# Large Language Models are Built-in Autoregressive Search Engines

Noah Ziems, Wenhao Yu, Zhihan Zhang, Meng Jiang
University of Notre Dame
{nziems2, wyu1, zzhang23<EMAIL_ADDRESS>
###### Abstract

Document retrieval is a key stage of standard Web search engines. Existing
dual-encoder dense retrievers obtain representations for questions and
documents independently, allowing for only shallow interactions between them.
To overcome this limitation, recent autoregressive search engines replace the
dual-encoder architecture by directly generating identifiers for relevant
documents in the candidate pool. However, the training cost of such
autoregressive search engines rises sharply as the number of candidate
documents increases. In this paper, we find that large language models (LLMs)
can follow human instructions to directly generate URLs for document
retrieval.

Surprisingly, when providing a few Query-URL pairs as in-context
demonstrations, LLMs can generate Web URLs where nearly 90% of the
corresponding documents contain correct answers to open-domain questions. In
this way, LLMs can be thought of as built-in search engines, since they have
not been explicitly trained to map questions to document identifiers.
Experiments demonstrate that our method can consistently achieve better
retrieval performance than existing retrieval approaches by a significant
margin on three open-domain question answering benchmarks, under both zero and
few-shot settings. The code for this work can be found at
https://github.com/Ziems/llm-url.

##  1 Introduction

Figure 1: Successful URL reconstructions by different size of GPT-3 as URL
generators. The models are prompted with the first 100 words of a Wikipedia
page then tasked with generating the URL of the page they came from. Tested on
10k Wikipedia pages sampled from the top 100k most frequent Wikipedia
entities.

Along with the success of deep learning, dual-encoder based retrievers have
become the dominant method for Web searching Zhu et al. (2021); Zhao et al.
(2022). For example, DPR Karpukhin et al. (2020) employs two independent
encoders to encode the question and the document respectively, then estimates
their relevance by computing a single similarity score between two
representations.

However, these methods suffer from two major drawbacks. First, the
representations of questions and documents are typically obtained
independently in modern dual-encoder dense retrieval models (Karpukhin et al.,
2020), allowing for only shallow interactions between them (Khattab et al.,
2021). Second, the question or document representation is embedded into a
single dense vector, potentially missing fine-grained information when
computing the similarity between the two vector representations (Khattab and
Zaharia, 2020).

Instead of computing similarity between question and document embeddings,
autoregressive search engines aim to directly generate document identifiers
then map them to complete documents in the predetermined candidate pool. This
approach has attracted increasing interest in information retrieval (IR) and
related fields Tay et al. (2022); Bevilacqua et al. (2022); Wang et al.
(2022). Compared to dual-encoder dense retrieval methods, autoregressive
search engines enjoy a number of advantages. First, autoregressive generation
models produce document identifiers by performing deep token-level cross-
attention, resulting in a better estimation than shallow interactions in dense
retrievers. Second, autoregressive search engines have been shown to have
strong generalization abilities, outperforming BM25 in a zero-shot setting Tay
et al. (2022). While it is theoretically possible to scale an autoregressive
search engine to the size of a large language model (LLM), such as GPT-3 with
175B parameters, in practice it is not feasible due to the computational
overhead of training such a large autoregressive search engine from scratch
Tay et al. (2022). To reduce the high training cost of autoregressive search
engine, a smaller model size is preferred. However, the results of our pilot
study in Figure 1 show smaller language models are significantly worse at
mapping passages to document identifiers than larger ones. Moreover, different
retrieval tasks can have unique retrieval requirements. One task may require a
model to retrieve factual evidence to support or refute a claim (i.e., fact
checking) Onoe et al. (2021) while another may require a model to retrieve
specific trivia information about an entity (i.e., entity linking) Petroni et
al. (2021); Zhang et al. (2022). It would be better if the retriever was
capable of generalizing to new retrieval tasks with only a few examples.

In this work, we explore the use of in-context demonstrations to prompt LLMs
to directly generate web URLs for document retrieval, namely LLM-URL.
Surprisingly, we find that by providing a few (query, URL) pairs as contextual
demonstrations, large language models (e.g. GPT-3) generate Web URLs where
nearly 90% of the corresponding documents contain answers to open-domain
questions. In this way, LLMs can be thought of as built-in search engines, as
they have not been explicitly trained to map questions or documents to
identifiers. Instead of using newly-created document identifiers, LLM-URL
leverages existing and widely used document identifiers directly, i.e., URLs.
We compare our approach to existing document retrieval methods on three
different open-domain question answering (QA) datasets: WebQ (Berant et al.,
2013), NQ (Kwiatkowski et al., 2019), and TriviaQA (Joshi et al., 2017).
Further, to avoid exceeding the limit on the number of input tokens of LLMs,
we employ an unsupervised passage filtering module to remove irrelevant
portions of supporting documents. To summarize, our main contributions are as
follows:

  1. 1.

We reveal that LLMs are built-in autoregressive search engines capable of
document retrieval by directly generating Web page URLs under both zero and
few-shot settings.

  2. 2.

We show retrieving documents by generating URLs with LLMs significantly
outperforms existing methods for document retrieval, as measured by Recall@K.
Further, we show that breaking the retrieved documents into passages then
using a ranker to filter the passages significantly reduces the number of
supporting passages while maintaining high recall.

  3. 3.

We show the retrieved documents improve downstream QA performance as measured
by EM when compared to baseline methods.

##  2 Related Work

###  2.1 Traditional Document Retrievers

Traditional methods such as TF-IDF and BM25 explore sparse retrieval
strategies by matching the overlapping contents between questions and passages
Robertson and Zaragoza (2009); Chen et al. (2017); Yang et al. (2019). DPR
Karpukhin et al. (2020) revolutionized the field by utilizing dense
contextualized vectors for passage indexing. It is first initialized as a
pretrained BERT model, then trained discriminatively using pairs of queries
and relevant documents, with hard negatives from BM25. Recent research has
improved DPR via better training strategies Xiong et al. (2020); Qu et al.
(2021); Zhang et al. (2023a) and passage re-ranking Mao et al. (2021); Yu et
al. (2021); Ju et al. (2022). However, representations of questions and
documents are typically obtained independently in modern dual-encoder dense
retrieval models (Karpukhin et al., 2020; Xiong et al., 2020), allowing for
only shallow interactions between them (Khattab et al., 2021).

Figure 2: The overall pipeline of our proposed LLM-URL. Given a question, LLM-
URL first generates a set of URLs which are extracted from the generated text.
The URLs are retrieved from the Internet then broken into passages which are
ranked and filtered such that only the most relevant are kept. Finally, these
passages are given as input to a reader model along with the original question
to generate a final answer.

###  2.2 Autoregressive Search Engines

Recent works have investigated the use of auto-regressive language models to
generate identifier strings for documents as an intermediate target for
retrieval Yu et al. (2022), such as Wikipedia page titles (De Cao et al.,
2020), root-to-leaf paths in a hierarchical cluster tree (Tay et al., 2022),
or distinctive n-grams that can be mapped to full passages (Bevilacqua et al.,
2022). Since the series of work was carried out almost simultaneously by
different research groups, they are often referred to multiple different names
in the literature, such as autoregressive search engine, differential search
index (DSI), and neural document indexers (NDI). Compared to traditional dense
document retrievers, these methods leverage a generation model to produce the
document indexes. By forcing the generation model to explain every token in
the question and document using cross-attention, the generation abilities of
the model significantly improve. Our work is closely related to these works,
showing experimentally that properly prompting pre-trained large language
models can achieve better performance than traditional dense retrieval models
Ouyang et al. (2022); Yu et al. (2023) .

##  3 Proposed Method

In this section we describe a new method, which we refer to as LLM-URL, that
employs a large language model (LLM) to perform effective and efficient web
document retrieval for knowledge-intensive NLP tasks such as open-domain
question answering (ODQA).

ODQA is a two step process consisting of a retriever and a reader. Given a
question $q$, the goal of the retriever is to find the top-$n$ passages
$\mathcal{P}_{n}$ relevant to answering $q$. Given $q$ and the top-$n$
relevant passages $\mathcal{P}_{n}$, the goal of the reader is to use internal
knowledge along with $\mathcal{P}_{n}$ to generate a correct answer $a$ to
question $q$. The passage retriever plays an essential role in this process.
When $\mathcal{P}_{n}$ contains more passages that have the correct answer,
the reader has a higher chance of finding it. Instead of _heavily training a
dedicated retriever_ , our LLM-URL solves the problem in a different way as
shown in Figure 2.

Given a question $q$, our LLM-URL should find a set of relevant passages to
$\mathcal{P}_{n}$ and give it to the reader. First, it prompts a LLM (e.g.,
GPT-3) to directly generate $m$ URLs for $q$. By default, it uses “Which $m$
Wikipedia URLs would have the answer?” as the instruction which is appended to
each input question as the prompt. We also append the beginning of the
Wikipedia URL (https://en.wikipedia.org/wiki) to the end of the prompt to
encourage the generation of URLs and restrict generation to the Wikipedia
article URL format. As LLM has the ability of in-context learning, we take
this advantage to enable the few-shot setting in the prompt. The prompt
described above also includes a series of in-context demonstrations. Each
demonstration contains a question sampled from the training set following the
prompt described above. At the end of each demonstration, $m$ URLs which point
to gold-labeled documents are listed. In the zero-shot setting, the original
prompt is used without any demonstrations. In the few-shot setting, the
original prompt appended to a series of $d$ demonstrations ($d$=10 in this
work).

Given the prompt, the LLM returns a generated sequence of tokens. Ideally
these tokens would construct a sequence of $m$ separated URLs. In practice,
the generated sequence often has extra information such as a proposed answer
that is unreliable and needs to be filtered. We use a regular expression to
extract all URLs from the sequence and discard all extra information. This
also filters out many URLs that are improperly formatted. After extraction,
GET requests are made using the extracted URLs and the contents of each
retrieval is used to create a set of fetched documents $\mathcal{D}_{f}$.
Often, $|\mathcal{D}_{f}|<m$ because some of the generated URLs do not follow
a correct format or do not point to real web pages on the Internet.

Method | Document Recall@1 | Document Recall@10
---|---|---
WebQ | NQ | TriviaQA | WebQ | NQ | TriviaQA
Contriever Izacard et al. (2021) | 63.8 | 53.2 | 60.6 | 63.8 | 80.8 | 82.5
BM25 Robertson and Zaragoza (2009) | 49.5 | 47.2 | 63.0 | 81.5 | 76.8 | 82.3
Google API | 61.1 | 55.5 | 51.4 | - | - | -
LLM-URL (Zero-Shot) | 76.8 | 61.7 | 71.3 | 87.7 | 83.2 | 85.5
LLM-URL (Few-Shot) | 79.7 | 62.6 | 73.5 | 89.9 | 83.9 | 86.8

Table 1: Document retrieval as measured by Recall@k. Google API Recall@10
results are left out due to high cost. Figure 3: We prompt LLM-URL to generate
$m$ documents and measure the recall as $m$ increases. Significant recall
improvements are seen when $m$ is small but as it increases the marginal
benefit decreases.

The set of fetched documents $\mathcal{D}_{f}$ can be passed directly to a
reader if $m$ is a small value or the reader being used can handle many large
documents. However, this is usually not the case. Often, $\mathcal{D}_{f}$
needs to be filtered such that only a small number of the most relevant
passages are given to the reader. To do this, our LLM-URL first breaks each
document $d\in\mathcal{D}_{f}$ into a set of small passages. The passages from
each document are collected into a new set, $\mathcal{P}_{f}$. A scoring
function is used to quantify the relevance of each passage with respect to the
question $q$, with high values indicating high relevance with respect to $q$
and low scores indicating low relevance. A simple scoring function such as
BM25 can be used or a more complex one such as DPR (Karpukhin et al., 2020)
can. The passages in $\mathcal{P}_{f}$ are then sorted from highest to lowest
and the top $n$ are kept as $\mathcal{P}_{n}$. Finally, $\mathcal{P}_{n}$ are
given to a reader along with $q$ to generate an answer.

##### Advantages of LLM-URL :

Existing autoregressive retrieval methods such as DSI and SEAL use a pre-
trained large language model then fine tune it to take questions as input and
generate relevant document identifiers as output Tay et al. (2022); Bevilacqua
et al. (2022). Both DSI and SEAL do extensive experiments on a variety of
document identifiers which are generated by a heavily trained language model.
Examples of these identifiers include unstructured atomic identifiers, naively
structured string identifiers, hierarchical document clustering, and others.
LLM-URL instead uses pre-existing document identifiers that exist on the
internet: URLs. Using URLs instead of the aforementioned identifiers has
multiple advantages. URLs often contain words related to the information they
link to, allowing for strong association of topics with their URLs. For
example, the title of each Wikipedia page is used in its URL, allowing the LLM
is able to directly generate the URL by leveraging semantic information from
the question. To validate the importance of URLs themselves, we also
experiment with prompting the LLM to generate Wikipedia titles instead of URLs
and find Recall@1 significantly reduces compared to prompting for URL
generation. We believe this is because the URL format itself helps prompt the
model for specific information in a specific format. Further, the use of URLs
allows us to simply obtain the evidence document via a HTTP request without
any need of training a model or building an index to find the mapping between
identifiers and documents.

##  4 Experiments

In this section, we present and discuss results from our experiments to
“directly” demonstrate that our LLM-URL is a strong retriever and “indirectly”
show that it achieves competitive performance on the ODQA task against state-
of-the-art solutions.

##### Large Language Model:

Following Figure 1, the large language model we use to generate URLs for our
experiments is GPT-3 text-davinci-003 with greedy decoding and a temperature
of 0. A variety of different prompts are tested for generating URLs, but
little difference in performance is observed, so we simply use the best
performing prompt which is discussed in Section 3.

##### Datasets:

We use three ODQA datasets including Web Questions, Natural Questions, and
Trivia QA. We use them to perform evaluation on both the task of document or
passage retrieval and ODQA itself.

###  4.1 Retrieval

We expect retrievers to find the most relevant documents and/or passages. We
conduct experiments on both document retrieval and passage retrieval.

Method Passage Recall@1 Passage Recall@10 Passage Recall@100 WebQ NQ TriviaQA
WebQ NQ TriviaQA WebQ NQ TriviaQA Contriever 18.2 18.8 34.0 55.7 54.8 67.9
79.8 79.6 83.3 BM25 19.1 22.8 46.2 51.8 55.6 71.7 76.6 79.6 83.9 LLM-URL
(Zero-Shot) 22.2 24.0 46.7 63.1 60.6 76.6 83.8 78.3 83.6 LLM-URL (Few-Shot)
22.3 25.5 49.1 64.8 60.8 77.8 85.9 79.0 84.8

Table 2: Passage retrieval as measured by Recall@1, Recall@10 and Recall@100.
Here LLM-URL is equipped with BM25 to perform the ranking task.

##### Evaluation metrics.

Recall@$k$ ($k$=1, 10, 100) is calculated by measuring the percentage of
documents or passages in the top-$k$ which contain one of the gold labeled
answers while exact match is calculated by the percentage of predicted answers
which match one of the gold labeled answers. While LLM-URL is not constrained
by which URLs can be generated for document retrieval, we restrict all
generations to Wikipedia URLs only for fair comparison, as discussed in
Section 3 All baseline models also use Wikipedia for retrieval, with some
fetching documents in real time and others fetching from an offline corpus.

####  4.1.1 Document Retrieval

##### Baselines:

Contriever Izacard et al. (2021) and BM25 Robertson and Zaragoza (2009) are
usually used for passage retrieval. Contriever is a dual encoder which uses a
dot product between dense representations of a question and passage to
calculate relevance. BM25 is a sparse retriever which uses the overlapping
contents between question and passage to calculate relevance. Because we use
the same passage size to chunk Wikipedia documents, we were able to map their
retrieved passages back to the original documents. We use Google API Brin and
Page (1998) restricted to Wikipedia as a third baseline to retrieve relevant
documents given a question.

Existing works such as DSI and SEAL have investigated the use of
autoregressive language models to generate identifier strings for documents as
an intermediate target for retrieval. DSI is a Transformer which has been
trained to map directly from question to document identifiers by memorizing
the contents of the entire corpus (Tay et al., 2022). SEAL is a variant of DSI
which uses ngrams as document ids to improve retrieval performance (Bevilacqua
et al., 2022). Neither DSI nor SEAL report retrieval results on full documents
and do not have publicly available implementations, so they are left out and
discussed in Table 3 and Section 4.1.2 on passage retrieval.

Unlike the baselines, our LLM-URL employs an LLM. It has two settings: zero-
shot and few-shot. In the zero-shot setting, no in-context demonstrations are
given whereas in the few-shot setting a few demonstrations are appended to the
prompt.

Figure 4: The percentage of valid URLs generated from LLM-URL as the total
number of generated URLs $m$ increases from 1 to 10. As $m$ increases, invalid
URL generations become more frequent.

##### Results:

The results of our document retrieval experiments are shown in Table 1. In
this setting Recall@$k$ is calculated directly after the documents are
retrieved with no intermediary steps. LLM-URL significantly outperforms
baseline methods on all datasets for both Recall@1 and Recall@10.
Specifically, zero-shot LLM-URL improves document Recall@1 relatively by
20.4%, 11.2%, and 13.2% over the strongest baseline on WebQ, NQ, and TriviaQA,
respectively. Few-shot LLM-URL further expands the improvement to 24.9%,
12.8%, and 16.7%, respectively. URLs can be extracted from the large-scale
parameters of LLMs, and these URLs can lead to more accurate documents than
what existing methods can retrieve. Both the LLM parameters and in-context
demonstrations are significantly useful in document retrieval.

Figure 3 shows Recall scores converge when the number of generated URLs $m$
increases. Due to the diminishing returns from increasing $m$, our experiments
do not explore values of $m$ greater than 10.

Method | Recall@1 | Recall@10
---|---|---
DSI${}^{\text{1}}$ | 25.1 | 56.6
SEAL${}^{\text{1}}$ | 26.3 | 74.5
LLM-URL (Zero-Shot) | 24.0 | 60.6
LLM-URL (Few-Shot) | 25.5 | 60.8
${}^{\text{1}}$explicitly trained for retrieval on NQ |

Table 3:  Passage retrieval as measured by Recall@1 and Recall@10. LLM-URL is
equipped with BM25 for passage ranking. Other datasets are left out due to not
being reported in either paper and no public implementations.

##### Are the generated URLs valid?

It is worth noting that the generated URLs are not always valid. Some
generated URLs do not have valid URL syntax and some point to Wikipedia pages
that do not exist. Rarely, URLs will be generated for domains aside from
Wikipedia. For fair comparison, all of these faulty URLs are discarded and
only documents coming from valid Wikipedia articles are kept.

Further analysis is done to measure the ratio of valid Wikipedia URLs while
the total number of generated URLs $m$ increases from 1 to 10, shown in Figure
4. The number of valid URL generations remains surprisingly high (i.e., higher
than 68%) as $m$ increases from 1 to 10. However, the rate of valid
generations appears to fall off as $m$ increases, indicating there are
diminishing returns from each marginal increase of $m$.

####  4.1.2 Passage Retrieval

##### Baselines:

Four methods, including Contriever, BM25, DSI (Tay et al., 2022), and SEAL
(Bevilacqua et al., 2022), were introduced in Section 4.1.1. Google API was
used for document retrieval and not applied to passages.

##### Results:

The results of our passage retrieval experiments are shown in Table 2. In this
setting Recall@$k$ is calculated on the top-$k$ passages ranked by the ranker
instead of just on the raw documents shown in Table 1. LLM-URL performs
slightly better than baseline methods for Recall@1 and Recall@10 and as well
as baseline methods for Recall@100. In the zero-shot setting, LLM-URL improves
relative Recall@1 by 16.2%, 5.3%, and 1.1% with respect to the strongest
baseline on WebQ, NQ, and TriviaA respectively. The few-shot setting of LLM-
URL expands the improvement to 16.8%, 11.8%, and 6.3%, respectively. For
Recall@10, similar improvements can be seen.

Method | Zero-Shot QA EM
---|---
WebQ | NQ | TriviaQA
Contriever + InstructGPT | 16.8 | 19.1 | 52.4
BM25 + InstructGPT | 16.0 | 20.5 | 53.3
Google + InstructGPT | 19.9 | 27.8 | 58.7
GenRead (InstructGPT) | 24.8 | 28.2 | 59.3
DSI${}^{\text{1}}$ \+ FiD | - |  31.4${}^{\text{2}}$ | -
SEAL${}^{\text{1}}$ \+ FiD | - | 43.6 | 41.8
InstructGPT (no docs.) | 18.6 | 20.9 | 52.6
LLM-URL (Zero-Shot) | 28.1 | 26.4 | 60.1
LLM-URL (Few-Shot) | 29.0 | 27.3 | 60.7
${}^{\text{1}}$explicitly trained for retrieval on NQ
${}^{\text{2}}$result from Bevilacqua et al. (2022)

Table 4: Zero-shot open-domain QA performance as measured by exact match (EM).
All LLM-URL models use InstructGPT as the reader unless otherwise stated.

For Recall@100, performance is better relative to baseline models for all
datasets except NQ. In the zero-shot setting, LLM-URL improves the relative
Recall@100 by 5.0% for WebQ and performs slightly worse than the best baseline
method on NQ and TriviaQA by 1.7% and 0.4% respectively. The few-shot setting
of LLM-URL for Recall@100 shows a slight improvement on WebQ and TriviaQA, but
performs slightly worse than the strongest baseline on NQ.

Despite being limited to only the passages from 10 documents, LLM-URL performs
better than baseline methods for smaller $k$ and performs as well as baseline
methods for higher values of $k$.

The comparison between LLM-URL and existing document identifier-based methods
such as DSI and SEAL are shown in Table 3. For Recall@1, zero-shot LLM-URL
performs slightly worse than the best baseline by 8.8%. This performance gap
is slightly smaller in the few-shot setting with LLM-URL performing 3.1% worse
than the best baseline. For Recall@10, zero-shot LLM-URL performs worse than
the best baseline by 18.7%. Few-shot LLM-URL performs only slightly better
than the zero-shot setting, performing worse than the best baseline by 18.4%.

###  4.2 Open-Domain Question Answering

##### Evaluation metric:

We use exact match (EM), which is short for exact string match with the
correct answer, because the goal of ODQA is to find an exact answer to any
question using Wikipedia articles.

##### Results:

Here we discuss the downstream QA performance of LLM-URL. In this setting, an
answer only has an exact match if the normalized generated text is within the
list of acceptable answers to a question. When combined with InstructGPT as a
reader, LLM-URL performs significantly better on WebQ and slightly better on
TriviaQA when compared with the best performing baseline methods. On NQ, LLM-
URL+InstructGPT performs worse than baseline NDIs and only slightly worse than
the best remaining baseline. In the zero-shot setting, LLM-URL+InstructGPT
improves upon the best baseline method by 13.3% and 1.3% on WebQ and TriviaQA
respectively. LLM-URL +InstructGPT performs worse than the best baseline
method by 39.5% on NQ. In the few-shot setting, LLM-URL+InstructGPT performs
better than the best baseline method by 16.9% and 2.3% on WebQ and TriviaQA
respectively. LLM-URL+InstructGPT performs worse than the best baseline method
by 37.4% on NQ.

Despite not being explicitly trained for retrieval, LLM-URL+InstructGPT
performs significantly better than baseline methods for WebQ, achieves on-par
performance with existing methods for TriviaQA, and performs slightly worse
than existing methods for NQ.

Our results indicate LLM-URL could be a promising solution to retrieval for a
wide range of knowledge intensive tasks with little to no training data
required.

###  4.3 Discussions

####  4.3.1 Time Sensitive Queries

There are a number of additional qualitative benefits that LLM-URL has over
existing methods. One large advantage of LLM-URL is that the documents are
retrieved in real time from the source. So long as the source stays up to date
without the URL itself changing, our proposed method is capable of answering
time sensitive queries without any extra modifications.

In contrast, existing dual encoder approaches such as Contriever require a
document to be re-encoded each time it changes. Existing methods such as SEAL
(Bevilacqua et al., 2022) and DSI are also tricky to keep up to date for time
sensitive queries as the LLM would have to be retrained to learn the new
content of the updated documents.

####  4.3.2 Frequent Entities Analysis

Figure 5: Common vs Uncommon entity recall@1. Common is defined as a question
containing entities in the top-1 million most common entities on Wikipedia.
LLM-URL performs much better when retrieving information about common
entities.

LLM-URL | Exists | Answer | Contriever | Answer | BM25 | Answer
---|---|---|---|---|---|---
wiki/Jellyfish | ✓ | ✓ | Smack (ship) | ✗ | Collective noun | ✗
wiki/Collective_noun | ✓ | ✗ | Collective noun | ✗ | Determiner | ✗
wiki/Smack_(group) | ✗ | ✗ | Cetacean intelligence | ✗ | Glass sea creatures | ✗
wiki/Cnidaria | ✓ | ✓ | Well smack | ✗ | Minotaur | ✗
wiki/Medusozoa | ✓ | ✓ | Plankton | ✗ | Mass noun | ✗
wiki/Scyphozoa | ✓ | ✓ | Sperm whale | ✗ | Well smack | ✗
wiki/Cubozoa | ✓ | ✓ | Loaded question | ✗ | Nomenclature | ✗
wiki/Hydrozoa | ✓ | ✓ | Jabberwocky | ✗ | Archomental | ✗
wiki/Staurozoa | ✓ | ✓ | Merrow | ✗ | Honey Smacks | ✗
wiki/Rhizostomeae | ✓ | ✓ | Loaded question | ✗ | Well smack | ✗

Table 5: Case study of retrieved documents from the question “A ’smack’ is a
collective noun for a group of which sea creatures?”. “Exists” means whether
the URL points to a valid Wiki page. “Answer” means whether the document
contains the answer. We omit the prefix of generated URLs for brevity
(https://en.wikipedia.org/). For BM25 and Contriever, we show the document
titles of the top-10 retrieved passages, respectively. The correct answer is
“jellyfish.”

Following Mallen et al. (2022), we analyze the retrieval performance of LLM-
URL when the gold-labeled answer entity is common versus when it is not. For
each question-answer pair in a given dataset we check to see if the labeled
entity exists within the top one-million common entities from Wikipedia. Using
this, we split our dataset into two distinct subsets: question-answer pairs
that contain a common entity and those that do not. In measuring the
performance of our model on these two distinct sets across Web Questions,
Natural Questions, and TriviaQA, we find LLM-URL performs significantly better
on common entity question-answer pairs. The results of our analysis are shown
in Figure 5. Across all three datasets, the recall of common-entity question-
answer pairs is many times greater than the recall from the rest of the
dataset.

Previous work has shown LLMs in the closed-book setting, where the model must
rely solely on the information contained within its weights, perform much
better on common-entities versus uncommon ones (Mallen et al., 2022). Our
results show this problem extends beyond the closed-book setting and also
applies to retrieval when using LLM-URL. This also could explain the high word
count from documents we found when evaluating LLM-URL. The average Wikipedia
article is 644 words, but the average word count from Wikipedia documents
retrieved via LLM-URL was 10k. We believe this discrepancy is caused by common
entities having much more detail in their Wikipedia articles and in turn
having much higher word count.

####  4.3.3 Case Study

In Table 5, we show a case study comparing LLM-URL with two baseline retrieval
methods, BM25 and Contriever, on the question “A ‘smack’ is a collective noun
for a group of which sea creatures?” which is in the TriviaQA test set. The
gold-labeled answer to this question is “jellyfish”.

In the closed-book setting, InstructGPT mistakenly predicts “dolphins” as the
answer. When using Contriever to retrieve 10 passages from Wikipedia given the
query, none of the passages contains the gold answer. For instance, Contriever
retrieves passages about “smack”, a kind of fishing vessel, along with other
passages about sperm whales, plankton, and other unrelated topics. Similar
results are found while using BM25 as the retriever.

In contrast, LLM-URL performs much better in this scenario, retrieving 7
documents which contain the answer. The top retrieved document is exactly
about the gold answer “jellyfish”. The fourth to the tenth documents all talk
about different types of jellyfish. After being chunked into passages then
sorted by the ranker, the top 10 passages are concatenated. Among them, it
contains “A group of jellyfish is called a smack,” which contains the answer
to the question and comes directly from the first retrieved document, titled
“jellyfish.” When InstructGPT is then prompted with these 10 passages along
with the question, the gold answer “jellyfish” is correctly generated.

This case study highlights multiple advantages of LLM-URL . First, LLM-URL
finds documents related to both the question and the answer. It directly
locates documents that talks about “jellyfish” instead while BM25 and
Contriever locate documents related to the question only–not the answer.
Second, LLM-URL is more precise than BM25 or Contriever. In this case, 7 out
of 10 generated URLs from LLM-URL point to a Wikipedia document that contains
the answer. However, both BM25 and Contriever fail to retrieve any documents
containing the answer. Third, the set of documents retrieved by LLM-URL are
complementary to each other, while in BM25 or contriever, each document in the
top-10 is selected independently. This is because the LLM is able to refer to
previous generated URLs before it generates the next one, allowing each newly
generated URL to be conditioned on all the previous URLs. This leads to a more
informative evidence context in open-domain question answering.

##  5 Conclusion and Future Work

In this paper, we explored whether large language models can generate URLs
prompted by human instructions for document retrieval. Surprisingly, we found
that by providing a few (query, URL) pairs as in-context demonstrations, large
language models (e.g. GPT-3) generated Web URLs where near 90% of the
corresponding documents contain correct answers to open-domain questions in
WebQ. Furthermore, by breaking the retrieved documents into passages then
ranking them with BM25, we showed a significant number of unnecessary passages
could be filtered out while retaining high recall, which outperformed baseline
methods by a significant margin.

There are numerous exciting directions for future work. While a number of
broad spectrum retrieval benchmarks such as BIER Thakur et al. (2021) exist,
it remains to be seen whether the few-shot demonstrations shown in this work
can be further tuned for specific retrieval tasks. Promptagator Dai et al.
(2022) shows significant performance improvements can be achieved by tuning
prompts in a similar way.

Further, it remains to be seen whether fine tuning the prompt for each
individual question can further improve the retrieval performance. As with
Promptagator, prior work has shown using clustering to select diverse
demonstrations for any given question further improves retrieval performance
as well as downstream QA performance.

## Limitations

Despite the strong performance on the presented datasets, our approach is
limited in its ability to update knowledge state and adapt to new domains. A
major feature of retrieve-then-read is the ability to swap in new documents
when new information is learned, such as temporally more recent documents, or
adding in documents from a new domain to quickly adapt to a new downstream
task. Our approach relies on a large language model to contain all this
knowledge and adding new knowledge would likely require some retraining. In
addition, large generation models still suffer from hallucination errors,
resulting in incorrect predictions. When tasked with generating 10 URLs, LLM-
URL may only generate 6 or 7 which link to valid documents. Finally, our
approach involves very large language models, slow web requests, and document
processing which may make it cumbersome to use in practice.

## Acknowledgements

This work was supported by NSF IIS-2119531, IIS-2137396, IIS-2142827,
CCF-1901059, and ONR N00014-22-1-2507. Wenhao Yu was partly supported by the
Bloomberg Data Science Fellowship.

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# Magnetic Relaxation in Two Dimensional Assembly of Dipolar Interacting
Nanoparticles

Manish Anand<EMAIL_ADDRESS>Department of Physics, Bihar National
College, Patna University, Patna-800004, India.

###### Abstract

Using the two-level approximation of the energy barrier, we perform extensive
kinetic Monte Carlo simulations to probe the relaxation characteristics in a
two-dimensional ($L_{x}\times L_{y}$) array of magnetic nanoparticle as a
function of dipolar interaction strength $h_{d}$, aspect ratio
$A_{r}=L_{y}/L_{x}$, and temperature $T$. In the case of weak dipolar
interaction ($h_{d}\approx 0$) and substantial temperature, the magnetic
relaxation follows the Néel Brown model as expected. Interestingly, the
dipolar interaction of enough strength is found to induce antiferromagnetic
coupling in the square arrangement of MNPs ($A_{r}=1.0$), resulting in the
fastening of magnetic relaxation with $h_{d}$. There is also a rapid increase
in relaxation even with $A_{r}<100$ above a particular dipolar interaction
strength $h^{\star}_{d}$, which gets enhanced with $A_{r}$. Remarkably, there
is a slowing down of magnetic relaxation with $h_{d}$ for the highly
anisotropic system such as linear chain of MNPs. It is because the dipolar
interaction induces ferromagnetic interaction in such a case. The thermal
fluctuations also affect the relaxation properties drastically. In the case of
weak dipolar limit, magnetization relaxes rapidly with $T$ because of
enhancement in thermal fluctuations. The effect of dipolar interaction and
aspect ratio on the magnetic relaxation is also clearly indicated in the
variation of Néel relaxation time $\tau_{N}$. In the presence of strong
dipolar interaction ($h_{d}>0.3$) and $A_{r}=1.0$, $\tau_{N}$ decreases with
$h_{d}$ for a given temperature. On the other hand, there is an increase in
$\tau_{N}$ with $h_{d}$ for huge $A_{r}$ $(>100)$. We believe that the
concepts presented in this work are beneficial for the efficient use of self-
assembled MNPs array in data storage and other related applications.

##  I Introduction

Magnetic nanoparticles (MNPs) have received significant attention in recent
years due to their unique magnetic properties and diverse technological
applications zhang2019 ; akbarzadeh2012 ; wang2019 ; tran2010 ; leo2018 ;
felix2019 ; wu2019 ; kechrakos2002 . For instance, MNPs have various
applications such as magnetic hyperthermia for cancer treatment, targetted
drug delivery, biosensors, data storage devices, etc. kumar2018 ; anand2016 ;
yu2016 ; malekzad2017 ; gu2016 ; reiss2005 . In these applications, the
dynamics of nanoparticles is primarily characterized by Néel relaxation time,
which depends on various vital parameters such as particle size, anisotropy
constant, temperature and magnetic interaction, etc. talantsev2018 ; ilg2020 ;
carrey2011 ; iacob2016 . Therefore, the study of magnetic relaxation in such a
system represents a topic of great interest.

The relaxation characteristics of non-interacting MNPs are well understood as
they are successfully explained using Néel and Brown theory of
superparamagnetism wernsdorfer1997 ; bedanta2013 . However, MNPs are found to
interact because of dipolar interaction. Therefore, dipolar interaction
affects the magnetic relaxation properties even for a low concentration of
nanoparticles garcia2000 . The dipolar interaction is long-ranged and
anisotropic. Consequently, it can induce ferromagnetic or antiferromagnetic
coupling between the magnetic moments depending on its relative position. The
dipolar interaction plays an essential role in determining various
thermodynamic and magnetic properties of crucial importance. For example, it
induces spin-glass like behaviour in randomly distributed MNPs majlis2018 . A
highly anisotropic system such as a linear chain favours the head to the tail
arrangement of magnetic moments cheng2009 . In these contexts, several works
have taken care of the role of dipolar interaction ovejero2016 ; shen2001 ;
tavares1999 ; chantrell1982 ; anand2018 . The dipolar interaction also
dictates the ground state morphology of the assembly of MNPs. Luttinger et al.
found the minimum energy state of a collection of classical dipoles to be
ferromagnetic in a face-centred cubic lattice. In contrast, it is
antiferromagnetic in a simple cubic lattice luttinger1946 . In two dimensions,
a square lattice of classical point dipoles exhibits an antiferromagnetic
arrangement of the moments, while for a triangular lattice, the arrangement is
ferromagnetic macisaac1996 ; politi2006 . The relaxation properties of
interacting MNPs depend not only on the strength of dipolar interaction but
also on the detailed MNPs arrangement and anisotropy axes orientations
dejardin2011 . For many decades, relaxation phenomena have been explored
because of their vital role in various technological applications and rich
physics. However, a complete understanding is still elusive due to the
frustrations induced by the dipolar interactions and the randomly oriented
anisotropy axes iglesias2002 ; labarta1993 ; balcells1997 .

Numerous works strengthen the fact that dipolar interaction strongly affects
magnetic relaxation properties vernay2014 ; morup2010 ; dormann1988 ;
jonsson2001 ; figueiredo2007 ; denisov2003 ; chamberlin2002 ; denisov2002 ;
shtrikman1981 ; kalmykov2004 ; osaci2015 ; jonsson2004 ; berkov2001 . For
example, Figueiredo et al. studied the magnetic relaxation using Monte Carlo
simulation in the triangular assembly of MNPs figueiredo2007 . They observed
an exponential decay of magnetization for weakly interacting MNPs. Denisov et
al. analyzed the relaxation properties in a two-dimensional assembly of MNPs
with perpendicular anisotropy axes denisov2003 . The dipolar interaction is
found to slow down magnetic relaxation. Chamberlin et al. studied the
relaxation in the dilute assembly of MNPs chamberlin2002 . They observed non-
exponential decay of magnetization even for weakly interacting MNPs. In
another study, Denisov et al. probed the magnetic relaxation in two-
dimensional assembly using mean-field approximations denisov2002 . It has been
shown that the magnetic relaxation in these ensembles is characterized by two
different relaxation times. Using mean-field approximations, Shtrikmann and
Wohlfarth studied the dependence of relaxation time on the dipolar interaction
shtrikman1981 . They observed an increase in relaxation time with an increase
in dipolar interaction strength. Using Fokker-Planck formalism, Yuri P.
Kalmykov derived an expression for relaxation time of uniaxial
superparamagnetic nanoparticles in the presence of an external magnetic field
applied at an arbitrary angle with respect to the anisotropy axis of the
particle, applicable for a wide range of damping parameter kalmykov2004 . By
taking into account the effect of damping, Osaci et al. derived an expression
for $\tau_{N}$ for dipolar interacting MNPs osaci2015 .
J$\ddot{\mathrm{o}}$nsson et al. also computed $\tau_{N}$ in an assembly of
dipolar interacting MNPs for a wide range of damping parameter jonsson2004 .
Using micromagnetic Langevin simulations, Berkov and Gorn observed that
uniaxial spins coupled via dipolar interaction exhibit damping effects such as
variation in blocking temperature with interaction strength berkov2001 .

In the context of a highly anisotropic system such as a linear chain of MNPs,
similar observations have been made laslett2015 ; laslett2016 ; iglesias2004 ;
anand2019 . For instance, Iglesias et al. found that the relaxation behaviour
changes from quasi logarithmic to power-law, increasing dipolar interaction
strength iglesias2004 . We have also shown that relaxation time crucially
depends on the dipolar interaction strength and anisotropy axes orientation
anand2019 . It is evident from above that the effect of dipolar interaction on
magnetic relaxation is not completely understood despite numerous work. An
accurate evaluation of relaxation time is also equally important as it plays a
vital role in various technological applications lenin2019 ; hergt2009 ;
kuncser2019 ; anand2021 ; rizzo1999 . For example, Kuncser et al. observed
that the amount of heat dissipation by the MNPs depends strongly on the
relaxation time of the system kuncser2019 . In a recent study, we have also
shown that relaxation time is dictated by the dipolar interaction in a linear
arrangement of MNPs, one of the essential quantifiers in determining the heat
dissipation in such a system anand2021 . Rizzo et al. also found that storage
capacity is primarily decided by the relaxation time rizzo1999 . Thus
motivated, we report the effect of dipolar interaction and temperature on the
magnetic relaxation in a two-dimensional assembly of MNPs with randomly
oriented anisotropy axes in the present work. In particular, we perform
extensive kinetic Monte Carlo simulation (kMC) to probe the relaxation
characteristics as a function of dipolar interaction strength, aspect ratio
and temperature in a two-dimensional array of nanoparticles.

The kMC simulation is now widely used to study the time-dependent properties
of the superparamagnetic assembly of magnetically interacting MNPs. In the kMC
simulation algorithm, there exists a linear relationship between the
simulation steps and the real time scale, which holds quite well for a wide
range of frequencies chantrell2000 . Therefore, the simulated relaxation
curves are obtained as a function of time, facilitating the precise evaluation
of relaxation time. In the Metropolis Monte Carlo method, on the other hand,
the relationship between simulation step and time is not well defined, so the
magnetic relaxation curves are obtained as a function of Monte Carlo steps,
precluding a quantitative comparison with analytical models melenev2012 . The
kMC simulation has been used to study the various quantities of interest in
MNPs assembly anand2019 ; tan2014 ; carrey2016 ; anand2020 . For instance, Tan
et al. studied the spatial dependence of heat dissipation due to hysteresis in
an assembly of dipolar interacting MNPs tan2014 . We have recently implemented
the kMC algorithm to analyze the hysteresis response in a one-dimensional
chain of MNPs as a function of dipolar interaction strength and anisotropy
axis orientation anand2020 . It has also been used to study transport
properties muller2011 ; verdes2002 .

The remaining of the paper is organized as follows: In Sec. II, we present the
model and discuss the various energy terms. We also discuss the kMC
simulations briefly. The simulation results will be discussed in Sec. III.
Finally, we summarize and conclude the work in Sec. IV.

##  II Model

We consider an assembly of spherically shaped and monodisperse magnetic
nanoparticle arranged in a two-dimensional lattice in the $xy$-plane with
system dimension $L_{x}\times L_{y}$. The particle has a diameter $D$, and the
lattice spacing is $a$, as shown in Fig. 1(a). Each particle has a magnetic
moment $\mu=M_{s}V$, $M_{s}$ being the saturation magnetization, and $V=\pi
D^{3}/6$ is the volume of the nanoparticle. We have also assumed that each
nanoparticle has uniaxial anisotropy constant $K_{\mathrm{eff}}$ is the
uniaxial anisotropy constant. The anisotropy or the easy axes are considered
to have random orientations to mimic the actual system. The energy of single
domain superparamagnetic nanoparticle due to uniaxial anisotropy is given by
anand2019 ; dattagupta2012 ; muscas2018

$E=K_{\mathrm{eff}}V\sin^{2}\theta.$ (1)

Here $\theta$ is the angle between the magnetic moment and the anisotropy
axis. It is evident from Eq. (1) that the energy function of a single
nanoparticle is a symmetric double well. There are two energy minima
$E^{0}_{1}$ and $E^{0}_{2}$ at $\theta=0$ and $\pi$, respectively. These
energy minima are separated by an energy barrier $E^{0}_{3}=K_{\mathrm{eff}}V$
at $\theta=\pi/2$ as depicted in Fig. 1(b). Due to thermal fluctuations,
magnetic moment changes its orientation within the particle by overcoming the
energy barrier $K_{\mathrm{eff}}V$. The mean time taken by the magnetic moment
to change its direction is known as Néel relaxation time $\tau^{0}_{N}$
defined as anand2019 ; carrey2011

$\tau^{0}_{N}=\tau_{o}\exp(K_{\mathrm{eff}}V/k_{B}T).$ (2)

Here $\tau_{o}=(2\nu_{o})^{-1}$, $\nu_{o}\approx 10^{10}$ $s^{-1}$ is the
attempt frequency. $k_{B}$ is the Boltzmann constant and $T$ is the
temperature. Eq. (2) is applicable for non-interacting MNPs.

In an assembly, MNPs primarily interact due to long-ranged dipolar
interaction. We can calculate the dipolar interaction energy
$E_{\mathrm{dip}}$ in such a case as odenbach2002 ; usov2017 ; anand2021hys

$E_{\mathrm{dip}}=\frac{\mu_{o}}{4\pi a^{3}}\sum_{j,\ j\neq
i}\left[\frac{\vec{\mu_{i}}\cdot\vec{\mu_{j}}}{(r_{ij}/a)^{3}}-\frac{3\left(\vec{\mu_{i}}\cdot\hat{r}_{ij}\right)\left(\vec{\mu_{j}}\cdot\hat{r}_{ij}\right)}{(r_{ij}/a)^{3}}\right].$
(3)

Here $\mu_{o}$ is the permeability of free space; $\vec{\mu}_{i}$ and
$\vec{\mu}_{j}$ are the magnetic moment vectors of $i^{th}$ and $j^{th}$
nanoparticle, respectively, and $r_{ij}$ is the center-to-center separation
between $\mu_{i}$ and $\mu_{j}$. $\hat{r}_{ij}$ is the unit vector
corresponding to $\vec{r}_{ij}$. The corresponding dipolar field
$\mu_{o}\vec{H}_{\mathrm{dip}}$ is given by the following expression
anand2021hys ; tan2014

$\mu_{o}\vec{H}_{\mathrm{dip}}=\frac{\mu\mu_{o}}{4\pi a^{3}}\sum_{j,j\neq
i}\frac{3(\hat{\mu}_{j}\cdot\hat{r}_{ij})\hat{r}_{ij}-\hat{\mu_{j}}}{(r_{ij}/a)^{3}}.$
(4)

As it is evident from Eq. (3) and Eq. (4) that dipolar interaction varies as
$1/r^{3}_{ij}$, we can define the strength of dipolar interaction
$h_{d}=D^{3}/a^{3}$ tan2010 . So $h_{d}=1.0$ is the largest dipolar
interaction strength, and $h_{d}=0$ can be termed as the non-interacting case.
Therefore, the total energy of the underlying system is given by tan2014 ;
anand2019

$E=K_{\mathrm{eff}}V\sum_{i}\sin^{2}\theta_{i}+\frac{\mu_{o}\mu^{2}}{4\pi
a^{3}}\sum_{j,\ j\neq
i}\left[\frac{\hat{\mu_{i}}\cdot\hat{\mu_{j}}-{3\left(\hat{\mu_{i}}\cdot\hat{r}_{ij}\right)\left(\hat{\mu_{j}}\cdot\hat{r}_{ij}\right)}}{(r_{ij}/a)^{3}}\right]$
(5)

Here $\theta_{i}$ is the angle between the anisotropy axis and the $i^{th}$
magnetic moment of the system.

The energy function given by Eq. (1) is modified due to the dipolar
interaction. Consequently, the energy barrier seen by the moments is also
altered. The single-particle energy function defined by Eq. (1) becomes
asymmetric, as depicted in Fig. 1(c). When the dipolar field is greater than
the single-particle anisotropy field $H_{K}=2K_{\mathrm{eff}}/M_{s}$
carrey2011 ; the energy function has only one minimum. The energy profile
displays two minima $E_{1}$ and $E_{2}$, and a maxima $E_{3}$ for each
magnetic moment when the dipolar field is less than $H_{K}$, as shown in Fig.
1(c). Therefore, the jump rate $\nu_{1}$ for the magnetic moment to go from
$E_{1}$ to $E_{2}$ via $E_{3}$ is given by hanggi1990

$\nu_{1}=\nu^{0}_{1}\exp\bigg{(}-\frac{E_{3}-E_{1}}{k_{B}T}\bigg{)}$ (6)

Similarly the jump rate $\nu_{2}$ for the magnetic moment to switch its
orientation from $E_{2}$ to $E_{1}$ is expressed as hanggi1990

$\nu_{2}=\nu^{0}_{2}\exp\bigg{(}-\frac{E_{3}-E_{2}}{k_{B}T}\bigg{)},$ (7)

where $\nu^{0}_{1}=\nu^{0}_{2}=\nu_{o}$.

The kMC procedure used in the present work is described in greater detail in
the work of Anand et al. and Tan et al. anand2019 ; tan2014 . Therefore, we do
not describe it here to avoid repetition. The kMC simulation has been used to
calculate effective Néel relaxation time $\tau_{N}$ for dipolar interacting
MNPs by studying magnetization $M(t)$ decay as a function of time $t$. For
this, we apply a very large external magnetic field
$\mu_{o}H_{\mathrm{max}}=20$ Tesla along the $y$-direction with respect to the
underlying system so that all the magnetic moments point along the applied
field direction. The total simulation time is divided into 2000 equal time
steps, and we switch off $\mu_{o}H_{\mathrm{max}}$ at $t=0$ to study the
magnetic relaxation as a function of time. As time passes, magnetic moments
relax to a state with zero or an extremely small magnetization. The resulting
magnetization-decay curve is then fitted to the form
$M(t)=M_{s}\exp(-t/\tau_{N})$ to extract $\tau_{N}$.

##  III Simulations Results

We consider spherical nanoparticles of Fe3O4 arranged in a two-dimensional
lattice ($L_{x}\times L_{y}$) with $D=8$ nm, $K_{\mathrm{eff}}=13\times
10^{3}$ Jm-3, and $M_{s}=4.77\times 10^{5}$ Am-1. The total number of
nanoparticles in the assembly is considered as $n=400$. We have considered
seven values of system sizes viz. $L_{x}\times L_{y}=20\times 20$, $16\times
25$, $10\times 40$, $8\times 50$, $4\times 100$, $2\times 200$ and $1\times
400$. The corresponding aspect ratio $A_{r}(=L_{y}/L_{x})$ of the underlying
system is $1.0$, 1.56, 4.0, 6.25, 25, 100 and 400, respectively. The dipolar
interaction strength $h_{d}$ is varied from 0 to 1.0. The temperature $T$ is
changed between 100 and 400 K. The anisotropy axes are assumed to have random
orientations. All the numerical data obtained have been averaged over several
independent runs to obtain good statistical averaging.

To validate our kMC method, we first study the magnetic relaxation in the
absence of dipolar interaction at room temperature. In Fig. 1(d), we plot the
simulated magnetization decay $M(t)/M_{s}$ versus $t$ curve of a square
arrangement of MNPs ($L_{x}\times L_{y}=20\times 20$) with $h_{d}=0.0$ and
$T=300$ K. The functional form of the curve is a perfectly exponential decay.
We fit it with $M(t)/M_{s}=\exp(-t/\tau^{0}_{N})$ which yields
$\tau^{0}_{N}=1.161\times 10^{-10}\pm 1.10\times 10^{-11}$ s. The theoretical
value calculated using Eq. (2) comes out to be $\tau^{0}_{N}=1.160\times
10^{-10}$ s, which is in perfect agreement with the simulated one. In the
absence of magnetic interaction, the magnetization decay curve is also found
to be independent of the aspect ratio $A_{r}$ as expected. Therefore, we have
not shown the corresponding curves to avoid duplication.

Next, we study the effect of dipolar interaction on the magnetic relaxation
with the square arrangement of MNPs at $T=300$ K. In Fig. (2), we plot
$M(t)/M_{s}$ versus $t$ curve with $L_{x}\times L_{y}=20\times 20$
($A_{r}=1.0$) for eight representative values of $h_{d}=$ 0.0, 0.1, 0.2, 0.3,
0.4, 0.6, 0.8 and 1.0. There is a smooth decay of magnetization in the
presence of weak dipolar interaction $h_{d}\leq 0.3$. The functional form of
the magnetization decay is perfectly exponential decaying in this case also.
Interestingly, there is a fastening in magnetization relaxation with an
increase in dipolar interaction strength for strongly interacting MNPs
($h_{d}>0.3$). It could be explained using the fact that the dipolar
interaction induces antiferromagnetic coupling between the magnetic moments in
a square lattice. The strength of antiferromagnetic coupling increases with an
increase in $h_{d}$, which results in the fastening of magnetization
relaxation. The dominance of antiferromagnetic coupling is in perfect
agreement with our recent work, where we have also found a characteristic
hysteresis curve of antiferromagnetic coupling predominance in a square
lattice anand2021hys . De’Bell et al. also observed dominance of
antiferromagnetic interaction with the Heisenberg spins arranged in square
array de1997 . The functional form of magnetization decay for weakly
interacting MNPs is also in qualitative agreement with the work of Figueiredo
et al. figueiredo2007 .

We then study the effect of the shape of the system, i.e., aspect ratio
$A_{r}$ on the magnetic relaxation. In Fig. (3), we plot the magnetization
decay $M(t)/M_{s}$ vs. $t$ curve for six representative values of
$A_{r}=1.56$, 4.0, 6.25, 25, 100 and 400. The dipolar interaction strength
$h_{d}$ has been varied between 0.0 to 1.0. Irrespective of $A_{r}$, the
functional form of the magnetization decay is perfectly exponentially decaying
for weakly interacting MNPs ($h_{d}\leq 0.3$). Remarkably, there is a
fastening of magnetization relaxation above certain dipolar interaction
strength $h^{\star}_{d}$, which also increases with $A_{r}$. It is seen that
there is a slowing down of magnetization relaxation below $h^{\star}_{d}$. On
the other hand, magnetization relaxes rapidly above this threshold value of
$h^{\star}_{d}$, which can be due to enhancement in antiferromagnetic
coupling. In the case of enormous $A_{r}$, the system behaves as highly
anisotropic. Consequently, the dipolar interaction promotes ferromagnetic
coupling, which slows down the magnetization reversal (slow decay of
magnetization) with an increase in $h_{d}$. The slowing down of magnetic
relaxation with dipolar interaction strength in the case of $A_{r}=400$ is in
perfect agreement with the work of Iglesias et al. iglesias2004 . These
observations indicate that one can tune the nature of dipolar interaction from
ferromagnetic to antiferromagnetic by just varying the aspect ratio of the
system. Consequently, the relaxation time can be manipulated in a more
controlled way, which is an essential aspect of data storage applications.

It is equally important to understand the effect of thermal fluctuations on
magnetic relaxation. In Fig. (4), we plot the magnetization decay curve as a
function of temperature $T$ for weak dipolar interaction. We have considered
the square arrangement of MNPs ($L_{x}\times L_{y}=20\times 20$, $A_{r}=1.0$)
with two values of dipolar interaction strength $h_{d}=0.0$ and 0.2. There is
a smooth decay of magnetization, and the functional form of the magnetic
relaxation curve is also perfectly exponential irrespective of $T$ and
$h_{d}$. There is a slow decay of magnetization in the absence of dipolar
interaction and smaller values of $T$. There is a fastening in magnetization
relaxation with an increase in temperature. It is due to the enhancement in
thermal fluctuations. The rate of magnetization decay becomes slower for
comparably large dipolar interaction strength ($h_{d}=0.2$). In the presence
of weak interaction, there is always an increase in magnetization decay with
increased thermal fluctuations. The functional form of magnetization decay is
also found to be independent of aspect ratio $A_{r}$ of the system for weak
dipolar interaction, as expected (curves not shown).

The magnetic relaxation should depend on the aspect ratio for strongly
interacting MNPs. Therefore, we now study the magnetization decay as a
function of temperature $T$ with various values of $A_{r}$ in the presence of
large dipolar interaction strength. In Fig. (5), we plot the magnetization
decay $M(t)/M_{s}$ vs. $t$ with $h_{d}=0.4$ and six values of $A_{r}=$1.0,
1.56,4.0,6.25, 100 and 400. The temperature has been varied between 100 and
400 K. It is seen that magnetization decay is faster as compared to the weakly
interacting MNPs (Fig. (4)) for a given temperature and $A_{r}<100$.
Magnetization also decays faster than an exponential function. It means that
the large dipolar interaction strength promotes antiferromagnetic interaction.
With an increase in thermal fluctuations, there is fastening in magnetization
relaxation. Interestingly, magnetization decays very slowly with $A_{r}>100$
and also has a weak temperature dependence. The dipolar interaction induces
ferromagnetic coupling between the magnetic moments for the highly anisotropy
system, i.e. for huge $A_{r}$. As a consequence, magnetization ceases to relax
for large dipolar interaction strength.

Next, we study the variation of magnetization decay $M(t)/M_{s}$ versus $t$
curve as a function of temperature for various values of $A_{r}$ and very
large dipolar interaction strength $h_{d}=0.6$ and 0.8 in Fig. (6) and Fig.
(7), respectively. All the other parameters are the same as that of Fig. (5).
For $A_{r}<100$, the magnetization decays very fast as compared to the weakly
interacting case. It is due to the fact the antiferromagnetic coupling is
enhanced because of dipolar interaction in such cases. Therefore,
magnetization tends to change its orientation very rapidly, resulting in rapid
decay of magnetization. On the other hand, the dipolar interaction of equal
strength induces ferromagnetic coupling when the aspect ratio is enormous
($A_{r}>100$). Consequently, magnetization relaxes very slowly or does not
relax at all for sizeable dipolar interaction strength. We observe weak
dependence of magnetization relaxation on thermal fluctuations provided
interaction strength is immense. The rapid fall of magnetization as a function
of time is in perfect qualitative agreement with the work of Volkov et al.
volkov2008 . We could not compare the entire range of dipolar interaction
strength and temperature as they have concentrated on a single value of
interaction strength temperature. Therefore our results can be used as a
benchmark in this context. We believe that these results will help the
experimentalist in choosing suitable values of system size, interaction
strength and other parameters of interest for desired relaxation
characteristics for better applications of these systems in data storage and
other related applications.

In Fig. (8), we plot the magnetization decay curve as a function of
temperature for the strongest dipolar interacting MNPs ($h_{d}=1.0$). All
other parameters are the same as that of Fig. (7). As the antiferromagnetic
coupling is the maximum, the decay rate of magnetization is the fastest for
$A_{r}<100$. Remarkably, the ferromagnetic interaction is also the largest for
highly anisotropic system ($A_{r}>100$). Therefore, magnetization ceases to
relax, which results in slowing down of magnetization relaxation for the
highly anisotropic system such as linear chain of MNPs. The temperature has a
negligible effect on the magnetic relaxation as the dipolar interaction
strength is the strongest. To study the effect of dipolar interaction strength
and aspect ratio on the magnetic relaxation quantitatively, we now plot the
variation of simulated Néel relaxation time $\tau_{N}$ as a function of
$h_{d}$ and $A_{r}$ at $T=300$ K in Fig. (9). $\tau_{N}$ is extracted from the
simulated magnetization decay $M(t)/M_{s}$ versus $t$ curve by fitting it with
$M(t)/M_{s}=\exp(-t/\tau_{N})$. There is a decrease in $\tau_{N}$ above a
particular value of dipolar interaction strength $h^{\star}_{d}$. There is
also an increase in $h^{\star}_{d}$ with $A_{r}$. It is quite evident that
$\tau_{N}$ increases with dipolar interaction strength below $h^{\star}_{d}$.
On the other hand, $\tau_{N}$ decreases very fast above this threshold value
of $h^{\star}_{d}$, which can be due to enhancement in antiferromagnetic
coupling. In the case of extremely large $A_{r}$, the system behaves as highly
anisotropic. Therefore, $\tau_{N}$ always increases with $h_{d}$ because of an
enhancement in ferromagnetic coupling in these cases.

Finally, we study the variation of $\tau_{N}$ as a function of temperature and
dipolar interaction strength for various values of $A_{r}$. In Fig. (10), we
plot the variation of $\tau_{N}$ as a function of $T$ and $h_{d}$ for six
representative values of $A_{r}=1.0$, 1.56, 4.0, 6.25, 100 and 400. It is
evident that $\tau_{N}$ decrease with $h_{d}$ for considerable dipolar
interaction strength ($h_{d}>0.3$) and $A_{r}<100$. It is because the dipolar
interaction promotes antiferromagnetic coupling for appreciable dipolar
interaction strength and $A_{r}<100$. As a consequence, there is a decrease in
$\tau_{N}$ with $h_{d}$. Remarkably, $\tau_{N}$ increases with $h_{d}$ for the
highly anisotropic system ($A_{r}>100$). It is due to the fact that dipolar
interaction induces ferromagnetic coupling in such cases. In the case of
weakly interacting MNPs, $\tau_{N}$ decreases rapidly with $T$ because of
enhancement in thermal fluctuations. While for strongly interacting MNPs,
there is weak dependence of $\tau_{N}$ on temperature.

##  IV Summary and Conclusion

Now, we summarize and discuss the main results presented in this work. We have
performed extensive kinetic Monte Carlo simulations to probe the magnetic
relaxation characteristics in a two-dimensional array of spherical and
monodisperse nanoparticle as a function of dipolar interaction strength, the
aspect ratio of the system and temperature. In the case of non-interacting and
weakly interacting MNPs, the functional form of the magnetization-decay curve
is a perfect exponential. The extracted Néel relaxation time from the
simulated curve is also in excellent agreement with the value calculated using
the Néel-Brown model. Interestingly, there exhibits fastening of magnetic
relaxation with an increase in dipolar interaction strength in a square
arrangement of MNPs, provided the interaction strength is significant. The
same is true even with the large aspect ratio of the system. It can be
explained using the fact that dipolar interaction promotes antiferromagnetic
coupling between the moments in the square arrangement of MNPs macisaac1996 .
Consequently, magnetization tends to reverse its orientation very rapidly with
an increase in dipolar interaction strength. Therefore, there is fast
magnetization relaxation in such cases. Remarkably, there is a slowing down of
magnetization relaxation as interaction strength increases in a highly
anisotropic system, i.e., when the aspect ratio is enormous. The dipolar
interaction of enough strength induces ferromagnetic interaction between the
MNPs in the highly anisotropic system such as a linear chain. Consequently,
magnetization ceases to relax in such a situation, resulting in extremely slow
magnetization reversal. The thermal fluctuation is also found to affect
magnetic relaxation drastically. There is a fastening of magnetization
relaxation with temperature in the weak dipolar limit. It is due to the fact
that as the temperature is increased, thermal fluctuation increases.
Therefore, magnetic moments tend to cross the energy barrier more frequently,
which leads to the fastening of magnetic relaxation.

The effect of dipolar interaction and aspect ratio on the magnetic relaxation
is also clearly manifested in the variation of Néel relaxation time
$\tau_{N}$. In the presence of strong dipolar interaction and square
arrangement of MNPs, $\tau_{N}$ decreases rapidly with $h_{d}$ for a given
temperature. A similar observation is made for the relatively large aspect
ratio $A_{r}$. It is because the dipolar interaction induces antiferromagnetic
coupling in these cases. Remarkably, the dipolar interaction of equal strength
promotes ferromagnetic coupling in the highly anisotropic system, such as a
linear arrangement of nanoparticles ($A_{r}$ is enormous). Consequently,
$\tau_{N}$ gets enhanced with $h_{d}$. There is a rapid increase in the value
of $\tau_{N}$ with an increase in thermal fluctuations for weakly interacting
MNPs. On the other hand, $\tau_{N}$ has a fragile dependence on temperature
provided dipolar interaction is strong.

In conclusion, we have studied thermal and dipolar interaction effects on the
magnetic relaxation in the two-dimensional arrangement of MNPs with randomly
oriented anisotropy axes using kinetic Monte Carlo simulation. Our results
suggest that magnetization relaxation can be tuned by just varying the dipolar
interaction and shape of the system. However, our results have been obtained
for monodisperse magnetic nanoparticles without any position disorder. We
believe that similar conclusions can be drawn for the system with the disorder
also as long as the dipolar interaction dictates their magnetic behaviour. We
also believe that the concepts presented in this work are incredibly relevant
for the efficient use of magnetic nanoparticles in data storage and other
related applications. These observations could also help the physicist
optimize various parameters of interest, such as frequency of the external
magnetic field, dipolar interaction strength and shape of the system in
magnetic hyperthermia applications.

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Figure 1: (a) Schematic of the two-dimensional array of magnetic
nanoparticles. $a$ is the lattice constant, and $D$ is the particle diameter.
(b) Schematic of the energy barrier in the absence of dipolar interaction.
There are the two energy minima at $\theta=0$ and $\pi$, respectively. There
is an energy maximum of strength $K_{\mathrm{eff}}V$ at $\theta=\pi/2$. (c)
Schematic of the energy barrier in the presence of dipolar interaction. The
modified energy minima are $E_{1}$ and $E_{2}$ and the maximum is $E_{3}$. (d)
Simulated magnetization decay $M(t)/M_{s}$ versus $t$ curve for $h_{d}=0.0$ at
temperature $T=300$ K. It has been fitted with
$M(t)=M_{s}\exp(-t/\tau^{0}_{N})$ and shown with the black line. The fitted
value comes out to be $\tau^{0}_{N}=1.161\times 10^{-10}\pm 1.10\times
10^{-11}$ s. The theoretical value of $\tau^{0}_{N}$ is $1.160\times 10^{-10}$
s. It shows perfect agreement between the theory and kMC simulation.

Figure 2: Magnetization decay curve as a function of dipolar interaction
strength $h_{d}$ for square assembly of MNPs at $T=300$ K. There is an
increase in magnetization relaxation with an increase in $h_{d}$ provided
dipolar interaction strength is appreciable ($h_{d}>0.3$). It can be
attributed to an enhancement in antiferromagnetic coupling with an increase in
interaction strength.

Figure 3: Magnetization decay $M(t)/M_{s}$ versus $t$ curves as a function of
dipolar interaction strength $h_{d}$ for various values of aspect ratio
$A_{r}$ at $T=300$ K. It is evident that there is an increase in magnetic
relaxation with $h_{d}$ even with the rectangular arrangement of MNPs
($A_{r}>1$), which can be due to an increased in antiferromagnetic coupling.
Interestingly, there is a slowing down in magnetization decay for huge aspect
ratio $A_{r}$ because of enhancement in ferromagnetic interaction.

Figure 4: Magnetization decay $M(t)/M_{s}$ versus $t$ curve as a function of
temperature $T$ for weakly interacting MNPs. We have considered two value of
dipolar interaction strength $h_{d}=0.0[(a)]$ and 0.2[(b)]. In the case of
non-interacting MNPs ($h_{d}=0.0$) and a given temperature, magnetization
relaxes faster as compared to h${}_{d}=0.2$. Magnetization-decay does not
depend on aspect ratio $A_{r}$ as expected (curves not shown).

Figure 5: Magnetization-decay curves as a function of temperature $T$ for
$h_{d}=0.4$. We have considered six values of aspect ratio $A_{r}=1.0$[(a)],
1.56[(b)], 4.0[(c)], 6.25[(d)], 100[(e)] and 400[(f)]. For $A_{r}<100$,
magnetization decays rapidly as compared with the non-interacting case. It is
because the nature of dipolar interaction changes from antiferromagnetic to
ferromagnetic for exceedingly large $A_{r}$. There is also an increase in
magnetization relaxation with temperature for $A_{r}<100$. While for a huge
aspect ratio, there is a weak dependence of relaxation on thermal
fluctuations.

Figure 6: Magnetization decay $M(t)/M_{s}$ versus $t$ curve as a function of
temperature $T$ for strongly interacting MNPs ($h_{d}=0.6$). We have
considered six values of aspect ratio $A_{r}=1.0$[(a)], 1.56[(b)], 4.0[(c)],
6.25[(d)], 100[(e)] and 400[(f)]. In the case of $A_{r}<100$, magnetization
relaxes very rapidly as compared to non-interacting case. It is because the
dipolar interaction promotes antiferromagnetic coupling between the MNPs in
such a case. On the other hand, MNPs ceases to relax with a huge aspect ratio
($A_{r}>100$) due to the enhanced ferromagnetic interaction. There is a weak
dependence of relaxation on thermal fluctuations.

Figure 7: Magnetization decay curves as a function of temperature $T$ for
$h_{d}=0.8$. We have considered six values of aspect ratio $A_{r}=1.0$[(a)],
1.56[(b)], 4.0[(c)], 6.25[(d)], 100[(e)] and 400[(f)]. There is a rapid fall
in magnetization as a function of time for $A_{r}<100$ because of
antiferromagnetic interaction indued by dipolar interaction. The dipolar
interaction promotes ferromagnetic coupling between the MNPs with $A_{r}>100$,
resulting in the slow decay of magnetization. Temperature does not affect the
relaxation characteristics because of the large dipolar interaction strength.

Figure 8: Magnetization-decay $M(t)/M_{s}$ versus $t$ curve as a function of
temperature $T$ for the strongest dipolar interacting MNPs ($h_{d}=1.0$). We
have considered six values of aspect ratio $A_{r}=1.0$[(a)], 1.56[(b)],
4.0[(c)], 6.25[(d)], 100[(e)] and 400[(f)]. Magnetization relaxes very rapidly
with time for $A_{r}<100$. On the other hand, magnetization relaxes slowly or
does relax at all for $A_{r}>100$ because of large ferromagnetic interaction.
Thermal fluctuations does not affect the relaxation as dipolar interaction
strength is the largest.

Figure 9: Variation of Néel relaxation time $\tau_{N}$ as a function of
dipolar interaction strength $h_{d}$ and aspect ratio $A_{r}$ at T=300 K.
$\tau_{N}$ decreases above a particular value of dipolar interaction strength
$h^{\star}_{d}$. The latter also increases with $A_{r}$. In the case of
enormous $A_{r}$, $\tau_{N}$ always increases with $h_{d}$ because of an
enhancement in ferromagnetic coupling.

Figure 10: Variation of Néel relaxation time $\tau_{N}$ as a function of
dipolar interaction strength $h_{d}$ and $T$ for six values of
$A_{r}=1.0$[(a)], 1.56[(b)], 4.0[(c)], 6.25[(d)], 100[(e)], and 400[(f)].
$\tau_{N}$ decreases with $h_{d}$ for appreciable dipolar interaction strength
($h_{d}>0.3$) and $A_{r}<100$. Remarkably, $\tau_{N}$ increases with $h_{d}$
for the highly anisotropic system ($A_{r}>100$). It is because the dipolar
interaction induces ferromagnetic coupling in such cases. There is a rapid
fall in $\tau_{N}$ with $T$ because of enhancement in thermal fluctuations.