remaining columns list
the number of utterances with a length of 1 character, 2 characters, 3
characters and so on, until 139 characters. Only the frequencies of “valid”
utterance lengths (1-139 characters) are included.

Dataset S2. Utterance length frequencies by US state. The rows of this comma-
separated file correspond to tweets posted from a US state. The first column
is the abbreviated US state (e.g., AK) and the remaining columns list the
number of utterances with a length of 1 character, 2 characters, 3 characters
and so on, until 139 characters. Only the frequencies of “valid” utterance
lengths (1-139 characters) are included.

## Tables

Table 1:  Slopes of utterance length quartiles temporal regression lines Subset | Q1 | Median (Q2) | Q3
---|---|---|---
| Slope | $r^{2}$ | Slope | $r^{2}$ | Slope | $r^{2}$
| (chars./year) |  | (chars./year) |  | (chars./year) |
All | -2.53 | 0.916 | -5.20 | 0.926 | -8.32 | 0.862
Sep–Dec | -5.29 | 0.812 | -7.85 | 0.894 | -9.97 | 0.889
Resampled | -2.54 | 0.918 | -5.25 | 0.927 | -8.23 | 0.860
URLs removed | -2.42 | 0.933 | -4.63 | 0.922 | -7.51 | 0.887
English only | -3.38 | 0.910 | -5.95 | 0.881 | -8.35 | 0.785
2010–2012 | -5.19 | 0.842 | -8.07 | 0.910 | -10.4 | 0.938
Trending topics | -3.41 | 0.153 | -6.83 | 0.294 | -7.61 | 0.440
Outside US | -5.57 | 0.838 | -8.15 | 0.904 | -10.4 | 0.909
Table 2:  Single-variable linear regression of median utterance length with
selected US Census Bureau QuickFacts variables

Independent variable | Parameter estimate (standard error)
---|---
| 1a | 1b | 1c | 1d | 1e | 1f | 1g
2011 resident Black population in percent $B$ | -3.411*** |  |  |  |  |  |
| (0.334) |  |  |  |  |  |
Persons 25 years and over who are high school graduates or higher from 2007 to 2011 in percent $H$ |  | 2.597*** |  |  |  |  |
|  | (0.462) |  |  |  |  |
Median household income from 2007 to 2011 in thousands of dollars $I$ |  |  | 1.074 |  |  |  |
|  |  | (0.574) |  |  |  |
Persons 25 years and over who has bachelor’s degree or higher from 2007 to 2011 in percent $C$ |  |  |  | 1.254** |  |  |
|  |  |  | (0.567) |  |  |
2010 population per square mile $D$ |  |  |  |  | -1.247** |  |
|  |  |  |  | (0.567) |  |
Owner-occupied housing units in percent of total occupied housing units from 2007 to 2011 $O$ |  |  |  |  |  | -0.846 |
|  |  |  |  |  | (0.582) |
Total value of manufacturing shipments in 2007 $M$ |  |  |  |  |  |  | -1.310**
|  |  |  |  |  |  | (0.564)
Constant | 35.40*** | 35.40*** | 35.40*** | 35.40*** | 35.40*** | 35.40*** | 35.40***
| (0.334) | (0.462) | (0.574) | (0.567) | (0.567) | (0.582) | (0.564)
$r^{2}$ | 0.685 | 0.397 | 0.068 | 0.093 | 0.092 | 0.042 | 0.101
adjusted $r^{2}$ | 0.678 | 0.384 | 0.049 | 0.074 | 0.073 | 0.022 | 0.082

Standard errors are presented in parentheses below the corresponding parameter
estimates. Bold indicates significance at the 5% level, $n=50$

** $p<0.05$

*** $p<0.01$


11institutetext:  Department of Physics, Tohoku University - Sendai 980-8578,
Japan
National Institute for Materials Science - Tsukuba, Ibaraki 305-0047, Japan

# Thermal Transport Imaging in the Quantum Hall Edge Channel

J. N. Moore 11 A. Kamiyama 11 T. Mano 22 G. Yusa 111122

###### Abstract

Research focused on heat transport in the quantum Hall (QH) edge channel has
successfully addressed fundamental theoretical questions surrounding the QH
physics. However, the picture of the edge channel is complicated by the
phenomenon of energy dissipation out of the edge, and theories treating this
dissipation are lacking. More experimental data is also needed to determine
the coupling mechanism by which energy leaves the edge channel. We developed a
method to map the heat transport in the QH edge to study the dissipation of
heat. We locally heated the QH edge and locally detected the temperature
increase while continuously varying the distance between heater and
thermometer. We thereby obtained the thermal decay length of the edge state.

##  1 Introduction

Heat transport in the quantum Hall (QH) edge channel has garnered much recent
attention. This interest began with the prediction and subsequent verification
of a charge-neutral mode in certain fractional QH edge states, which carries
heat counter to the direction of the charge mode [1, 2]. The discovery of the
charge-neutral mode was followed by further effort to understand the complex
structure of the QH edge state. For this, charge transport measurements alone
do not provide sufficient detail; but measurements of heat transport and
thermal conductance in the edge have generated significant progress in
identifying the composition of compound edge channels and revealing the
process of equilibration between edge modes [3, 4, 5, 6]. Thermal conductance
has also been critical to testing the order of the $\nu=5/2$ QH state and
providing evidence for the non-Abelian statistics of its quasiparticles [7].
$\nu=\frac{h}{eB}n_{\text{e}}$ is the Landau-level filling factor, where $h$
is the Planck constant, $e$ is the elementary charge, $B$ is the perpendicular
magnetic field and $n_{\text{e}}$ is the 2D electron density.

Theoretical proposals have been made to use the chiral heat transport of the
QH edge to produce several types of mesoscopic quantum heat devices such as a
refrigerator, heat engine, heat rectifier and heat diode [8, 9, 10, 11]. These
devices should operate very efficiently if heat travels ballistically in the
edge. An early theoretical assumption of the edge channel was that, just as
electrons do not leave the edge channel due to an absence of available states
in the bulk, heat will also remain inside the edge channel [12]. However, a
thermal decay length of $\sim 20$ $\mu$m was measured in the $\nu=1$ [13] and
$\nu=2$ QH states [14]. This is significantly shorter than the decay length of
charge transport, as can be inferred from $\sim 1$-mm-long samples which
display vanishingly small longitudinal resistance. The cause of heat
dissipation from the QH edge is unknown, but is likely due in part to coupling
with the two-dimensional (2D) QH bulk; heat propagation through the bulk was
detected by experiments using three different methodologies [4, 15, 16].
Mechanisms considered were 1) long-rang Coulomb interaction with localized
bulk states, 2) coupling with bulk spin states 3) and coupling with the
lattice.

We developed an optical heating technique to investigate QH edge heat
transport and heat dissipation. Illuminating a sample with a focused laser
beam allows for the local heating of electrons at any arbitrary position,
which can be changed in situ. Laser heating also offers the potential of
modulating the heating power to measure thermal relaxation times as short as
$\sim 1$ ps [17]. We combined optical heating with local thermometry of the QH
edge to acquire spatial information about heat transport. We succeeded in
measuring the thermal decay length of the edge channel by measuring the
temperature response as a function of the distance between the heater and
thermometer. When temperature response is spatially resolved in sufficient
detail, it can inform the creation of new models which incorporate energy
dissipation out of the edge [18]. With this goal, we performed the first
continuous real-space imaging of heat transport in the QH edge.

##  2 Experiment

The 2D electron system (2DES) was contained in a
Al0.3Ga0.7As/GaAs/Al0.3Ga0.7As 15-nm quantum well (QW) grown on a Si-doped
(100) GaAs substrate [19, 20, 21]. The conducting substrate was insulated from
the 2DES by a GaAs/AlAs superlattice; this allowed us to use the substrate as
a global back gate of the 2DES. We modulation doped the QW by a unique
superlattice $\delta$-doping structure designed to suppress photoconductivity
in channels parallel to the QW so that our transport measurements will
primarily detect conduction in the 2DES (see Superlattice Doping Structure in
Supplementary Material for details). Figure 1 shows the band energy and 3D
electron density $n_{\text{3D}}$ of the sample self-consistently simulated as
a function of distance $z$ from the sample surface. Residual accepters which
were unintentionally doped during the wafer growth are ignored in the
calculation. The experimentally measured $n_{\text{e}}$ and mobility of the QW
in this structure were $1.7\times 10^{11}$ cm-2 and $0.6\times 10^{6}$ cm2/V s
respectively.

[scale=0.45]waferstructurePL.eps

Figure 1: (a) The conduction band energy $E_{\text{C}}$, heavy hole valence
band energy $E_{\text{V}}$, Fermi energy $E_{\text{F}}$, and 3D electron
density $n_{\text{3D}}$ self-consistently simulated at the gamma point and
plotted as a function of distance $z$ from the surface at a location $1$
$\mu$m from the etched mesa edge. The vertical dashed lines indicated as
$\delta①$ ($\delta②$) are $\delta$-dopes having Si doping concentration
$6\times 10^{11}$ cm-2 ($3\times 10^{11}$ cm-2). (b) The
$\sigma^{-}$-polarized photoluminescence spectrum, corresponding to
recombination in the bottom Zeeman level, collected at two different magnetic
field strengths $B$.

We conducted the following experiments at a lattice temperature $T_{\text{L}}$
of 1.7 K, unless otherwise stated, while the 2DES was tuned to the $\nu=2$ QH
state by adjusting the global back gate voltage and $B$ to the center of the
QH resistance plateau. To create local heating, we illuminated the sample by a
linearly polarized 785-nm diffraction-limited continuous wave laser spot of
diameter $\sim 1$ $\mu$m[19, 20, 21, 22]. The laser photon energy,
$E_{\text{ph}}=1.579$ eV, was greater than the ground state energy of an
electron-hole pair in the QW, $E_{\text{rec}}=1.532\sim 1.537$ eV, which we
measured in this sample as described below. $E_{\text{ph}}$ was also less than
the bandgap of AlGaAs ($\sim 1.9$ eV)[23]. Thus, photon energy was absorbed in
the QW, but not in the neighboring AlGaAs. Photon energy was, however,
absorbed by the GaAs substrate, which is separated from the QW by a 200-period
GaAs/AlAs (2 nm/2 nm) superlattice. Some of the energy absorbed in the
substrate was converted to heat which was diffused through the sample by
phonons. We numerically simulated the resulting phonon temperature throughout
the sample and found that it increased by only $\sim 0.1$ $\mu$K (see 3D
Simulation of Phonon Heating in Supplementary Material for details), which is
due to the large phonon thermal conductivity and the freedom of phonons to
diffuse heat in three dimensions.

The QH edge, in contrast, conducts heat in only one dimension with a thermal
conductivity
$K_{\text{H}}=\nu_{\text{Q}}\frac{\pi^{2}}{3}\frac{k_{\text{B}}^{2}}{h}T$,
where $\nu_{\text{Q}}=2$ is the number of downstream heat conducting edge
modes, $T=1.7$ K is the edge temperature, and $k_{\text{B}}$ and $h$ are the
Boltzmann and Planck constants respectively [12]. Now we estimate the
temperature increase of the QH edge, $\Delta$$T_{\text{edge}}$. We take the
absorption coefficient of the QW to be $1.7\times 10^{4}$ cm-1 [24], from
which we calculate that $2.5\%$ of the incident optical power will be absorbed
in the QW. Most of the absorbed energy is consumed in generating an electron-
hole pair across the bandgap. The maximum energy available for heat generation
is $E_{\text{ph}}-E_{\text{rec}}$. We measured $E_{\text{rec}}$ from the peak
energy in the photoluminescence spectrum collected during the experiment (Fig.
1b). We interpret this peak to correspond to recombination of singlet-state
trions, which are a bound state of a hole with two electrons [25], because as
a function of $n_{\text{e}}$ this peak is continuous with the singlet trion
peak identified at $\nu<1$ [27, 26]. The maximum fraction of absorbed energy
that can be converted to heat is therefore
$(E_{\text{ph}}-E_{\text{rec}}$)/$E_{\text{ph}}=2.9\%$. Using the incident
illumination of 3 nW in our experiment, we calculate
$\Delta$$T_{\text{edge}}=830$ mK, which is an upper estimate assuming the
stated heat conversion efficiency and neglecting any dissipation of heat out
of the edge into the environment.

Following Ref. [13], we used thermoelectric thermometry to locally detect the
increase in the QH edge temperature. This method of thermometry measures the
thermoelectric voltage $V_{\text{th}}$ that is generated across a gradient in
electron temperature. Because $T_{\text{L}}$ does not increase significantly,
the phonon drag contribution to $V_{\text{th}}$ can be ignored [28]. In our
device, shown in Fig. 2, $V_{\text{th}}$ appeared across a gated $3$-$\mu$m
constriction, labeled “C”, which we created by photolithography and etching.
We generated a temperature gradient across this constriction by laser heating
the 2DES on the lower side of the constriction. Across the constriction,
$V_{\text{th}}=-\int_{T_{\text{1}}}^{T_{\text{2}}}\;S\text{d}T$, where
$T_{\text{1}}$ and $T_{\text{2}}$ are the electron temperatures on the lower
and upper side of the constriction respectively, and $S$ is the Seebeck
coefficient depending on temperature and $n_{\text{e}}$. For small thermal
gradients, $V_{\text{th}}$ is proportional to $T_{\text{1}}-T_{\text{2}}$,
with $S$ determining the sensitivity of the thermometer.

Under the gated region of the constriction, $S$ is modified by the front gate
voltage $V_{\text{fg}}$ and is determined by the Mott relation to exhibit
peaks corresponding to the steps between plateaus in the constriction
conductance [29, 14]. Past work at $T_{\text{L}}=1.65$ K used a peak in $S$ of
$-20$ $\mu$V/K to achieve a thermometer sensitivity of $\sim 2$ mK [30].
Because of the incompressibility of the 2DES in the $\nu=2$ QH state,
$S_{xx}=S_{xy}=0$ outside the gated region [31]. Therefore, although the laser
heating creates a temperature gradient along the QH edge, no thermoelectric
voltage is generated there. This allowed us to use Ohmic contacts (labeled (1)
and (2)) connected to the 2DES on the lower and upper sides of the
constriction to measure $V_{\text{th}}$ generated in the gated region alone.
Note that under the illumination intensity of our experiment ($\sim 3$ nW), we
estimate the ratio of photoexcited carriers to native 2D electrons to be $\sim
10$ to 1 million. This indicates that the photocarriers cannot significantly
modify $S$ or independently generate photovoltage.

Simultaneously with the measurement of $V_{\text{th}}$, we measured the
source-drain current $I_{\text{sd}}$ and four-terminal conductance $G$ through
the constriction using a constant source-drain excitation voltage
$V_{\text{sd}}$ of 10 $\mu$V. We measured $V_{\text{th}}$ and $G$ by lock-in
detection with modulation of $V_{\text{sd}}$ and the laser power at 13.1 Hz
and 5.94 kHz respectively. We performed all measurements at fixed phase and
plotted the amplitude of the in-phase component, denoted by
$\left|\cdot\right|$. We measured $V_{\text{th}}$ at a phase of $67^{\circ}$,
where it’s amplitude was maximum.

[scale=0.45]device.eps

Figure 2: Optical microscope image of the experimental device. C: 3.0 $\mu$m
constriction in the 2DES covered by a gold gate. Red dotted outline: the
region mapped in Fig. 3. Blue solid line: The path of the QH edge when $B$ is
oriented as indicated in Fig. 3(b). Ohmic electrodes (colored orange): (S)
voltage source, (D) current drain, (1) and (2) voltage probes.

##  3 Results and Discussion

We first investigated the spatial distribution of $V_{\text{th}}$ at $B=6.3$
T. We found that a voltage signal is detectable even when the laser
illumination does not fall on the region of the sample containing the 2DES.
This voltage signal, which we will call $V_{\text{b}}$, persists at 33 mK
where we mapped it by scanning the laser in the region indicated by the dotted
red box in Fig. 2. This measurement of $V_{\text{b}}$ is shown in Fig. 3(a).
During this measurement we depleted the electrons under the front gate to
prevent any generation of $V_{\text{th}}$ across the gate. We see that
$V_{\text{b}}$ has no notable spatial variation other than its diminution
where the front gate blocks light from reaching the semiconductor. The voltage
signal also appears spatially uniform at 1.7 K at locations far from the front
gate (see Mapping Voltage Response to Illumination at the Ohmic Contacts in
Supplementary Material for details). We conclude that a background signal
independent of $V_{\text{th}}$ exists that is spatially uniform over the
length scale that we map. We will later discuss the possible origin of
$V_{\text{b}}$. We mapped $V_{\text{th}}$ at 1.7 K using a gate voltage
$V_{\text{g}}=-0.5$ V. Figure 3(b) shows this $V_{\text{th}}$ obtained after
subtracting $V_{\text{b}}$ for this condition, which we assume to have the
same spatial distribution as in Fig. 3(a).

To interpret the map of $V_{\text{th}}$, we compare it to a map of reflected
laser intensity measured simultaneously (Fig. 3(c)). Matching this intensity
map to the optical image of the sample in this region (Fig. 3(d)), we located
the etched boundaries of the QW and the gated region, which are shown as
dotted outlines in Figs. 3(a) and (b). It is apparent that $V_{\text{th}}$ is
strongest when illumination is at the QH edge closest to the gate. And there
is a decay with distance along the edge in the direction upstream from the
gate. This observation can be understood from the chirality of heat transport
in the QH edge channel. Heat injected upstream of the gate propagates
clockwise and reaches the gate location, which creates the thermal gradient
across the gate detected by $V_{\text{th}}$. Heat injected downstream of the
gate only propagates away from the gate, so no thermal gradient is produced
across the gate. The map of $V_{\text{th}}$ remained the same regardless of
whether or not we applied source-drain current.

[scale=0.31]6.3Tmap.eps

Figure 3: (a) Spatial map of voltage measured across Ohmic electrodes (1) and
(2) while the scanned laser power was $3.5$ nW, $V_{\text{fg}}$ was $-1.4$ V,
and $T_{\text{L}}$ was 33 mK. (b) Spatial map of voltage measured across Ohmic
electrodes (1) and (2) after subtracting a background component with the
spatial distribution in (a). Laser power was $3.0$ nW, $V_{\text{fg}}=-0.5$ V,
$T_{\text{L}}=1.7$ K, and $\nu=2$. Light blue dashed line and blue square
correspond to measurements in Figs. 4 and 5 respectively. Arrows indicate the
propagation direction of QH edge. (c) Spatial map of reflected laser intensity
measured simultaneously with the measurement in (b). (d) Optical microscope
image of the sample in the region mapped in (a) and (b). Outlines of the gate
and etched 2DES boundary in the image are overlayed as dasshed lines on (a)
and (b). $B=6.3$ T during the measurements ((a), (b) and (c)).

In Fig. 4 we plot $V_{\text{th}}$ as a function of distance $d$ from the gate
along the edge upstream of the gate. The data plotted are taken from the
points in Fig. 3(b) overlayed by the blue dashed line which is a segment of
the etching boundary visible in Fig. 3(d). We fit these data to an exponential
decay in order to extract a decay length of $10\pm 1$ $\mu$m. We interpret
this decay length as the distance over which the QH edge at elevated
temperature cools into the environment at $T_{\text{L}}$. Our measured thermal
decay length is shorter than the thermal decay length of $\sim 20$ $\mu$m
measured previously in the $\nu=1$ [13] and $\nu=2$ QH states [14] at a
significantly lower temperature of 0.1 K.

[scale=0.35]Vthdecay.eps

Figure 4: $V_{\text{th}}$ as a function of upstream distance from the gold
gate along the etched 2DES boundary, obtained from the points overlayed by the
blue dashed line in Fig. 3(b). The fit of these data is $[64\pm
6\;\text{nV}]\text{exp}(-d/[10\pm 1\;\mu\text{m}])$, where the shaded region
is the $68\%$ confidence interval.

We next investigated how $V_{\text{th}}$ depends on $V_{\text{fg}}$. In Fig.
5(a) we plot against $V_{\text{fg}}$ the voltage
$V_{\text{th}}^{\prime}=V_{\text{th}}+V_{\text{b}}$ obtained before
subtracting the background signal, which we measured while illuminating the
2DES edge at the location indicated by the blue square in Fig. 3(b). Together
we plot $V_{\text{b}}$ obtained by averaging the signal measured at the four
corners of the region mapped in Fig. 3; we assume that no significant
thermoelectric signal was generated by illumination at these four corners
because of their distance from the QH edge. Notably, $V_{\text{b}}$ exhibits a
dependence on $V_{\text{fg}}$ which is opposite of $G$, plotted in Fig. 5(b).
A possible reason is that $V_{\text{b}}$ arises from the excitation of charged
carriers in the GaAs substrate of the sample which modify the potential in the
QW. When $G$ in the constriction is large, potential differences between Ohmic
electrodes (1) and (2) can equilibrate, so $V_{\text{b}}$ is suppressed;
whereas, when the constriction is depleted, no equilibration occurs and
$V_{\text{b}}$ is maximum. In Fig. 5(b) we plot $V_{\text{th}}$ obtained from
the difference of the curves in Fig. 5(a). $V_{\text{th}}$ has a peak where
$G$ is in transition between zero and its quantized value of $e^{2}/h$. This
is consistent with the work previously mentioned verifying the Mott relation
in this type of device [29, 14]. The $\nu=1$ plateau in $G$ is absent, and the
$\nu=2$ plateau is lower than the correct quantized value. These may be caused
by backscattering across the constriction induced by potential fluctuations.

[scale=0.4]VthG.eps

Figure 5: (a) Left axis: Voltage measured as a function of $V_{\text{fg}}$
while illuminating at the location of the blue square in Fig. 3(b). Right
axis: Average voltage measured as a function of $V_{\text{fg}}$ while
illuminating at the four courners of the region mapped in Fig. 3(b).
Illuminating power was $3.5$ nW , $T_{\text{L}}=33$ mK, and $\nu=2$. (b) Left
axis: The voltage obtained from the difference of the two curves in (a). Right
axis: The four-terminal conductance of the constriction measured
simultaneously with the voltage plotted on the right axis of (a).

Next, we repeated the mapping of $V_{\text{th}}$, still at $\nu=2$, while
applying a weaker field of $B=3.6$ T directed both upward and downward
perpendicular to the 2DES (Figs. 6(a) and (b) respectively). At this weaker
$B$, $V_{\text{b}}$ became undetectably small. The difference in the magnitude
of $V_{\text{th}}$ between the two plots is due to the different conditions
$V_{\text{fg}}$ of the gate during the measurements. For both $B$ directions,
the $V_{\text{th}}$ signal shows the same behavior seen at 6.3 T, which is
consistent with chiral downstream heat propagation in the QH edge. As in Fig.
4, in Figs. 6(c) and (d) we plot $V_{\text{th}}$ as a function of upstream
distance $d$ from the gate. Data are taken along the blue dashed lines in
Figs. 6(a) and (b) respectively, and fit to extract thermal decay lengths of
$5\pm 1$ $\mu$m and $5.6\pm 0.3$ $\mu$m respectively. The measured thermal
decay length has become shorter, possibly caused by the decrease in $B$ or the
decrease in $n_{\text{e}}$ throughout the device by the same proportion. A
more systematic study of how thermal decay length depends on experimental
conditions is needed to understand the mechanism of heat dissipation from the
QH edge.

[scale=0.33]3.56Tmapping.eps

Figure 6: (a) and (b) Spatial maps of the voltage obtained as in Fig. 3(b)
while $B=3.6$ T was directed downward and upward respectively and
$V_{\text{fg}}$ was $-458$ mV and $-520$ mV respectively. Illuminating power
was $2$ nW, $T_{\text{L}}=1.7$ K, and $\nu=2$. Dotted lines are outlines of
the gate and etched 2DES boundary. Arrows indicate the propagation direction
of QH edge. (c) and (d) $V_{\text{th}}$ as a function of distance upstream
from the gate obtained from points overlayed by the light blue dashed lines in
(a) and (b) respectively. The fits of these data are (c) $[28\pm
7\;\text{nV}]\text{exp}(-d/[5\pm 1\;\mu\text{m}])$ and (d) $[112\pm
5\;\text{nV}]\text{exp}(-d/[5.6\pm 0.3\;\mu\text{m}])$, where the shaded
regions are the $68\%$ confidence interval.

We will now discuss possible challenges to our interpretation of the measured
$V_{\text{th}}$ signal as thermoelectric voltage. Photovoltage is a phenomenon
which requires consideration in our experiment. Photovoltage generated by
microscopic laser photoexcitation of electrons into the QH edge was measured
in single-heterojunction QWs [32, 33]. In that work, spatial maps of
photovoltage with peak magnitudes of $\sim 200$ nV and $\sim 1$ $\mu$V were
detected for illumination powers of $\sim 1$ nW and $\sim 1$ $\mu$W
respectively. The photovoltage occurred in the absence of any gate or
constriction of the 2DES and had a localized response to illumination at the
2DES edge. In Ref. [33], the photovoltage could be seen to decay with distance
from the Ohmic contacts used for detection with a decay length of $\sim 1000$
$\mu$m at 1.4 K.

We mapped the voltage response to illumination while scanning the laser at the
Ohmic contacts because the photovoltage should be maximized there (see Mapping
Voltage Response to Illumination at the Ohmic Contacts in Supplementary
Material for details). However, we were unable to detect any signal with the
characteristics of photovoltage. We observed only the spatially uniform
background voltage discussed above, so we conclude that any photovoltage that
may be generated is too small to be detected using the illumination power of
our experiment. This is probably because the QW in our device is square in the
growth direction, which causes photoexcited holes to remain in the QW rather
than vertically drifting out of the well as in previous experiments. The
presence of photoholes at the edge of the 2DES, which have the same chirality
as electrons, can result in a reduction or complete negation of the
photoexcited charge.

It is also important to consider the thermalization of photoexcited carriers
in our experiment. Here, ”thermalization” entails the processes in which
electrons occupying energy states above the Fermi distribution relax into the
Fermi distribution while raising the Fermi temperature. The QH edge is a one-
dimensional Tomonaga-Luttinger liquid [35, 34] which, in an ideally clean
system at sufficiently low excitation energy, lacks the thermalization
processes that are present in Fermi liquids [36]. Confirmation of this was
provided by quantum-dot energy spectroscopy measurements identifying a
nonequilibrium energy distribution as far away as $15$ $\mu$m from the point
of electrical excitation of the edge state [37]. However, the QH edge in that
work was defined by electrostatic gating, which creates a smooth confinement
potential. It was shown that when electrons having several times the
longitudinal optical (LO) phonon energy (36 meV) were injected into the QH
edge, electrostatically broadening the edge potential suppressed scattering
with these LO phonons, which hindered efficient equilibration [38]. Further
results, also using excitation greater than the LO phonon energy, indicated
that relaxation by electron-electron scattering is also suppressed by edge
potential broadening and increased excitation energy [39].

The photoexcited electron-hole pairs in our experiment have an initial
combined excitation energy of $\sim 45$ meV, meaning that LO phonon scattering
is important for thermalization. However, because the QH edge in our device is
defined by etching, the confinement potential should be relatively sharp
compared to a gate-defined potential [4]. It is conceivable, therefore, that
the LO phonon scattering length is among the lowest values measured in Ref.
[39], i.e. $<1$ $\mu$m. This scattering length is a lower bound for the
thermalization length, i.e. the distance over which thermalization occurs, in
our experiment. If the thermalization length is larger than the laser spot
size, the heating region will be larger than we initially assumed. Considering
this possibility, the thermal decay lengths we measure should be interpreted
as an upper bound. Lastly, we note that because the constriction width is
comparable to the measured thermal decay lengths, details of the potential
landscape inside the constriction can significantly impact the effective
distance between temperature probe and heating point. To address this
limitation, a quantum point contact gate structure could be used as the
temperature sensor.

In conclusion, we have demonstrated measurement of the thermal decay length of
the QH edge by real-space optical mapping of thermovoltage generated at a
local thermometer. We find the thermal decay length to change with
experimental parameters, which calls for further investigation of the
mechanism of energy dissipation from the QH edge.

###### Acknowledgements.

The authors are grateful to T. Fujisawa for fruitful discussions. This work
was supported by Grants-in-Aid for Scientific Research (Grant Nos. 21F21016,
19H05603, 21J14386 and 21H05188) from the Ministry of Education, Culture,
Sports, Science, and Technology (MEXT). J. N. M. and A. K. were supported by
Grants-in-Aid from MEXT. Data availability statement: The data that support
the findings of this study are available upon reasonable request from the
author.

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