11institutetext: Indian Institute of Technology, Madras
11email<EMAIL_ADDRESS>
# Introducing Inter-Relatedness between Wikipedia Articles in Explicit
Semantic Analysis

Naveen Elango 11 ME16B077 Pawan Prasad K 11 ME16B179

###### Abstract

Explicit Semantic Analysis (ESA) is a technique used to represent a piece of
text as a vector in the space of concepts, such as Articles found in
Wikipedia. We propose a methodology to incorporate knowledge of Inter-
relatedness between Wikipedia Articles to the vectors obtained from ESA using
a technique called Retrofitting to improve the performance of subsequent tasks
that use ESA to form vector embeddings. Especially we use an undirected Graph
to represent this knowledge with nodes as Articles and edges as inter
relations between two Articles. Here, we also emphasize how the ESA step could
be seen as a predominantly bottom-up approach using a corpus to come up with
vector representations and the incorporation of top-down knowledge which is
the relations between Articles to further improve it. We test our hypothesis
on several smaller subsets of the Wikipedia corpus and show that our proposed
methodology leads to decent improvements in performance measures including
Spearman’s Rank correlation coefficient in most cases.

###### Keywords:

Wikipedia Graph ESA Retrofitting.

##  1 Introduction

Methods of capturing semantic relatedness in words/documents have been
explored to a significant extent by the NLP community. Major approaches
include data-driven learning of vector representations [7, 8, 9] from a very
large corpora. While such bottom-up approaches have proven to be effective in
recent times, the outcomes of such models can be further amplified by using
some form of top-down knowledge gained over time, by humans. Such information
has been crystallized into structured databases since as early as 1995 [5].
These top-down resources can be utilized to guide bottom-up expeditions to
identify relations among words/documents.

Several works in this domain have surfaced recently. One such elegant idea,
arrived at by Faruqui et al., is retrofitting. We build our proposal on their
paper [3], utilizing the technique rather unconventionally, in Explicit
Semantic Analysis (ESA) [1], for improving word representations using top-down
knowledge from Wikipedia.

Explicit Semantic Analysis is a popular technique that represents the meaning
of texts in a high-dimensional space of concepts derived from Wikipedia and
using machine learning techniques explicitly represent the meaning of any text
as a weighted vector of Wikipedia-based concepts. The information used here
from the Wikipedia corpus are essentially the Article titles and the text or
body of the Article. We suggest that by using an additional information which
is often disregarded, such as the way in which one Wikipedia Article
references to another, we could improve upon the earlier work as performed in
ESA. One could imagine such references made back and forth between several
Wikipedia Articles as a network of useful information. We incorporate this on
top of ESA and see some minor improvements on several Benchmark tasks such as
Word Similarity tasks like WS-353, RG-65 and MEN3k [10, 11, 12]. Especially we
see improvements of around 2% on average in performance measured using
Spearman’s Rank Correlation Coefficient. To represent the knowledge in a
useful manner we take advantage of the Graph data structure, specifically an
undirected Graph, wherein the nodes are the Articles and edges between nodes
correspond to Inter-Relatedness between corresponding Articles.

In addition our work also comprises of implementing the paper - Retrofitting
word vectors to semantic lexicons on a number of evaluation benchmarks across
several data sets and cross-validating the results cited in the paper [3]. In
section 2 we discuss a few papers as background and related work that we
consider and take motivation from, while forming our hypothesis. In section 3
we discuss our implementation of the paper that introduces Retrofitting [3]
which is a concept that we use to apply the Inter-Relatedness knowledge on top
of ESA. In section 4 we introduce our main contributions and hypothesis to
improve upon ESA using the discussed knowledge of Inter-Relatedness between
Articles and present the results in doing so by drawing comparison before and
after incorporating that knowledge.

##  2 Background and Related Work

###  2.1 Retrofitting

Faruqui et al. introduced the novel method of retrofitting for capturing
relational information among vector representations of words [3]. Retrofitting
is applied on word vectors as a post-processing step such that, semantically
related words will be nearer in their vector embedding space. This includes
synonyms, hypernyms, hyponyms and paraphrases for the given word. Such
relations among semantic lexicons are captured in resources including the
Paraphrase Database [4], WordNet [5] and FrameNet [6].

Although retrofitting is agnostic to the manner in which the vectors were
derived, a key point to note is the type of top-down knowledge database to be
used for retrofitting. This in fact, depends on the nature of vectors at hand.
That is, Faruqui et al. consider word vector representations and hence any
database which captures the many relations among words would suffice (for
example, WordNet, FrameNet). However if one were to refine the vector
representations of documents for example, we need to consider a database of
similar text documents to construct a graph wherein edges connecting two nodes
correspond to two documents having similar content. Our proposal exploits this
observation to build a customized graph of Wikipedia articles for
retrofitting, as elaborated further in Section 4.

###  2.2 Explicit Semantic Analysis (ESA)

ESA is a novel method in which semantic relatedness between words or documents
are captured by representation of meaning in a high-dimensional space of
natural concepts derived from Wikipedia [1]. Each concept is represented as an
attribute vector of words that occur in the corresponding wikipedia article.
Entries of these concept vectors are assigned weights using TF-IDF scheme. We
should note that concepts are merely the titles of articles across which TF-
IDF scores are calculated, for example, Computer Science, Automobile or India.

An inverted index is used to retrieve the relevant concepts for a given word.
In the words of Markovitch et al., Each word appearing in the Wikipedia corpus
can be seen as triggering each of the concepts it points to in the inverted
index [2]. Moreover, a piece of text (string of words) is simply represented
as the centroid of the vectors representing its words in the concept space.

###  2.3 Non-Orthogonal Explicit Semantic Analysis (NESA)

One of the drawbacks in using ESA to obtain vector embeddings is that ESA
inherently assumes that Wikipedia concepts are orthogonal to each other.
Therefore, it considers that two words are related only if they co-occur in
the same articles. However, two words can be related to each other even if
they appear separately in related articles rather than co-occurring in the
same articles. In this paper [18], the authors come up with a technique to
successfully overcome this issue leading to improvements in several of the
Benchmarks and achieving state-of-the-art results.

To compute text relatedness, NESA uses relatedness between the dimensions of
the distributional vectors to overcome the orthogonality in ESA model. This is
accomplished by utilizing a square matrix $C_{n,n}$ where n is the total
number of dimensions or concepts containing the correlation weights between
the dimensions. While obtaining relatedness score between two words, instead
of using something like a cosine similarity measure which is multiplying the
two word vectors and normalizing them, they first multiply the first word
vector with the square matrix $C_{n,n}$ and then later multiply this with the
second word vector. What this essentially boils down to is that each concept
dimension is made to further constitute a semantic vector built in another
vector space. To construct $C_{n,n}$ consisting of correlation scores between
Articles they provide four different methods and also discuss the performance
improvements achieved by each one of them.

##  3 Paper Implementation

The objective of retrofitting is to reduce the Euclidean distance between a
pair of similar vectors and in doing so we end up with a richer representation
of words using vectors. Therefore, we want our word vector in question $q_{i}$
to be close to the observed value $\hat{q_{i}}$ and close to its neighbours
$q_{j},\forall j$ such that $(i,j)\in E$, that is they constitute an edge in
the graph constructed from lexicon database. Thus, the optimization criteria
is [3]:

$\Psi(Q)=\sum_{i=1}^{n}\left(\alpha_{i}||q_{i}-\hat{q_{i}}||^{2}+\sum_{(i,j)\in
E}\beta_{ij}||q_{i}-q_{j}||^{2}\right)$ (1)

$\alpha\ and\ \beta$ are hyper-parameters controlling the relative strengths
of associations between two vectors
$n=$ Number of word vectors to be retrofitted
Differentiating $\psi$ with respect to $q_{i}$ gives the following online
update equation (refer paper by Faruqui et al. for details [3]):

$q_{i}=\frac{\sum_{j:(i,j)\in
E}\beta_{ij}q_{j}+\alpha_{i}\hat{q_{i}}}{\sum_{j:(i,j)\in
E}\beta_{ij}+\alpha_{i}}$ (2)

Following the implementation details stated in [3], we apply retrofitting on 3
different sets of word vectors and by using 4 different semantic lexicon
resources. We assess the performance of each combination with and without
retrofitting on standard lexical semantic evaluation tasks. An analysis of the
results obtained (see Table 4) as well as comparison with the results reported
in [3] are detailed in Section 3.4. Refer Section 3.3 for details regarding
evaluation tasks.

###  3.1 Semantic Lexicons

We use Paraphrase Database (PPDB), WordNetsyn (${WN}_{syn}$, captures only
synonymous relations between words), WordNetall (${WN}_{all}$, captures
relations including synonyms, hypernyms and hyponyms) and FrameNet [4, 5, 6].
The size of the graphs obtained from the different top-down knowledge
databases are shown in Table 1.

Table 1: Size of graphs obtained from lexicon databases Lexicon | Words | Edges
---|---|---
PPDB | $102,90$ | $374,555$
WordNetsyn | $148,730$ | $304,856$
WordNetall | $148,730$ | $934,705$
FrameNet | $10,822$ | $417,456$
Table 2: Word Vector Characteristics Word Vector |  Dimension | Training Corpus
---|---|---
Glove Vectors | $300$ |  $6$ Billion tokens
Skip-Gram Vectors (SG) | $300$ |  $100$ Billion tokens
Global Context Vectors (GC) | $50$ |  $1$ Billion tokens
Multilingual Vectors (Multi) | $300$ |  $100$ Billion tokens

###  3.2 Word Vectors

We use publicly available pre-trained English word vectors for implementation.
These include the Glove, Skip-Gram (SG) and Global Context (GC) vectors[7, 8,
9]. The dimensions and training corpus characteristics for each set of word
vectors are enlisted in Table 2. We once again emphasise here that
retrofitting is independent of the training models used for extracting vector
representations [3].

###  3.3 Retrofitting: Evaluation Benchmarks

The resultant vector representations are evaluated on tasks that test how well
they capture both semantic and syntactic aspects of the representations along
with an extrinsic sentiment analysis task. A summary of the different tasks
are provided in Table 3. For the sake of comparison of results, we perform the
same evaluation tasks as given in the base paper [3].

We should note that the Syntactic Relations task is computationally expensive
since for each list of 4 words (refer Section $3.4$), we find cosine
similarity with each of the word types in the vector representations.
Moreover, since the Skip-Gram vector set is enormous ($\approx$ 3GB compressed
and $\approx$ 8GB in .txt file) the estimated time was large($\approx$ 22
hours) and we were facing issues with physical memory.

Table 3: Evaluation metrics and Dataset characteristics Evaluation Task | Metric | Dataset
---|---|---
Word Similarity: MEN-3k |  Spearman’s rank correlation coefficient | 3000 word pairs
Word Similarity: RG-65 | 65 word pairs
Word Similarity: WS-353 | 353 word pairs
Synonym Selection (TOEFL) | Accuracy | 80 sets of 5 words
Sentiment Analysis (SA) | 6920 + 872 + 1821 sentences

####  3.3.1 Word Similarity

We evaluate on 3 different benchmarks commonly used for measuring word
similarity. These are the WS-353 dataset [10] containing 353 pairs of English
words that have been assigned similarity ratings by humans, the RG-65 dataset
[11] that contain 65 pairs of nouns and the MEN dataset [12] consisting of
3,000 word pairs. Assessing the similarity of words amounts to comparing the
corresponding vectors using the cosine metric. We report Spearman’s rank
correlation coefficient between the rankings produced by our model against the
human rankings.

####  3.3.2 Synonym Selection (TOEFL)

The TOEFL synonym selection task is to select the semantically closest word to
a target word from a list of four candidates [13]. The dataset contains 80
such questions. An example is “rug $\rightarrow$ sofa, ottoman, carpet,
hallway”, with carpet being the most synonym-like candidate to the target
word.

####  3.3.3 Sentiment Analysis (SA)

Socher et al. created a treebank containing sentences annotated with labels on
phrases and sentences from movie review excerpts [14]. The coarse grained
treebank of positive and negative classes has been split into training,
development, and test datasets containing 6,920, 872, and 1,821 sentences,
respectively. We train an L2-regularized logistic regression classifier on the
average of the word vectors of a given sentence to predict the coarse-grained
sentiment tag at the sentence level, and report the test-set accuracy of the
classifier.

####  3.3.4 Syntactic Relations (SYN-REL)

The dataset is composed of analogous word pair that follow a common syntactic
relation [15]. For example, walking and walked; The task is to find a word
(word vector) $q_{d}$ that best fits the relation $q_{a}:q_{b}::q_{c}:q_{d}$,
given $q_{a},q_{b},q_{c}$. An estimate of $q_{d}$ is obtained by using the
vector offset method [8, 16] as follows:

$\displaystyle q_{a}:q_{b}$ $\displaystyle::q_{c}:q_{d}$
$\displaystyle\Rightarrow q_{a}-q_{b}=q_{c}-q_{d}$ $\displaystyle\Rightarrow
q_{d}=q_{b}–q_{a}+q_{c}$

Therefore, we find the vector q which has the highest cosine similarity to the
estimate.

###  3.4 Results of Retrofitting

In table 4, the results of our implementation of retrofitting are presented.
We present the absolute scores in each case of retrofitting. Note that the
first three columns represent Spearman’s rank correlation coefficient and the
last two represent Accuracy. Note that all absolute values of performance
measures are scaled to $100$ and reported as percentages. Key observations
based on the model performance are enlisted below:

Table 4: Results of absolute performance with retrofitting obtained by us; Spearman’s correlation (columns 2,3 and 4 from left) and accuracy (columns 5,6 and 7 from left) on different tasks. Higher scores are always better. Word Vector+Lexicon | MEN-3k | RG-65 | WS-353 | TOEFL | SA | SYN REL
---|---|---|---|---|---|---
GLOVE | 73.75 | 76.95 | 60.85 | 88.75 | 78.31 | 78.1
GLOVE+PPDB | 76.36 | 82.49 | 63.08 | 93.75 | 79.24 | 77.1
GLOVE+${WN}_{syn}$ | 73.71 | 79 | 61.34 | 93.75 | 78.97 | 66.8
GLOVE+${WN}_{all}$ | 75.94 | 85 | 61.52 | 91.25 | 79.24 | 66.1
GLOVE+FN | 70.09 | 75.62 | 56.14 | 91.25 | 77.76 | 69.0
SG | 73.22 | 76.08 | 65.9 | 87.5 | 81.82 | 74.2
SG+PPDB | 75.25 | 81.5 | 71.28 | 95 | 82.59 | 77.7
SG+${WN}_{syn}$ | 72.81 | 76.35 | 66.47 | 93.75 | 82.31 | 59.9
SG+${WN}_{all}$ | 73.65 | 84.39 | 67.4 | 93.75 | 81.82 | 59.1
SG+FN | 69.88 | 79.02 | 61.84 | 87.5 | 81.38 | 64.64
GC | 31.4 | 62.99 | 62.58 | 57.5 | 67.38 | 18.8
GC+PPDB | 40.31 | 68.99 | 64.58 | 71.25 | 68.92 | 27.3
GC+${WN}_{syn}$ | 34.87 | 69.3 | 63.18 | 63.75 | 68.53 | 13.8
GC+${WN}_{all}$ | 38.01 | 73.23 | 64.8 | 62.5 | 68.7 | 13.9
GC+FN | 33.15 | 67.01 | 62.57 | 61.25 | 67.49 | 14.7

  * •

Firstly we note that with similar hyperparameter settings, we obtain an
equivalent performance corresponding to the results given in base paper [3].

  * •

We obtain major improvements in word similarity tasks (columns 1-3 in table 4)
when implementing retrofitting with all the top-down knowledge databases with
the exception of FrameNet. In fact, with the use of FrameNet on any evaluation
task, we seldom observe any improvements (in some cases, we observe decrease
in performance) with retrofitting. Faruqui et al. cite the reason being that
the nature of top-down knowledge captured by FrameNet between words is very
abstract and distantly related, (for example, prominent and eye-catching).

  * •

In the case of TOEFL task, we observe very high improvements in accuracy, the
highest increment $(+13.5\%)$ occurring in the case of GC vectors with PPDB
retrofit.

  * •

We observe modest improvements in the case of Sentiment Analysis task (SA),
the highest increment $(+1.5\%)$ occurring in the case of GC vectors with PPDB
retrofit.

  * •

For the SYN-REL task, we observe improvement in case of GC $(+8.5\%)$ and SG
$(+3.5\%)$ with PPDB retrofit. For all other cases, we always observe drop in
accuracy, the highest reduction occurring in the case of SG vectors with
${WN}_{all}$ retrofit $(-15.1\%)$. As stated by Faruqui et al. in the base
paper [3], the poor performance can be attributed to the fact that SYN-REL is
inherently a syntactic task and by retrofitting, we only incorporate semantic
information in the vector representations.

  * •

Finally, we note that retrofitting with PPDB database gives the best
improvement across a majority of the combinations (Word vector set $+$
Evaluation task), followed by ${WN}_{all}$, same as in the case of results
provided in the base paper [3].

##  4 Retrofitting in ESA

We aim to incorporate the knowledge of Inter-Relatedness between Wikipedia
Articles into ESA to improve the quality of vector representation of
words/texts and we use the method of retrofitting to accomplish this.
Specifically, we retrofit the concept vectors which are vector embedding in
the space of words using a graph like network consisting of the inter-
relatedness knowledge. So, once we arrive at the representation of Wikipedia
Articles as an attribute vector of words by using the TFIDF scheme, we
incorporate semantic relatedness in vector embedding of Articles by means of
an information which is the inter-relatedness of documents/articles in
Wikipedia.

There might be cases where two words can be related to each other even if they
appear separately in related articles rather than co-occurring in the same
articles which would then indicate that the word vectors have different
representations and thus not similar. This is one of the drawbacks of ESA
which we hope to overcome by incorporating our proposed methodology to enhance
the word vectors obtained through ESA. Note that in the paper Non-Orthogonal
Explicit Semantic Analysis (NESA) [18] the authors also try to overcome a
similar problem but one of the key difference is that they accomplish this by
making each concept dimension further constitute a semantic vector built in
another vector space. Whereas we aim to solve this by retaining the concept
space by using retrofitting as a means to enhance the vector representations
using knowledge from Inter-Relatedness Graph obtained from Wikipedia corpus.

If we view an article in Wikipedia, in its description or content, there are
links to other related Wikipedia articles in sections such as “References”,
“See also” etc. and also as hyperlinks within the body of the Article. This
type of information is particularly disregarded when working with plain text
in ESA. We incorporate this form of knowledge so that closely interrelated
concepts are nearer in their corresponding weighted representation of words.
For the purpose of retrofitting, we build a graph-like data structure
representing the inter related concepts explained in detail in Section 4.1.
Then we modify the weighted representation of words for each concept to be
nearer to itself as well as its related concepts using the same optimization
criteria arrived at by Faruqui et al. (refer Equations 1 and 2).

###  4.1 Wikipedia Inter-Relatedness Graph (Wiki Graph)

The information that we are willing to retrofit the tf-idf matrix in ESA with
is the inter relatedness between Wikipedia Articles. To do so, we construct an
undirected graph with nodes as Article Titles and edges are constructed
between two inter-related Articles. Inter-related articles could be viewed as
the Top-Down Knowledge that we are incorporating into ESA while ESA is
primarily a Bottom-Up approach. The main source of inter-relatedness is the
hyperlinks present in each article which in Wikipedia parlance is called as
internal links or wikilinks.

Figure 1: A snapshot of a tiny portion of Wikipedia inter-relatedness graph
(undirected) showing nodes as Wikipedia Articles and edges between two related
Articles.

There are two ways that we have come up with to generate the Graph. One is
based on prior work by [17] where they extract the hyperlinks from main text
in each article which leads to another article within Wikipedia. The graph
constructed this way is done so for the entire Wikipedia corpus at specific
timestamps across history and we’ll be using the latest one for our task in
hand. Based on the corpus or the subset of the Wikipedia Articles that we are
working with in ESA we extract the corresponding subgraph containing relations
between only these Articles. Another method that we have come up with and is
more efficient is directly parsing them from the raw data that we extract as
dump from Wikipedia111https://en.wikipedia.org/wiki/Special:Export. Each
Article is represented as a json object and the key links_out contains as
value a list of all Wikipedia Articles that have been referred to from the
original Article. Using this we reconstruct the graph from scratch specific to
the dataset or corpus in hand. This is faster in computation time since we are
constructing from base-up only what we need instead of having to reduce a much
bigger graph. Note that for purposes of reusing the graph, we save them in the
form of an adjacency list in text file in local storage.

Owing to space complexity, as discussed in later sections for storing the tf-
idf matrix, we are working with a smaller corpus derived for each specific
Evaluation task and the structure for the corresponding graphs is summarized
in Table 5. A tiny portion of the graph is illustrated in Figure 1 to show how
Articles may be inter-related with each other.

Table 5: Summary of the extracted Graphs corresponding to each corpus that we use in ESA. Articles per word denotes the number of articles extracted to form the corpus for each word in the dataset corresponding to the Evaluation Task Evaluation Task |  | Articles
---
per word
Articles | Nodes | Edges |  | % Wikipedia
---
x$(10^{-2})$
WS-353 | 5 | 2133 | 1459 | 4637 | 5.3325
6 | 2555 | 1768 | 5626 | 6.3875
RG-65 | 10 | 498 | 150 | 179 | 1.245
20 | 993 | 338 | 445 | 2.4825
MEN-3k | 3 | 2337 | 1775 | 8225 | 5.8425
5 | 3972 | 3026 | 13244 | 9.93
WikiLink Graphs | - | 13,685,337 | 13,685,337 | 163,380,007 | -

###  4.2 Implementation

In Appendix 1, we present a detailed example for implementation of our
proposal. An overview is provided in the subsequent paragraphs.

Firstly, the Wikipedia concepts are represented as vector embedding using
words as dimensions with the TF-IDF scheme following the procedure for
representing concepts in word space given in [1]. Then, the graph required for
representing the inter-relatedness of Wikipedia concepts are extracted.
(Section 5.1).

The TF-IDF vectors for each concept are now retrofitted in the same fashion as
in [3] wherein the objective function (Eq. 1) is minimized thus resulting in
the update step (Eq. 2).

Note that owing to the limited availability of compute resources at the time,
the TF-IDF matrix for the entire Wikipedia corpus as well as an alternate
(simple-Wikipedia) were unable to fit in the physical memory. Hence, we work
on a much smaller corpora and extract a subset of Wikipedia pertaining to each
Evaluation task (see Section 4.3) we consider at hand. To do so we take the
top n articles obtained by searching for each word used in the Evaluation
dataset and then compiled them together.

###  4.3 Evaluation

We evaluate our model on word similarity tasks using the standard benchmarks
WS-353, RG-65, and MEN-3k dataset [10, 11, 12]. A summary of vector
characteristics along with performance of our model on the benchmarks are
given in Table 6.

####  4.3.1 Word Vectors

Owing to the computational constraints for testing our hypothesis on the
entire Wikipedia corpus we derive much smaller subsets of Articles from
Wikipedia separately for each corresponding Evaluation Benchmarks that we
perform. For a given evaluation dataset, we extract the relevant concepts for
each of the words present in the dataset from Wikipedia corpus using the
Wikipedia library by uniformly extracting a fixed number of concepts for each
word. We then construct our corpus wherein words are expressed in the space of
the extracted concepts alone. For example, RG-65 (10) implies that for each
word in RG-65 dataset top 10 concepts are extracted (See Column 1, Table 6).
Thus,in the resulting corpus, words are expressed in the $498$ dimensional
space of concepts (Total extracted concepts = $2*65*10=1300$ of which $498$
are unique, see Column 2, Table 6). As in ESA [1], we express a concept in
terms of words occurring in corresponding Wikipedia article using TF-IDF
scheme. Refer Columns 1-3, Table 6 regarding the vector characteristics used
for evaluation.

Once we obtain the Word/Concept matrix, we apply retrofitting on concept
vectors using the top-down knowledge graph specific to the evaluation dataset
(refer Table 5 and Section 4.1). Results of our model implementation are
summarized in the next section.

###  4.4 Results

We ran experiments for each of the evaluation task coupled with it’s
characteristic corpus as shown in Table 5 and report the Spearman’s Rank
correlation coefficient. To validate our hypothesis we compare the before and
after results by first performing Explicit Semantic Analysis for each task
followed by introducing inter-relatedness between Wikipedia Articles using
Retrofitting. These are summarized in Table 6.

Note that all absolute values of performance measures are scaled to $100$ and
reported as percentages. Overall from the several Benchmarks that we ran we
are able to show that on average using our method results in improvements in
the performance measure.

  * •

For the WS-353 Benchmark, we observe an increase in performance by 2% and 1%
for WS-353 (5) and WS-353 (6) variants where the number inside brackets denote
the number of Articles used to form the corpus per word in the Benchmark
dataset.

  * •

We observe the highest improvement $(+3.01\%)$ in case of RG-65 (20) (RG-65
Benchmark with top 20 concepts extracted for each word).

  * •

Whereas, in the case of MEN-3k Evaluation task we observe a slight improvement
$(+0.15\%)$ as well as a slight decrement $(-0.39\%)$ in performance compared
to vanilla ESA for the MEN-3k (5) and MEN-3k (3) variants respectively. Hence,
we can say that there is not enough evidence that our methodology provides
performance improvements when measured using the MEN-3k Benchmark task.
Furthermore for most of the tasks the performance increases by $1-2\%$ unless
when there is no clear improvements.

Next, we compare our model performance against results obtained with Glove,
Skip Gram (SG) and Global Context (GC) word vectors on word similarity tasks
before and after retrofitting with different semantic lexicon databases (See
Table 4 and refer Section 3.4). Note that we use a much smaller percentage of
Wikipedia compared to the case of Glove and GC vectors owing to computational
constraints (See Table 5) and has to be taken into account while comparing the
results.

  * •

On the MEN-3k benchmark, the highest correlation obtained by our methodology
is $66.06\%$ (MEN-3k (3) $+$ Wiki Graph). Our results (ESA with Wiki Graph
retrofit) are better than the performance of GC with/without retrofit by a
margin of atleast $+22.55\%$. However, both SG and Glove vectors with/without
retrofit give better results compared to our model (minimum and maximum
margins of $+3.82\%$ and $+10.3\%$ respectively).

  * •

Evaluating our model on RG-65 benchmark, the highest correlation obtained is
$81.30\%$ (RG-65 (10) $+$ Wiki Graph). Once again, we produce better results
with our model (ESA with Wiki Graph retrofit) than GC with/without retrofit by
a margin of atleast $+8.07\%$. Performances of SG and Glove are similar to our
model except in the case of retrofit with either PPDB or ${WN}_{all}$, wherein
large improvements and higher correlations $(85\%)$ are observed (refer
Section 3.4).

  * •

On the WS-353 benchmark, we obtain $56.84\%$ correlation (WS-353 (5) $+$ Wiki
Graph). With the exception of Glove $+$ FrameNet retrofit, all combinations of
word vectors $+$ top-down database retrofit result in superior performance
than our model (minimum and maximum decrements of $-4.01\%$ and $-14.44\%$
respectively).

Table 6: Results of absolute performance obtained using ESA and with our proposed methodology. The metric for performance is Spearman’s Rank correlation coefficient (Higher scores are always better). Note that for a given evaluation task, we use different corpora. For example, in case of WS-353 (5), words are expressed in the $2133$ dimensional space of concepts (Column 2, Row 1 ), which is a consequence of constructing corpora by extracting only the top 5 concepts for each word. | Evaluation
---
Task
| Dimension
---
(Word Vector )
| Dimension
---
(Concept Vector)
| Vector
---
Representation
Performance
WS-353 (5) | 2133 | 162080 | ESA | 54.61
(+) Wiki Graph | 56.84
WS-353 (6) | 2555 | 183592 | ESA | 55.12
(+) Wiki Graph | 55.97
RG-65 (10) | 498 | 44401 | ESA | 79.84
(+) Wiki Graph | 81.30
RG-65 (20) | 993 | 67881 | ESA | 78.09
(+) Wiki Graph | 81.10
MEN-3k (3) | 2337 | 189661 | ESA | 66.45
(+) Wiki Graph | 66.06
MEN-3k (5) | 3972 | 253207 | ESA | 62.86
(+) Wiki Graph | 63.01

###  4.5 Conclusion & Future Work

Through our work in this project we first try to implement a paper conveying
an elegant solution for enhancing Word Vectors by means of Retrofitting
Semantic Lexicons to gain a deeper understanding on the subject and how
research ideas are formed, hypothesized and evaluated. Then we propose our
idea of improving the vector embeddings obtained through Explicit Semantic
Analysis by introducing top-down knowledge of Inter-Relatedness between
Wikipedia Articles. We do so by constructing this knowledge using the network
of hyperlinks in each Article linking other similar Articles to itself and
formally storing it as an undirected graph with nodes as Articles and Edges
showing Inter-relatedness between two such nodes.

Based on our experimentation with different datasets, each evaluated using
different Benchmarks, we show that this methodology in fact leads to decent
improvements in performance of the Word vectors obtained after introducing
this top-down knowledge by means of Retrofitting to ESA. In the future this
could be extended to much larger dataset like the entire Wikipedia corpus to
possibly realize state of the art results since we had to limit ourselves to
smaller subsets of Wikipedia corpus owing to the computational constraints
present at the moment.

## References

  * [1] E. Gabrilovich and S. Markovitch. Computing semantic relatedness using Wikipedia-based explicit semantic analysis. In Proceedings of the 20th International Joint Conference onArtiﬁcial Intelligence (IJCAI 07), pages 1606–1611. MorganKaufmann Publishers Inc., 2007.
  * [2] Egozi, O., Markovitch, S., Gabrilovich, E.: Concept-Based Information Retrieval using Explicit Semantic Analysis. ACM Transactions on Information Systems 29(2), 8:1–8:34 (2011)
  * [3] Manaal Faruqui, Jesse Dodge, Sujay K. Jauhar, Chris Dyer, Eduard Hovy, and Noah A. Smith. 2015a. Retrofitting word vectors to semantic lexicons. In Proceedings of NAACL.
  * [4] Juri Ganitkevitch, Benjamin Van Durme, and Chris Callison-Burch. 2013. PPDB: The paraphrase database. In Proceedings of NAACL.
  * [5] George A Miller. 1995. Wordnet: a lexical database for english. Communications of the ACM.
  * [6] Collin F. Baker, Charles J. Fillmore, and John B. Lowe. 1998. The berkeley framenet project. ACL ’98.
  * [7] Jeffrey Pennington, Richard Socher, and Christopher D. Manning. 2014. Glove: Global vectors for word representation. In Proceedings of EMNLP.
  * [8] Tomas Mikolov, Kai Chen, Greg Corrado, and Jeffrey Dean. 2013a. Efficient estimation of word representations in vector space. arXiv preprint arXiv:1301.3781.
  * [9] Eric H Huang, Richard Socher, Christopher D Manning, and Andrew Y Ng. 2012. Improving word representations via global context and multiple word prototypes. In Proceedings of ACL.
  * [10] Lev Finkelstein, Evgeniy Gabrilovich, Yossi Matias, Ehud Rivlin, Zach Solan, Gadi Wolfman, and Eytan Ruppin. 2001. Placing search in context: the concept revisited. In WWW, New York, NY, USA.
  * [11] Herbert Rubenstein and John B. Goodenough. 1965. Contextual correlates of synonymy. Commun. ACM, 8(10):627–633, October.
  * [12] Elia Bruni, Gemma Boleda, Marco Baroni, and Nam-Khanh Tran. 2012. Distributional semantics in technicolor. In Proceedings of ACL.
  * [13] Thomas K Landauer and Susan T. Dumais. 1997. A solution to plato’s problem: The latent semantic analysis theory of acquisition, induction, and representation of knowledge. Psychological review.
  * [14] Richard Socher, Alex Perelygin, JeanWu, Jason Chuang, Christopher D. Manning, Andrew Y. Ng, and Christopher Potts. 2013. Recursive deep models for semantic compositionality over a sentiment treebank. In Proceedings of EMNLP
  * [15] Tomas Mikolov, Wen-tau Yih, and Geoffrey Zweig. 2013b. Linguistic regularities in continuous space word representations. In Proceedings of NAACL.
  * [16] Omer Levy and Yoav Goldberg. 2014. Linguistic regularities in sparse and explicit word representations. In Proceedings of CoNLL.
  * [17] Consonni, C., Laniado, D. and Montresor, A., 2019, July. WikiLinkGraphs: A complete, longitudinal and multi-language dataset of the Wikipedia link networks. In Proceedings of the International AAAI Conference on Web and Social Media (Vol. 13, No. 01, pp. 598-607).
  * [18] N. Aggarwal, K. Asooja, G. Bordea, and P. Buitelaar. Non-orthogonal explicit semantic analysis. Lexical and Computational Semantics (* SEM 2015), pages 92–100, 2015

##  5 Appendix 1

In this section, we provide a step by step walkthrough for implementing our
proposal (refer Section 4). Consider the piece of text, ”Information Retrieval
in Search Engines”. After pre-processing the query we are left with the
following list of tokens:

[’information’, ’retrieval’, ’search’, ’engines’]

In short, we are left with 4 word tokens:

[$w_{1}$, $w_{2}$, $w_{3}$, $w_{4}$]

In ESA [1], we represent each of these words as vectors in the concept space.
We first retrieve the concepts corresponding to the words using inverted
indexing. We then represent the concepts themselves as a vector embedding of
the words which occur in the article corresponding to the concept (concept is
just the title of article).

$\displaystyle w_{1}$ $\displaystyle\rightarrow doc_{1},\ doc_{2},\ doc_{3}$
$\displaystyle w_{2}$ $\displaystyle\rightarrow doc_{1},\ doc_{2}$
$\displaystyle w_{3}$ $\displaystyle\rightarrow doc_{1},\ doc_{3}$
$\displaystyle w_{4}$ $\displaystyle\rightarrow doc_{2},\ doc_{3}$

Thus, $w_{2}$ occurs in ${doc}_{1}$, ${doc}_{2}$, words
$w_{1}$,$w_{2}$,$w_{3}$ occur in ${doc}_{1}$ and so on. This relationship is
captured in the word/concept matrix (refer 7).

Table 7: Word/Concept Matrix. Concepts can be expressed in 4 dimensional space of words occuring in corresponding articles. Consequently, words can be expressed in 3 dimensional space of concepts. Word/Concept | $c_{1}$ | $c_{2}$ | $c_{3}$
---|---|---|---
$w_{1}$ | ${tf}_{1}^{1}$ | ${tf}_{1}^{2}$ | ${tf}_{1}^{3}$
$w_{2}$ | ${tf}_{2}^{1}$ | ${tf}_{2}^{2}$ | $0$
$w_{3}$ | ${tf}_{3}^{1}$ | $0$ | ${tf}_{3}^{3}$
$w_{4}$ | $0$ | ${tf}_{4}^{2}$ | ${tf}_{4}^{3}$

The concept vectors $\vec{c_{1}},\ \vec{c_{2}},\ \vec{c_{3}}$ corresponding to
${doc}_{1},\ {doc}_{2},\ {doc}_{3}$ respectively are vectors of words:

$\displaystyle\vec{c_{1}}$ $\displaystyle={tf}_{1}^{1}\
\hat{w_{1}}+{tf}_{2}^{1}\ \hat{w_{2}}+{tf}_{3}^{1}\ \hat{w_{3}}$
$\displaystyle\vec{c_{2}}$ $\displaystyle={tf}_{1}^{2}\
\hat{w_{1}}+{tf}_{2}^{2}\ \hat{w_{2}}+{tf}_{4}^{2}\ \hat{w_{4}}$
$\displaystyle\vec{c_{3}}$ $\displaystyle={tf}_{1}^{3}\
\hat{w_{1}}+{tf}_{3}^{3}\ \hat{w_{3}}+{tf}_{4}^{3}\ \hat{w_{4}}$

$c_{i}\implies i^{th}\ concept$
$w_{i}\implies i^{th}\ word$
$tf_{i}^{j}\implies TF-IDF\ of\ i^{th}\ word\ occurring\ in\ j^{th}\ document$

Applying retrofitting (using Eqn. 1 and 2 from Section 3) to these concept
vectors based on the Wikipedia graph, we get,

$\displaystyle\vec{c_{1}}$ $\displaystyle={{tf}_{1}^{1}}^{R}\
\hat{w_{1}}+{{tf}_{2}^{1}}^{R}\ \hat{w_{2}}+{{tf}_{3}^{1}}^{R}\ \hat{w_{3}}$
$\displaystyle\vec{c_{2}}$ $\displaystyle={{tf}_{1}^{2}}^{R}\
\hat{w_{1}}+{{tf}_{2}^{2}}^{R}\ \hat{w_{2}}+{{tf}_{4}^{2}}^{R}\ \hat{w_{4}}$
$\displaystyle\vec{c_{3}}$ $\displaystyle={{tf}_{1}^{3}}^{R}\
\hat{w_{1}}+{{tf}_{3}^{3}}^{R}\ \hat{w_{3}}+{{tf}_{4}^{3}}^{R}\ \hat{w_{4}}$

${tf_{i}^{j}}^{R}\implies TF-IDF\ of\ i^{th}\ word\ occurring\ in\ j^{th}\
document\ after\ retrofit$
For example, ${{tf}_{1}^{3}}^{R}$ is the vector weight of $w_{3}$ occurring in
$doc_{1}$ after retrofit.

Ultimately, we arrive at the word vectors
$\vec{w_{1}},\vec{w_{2}},\vec{w_{3}},\vec{w_{4}}$ which are vector embedding
in the concept space, where the weight (magnitude) corresponding to concept
$\vec{c_{j}}$ for $i^{th}$ word vector $\vec{w_{i}}$ is given by
${tf_{i}^{j}}^{R}$. Thus,

$\displaystyle\vec{w_{1}}$ $\displaystyle={{tf}_{1}^{1}}^{R}\
\hat{c_{1}}+{{tf}_{1}^{2}}^{R}\ \hat{c_{2}}+{{tf}_{1}^{3}}^{R}\ \hat{c_{3}}$
$\displaystyle\vec{w_{2}}$ $\displaystyle={{tf}_{2}^{1}}^{R}\
\hat{c_{1}}+{{tf}_{2}^{2}}^{R}\ \hat{c_{2}}$ $\displaystyle\vec{w_{3}}$
$\displaystyle={{tf}_{3}^{1}}^{R}\ \hat{c_{1}}+{{tf}_{3}^{3}}^{R}\
\hat{c_{3}}$ $\displaystyle\vec{w_{4}}$ $\displaystyle={{tf}_{4}^{2}}^{R}\
\hat{c_{2}}+{{tf}_{4}^{3}}^{R}\ \hat{c_{3}}$

Therefore, the resultant vector embedding in the concept space for
”Information Retrieval in Search Engines” is given by,

$\displaystyle\vec{W}$
$\displaystyle=\vec{w_{1}}+\vec{w_{2}}+\vec{w_{3}}+\vec{w_{4}}$
$\displaystyle\vec{W}$
$\displaystyle=({{tf}_{1}^{1}}^{R}+{{tf}_{2}^{1}}^{R}+{{tf}_{3}^{1}}^{R})\hat{c_{1}}$
$\displaystyle+({{tf}_{1}^{2}}^{R}+{{tf}_{2}^{2}}^{R}+{{tf}_{4}^{2}}^{R})\hat{c_{2}}$
$\displaystyle+({{tf}_{1}^{3}}^{R}+{{tf}_{3}^{3}}^{R}+{{tf}_{4}^{3}}^{R})\hat{c_{3}}$


# Random singlet-like phase of disordered Hubbard chains

Josephine Yu1, Hong-Chen Jiang2, Rahul Nandkishore3,4, S. Raghu4,2 1
Department of Applied Physics, Stanford University, Stanford, CA 94305, USA  2
Stanford Institute for Materials and Energy Sciences, SLAC National
Accelerator Laboratory, Menlo Park, CA 94025, USA  3 Department of Physics and
Center for Theory of Quantum Matter, University of Colorado, Boulder, CO,
80309, USA  4 Department of Physics, Stanford University, Stanford, CA 94305,
USA

###### Abstract

Local moment formation is ubiquitous in disordered semiconductors such as
Si:P, where it is observed both in the metallic and the insulating regimes.
Here, we focus on local moment behavior in disordered insulators, which arises
from short-ranged, repulsive electron-electron interactions. Using density
matrix renormalization group and strong-disorder renormalization group
methods, we study paradigmatic models of interacting insulators: one
dimensional Hubbard chains with quenched randomness. In chains with either
random fermion hoppings or random chemical potentials, both at and away from
half-filling, we find exponential decay of charge and fermion 2-point
correlations but power-law decay of spin correlations that are indicative of
the random singlet phase. The numerical results can be understood
qualitatively by appealing to the large-interaction limit of the Hubbard
chain, in which a remarkably simple picture emerges.

##  I Introduction

A fundamental challenge in condensed matter physics has been to understand the
implications of local moment formation in disordered electronic systems. Local
moments have long been observed in disordered semiconductors, such as Si:P and
Si:P,B, where they crucially affect the thermodynamicPaalanen  _et al._ (1986,
1988) and dynamicalMahmood  _et al._ (2021) properties in both the metallic
and the insulating regimes Bhatt  _et al._ (1988); Belitz and Kirkpatrick
(1994); Miranda and Dobrosavljević (2005). If the properties of the phases
themselves are altered by local moments, it follows that at least in
principle, local moments can influence the universal behavior near metal-
insulator quantum phase transitionsRosenbaum  _et al._ (1980); Sarachik  _et
al._ (1998); Kravchenko and Sarachik (2003).

Much work has been done in understanding magnetic excitations in the metallic
regime of disordered systemsFinkelstein (1990); Belitz and Kirkpatrick (1994).
In two spatial dimensions, for instance, there is a tendency towards a
magnetic instability even in the weak disorder limit, far from a putative
metal-insulator transitionFinkel’stein (1983); Andreev and Kamenev (1998);
Chamon  _et al._ (1999); Kamenev and Andreev (1999); Chamon and Mucciolo
(2000); Nosov  _et al._ (2020). By contrast, since the early seminal work of
Bhatt, Lee and Co-workersBhatt and Lee (1982); Zhou  _et al._ (2009),
considerably less attention has been devoted to local moment behavior in the
insulating regime.

Assuming the existence of local moments in the insulator, the theory of Bhatt
and Lee Bhatt and Lee (1982) establishes the tendency towards random singlet
formation, due to an exponentially broad distribution of antiferromagnetic
exchange interactions among the local moments. Later work by Bhatt and Fisher
Bhatt and Fisher (1992) pushed this picture further, into the metallic regime,
arguing that local moments essentially decouple from the metallic electrons,
due to vanishingly small Kondo temperatures. Nevertheless, it remains unclear
how such behavior emerges from electrons in a random landscape, in the
presence of short-ranged interactions.

In this paper, we report some progress in this direction and analyze models of
electrons in the presence of both short-range interactions and strong
disorder. Given our focus on the insulating state, we study one dimensional
models, in which the tendencies towards insulating ground states are
strongest. We are especially interested in the behavior away from half-
filling, where at least microscopically, a description in terms of local
moments alone is not justified a priori.

Using a combination of density matrix renormalization group simulations and
real space renormalization group techniques, we demonstrate that the Hubbard
chain exhibits random singlet behavior both at and away from half-filling. Our
conclusion holds for both random potentials (site disorder) and random
hoppings (bond disorder). To make intuitive sense of our results, we appeal to
the strong-interaction limit of the Hubbard model and account for our results
in terms of spin-charge separationOgata and Shiba (1990): nearly-free holes
exhibit Anderson localization, while spins experience random Heisenberg
exchange, resulting in random singlet formation along the lines of Bhatt and
Lee.

##  II Model

The simplest effective Hamiltonian governing electrons with disorder and
short-range interactions is the Hubbard model with randomness:

$H=-\sum_{i,\sigma}t_{i}(c_{i,\sigma}^{\dagger}c_{i+1,\sigma}+\text{h.c.})+\sum_{i}\mu_{i}n_{i}+U\sum_{i}n_{i\uparrow}n_{i\downarrow},$
(1)

where $c^{\dagger}_{i\sigma}(c_{i\sigma})$ creates(destroys) an electron with
spin $\sigma=\uparrow,\downarrow$, on lattice site $i$, the density operator
on site $i$ is $n_{i}=\sum_{\sigma}c^{\dagger}_{i\sigma}c_{i\sigma}$. Onsite
interactions are taken to be repulsive: $U>0$, $t_{i}$ are nearest-neighbor
hopping amplitudes, and $\mu_{i}$ are on-site chemical potentials.

Figure 1:  Density-density, fermion 2-point and pair-field correlation
functions on $L=144$ Hubbard chains at electron filling $n=11/12$ with random
potential, random hopping as well as for the disorder-free systems.
Additionally, results for the non-interacting ($U=0$) but site-disordered
system are shown. All data is shown for $r\leq L/2$. The random-potential
system has parameters $U=6\overline{t}$, $W_{\mu}/2=3\overline{t}$, $W_{t}=0$;
the random-hopping system has parameters $U=12\overline{t}$, $W_{\mu}=0$,
$W_{t}/2=0.65\overline{t}$; the clean system has parameters $U=6\overline{t}$,
$W_{\mu}=W_{t}=0$; and the Anderson system has $U=0$,
$W_{\mu}/2=3\overline{t}$, $W_{t}=0$. (a) Disorder-averaged density-density
correlations $\overline{C_{n}(r)}$ (Eq. 2) decay exponentially with distance
$r$ for the disordered systems, as opposed to decaying with a power law as in
the clean system. (b) Electron 2-point functions decay exponentially in all
disordered systems. (c) Superconducting pair-field correlations decay
exponentially in the disordered systems, in direct contrast to the power-law
decay in the non-disordered Luttinger liquid.

We study two types of quenched randomness: site disorder, where the local
chemical potentials $\mu_{i}$ are random, and bond disorder, where $t_{i}$ are
random. We choose $\mu_{i}$ from a uniform distribution of mean
$\overline{\mu}=0$ and width $W_{\mu}$. Similarly, $t_{i}$ are chosen from a
uniform distribution of mean $\overline{t}$ and width $W_{t}$. We choose
$\overline{t}=1$, setting this to be the unit of energy.

In the absence of randomness, the model is integrable and has of course been
thoroughly studiedArovas  _et al._ (2021); the half-filled system is well-
described by a $S=1/2$ Heisenberg antiferromagnet, and the doped chain
exhibits Luttinger liquid behavior over a range of electron concentrations and
interaction strengths. With perturbatively weak disorder, it is known that the
Luttinger liquid tends towards localization and that repulsive interactions
enhance this tendency Giamarchi and Schulz (1987). At half-filling, bond
disorder preserves particle-hole symmetry, and a spin chain with random
antiferromagnetic exchange accurately captures the low energy behavior of the
system. It has been well-established that the latter results in an infinite
randomness fixed point with random singlet behavior Ma  _et al._ (1979);
Fisher (1994). We wish to explore the fate of the Hubbard chain with site and
bond randomness, both at and away from half-filling where such a description
in terms of spin alone is not necessarily valid.

##  III DMRG Results

We analyze Hubbard chains described by Eq. 1 at two fixed electron filling
fractions, $n=1$ (half-filling) and $n=11/12$, using the density matrix
renormalization group (DMRG)White (1992, 1993) procedure. We perform all of
the simulations in the strong-interaction regime, $U=12\overline{t}$ for the
random-hopping chains or $U=6\overline{t}$ for the random-potential chains.
The DMRG algorithm works well for both clean and weakly disordered systems.
For a more reliable study, we first obtain the ground state of a system with
weak disorder, then quasi-adiabatically increase the disorder strength,
adaptively increasing sweep number and the number of basis states kept, until
the resulting well-converged ground state has been obtained. A similar
procedure has been used before in DMRG to treat disordered systems Xavier  _et
al._ (2018). In the present study, we perform up to 50 sweeps and keep up to
$m=1024$ states with a typical truncation error $\epsilon\sim 10^{-9}$. For
all parameters, we sample at least 300 independent disorder realizations.

To characterize the ground state properties, we calculate various equal-time
correlation functions over the interior half of the chain (from site L/4 to
3L/4), to minimize the boundary effects. We focus on measures of the charge
and spin behaviors, probed through the charge density-density fluctuation
correlation function

$C_{n}(r)\equiv\expectationvalue{(n(x)-\expectationvalue{n(x)})(n(x+r)-\expectationvalue{n(x+r)})}$
(2)

and the spin-spin correlation function

$C_{\sigma}(r)\equiv\expectationvalue{S(x)S(x+r)}$ (3)

respectively, where $r$ is the displacement between two sites along the chain
and $x=L/4$ is a fixed reference point. We also measure the fermion 2-point
function

$G(r)\equiv\expectationvalue{c_{\uparrow}^{\dagger}(x)c_{\uparrow}(x+r)},$ (4)

where the choice of the up-spin does not matter due to the spin $SU(2)$
symmetry.

Lastly, the superconducting pair-field correlation function is defined as

$D(r)\equiv\expectationvalue{\Delta^{\dagger}(x)\Delta(x+r)}$ (5)

where
$\Delta(y)\equiv(c_{\downarrow}^{\dagger}(y)c_{\uparrow}^{\dagger}(y)-c_{\uparrow}^{\dagger}(y)c_{\downarrow}^{\dagger}(y))/\sqrt{2}$
is the spin-singlet pair creation operator on bond $y$.

We start by presenting evidence for charge localization in the disordered
systems away from half-filling. In the absence of disorder, the $n=11/12$
system has a Luttinger liquid ground state. We observe that disorder,
regardless of type, localizes the charges. Fig. 1 shows exponential decay of
the disorder-averaged density-density correlation function
$\overline{C_{n}(r)}$ (Eq. 2), the fermion 2-point function $\overline{G(r)}$
(Eq. 4), and the superconducting pair-field correlation function
$\overline{D(r)}$ (Eq. 5) of the ground state in the disordered systems as a
function of distance $r$, indicating a gap to charge excitations.
Qualitatively, the behaviors of these correlation functions in the random-
potential and random-hopping systems are very similar, both differing
significantly from the power-law correlations expected for the disorder-free
system. These correlation functions for both site and bond randomness resemble
those of an Anderson insulator.

Figure 2:  Disorder-averaged spin-spin correlation $\overline{C_{\sigma}(r)}$
(Eq. 3) for $L=144$-site chains. Data is shown for distances $r\leq L/3$ , as
fluctuations increase with distance. Black dashed lines indicate $~{}r^{-2}$
decay. (a) Random Heisenberg chain with spin exchange couplings derived
perturbatively from the Hubbard parameters (yellow) and random-hopping Hubbard
chain at half-filling (purple) (see main text). In both chains,
$\overline{C_{\sigma}(r)}$ exhibits long-distance $r^{-2}$ behavior beyond
around 10 lattice sites. (b) Spin correlations in random-hopping and random-
potential systems at $n=11/12$ electron filling also exhibit decay close to
$r^{-2}$. Insets show $r^{2}\cdot\overline{C_{\sigma}(r)}$ exhibiting stable
long-distance random-singlet-like behavior, and solid lines in the insets are
average values of $r^{2}\cdot\overline{C_{\sigma}(r)}$ at long distances.

We now turn to analyzing the disorder-averaged spin-spin correlation functions
$\overline{C_{\sigma}(r)}$ (Eq. 3) for different chains: random-potential and
random-hopping Hubbard chains as well as the random-exchange Heisenberg
antiferromagnetic chain. Despite the presence of disorder, we find that the
spin $SU(2)$ symmetry is preserved in $C_{\sigma}(r)$ of each disorder
realization and therefore in the disorder-averaged correlation function
$\overline{C_{\sigma}(r)}$.

The half-filled system is unsurprisingly a Mott insulator, since the repulsive
interaction $U$ is the dominant energy scale. The effective low-energy
description of the half-filled, large-interaction systems should then be
equivalent to the random Heisenberg antiferromagnet. The results for the
random Heisenberg chain and the random-hopping Hubbard chain at half-filling
are shown in Fig. 2a, and the agreement between them reflects this intuition.
Our results agree also with previous studies of the disordered Hubbard chain
at half-filling Mélin and Iglói (2006). We note that the random-potential
chain at half-filling exhibits some curious charge behavior at weak disorder
that can be understood through a particle-hole transformation (see Appendix).
At large distances $r$, the spin correlations in both the half-filled Hubbard
chain with random hoppings and the random Heisenberg chain exhibit decays
close to $1/r^{2}$, the expected behavior in a random singlet phase. More
surprisingly, Fig. 2b shows that $\overline{C_{\sigma}(r)}$ in the random
hopping and random potential systems away from half-filling decay at large
distances $r$ as a power law close to $1/r^{2}$, indicating that the spin
order, both at and away from half-filling, are random-singlet-like. Away from
half-filling, statistical fluctuations decrease more slowly with sample
number, as electron configurations must now be taken into account.

Another hallmark of the random singlet phase is that the physics is dominated
by rare, long-range singlets. We further characterize the systems studied here
by comparing the disorder-averaged spin-spin correlations
$\overline{C_{\sigma}(r)}$ to the root-mean-square (RMS) spin-spin
correlations $\sqrt{\overline{(C_{\sigma}(r))^{2}}}$. In the random-singlet
phase, both $\overline{C_{\sigma}(r)}$ and $\overline{(C_{\sigma}(r))^{2}}$
are dominated by the probability of forming a singlet of length $r$, which
scales as $1/r^{2}$ at large $r$. One then expects the disorder-averaged spin-
spin correlations to scale as $1/r^{2}$ and the RMS spin-spin correlations to
scale as $1/r$ in the random-singlet phase. By contrast, the two quantities
should agree in the weak-disorder limit, in which rare-region effects can be
ignored. Fig. 6 shows $\overline{C_{\sigma}(r)}$ and
$\sqrt{\overline{(C_{\sigma}(r))^{2}}}$ with behaviors consistent with random-
singlet physics for a random-potential system away from half-filling. Similar
rare region-dominated behavior is found for the random-hopping system.

##  IV Large-Interaction Limit and Numerical SDRG

Our results thus far can be understood qualitatively through the simple
picture of the Bethe Ansatz solution of the (clean) Hubbard chain in the
$U/t\rightarrow\infty$ limit. In this limit, the spins and charges are
decoupled: the holes are free to order $t/U$, and the spins form a Heisenberg
antiferromagnet on electron coordinatesOgata and Shiba (1990). One could thus
expect the decoupled spins and charges to respond independently to the
disorder. Then, the holes undergo Anderson localization, while the spins form
a random singlet on electron coordinates.

Quantitatively, we can explore this perspective by numerically implementing a
strong-disorder renormalization group (SDRG) decimation procedure. We again
study the $n=11/12$ chain with random potential, imposing now that the holes
are localized (as justified by the evidence from DMRG, Fig. 1) at local maxima
in the potential of a given disorder realization so that we are left with an
effective spin model. The effective spin model, a random Heisenberg chain on
electron coordinates, can be computed using perturbation theory in $t/U$. We
then numerically implement the SDRG decimation of the effective random spin
model (see Appendix). Averaging the resulting ground states of many
realizations is equivalent to averaging over all possible positions of the
localized holes, and one recovers random singlet behavior.

Figure 3: Spin correlation statistics in a $L=144$ site chain with random
potential, $W_{\mu}/2=3\bar{t}$. Disorder-averaged spin correlations
$\overline{C_{\sigma}(r)}$ and the root-mean-square spin correlations
$\sqrt{\overline{(C_{\sigma}(r))^{2}}}$ shown for $r\leq L/3$. In the absence
of disorder, the two quantities should be equivalent, but the difference in
the two quantities indicates the importance of rare regions indicative of a
strong disorder fixed point. A dotted line shows a $1/r^{2}$ decay, and a
dashed line shows a $1/r$ decay.  Figure 4:  Results of numerically
implementing bond decimation, assuming charge localization, for a $L=600$-site
chain and $L=6000$-site chain, both with electron density $n=11/12$, averaged
over 5000 realizations. (a) Probability $P_{s}(r)$ of forming a singlet pair
over $r$ lattice sites exhibits $r^{-2}$ scaling. Data is shown for $r\leq
2000$. (b) Probability distribution of $J$, spin exchange energies, in units
of the energy scale $J_{0}=\text{max}(\\{J\\})$ when $50$ free spins remain.
The black line is a $1/J$ distribution normalized over the window of energies
shown. There is strong agreement between the bond distribution found in the
numerical SDRG procedure for the longer chain and the $1/J$ distribution of
spin exchange energies predicted by the infinite-randomness fixed point
corresponding to the random singlet phase.

From the numerical decimation, we obtain the probability $P_{s}(r)$ of forming
a singlet pair separated by a distance of $r$ lattice sites, shown in Fig. 4a
for a $L=600$-site chain and a $L=6000$-site chain at $n=11/12$ electron
density. This probability distribution ultimately determines the long-distance
behavior of the disorder-averaged spin-spin correlations
$\overline{C_{\sigma}(r)}$Hoyos  _et al._ (2007); Fisher (1994). The
probability of forming a singlet pair clearly decays as $r^{-2}$ (dashed black
line) at large $r$, which is consistent with the observations from DMRG.
Furthermore, the probability distribution of bond energies at late times in
the decimation (when only 50 free spins remain) approaches a power-law
distribution agreeing with analytical predictions on the vacancy-free
Heisenberg chain Fisher (1994), as shown in Fig. 4b. Our numerical SDRG
analysis thus indicates that a disordered system with a finite density of
localized holes still exhibits random-singlet-like behavior, corroborating the
results from DMRG and the intuitive picture offered by the large-interaction
limit of the Bethe Ansatz solution.

##  V Discussion

We have explored the ground state properties of Hubbard chains in the presence
of quenched bond and site randomness, both at and away from half-filling. We
find in all cases that disorder localizes charges and gives rise to random
antiferromagnetic spin interactions, ultimately driving the system to a
random-singlet-like phase. These results are consistent with the simple
picture offered by the large-interaction limit of the Bethe Ansatz solution
for the Hubbard chain, in which charges and spins are decoupled and respond
independently to disorder.

Our analysis here is specific to one dimension. In higher dimensions, one has
to also consider the nontrivial effects of lattice geometry, particularly
geometric frustration. Studies of (quasi) two-dimensional disordered spin
systems suggest that geometry, alongside disorder, plays an important role in
determining the spin state Laflorencie (2005); Liu  _et al._ (2018); Ren  _et
al._ (2020); in this case, possibilities include short-range antiferromagnetic
order, random-singlet, and spin glass order.

Since all single particle states are localized in the random Hubbard chain,
the systems we have considered here offer valuable insight to the nature of
the interacting insulator. For the same reason, these models do not allow us
to make contact with the physics of local moments on the metallic side, or
even near the metal-insulator transition. While most one-dimensional models
suffer from the same affliction, higher-dimensional models might allow for
study of metal-insulator transitions, but they prove significantly more
difficult to solve without employing physically-motivated
approximationsMilovanović  _et al._ (1989); Tusch and Logan (1993); Pezzoli
and Becca (2010). Fortunately, one dimensional models with quasiperiodicity as
a proxy for disorder, exhibit single-particle mobility edgesDas Sarma  _et
al._ (1990, 1988); Ganeshan  _et al._ (2015), and remain solvable in the
presence of interactionsLi  _et al._ (2015). Thus, the extent to which their
low energy behavior carries over to their disordered, higher dimensional
counterparts, is an open question that can be investigated with a fair degree
of rigor. We shall report on this in future studies.

##  VI Acknowledgments

We thank V. Dobrosavljevic, S. Kivelson, P. Kumar, E. Miranda, C. Murthy, P.
Nosov, and J.-H. Son for helpful discussions. SR and HCJ were supported by the
Department of Energy, Office of Basic Energy Sciences, Division of Materials
Sciences and Engineering, under Contract No. DE-AC02-76SF00515. JJY was
supported by the National Science Foundation Graduate Research Fellowship
under Grant No. DGE-1656518. RN acknowledges support from the Simons
Foundation through a Simons Fellowship in Theoretical Physics, and from the
Sloan Foundation through a Sloan Research Fellowship. Some of the computing
for this project was performed on the Sherlock cluster.

##  VII Appendix

###  VII.1 System with a random potential at half-filling

At half-filling, the clean Hubbard chain is a Mott insulator due to the
presence of strong interactions, which force the ground state to have only
singly-occupied sites. At half-filling, particle-hole symmetry of the Hubbard
model allows for an interesting distinction between the effects of random
hopping and random potential. The difference is demonstrated in the behaviors
of the charge density-density correlation functions (Fig. 5a) at weak and
strong disorders. While the weak random hopping model is still interaction-
dominated and thus remains similar to the non-random half-filled Hubbard
model, the weak random chemical potential model appears to promote charge
density fluctuation correlations. For a range of $W_{\mu}$ sufficiently small,
the density fluctuation correlations appear to have contrasting short- and
long-distance behaviors. At sufficiently strong disorder, both the random
potential and random hopping models yield exponentially decaying charge
density-density correlations, although with different correlation lengths.

Figure 5: Density-density correlation functions in half-filled Hubbard chains
of length $L=72$ with (a) weak and (b) strong site and bond disorder. In both
cases, density correlations in the chain with random hopping remain dominated
by interactions and exhibit little change as the disorder is increased. In
contrast, the chain with random potential undergoes a transition in behavior
as the disorder increases. The density fluctuation correlations in the chain
with weak random potential appear to have a short-range, interaction dominated
region joined to a disorder-dominated region at large distances. At strong
disorder, the same system becomes dominated by disorder.

This weak-to-intermediate disorder behavior of the half-filled chain may be
understood by analyzing the negative-$U$ Hubbard model. The particle-hole
transformation

$c_{i,\uparrow}^{\dagger}\rightarrow(-1)^{i}d_{i,\uparrow}\quad,\quad
c_{i,\downarrow}\rightarrow d_{i,\downarrow},$ (6)

maps the original Hubbard Hamiltonian in Eq.1 to

$\tilde{H}=-\sum_{i,\sigma}t_{i}\left(d_{i,\sigma}^{\dagger}d_{i+1,\sigma}+\text{h.c.}\right)-\sum_{i}\mu_{i}S_{i}^{z}-U\sum_{i}\tilde{n}_{i\uparrow}\tilde{n}_{i\downarrow},$
(7)

where $\tilde{n}_{i,\sigma}=d_{i,\sigma}^{\dagger}d_{i,\sigma}$. Comparing Eq.
7 and Eq. 1, we see that the spin and charge sectors effectively swap roles
($n_{i}\rightarrow S_{i}^{z}$) and that the sign of the interaction has
flipped ($U\rightarrow-U$). The disorder in hopping remains bond disorder, but
the disorder in chemical potential is transformed to a random magnetic field.
Correspondingly, the charge density-density correlator (Eq. 2) of the $U>0$
model is transformed into a spin-spin correlator
$\expectationvalue{S_{z}(x)S_{z}(x+r)}$ in the $U<0$ model.

In the large interaction limit $|U/t|\gg 1$, the ground state of the $U<0$
Hubbard model is superconducting. The $s$-wave superconducting state has a gap
to spin excitations (interaction-dominated) and is stable to bond disorder.
However, it is unstable to magnetic disorder. In the presence of weak magnetic
disorder, the superconducting state retains the interaction-dominated behavior
at short distances but shows evidence of pair-breaking at longer lengths,
shown in Fig. 5a. For stronger magnetic disorder (Fig. 5b), the spins of the
$U<0$ problem are completely locally polarized by the strong random fields,
meaning the charges of the $U>0$ problem are localized by deep wells and high
barriers in the random potential.

###  VII.2 Numerical SDRG Of the Hubbard Chain

To corroborate the results from our DMRG calculations, we implement the real-
space strong-disorder renormalization group (SDRG) procedure used to
characterize the ground state and infinite-randomness fixed point of the
random antiferromagnetic Heisenberg chainFisher (1994). We cannot apply this
procedure directly, since we start with a Hubbard chain. Rather, we tackle the
problem using perturbation theory in $t/U$ to find the effective low-energy
(spin) Hamiltonian corresponding to the system, assuming the charges are
localized. The initial Hamiltonian is Eq. 1.

At half-filling the procedure is straightforward as the physical system is a
Mott insulator, so the low-energy description is a Heisenberg antiferromagnet
with no vacancies. The spin interaction between the $i$th and $i+1$th spins is
found at second order in $t/U$ as $J_{eff}^{i}=4t_{i}^{2}/U$ in the random-
hopping chain. In the random-potential chain, we perturb the eigenstates of
$H_{0}=H_{U}+H_{\mu}$ with the correction $V=H_{t}$ to second order. Starting
with
$\ket{\psi_{0}}=(\ket{\uparrow,\downarrow}-\ket{\downarrow,\uparrow})/\sqrt{2}$,
we see that acting with $V$ brings this to two intermediate states:

$\displaystyle\ket{\uparrow\downarrow,0}\quad$ $\displaystyle\quad
E=U+\mu_{i}-\mu_{j}$ $\displaystyle\ket{0,\uparrow\downarrow}\quad$
$\displaystyle\quad E=U+\mu_{j}-\mu_{i}$

Then, the total second-order energy correction to the singlet state is

$\displaystyle\Delta
E^{(2)}=-\frac{2t^{2}}{U+(\mu_{i}-\mu_{j})}-\frac{2t^{2}}{U-(\mu_{i}-\mu_{j})}$

From this, we see that the effective spin interaction between two neighboring
sites is

$\displaystyle J_{eff}^{ij}$
$\displaystyle=\frac{2t^{2}}{U+(\mu_{i}-\mu_{j})}+\frac{2t^{2}}{U-(\mu_{i}-\mu_{j})}$
$\displaystyle=\frac{4t^{2}}{U}\left[\frac{1}{1-(\Delta\mu_{ij}/U)^{2})}\right],$

where $\Delta\mu_{ij}=\mu_{i}-\mu_{j}$. This interaction remains
antiferromagnetic so long as $W_{\mu}$ never exceeds $U$.

Away from half-filling, the evidence from DMRG suggests that the charges are
still localized. We assume then that the holes sit at the local maxima of the
chemical potential, and we can again recover a description the low-energy
physics in terms of purely spin. In this case, spin interactions of
neighboring spins with no vacancy separating them still take the form
described above, but the spin interaction $J_{eff}^{i}$ between spins
separated by a hole, at sites $i$ and $i+2$, is now found at fourth order in
$t/U$ using Rayleigh-Schrodinger perturbation theory. Note that we must go to
fourth order because the singlet and triplet energies are split only when the
intermediate states involve double occupancy.

In general, the fourth-order correction to the $n$-th energy in perturbation
has the form:

$\Delta E_{n}^{(4)}=\sum_{i\neq n}\sum_{j\neq n}\sum_{k\neq
n}\frac{\bra{n}V\ket{i}\bra{i}V\ket{j}\bra{j}V\ket{k}\bra{k}V\ket{n}}{(E_{n}^{(0)}-E_{k}^{(0)})(E_{n}^{(0)}-E_{j}^{(0)})(E_{n}^{(0)}-E_{i}^{(0)})}.$
(8)

We consider a three-site system with a hole on site 2 (by construction, this
implies $\mu_{2}>\mu_{1},\mu_{3}$). Starting with the singlet state
$\ket{S_{13}}=(\ket{\uparrow,0,\downarrow}-\ket{\downarrow,0,\uparrow})/\sqrt{2}$
which has energy $E=\mu_{1}+\mu_{3}$, one can identify six possible
contributions to the fourth-order energy correction, shown in Table 1. Let
$\ket{S_{ij}}$ be the singlet state between spins on sites $i$ and $j$ and
$V_{ij}=\bra{i}V\ket{j}$. Again, we consider the case of a random potential,
so $H_{0}=H_{U}+H_{\mu}$ and $V=H_{t}$.

Int. state 1 | $V_{01}$ | Int, state 2 | $V_{12}$ | Int. state 3
---|---|---|---|---
$\ket{S_{12}}$, $E=\mu_{1}+\mu_{2}$ | $t$ |  $\ket{0,\uparrow\downarrow,0}$, $E=2\mu_{2}+U$ | $\sqrt{2}t$ | $\ket{S_{12}}$
$\ket{S_{12}}$, $E=\mu_{1}+\mu_{2}$ | $t$ |  $\ket{0,\uparrow\downarrow,0}$ , $E=2\mu_{2}+U$ | $\sqrt{2}t$ | $\ket{S_{23}}$
$\ket{S_{12}}$ , $E=\mu_{1}+\mu_{2}$ | $t$ |  $\ket{\uparrow\downarrow,0,0}$ , $E=2\mu_{1}+U$ | $\sqrt{2}t$ | $\ket{S_{12}}$
$\ket{S_{23}}$ , $E=\mu_{3}+\mu_{2}$ | $t$ |  $\ket{0,\uparrow\downarrow,0}$, $E=2\mu_{2}+U$ | $\sqrt{2}t$ | $\ket{S_{23}}$
$\ket{S_{23}}$ , $E=\mu_{3}+\mu_{2}$ | $t$ |  $\ket{0,\uparrow\downarrow,0}$, $E=2\mu_{2}+U$ | $\sqrt{2}t$ | $\ket{S_{12}}$
$\ket{S_{23}}$, $E=\mu_{3}+\mu_{2}$ | $t$ |  $\ket{0,0,\uparrow\downarrow}$ , $E=2\mu_{3}+U$ | $\sqrt{2}t$ | $\ket{S_{23}}$
Table 1: Contributions involving double occupancy of virtual states in the
fourth-order energy correction of a singlet state across a hole.

Defining $\mu_{ij}=\mu_{i}-\mu_{j}$, we can write the effective spin
interaction between spin 1 and 3 as:

$\displaystyle\begin{split}J_{eff}^{1,3}&=\frac{2t^{4}}{U^{3}}\bigg{[}\frac{1}{(\mu_{12}/U)^{2}\cdot(1+\mu_{31}/U)}\\\
&+\frac{1}{(\mu_{32}/U)^{2}\cdot(1-\mu_{31}/U)}\\\
&+\frac{1}{(\mu_{12}/U)^{2}\cdot(1-\mu_{12}/U-\mu_{32}/U)}\\\
&+\frac{1}{(\mu_{32}/U)^{2}\cdot(1-\mu_{12}/U-\mu_{32}/U)}\\\
&+\frac{1}{(\mu_{12}\mu_{32}/U^{2})\cdot(1-\mu_{12}/U-\mu_{32}/U)}\\\
&+\frac{1}{(\mu_{12}\mu_{32}/U^{2})\cdot(1-\mu_{12}/U-\mu_{32}/U)}\bigg{]}\end{split}$
(9)

Note that because $\mu_{2}>\mu_{1},\mu_{3}$ by construction, the last four
terms in the effective interaction are guaranteed to be positive. So long as
$W_{\mu}<U$, all terms in $J_{eff}$ are positive and therefore $J_{eff}$
remains antierromagnetic. The expression in Eq. 9 holds for the case of
$\ell=1$ holes in a row. If there are $\ell\geq 2$ holes in a row, we
approximate the effective spin interaction between the spins sandwiching the
holes by the correct order of magnitude in $t/U$: $U(t/U)^{2\ell}$. Note that
the interactions also remain antiferromagnetic for the same reason.

Given the similarity in the effects of the random hopping and random potential
away from half-filling, we implement the SDRG procedure only for the case of
the random potential, but we expect no qualitative difference if considering
the system with random hoppings. We use a $L=600$ site chain, with $n=11/12$
electron filling.

The SDRG decimation procedure then proceeds as described for a random
Heisenberg antiferromagnetic chain with no vacancies. At a given step, say the
strongest bond $J_{j}$ connects the $j$-th spin to the $j+1$th spin (note that
these may not reside on sites $j$ and $j+1$ away from half-filling). This bond
is decimated, as spins $j$ and $j+1$ are locked into a singlet, giving rise to
an effective bond of $\tilde{J}=J_{j-1}J_{j+1}/(2J_{j})$. In order to
reconstruct the probability $P_{s}(r)$ of forming a singlet bond with spins
separated by $r$ sites, we track the spatial indices of the singlets that are
formed at each step and find the distribution of their separations $r$ over
many realizations.

To find the distribution of couplings near the end of the decimation
procedure, we consider the couplings of each chain when there are $50$
remaining free spins. These couplings are normalized by the energy scale at
each SDRG step (i.e., by the largest bond in the system at each step). Fig. 4b
in the main text shows the resulting histogram of normalized couplings across
many realizations for each chain length.

###  VII.3 Typical spin correlations

We analyze the typical (rather than average) spin correlations,
$\overline{\ln|C_{\sigma}(r)|}$, which has a long-distance behavior
$\overline{\ln|C_{\sigma}(r)|}\sim r^{0.5}$ in the random singlet phase Fisher
(1994). We find typical spin correlations $\overline{\ln|C_{\sigma}(r)|}\sim
r^{p}$, with $p$ between $0.42$ and $0.48$ for the region $L/4<r<L/2$.

Figure 6: Typical spin correlations $\overline{\ln|C_{\sigma}(r)|}$ in a
$L=144$ random potential chain, at $n=11/12$ electron filling, with
$W_{\mu}=3\bar{t}$.

## References

  * Paalanen  _et al._ (1986) M. A. Paalanen, S. Sachdev, R. N. Bhatt, and A. E. Ruckenstein, Spin dynamics of nearly localized electrons, Phys. Rev. Lett. 57, 2061 (1986).
  * Paalanen  _et al._ (1988) M. A. Paalanen, J. E. Graebner, R. N. Bhatt, and S. Sachdev, Thermodynamic behavior near a metal-insulator transition, Phys. Rev. Lett. 61, 597 (1988).
  * Mahmood  _et al._ (2021) F. Mahmood, D. Chaudhuri, S. Gopalakrishnan, R. Nandkishore, and N. P. Armitage, Observation of a marginal Fermi glass, Nature Physics 17, 627 (2021).
  * Bhatt  _et al._ (1988) R. Bhatt, M. Paalanen, and S. Sachdev, MAGNETIC PROPERTIES OF DISORDERED SYSTEMS NEAR A METAL-INSULATOR TRANSITION, Journal de Physique Colloques 49, C8 (1988).
  * Belitz and Kirkpatrick (1994) D. Belitz and T. R. Kirkpatrick, The Anderson-Mott transition, Reviews of Modern Physics 66, 261 (1994).
  * Miranda and Dobrosavljević (2005) E. Miranda and V. Dobrosavljević, Disorder-driven non-Fermi liquid behaviour of correlated electrons, Reports on Progress in Physics 68, 2337 (2005).
  * Rosenbaum  _et al._ (1980) T. F. Rosenbaum, K. Andres, G. A. Thomas, and R. N. Bhatt, Sharp metal-insulator transition in a random solid, Phys. Rev. Lett. 45, 1723 (1980).
  * Sarachik  _et al._ (1998) M. P. Sarachik, D. Simonian, S. V. Kravchenko, S. Bogdanovich, V. Dobrosavljevic, and G. Kotliar, Metal-insulator transition in si:x (x=p,b): Anomalous response to a magnetic field, Phys. Rev. B 58, 6692 (1998).
  * Kravchenko and Sarachik (2003) S. V. Kravchenko and M. P. Sarachik, Metal–insulator transition in two-dimensional electron systems, Reports on Progress in Physics 67, 1 (2003).
  * Finkelstein (1990) A. M. Finkelstein, Electron liquid in disordered conductors  (1990).
  * Finkel’stein (1983) A. Finkel’stein, ON THE FREQUENCY AND TEMPERATURE-DEPENDENCE OF THE CONDUCTIVITY NEAR A METAL-INSULATOR-TRANSITION-Web of Science Core Collection, Journal of Experimental and Theoretical Physics Letters 37, 517 (1983).
  * Andreev and Kamenev (1998) A. V. Andreev and A. Kamenev, Itinerant ferromagnetism in disordered metals: A mean-field theory, Phys. Rev. Lett. 81, 3199 (1998).
  * Chamon  _et al._ (1999) C. Chamon, A. W. W. Ludwig, and C. Nayak, Schwinger-keldysh approach to disordered and interacting electron systems: Derivation of finkelstein’s renormalization-group equations, Phys. Rev. B 60, 2239 (1999).
  * Kamenev and Andreev (1999) A. Kamenev and A. Andreev, Electron-electron interactions in disordered metals: Keldysh formalism, Phys. Rev. B 60, 2218 (1999).
  * Chamon and Mucciolo (2000) C. Chamon and E. R. Mucciolo, Nonperturbative saddle point for the effective action of disordered and interacting electrons in 2d, Phys. Rev. Lett. 85, 5607 (2000).
  * Nosov  _et al._ (2020) P. A. Nosov, I. S. Burmistrov, and S. Raghu, Interaction-induced metallicity in a two-dimensional disordered non-fermi liquid, Phys. Rev. Lett. 125, 256604 (2020).
  * Bhatt and Lee (1982) R. N. Bhatt and P. A. Lee, Scaling Studies of Highly Disordered Spin-$\frac{1}{2}$ Antiferromagnetic Systems, Physical Review Letters 48, 344 (1982).
  * Zhou  _et al._ (2009) S. Zhou, J. A. Hoyos, V. Dobrosavljevic, and E. Miranda, Valence-bond theory of highly disordered quantum antiferromagnets, EPL (Europhysics Letters) 87, 27003 (2009), arXiv:0810.3043 .
  * Bhatt and Fisher (1992) R. N. Bhatt and D. S. Fisher, Absence of spin diffusion in most random lattices, Physical Review Letters 68, 3072 (1992).
  * Ogata and Shiba (1990) M. Ogata and H. Shiba, Bethe-ansatz wave function, momentum distribution, and spin correlation in the one-dimensional strongly correlated Hubbard model, Physical Review B 41, 2326 (1990).
  * Arovas  _et al._ (2021) D. P. Arovas, E. Berg, S. Kivelson, and S. Raghu, The Hubbard Model, arXiv:2103.12097 [cond-mat] (2021), arXiv:2103.12097 [cond-mat] .
  * Giamarchi and Schulz (1987) T. Giamarchi and H. J. Schulz, Localization and Interaction in One-Dimensional Quantum Fluids, Europhysics Letters (EPL) 3, 1287 (1987).
  * Ma  _et al._ (1979) S.-k. Ma, C. Dasgupta, and C.-k. Hu, Random antiferromagnetic chain, Phys. Rev. Lett. 43, 1434 (1979).
  * Fisher (1994) D. S. Fisher, Random antiferromagnetic quantum spin chains, Physical Review B 50, 3799 (1994).
  * White (1992) S. R. White, Density matrix formulation for quantum renormalization groups, Physical Review Letters 69, 2863 (1992).
  * White (1993) S. R. White, Density-matrix algorithms for quantum renormalization groups, Physical Review B 48, 10345 (1993).
  * Xavier  _et al._ (2018) J. C. Xavier, J. A. Hoyos, and E. Miranda, Adaptive density matrix renormalization group for disordered systems, Physical Review B 98, 195115 (2018).
  * Mélin and Iglói (2006) R. Mélin and F. Iglói, Strongly disordered Hubbard model in one dimension: Spin and orbital infinite randomness and Griffiths phases, Physical Review B 74, 155104 (2006), arXiv:cond-mat/0607269 .
  * Hoyos  _et al._ (2007) J. A. Hoyos, A. P. Vieira, N. Laflorencie, and E. Miranda, Correlation amplitude and entanglement entropy in random spin chains, Physical Review B 76, 174425 (2007), arXiv:0704.0951 .
  * Laflorencie (2005) N. Laflorencie, Scaling of Entanglement Entropy in the Random Singlet Phase, Physical Review B 72, 140408 (2005), arXiv:cond-mat/0504446 .
  * Liu  _et al._ (2018) L. Liu, H. Shao, Y.-C. Lin, W. Guo, and A. W. Sandvik, Random-Singlet Phase in Disordered Two-Dimensional Quantum Magnets, Physical Review X 8, 041040 (2018).
  * Ren  _et al._ (2020) H.-D. Ren, T.-Y. Xiong, H.-Q. Wu, D. N. Sheng, and S.-S. Gong, Characterizing random-singlet state in two-dimensional frustrated quantum magnets and implications for the double perovskite Sr$_2$CuTe$_{1-x}$W$_{x}$O$_6$, arXiv:2004.02128 [cond-mat] (2020), arXiv:2004.02128 [cond-mat] .
  * Milovanović  _et al._ (1989) M. Milovanović, S. Sachdev, and R. N. Bhatt, Effective-field theory of local-moment formation in disordered metals, Phys. Rev. Lett. 63, 82 (1989).
  * Tusch and Logan (1993) M. A. Tusch and D. E. Logan, Interplay between disorder and electron interactions in a d=3 site-disordered anderson-hubbard model: A numerical mean-field study, Phys. Rev. B 48, 14843 (1993).
  * Pezzoli and Becca (2010) M. E. Pezzoli and F. Becca, Ground-state properties of the disordered Hubbard model in two dimensions, Physical Review B 81, 075106 (2010).
  * Das Sarma  _et al._ (1990) S. Das Sarma, S. He, and X. C. Xie, Localization, mobility edges, and metal-insulator transition in a class of one-dimensional slowly varying deterministic potentials, Physical Review B 41, 5544 (1990).
  * Das Sarma  _et al._ (1988) S. Das Sarma, S. He, and X. C. Xie, Mobility Edge in a Model One-Dimensional Potential, Physical Review Letters 61, 2144 (1988).
  * Ganeshan  _et al._ (2015) S. Ganeshan, J. H. Pixley, and S. Das Sarma, Nearest Neighbor Tight Binding Models with an Exact Mobility Edge in One Dimension, Physical Review Letters 114, 146601 (2015).
  * Li  _et al._ (2015) X. Li, S. Ganeshan, J. H. Pixley, and S. Das Sarma, Many-Body Localization and Quantum Nonergodicity in a Model with a Single-Particle Mobility Edge, Physical Review Letters 115, 186601 (2015).