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KR4HI’22: International Workshop on Knowledge Representation for Hybrid
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[orcid=0000-0002-9622-4142<EMAIL_ADDRESS>url=https://rfguimaraes.github.io, ]

[orcid=0000-0002-3889-6207<EMAIL_ADDRESS>url=https://www.uib.no/en/persons/Ana.Ozaki, ]

# On the Effectiveness of Knowledge Graph Embeddings: a Rule Mining Approach

Johanna Jøsang  Department of Informatics, University of Bergen  Ricardo
Guimarães  Ana Ozaki

(2022)

###### Abstract

We study the effectiveness of Knowledge Graph Embeddings (KGE) for knowledge
graph (KG) completion with rule mining. More specifically, we mine rules from
KGs before and after they have been completed by a KGE to compare possible
differences in the rules extracted. We apply this method to classical KGEs
approaches, in particular, TransE, DistMult and ComplEx. Our experiments
indicate that there can be huge differences between the extracted rules,
depending on the KGE approach for KG completion. In particular, after the
TransE completion, several spurious rules were extracted.

###### keywords:

Rule Mining Knowledge Graphs Knowledge Graph Embeddings

##  1 Introduction

Nowadays, Knowledge Graphs (KGs) are an increasingly popular way to represent
data [1]. A KG can be often seen as a labelled directed graph in which the
nodes represent the elements in the domain of interest (e.g. people) and the
edges represent a relation between two elements. For instance, a KG such as
Wikidata, might include the node “Paris” with an outgoing edge labelled
“capital of” to the node “France”.

Knowledge graph embeddings (KGEs) attempt to capture patterns present in KGs
and generalize them so as to infer new data (commonly, in the format of RDF
triples). Such data can then be used to “complete” the information in the KG.
Since the publication of the very classical KGE approach known as TransE [2],
several authors proposed alternative approaches for KGEs. KGEs can be
effective methods for KG completion if they capture patterns in the data and
can generalize such patterns in a uniform way. That is, there should not be
much distortions on the facts classified as plausible by a KGE method.

A main challenge is to determine and evaluate how effective they are for KG
completion. In other words, how well such KGEs can generalize from the data
and whether there are biases in the process that significantly affect KG
completion based on such models. Understanding the power of KGEs is vital to
enable human comprehension of the power and limitation of these models.
Example 1 illustrates how a KGE can be used for KG completion.

######  Example 1.

If most individuals in the data who are siblings are also relatives, then we
would expect a KGE approach to capture such patterns. The KGE would give a
high rank to triples expressing two individuals known to be siblings as
relatives.

However, the underlying _rules_ captured by KGEs are hidden in the model. To
study the effectiveness of KGEs for KG completion, we apply _rule mining_ : a
data mining based approach for extracting rules that represent the main
patterns in the data [3]. Such rules can be useful to identify implicit
knowledge in the domain associated with the data. The following Example
demonstrates this.

######  Example 2.

In the context of Example 1, such rule could be ‘if two persons are siblings
then they are (family) relatives”. In symbols, $sibling(x,y)\Rightarrow
relative(x,y)$ (as usual, e.g. [3], the universal quantifier over the
variables has been omitted).

Figure 1: KGEs can be applied for KG completion and impact the rules that hold
with high confidence in the extended (completed) KG. Rule Mining can be
applied to induce new links and change KGEs results, however, we focused on
the first case since our goal is to investigate the effectiveness of KGEs.

Rule mining and KGEs can complement each other as methods for identifying
patterns in KGs. Fig. 1 illustrates the interplay between KGEs and rule
mining. In this work, we mine rules from KGs before and after they have been
completed by a KGE to compare possible differences in the rules extracted. As
we discuss in this work, there can be significant distortions for KG
completion depending on the chosen KGE approach. Our work gives some insights
about how to detect such distortions. We apply our method to classical KGEs
approaches, in particular, TransE, DistMult and ComplEx.

In the next sections, we discuss related work (Section 2), introduce the most
relevant basic notions from the literature (Section 3), describe our method
(Section 4), present the experimental results (Section 5), and conclude
(Section 6).

##  2 Related Work

In this Section, we discuss the approaches which relate more closely to this
work. We focus in particular on rule mining methods which also employ KGEs, or
similar approaches. KALE [4], for example, learns rules in a two-step process:
first by mining rules from TransE embeddings and then retraining the
embeddings on a joint model for triples and rules. While KALE unifies KGEs and
rule learning, our approach is agnostic regarding the KGE model. RUGE [5] is
based on KALE and employs a similar procedure where soft rules are obtained
from the data, which will guide the embeddings learned. In both KALE and RUGE,
only the embeddings are updated, the rules remain the same from the moment
they are extracted. In contrast, IterE [6] iteratively updates both embeddings
and the rules. The score of the rules in IterE depends directly on the
embeddings of the relations that it uses. ExCut [7] also employs KGEs and rule
mining, however, with the purpose of explaining clusterings of entities.

Meilicke et al. [8] combines rule-based approaches and KGEs using an _ensemble
approach_. In ensemble learning, the idea is that each model provides an
answer to a problem, and then these individual answers are aggregated to give
a final verdict [9]. Since rule-based and KGE-based methods often offer
complimentary performance on different instances, the authors devised a system
that performs well in different settings for KG completion [10]. The key
distinction between the mentioned studies and ours is that they aim at finding
true triples to perform KG completion, while we focus our investigation on the
effectiveness of KG completion via KGEs by mining rules before and after the
completion and then analysing the results.

##  3 Background

Here we provide basic notions about KGEs and rule mining relevant for this
paper.

###  3.1 KG Embeddings

KGEs are functions that map entities and relations in KG to a vector space,
usually with low-dimension. The main goal of KGE methods is to represent
entities and relations in such a way that the patterns of the data are
preserved. In particular, one of the main applications of these methods is KG
completion: the task of identifying missing triples of a KG.

The mathematical representation assigned to an entity and/or a relation is
called its embedding and each KGE approach determines constraints on how the
embeddings of entities and relations relate to each other. For instance, in
the TransE model [2], every entity and relation is mapped to a vector in
$\mathbb{R}^{n}$. Given a triple $(s,r,o)$ in the KG, the TransE model aims at
minimising the distance between $\mathbf{s}+\mathbf{r}$ and $\mathbf{o}$,
where $\mathbf{s}$, $\mathbf{r}$ and $\mathbf{o}$ are the embeddings of $s$,
$r$ and $o$111The letters ‘s’, ‘r’, and ‘o’ stand for _subject_ , _relation_ ,
and _object_ , respectively The letters ‘h’ and ‘t’ are also often used
instead of ‘s’ and ‘o’, meaning _head_ and _tail_ , respectively..

The computation of KGEs follows the standard machine learning pipeline: the
embeddings start with randomly assigned values, which are then updated using
stochastic gradient descent or similar measures while minimising a loss
function that encodes the desired property imposed by the model. The loss
function usually depends on a score (or energy) function, which determines,
for a given triple, how likely it is to be true, given the embeddings of its
elements. In the case of TransE, the score function is
$f_{\text{TransE}}(s,r,o)={||\mathbf{s}+\mathbf{r}-\mathbf{o}||}_{\ell}$, in
which ${||.||}_{\ell}$ denotes the $\ell$-norm (usually $L_{1}$ or
$L_{2}$-norm).

Nowadays, there are numerous approaches, including more sophisticated versions
of TransE [11] which are also based on geometric formulations. There are also
approaches such as DistMult [12] which is based on tensor factorisation.
Embedding models vary concerning the number of parameters learned per entity
and relation, and which patterns they can capture from data. For instance,
while DistMult is computationally cheap, it is unable to represent asymmetric
relations such as _sibling_ , while ComplEx demands more parameters, but can
represent both symmetric and asymmetric relations [13].

###  3.2 Rule Mining

An _atom_ is an expression of the form $r(x,y)$, where $r$ is a relation and
$x,y$ are variables. Rules in this work are expressions of the form
$\mathbf{B}\Rightarrow H$ where $\mathbf{B}$ (called the _body_ of the rule)
is a conjunction of atoms and $H$ is an atom (called the _head_ of the rule).
A rule is _closed_ if all variables appear in at least two atoms. A closed
rule is always _safe_ , i.e. all head variables appear also in at least one
body atom. From now on, assume all rules we speak of are closed. Regarding the
measure used for ranking the rules, we use the _Partial Completeness
Assumption (PCA) confidence_ (see Appendix A for details). This measure is
more elaborate than the classical notion of confidence from rule mining in KGs
[3], which is defined as the proportion of true predictions out of the true
and false predictions. The PCA lies between the open and the closed world
assumptions. The main intuition is that if a node is a parent of another node
via a relation $r$ then it is assumed that all the information regarding
childhood of this node by $r$ is complete. As an example, if we have the
information that the mathematician Artur Ávila was born in Brazil, we assume
that an assertion that he was born in Argentina is false. On the other hand,
if we do not know where the mathematician Graciela Boente was born, then we do
not assume that an assertion that she was born in Argentina is false.

##  4 Comparing KGEs with Rule Mining

Here we describe our approach for investigating the effectiveness of KGEs for
KG completion. By KG completion we mean the process of adding new, potentially
true triples to the original KG. As the resulting KG may still be incomplete,
we will refer to the new versions generated through this process as
_extensions_ or _extended KGs_. In other words, we study how KGEs can be
applied to extend KGs with new triples which are associated with high PCA
confidence.

We compare the results of rule mining before and after the KG has been
extended by new triples. First we create the triples using an entity selection
method and then check what is the confidence (w.r.t. the KGE). In summary, the
main steps of our pipeline are as follows.

  1. 1.

Rule Mining on Original KG: We apply rule mining to the KG.

  2. 2.

KG Completion: We extend the KG with different KGEs and entity selection
methods.

  3. 3.

Rule Mining on Extended KG: We apply rule mining to the extended KG.

  4. 4.

Analysis: To study the effectiveness of the KG completion, we compare the
rules mined before and after extending the KG.

In Step 1, we apply PCA confidence (see definition in Appendix A) from rule
mining. In Step 2, we consider three KGEs and three entity selection methods,
which we detail in the next paragraph. Finally, in Steps 3 and 4 we apply rule
mining in the extended KG (using PCA confidence to score the rules) and
compare the rules.

In Step 2 of our pipeline, we attempt to extend the KG with new triples that
received a high ranking according to a KGE. The challenge here is that it is
not feasible to check for all possible triples. In practice, one needs to
apply some heuristics to find “good triples”, that is, new triples that are
good candidates to receive a high ranking. To address this issue, we
considered the strategies already implemented in AmpliGraph [14]. However, we
had technical difficulties when trying to use them222 In the current version
of AmpliGraph (1.4.0) it is not recommended to use exhaustive search (e.g. for
a random selection) due to a large amount of computation required to evaluate
all triples
https://docs.ampligraph.org/en/1.4.0/generated/ampligraph.discovery.discover_facts.html.
There were technical difficulties in using the other strategies.. The most
effective and efficient strategy employed by AmpliGraph to find “good triples”
is to search for less frequent entities333The AmpliGraph team say that this
assumption has been true for their empirical evaluations, but is not
necessarily true for all datasets [15].. We implemented this strategy in this
work. To study the effect of frequency, we also implemented an entity
selection method that selects the most frequent and (uniform) random
selection. In addition, we included a probabilistic selection method based on
the frequency of the entities in the dataset, where the least frequent
entities are most likely to be selected. The latter served as a method between
the least frequent and the random selection methods.

##  5 Experiments

In this Section, we discuss the implementation of the pipeline and methodology
presented in Section 4 and present the results of the experiments. The main
parameters of our experiments are the KGE, the entity selection method, and
the cutoff of the ranking of the rules (a high ranking is associated with low
number, where $1$ is the highest rank). The model selection, candidate
generation and extension were implemented in Python, relying mostly on
AmpliGraph. We mined the rules from the extended KGs using AMIE3 [3]. More
precisely, we used the following command to mine rules: java -jar amie-
milestone-intKB.jar -bias lazy -full -noHeuristics -ostd [TSV file], in which
the TSV file contains the input KG. The experiments were run on a server with
64 GB of RAM and an Intel Core i9-7900X 3.3GHz processor. In the following, we
discuss the other important aspects and results.

###  5.1 Experimental Setup

Next, we discuss the datasets and KGE models that we employed in our
experiments.

####  5.1.1 Datasets and Model Selection

We considered two KGs as datasets in this paper. However, instead of using
them directly, we restricted our attention to six types of relations in each
KG, removing the triples that use other relations. We imposed this constraint
to cope with the volume of data, and to control the number of resulting rules.
Moreover, as the we have a fixed limit on the number of candidate triples,
limiting the number of relations increases the overall rate of triples per
relation, which facilitates rule mining and gives a clearer picture of the
influence of each embedding method.

The first dataset is the well-known WN18RR, a KG derived from WordNet and that
contains roughly 93000 triples and 11 relations [16]. We choose this KG mostly
due to its popularity as a benchmark dataset for KG completion and its
relatively small number of relation types.

In this KG, the entities are sets of words called _synsets_ and the relations
connect synsets depending on their meaning. We selected the six most frequent
relations from this KG, more specifically: $hypernym$, $derivationally\
related\ form$, $member\ meronym$, $has\ part$, $synset\ domain\ topic\ of$
and $instance\ hypernym$. As a result, our version of the WN18RR KG has 88227
triples, corresponding to 95% of the original KG.

The second is the Family KG, a portion of Wikidata5M obtained from the PyKEEN
library444https://pykeen.readthedocs.io/en/stable/api/pykeen.datasets.Wikidata5M.html#pykeen.datasets.Wikidata5M.
Wikidata5M is a KG based on Wikidata with over 20 million triples over 800
types of relations [6]. As with WN18RR, Wikidata5M is also commonly employed
when evaluating methods for KG completion. We focused on relations of the
“family” domain, that is: P22 ($father$), P25 ($mother$), P26 ($spouse$), P40
($child$), P1038 ($relative$) and P3373 ($sibling$). From now on, we will use
the natural language readings (e.g. $child$) instead of the original property
codes (e.g. P40) for readability. We selected this particular subset of
properties because the structure of family relations is well-known and easy to
understand. Therefore, it will facilitate the analysis of the rules obtained
according to the KGEs. After restricting to these six relations, the Family KG
contains around 250000 triples.

Due to limited computational resources, we used a random search (instead of
grid search) over all combinations of values for the hyperparameters.
Additionally, for each hyperparameter combination, we selected randomly and
uniformly a learning rate between $10^{-4}$ and $10^{-2}$. Table 1 depicts how
the resulting models of each KG embedding method performed on each dataset
regarding KG completion. We evaluated the different embedding methods on
standard metrics for KG completion: mean rank (MR), the lower the better; mean
reciprocal rank, which ranges from 0 (worst) to 1 (best); and hits@k which
goes from 0 (worst) to 1 (best).

Dataset | WN18RR KG | Family KG
---|---|---
Model | MR | MRR | Hits@K | MR | MRR | Hits@K
1 | 3 | 10 | 1 | 3 | 10
Random | 495.32 | 0.01 | 0.00 | 0.00 | 0.01 | 498.72 | 0.00 | 0.00 | 0.00 | 0.10
TransE | 34.29 | 0.60 | 0.51 | 0.66 | 0.76 | 2.59 | 0.93 | 0.88 | 0.97 | 0.99
DistMult | 152.37 | 0.62 | 0.59 | 0.63 | 0.66 | 7.45 | 0.98 | 0.99 | 0.99 | 0.99
ComplEx | 139.36 | 0.59 | 0.57 | 0.60 | 0.63 | 4.64 | 0.99 | 0.98 | 0.99 | 0.99
Table 1: Results of selected models evaluated on test set. The best results
per column are highlighted.

In Table 1, we can see that the Random Baseline performed poorly in all
metrics for every dataset. This method simply assigns to each triple a pseudo-
random number between 0 and 1, chosen uniformly. While TransE has a number of
limitations with relation that are symmetric or one-to-many, it was the best
method regarding mean rank and Hits@10 on both datasets. DistMult and ComplEx
performed similarly in both datasets, even though both include asymmetric
relations, which DistMult struggles to represent.

####  5.1.2 KG Completion

We generated one extension of each dataset for each of the 48 combination of
parameters: 4 embedding models $\times$ 4 candidate selection strategies
$\times$ 3 rank cutoff values. Also, we created each extension in two steps:
candidate triple generation and candidate ranking.

In the first step, we generate the triples that are going to be added to the
original dataset. Given one of the four entity selection methods we discussed
in Section 4, we select the top 1000 entities according to the method. For
example, if the extension has _least frequent_ as its candidate selection
method, we picked the 1000 least frequent entities in the original KG. We
limited the number of entities to 1000 due to computational limitations. As a
consequence of this limit, we need to consider at least $10^{3}\times 6\times
10^{3}=6\times 10^{6}$ distinct triples for each extension, among those
already in the original KG and the new candidates.

In the second step, we use the KGE models to decide which of the candidate
triples should be added in the extension of a KG. As the score functions in KG
embedding are more useful for comparison rather than for deciding the
plausibility of single triple in general, it is difficult to set an acceptance
threshold based on score alone. Therefore, we decided to use the rank of a
triple instead. Precisely, each extension has an associated value for the rank
cutoff: if a candidate triple had a worse (higher) rank than the cutoff, it
was not included in the extension. The values which we considered, from the
most to the least strict were 1, 4 and 7 because they linger in the usual
ranges adopted for the Hits@K metric, which is commonly applied to evaluate
performance in KG completion.

###  5.2 Results

In this Subsection, we discuss the results of the experiments. We focus on the
contrast in number and quality of rules mined according to the embedding
model, as it impact the results much more significantly, and later we analyse
the effect of the other parameters.

Dataset | WN18RR | Family
---|---|---
Model |  | Orig. Rules
---
Found
| Orig. Rules
---
Missed
| New Rules
---
Found
| Orig. Rules
---
Found
| Orig. Rules
---
Missed
| New Rules
---
Found
Random | 9 | 1 | 0 | 85 | 9 | 0
TransE | 10 | 0 | 659 | 94 | 0 | 986
DistMult | 10 | 0 | 5 | 93 | 1 | 90
ComplEx | 10 | 0 | 7 | 94 | 0 | 43
Table 2: Distribution of rules over the extended KGs. Rules mined from
multiple extensions count only once

Table 2 depicts the number of unique rules mined per embedding model
considering all combinations of parameters on the different datasets. There
were no new rules obtained from KGs extended via the random baseline, which is
expected, as it effectively adds triples randomly. Additionally, the random
baseline was the worst regarding the extraction of original rules: one of the
ten original rules of the WN18RR KG and 9 out of 94 for the Family KG were
never mined from extensions with random baseline. TransE extensions resulted
in a large number of rules, when compared with the other methods, the number
of new rules is over an order of magnitude higher than DistMult or ComplEx.
Also, when using DistMult-extended KGs, AMIE3 could not find the rule
$child(x,y)\Rightarrow mother(x,y)$ in the Family KG. Even so, DistMult
enabled more new rules than ComplEx on both KGs. This difference is a
consequence of ComplEx adding fewer candidate entities when compared with
DistMult. For instance, when using the _probabilistic_ selection method,
DistMult assigned rank 1 to 1468 candidates on the WN18RR KG, while ComplEx
gave rank 1 to only 866. These results show that extending KGs in this way can
give access to new rules, while preserving the old ones. Furthermore, Table 2
conforms with our intuition that the differences between the results with
DistMult and ComplEx should be smaller in comparison to the difference to the
results with TransE.

((a)) Original WN18RR KG

((b)) Extended WN18RR KG

((c)) Original family KG

((d)) Extended family KG

Figure 2: Distribution of PCA confidence of mined rules by KG embedding
models. PCA confidence scores are calculated on the original KG and the
extended KGs from which the rules are mined. The dashed line represents the
median PCA confidence of the rules mined from the original KG.

Next, we discuss the PCA confidence of the rules mined. Figure 2 displays the
distribution of PCA confidence of the rules discovered in the extensions,
grouped by embedding method. The plots contain standard boxplots with whiskers
indicating the variability outside the upper and lower quartiles, as well as
outliers. The dashed horizontal line denotes the median of the PCA confidence
for the original rules. Figures 2(a) and 2(c) consider the PCA confidence
calculated on the original KGs, therefore those boxplots do not need to
consider duplicates. In Figs. 2(b) and 2(d) instead, each occurrence of a rule
is treated as a separate instance (therefore, the same rule may be counted
multiple times, if it was mined in different extensions of the same method).
The random baseline method produced a better PCA distribution than the others,
but without new rules. In contrast, with TransE produced a large number of
rules, however most of these rules have low PCA values, as shown in Fig. 2.

In the Family KG, DistMult and ComplEx has similar performance regarding PCA
confidence, with a slight lead by DistMult. On the WN18RR the difference is
more pronounced: on the original data, ComplEx has better PCA confidence (the
median is much higher than for DistMult in Fig. 2(a)), in contrast, DistMult
has better performance on its respective extensions (as shown by the overall
position of the boxplots in Fig. 2(b)). As the PCA confidence in the
extensions are biased towards the method themselves, this may indicate that
ComplEx had a better performance than DistMult.

To understand what happens in the TransE case, let us consider the family
dataset. Since the relations _sibling_ , _spouse_ , and _relative_ are
symmetric, they all collapse to the null vector. Furthermore, in this dataset
there is nothing that differentiates the relation _mother_ from _father_ ,
thus they are mapped to the same vector. As _father_ and _mother_ have the
same embedding and are both inverses of _child_ , the embedding for _child_ is
the opposite vector of the former. Therefore, while TransE ranked existing
triples well, as shown in the results for MR and MRR, it probably ranked many
wrong triples as correct, generating a poor and biased set of candidate
entities. As a result, AMIE3 mined the rules $spouse(x,y)\Rightarrow
father(x,y)$ and $child(z,x)\wedge mother(z,y)\Rightarrow spouse(x,y)$ solely
from TransE-extended KGs. The resulting embeddings, computed in practice (see
LABEL:TransE_embedding_family and LABEL:TransE_embedding_WN18RR in the
appendix) reflect this situation. Not only that, although TransE mined many
more rules, the mean PCA confidence of rules mined from TransE extensions is
around 0.1, while the mean is much closer to 0.5 with the other embedding
models. That is not to say that all rules mined with TransE were bad or that
all other methods produced plausible rules exclusively. For instance, both
TransE and DistMult extensions enabled the rule $mother(x,y)\Rightarrow
child(x,y)$. In the DistMult case, this is likely a consequence of the fact
that $mother(x,y)$ and $mother(y,x)$ will always be equality plausible
according to DistMult (and the same happens for $child(x,y)$ and
$child(y,x)$).

((a)) Family KG

Figure 3: Venn diagrams indicating the number of rules learned per method and
overlap for each KG. Rules mined from different extensions count only once

Figure 3 shows how much the new rules extracted per KG embedding method
overlap. As expected, the majority of the rules derived from TransE-extended
KGs are unique. Also, every rule found with ComplEx was also found with
TransE. Surprisingly, the overlap between DistMult and ComplEx was relatively
small despite the two methods being similar. In fact, every rule that was
found both in ComplEx and DistMult-extended KGs was also found in KGs extended
with TransE. Besides, only DistMult-extended KGs generated rules that were not
found in any TransE-extended KG.

As the random baseline method did not find new rules, we can conclude that the
KG embedding methods identified non-trivial patterns in the data. In Fig. 3,
we can observe that the choice of model can have a significant impact in the
rules learned if we complete a KG using the embedding scores. Furthermore, the
results also highlight how limitations of the embedding models may impact the
resulting rules. However, we remark that the performance of the methods on KG
completion tasks (discussed in Subsection 5.1.1) do not work well as
predictors of the number or PCA confidence of the rules learned.

####  5.2.1 Effect of Entity Selection Method and Rank Cutoff

Next, we discuss the impact of the other two parameters, entity selection
strategy and rank cutoff value, on the quantity and PCA confidence of the
rules mined by AMIE3. In our analysis, we exclude rules that were only
discovered by TransE, due to the large number of rules with low PCA confidence
and the other issues due to TransE’s ability to capture the semantics of the
relations in KGs, as we discussed before.

The _probabilistic_ , _random_ and _least frequent_ entity selection
strategies have similar performance regarding PCA confidence (see Fig. C.1).
In comparison, the _most frequent_ strategy performs significantly worse.
However, in terms of number of rules learned, the _most frequent_ approach
stands out as the best one. This effect was more noticeable in the Family KG,
in which AMIE3 mined a total of 105 new rules in KGs extended using the _most
frequent_ strategy, while it mined only 36 new rules considering the second
best, the _random_ approach. We will investigate the causes of this behaviour
in future works. In the WN18RR KG all strategies allowed AMIE3 extract the
original rules, but in the Family KG AMIE3 could find all original rules only
when in KGs extended with the _least frequent_ method (see Tables C.5 and C.4
in the appendix).

The other relevant parameter is the rank cutoff value, which determines how
many candidates were selected for each completion query. Our initial
assumption was that increasing the cutoff would worsen the overall PCA
confidence of the new rules however it would increase the number of new rules
learned. Both in the Family and WN18RR KGs, we observed that the number of
original rules extracted decreased as we increased the rank cutoff value, but
the number of new rules mined increased, which is consistent with our
assumption (see Tables C.6 and C.7 in the appendix).

However, the impact of the rank cutoff on the distribution of PCA rules was
not as consistent as we expected (see Fig. C.2 in the appendix). In our
experiments, the PCA confidence on the original data had similar medians on
the three cutoff values in the WN18RR KG, but the proportion with low PCA
increased when going from rank 1 to rank 4, and when going from rank 4 to rank
7 it was the proportion of rules with high PCA that increased instead. When
evaluating each rule on its source extension in the WN18RR case, the median of
PCA confidence was significantly lower when the cutoff was rank 1, and even
worse when the cutoff was 4. Yet, the whole distribution of PCA confidence
improved considerably when increasing the rank to 7. One possible explanation
for this behaviour is that with large cutoffs the new data ends up dominating
the dataset, therefore, biasing the rules significantly. In the case of the
Family KG, increasing the rank cutoff improved the PCA confidence in general
on the original dataset. In fact, the whole distribution improves, including
the median w.r.t. the median of the original rules. On the extended data for
the Family KG, we observed a slight improvement from rank 1 to rank 4, and
rank 7 performed slightly worse than the others. Therefore, it seems that the
effects of rank cutoff depends more on the KG at hand.

####  5.2.2 Rule Comparison

We now discuss some of the rules mined by AMIE3 and their PCA confidence on
the original KGs and extensions. Table 3 displays the original rules extracted
by AMIE3 from the original WN18RR KG. The Table indicates in how many of the
48 possible extensions (one for each combination of parameters) each rule was
mined (in the column “Frequency”) and its PCA Confidence in the original KG.
Due to space constraints, we placed an analogous
LABEL:family_original_rules_table_PCA for the Family KG in the appendix
(LABEL:family_original_rules_table_PCA).

Rule | Frequency | PCA Confidence
---|---|---
DRF(y, x) $\Rightarrow$ DRF(x, y) | 48 | 1.000000
IH(x, z) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 24 | 0.886525
hypernym(x, z) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 36 | 0.828871
has_part(x, z) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 24 | 0.809524
has_part(z, x) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 24 | 0.771739
hypernym(z, x) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 34 | 0.726667
has_part(x, z) $\wedge$ IH(y, z) $\Rightarrow$ has_part(x, y) | 19 | 0.532544
DRF(x, z) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 24 | 0.492857
DRF(z, x) $\wedge$ SDTO(z, y) $\Rightarrow$ SDTO(x, y) | 24 | 0.492857
has_part(x, z) $\wedge$ hypernym(y, z) $\Rightarrow$ has_part(x, y) | 21 | 0.104016
Table 3: Rules mined from the original WN18RR KG, with their corresponding PCA
confidences and how many times they were mined from the 48 extensions. DRF =
“derivationally related form", STDO = “synset domain topic of" and IH =
“instance hypernym".

In both KGs, there was a strong correlation between the PCA confidence in the
original dataset and the frequency a rule in it was mined from the extensions.
Using Spearmann’s rank correlation, we get a coefficient of 0.65 (with p-value
0.04) for the WN18RR KG and 0.86 (with p-value $5.85e^{-29}$) for the Family
KG. In the Family KG, the only rules which were not mined in all extensions
were those that use the predicate $relative$. There are two main possible
causes: (1) this predicate is the least frequent relation in the Family KG and
(2) the nature of the “relative” relation, which not only is symmetric and
transitive, but also a 1-N relation that subsumes many of the other relations
in the dataset, such as $father$, $mother$, and $sibling$.

##  6 Conclusion

We investigated the influence of distinct KGEs for KG completion by using rule
mining to identify hidden patterns in the original KGs and after completion
with a given KGE. In most cases, no rules were lost but several new rules were
added for the TransE approach. We conclude that the choice of KGE model
strongly influences the number and quality of the rules mined. Interestingly,
the results for Link Prediction do not necessarily reflect on the confidence
of the new rules obtained from extended KGs. However, the PCA confidence of a
rule in the original data can indicate how likely it is to be mined in an
extended (or “completed”) KG. Moreover, even models that are similar in
formalisation and results in KG completion tasks (DistMult and ComplEx in our
case) may differ considerably in the rules that they enable. As future work,
we mention an extension of this study with large KGs and fewer limitations on
the number of candidates generated.

## Acknowledgements

The first author is supported by the ERC project “Lossy Preprocessing”
(LOPRE), grant number 819416, led by Prof. Saket Saurabh. The second author is
supported by the NFR project “Learning Description Logic Ontologies”, grant
number 316022.

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##  Appendix A Definition of PCA Confidence

For convenience of the reader, we provide the formal definition of PCA
confidence [3]. We model a _knowledge base_ (KB) $\mathcal{K}$ as a set of
assertions $r(s,o)$, also called _facts_ , where $r$ is a relation and $s,o$
(subject, object) are constants. A (Horn) _rule_ is an expression of the form
$\mathbf{B}\Rightarrow H$ where $\mathbf{B}$ is a conjunction of atoms and $H$
is an atom (called _head_ of the rule). As already mentioned, a rule is closed
if all variables appear in at least two atoms. A substitution $\sigma$ is a
partial mapping from variables to constants. Substitutions can be
straightforwardly extended to atoms and conjunctions. Given a rule
$R=B_{1}\wedge\ldots\wedge B_{n}\Rightarrow H$ and a substitution $\sigma$, we
call $\sigma(R)$ an _instantiation_ of $R$. If $\sigma(B_{i})\in\mathcal{K}\
\forall i\in\\{1,\ldots,n\\}$, we call $\sigma(H)$ a prediction of $R$ from
$\mathcal{K}$, and we write $\mathcal{K}\wedge R\models\sigma(H)$. If
$\sigma(H)\in\mathcal{K}$, we call $\sigma(H)$ a _true prediction_. The
_functionality score_ of a relation $r$ is

$fun(r)=\frac{|\\{s:\exists
o:r(s,o)\in\mathcal{K}\\}|}{|\\{(s,o):r(s,o)\in\mathcal{K}\\}|}$ (1)

If we have $r(s,o)$ in the KB $\mathcal{K}$, and if $fun(r)\geq fun(r^{-})$,
then we assume that all $r(s,o^{\prime})\in\mathcal{K}$ do not hold in the
real world. If $fun(r)<fun(r^{-})$, then the PCA says that all
$r(s,o^{\prime})\in\mathcal{K}$ do not hold in the real world. These
assertions are _false predictions_. The _support of a rule $R$ in a KB
$\mathcal{K}$_ is the number of true predictions $p$ (of the form $r(x,y)$)
that the rule makes in the KB:

$support(R)=|\\{p:(\mathcal{K}\wedge R\models p)\wedge p\in\mathcal{K}\\}|$
(2)

The _Partial Completeness Assumption_ can be defined as follows. If we have
$r(s,o)$ in the KB $\mathcal{K}$, and if $fun(r)\geq fun(r^{-})$, then we
assume that all $r(s,o^{\prime})\in\mathcal{K}$ do not hold in the real world.
If $fun(r)<fun(r^{-})$, then the PCA says that all
$r(s^{\prime},o)\in\mathcal{K}$ do not hold in the real world. The formula for
_PCA confidence_ is

$pca\text{-}conf(\mathbf{B}\Rightarrow
r(x,y))=\frac{support(\mathbf{B}\Rightarrow r(x,y))}{|\\{(x,y):\exists
z:\mathbf{B}\wedge r(x,z)\\}|}$ (3)

for the case where $fun(r)\geq fun(r^{-})$. If $fun(r)<fun(r^{-})$, the
denominator becomes $|\\{(x,y):\exists x:B\wedge r(x,y)\\}|$.

##  Appendix B Additional Data on Model Selection

Table B.1 displays the ranges of values given to the hyperparameters during
model selection.

Hyperparameter | Values
---|---
Batches count | 50, 100
Epochs | 50, 100
Dimensions | 50, 100, 200
Negative samples per positive | 5, 10, 15
Loss function | pairwise, negative log likelihood
Pairwise margin loss | 0.5, 1, 2
Table B.1: Hyperparameter values selected for model selection

##  Appendix C Additional Data for Subsection 5.2

This section includes additional tables and plots that support the discussion
in Subsection 5.2.

Table C.2: TransE’s 50-dimensional embedding vectors for the different relations in the family dataset, rounded to the third decimal. This table shows that the vectors for all relations are close to the zero-vector. child | father | mother | relative | sibling | spouse
---|---|---|---|---|---
0.009 | -0.010 | -0.010 | -0.001 | 0.000 | 0.000
0.002 | -0.002 | -0.002 | 0.000 | 0.000 | 0.000
-0.001 | 0.001 | 0.002 | 0.001 | 0.000 | 0.000
-0.005 | 0.005 | 0.005 | 0.000 | 0.000 | 0.000
-0.012 | 0.012 | 0.012 | 0.001 | 0.000 | 0.001
-0.007 | 0.008 | 0.007 | 0.001 | 0.000 | -0.001
0.006 | -0.006 | -0.006 | -0.001 | 0.000 | 0.000
-0.004 | 0.005 | 0.004 | 0.001 | 0.000 | 0.000
0.003 | -0.002 | -0.002 | 0.001 | 0.000 | 0.000
0.016 | -0.016 | -0.015 | 0.000 | 0.000 | 0.000
-0.002 | 0.002 | 0.002 | 0.000 | 0.000 | 0.000
-0.001 | 0.001 | 0.001 | 0.001 | 0.000 | 0.000
0.006 | -0.005 | -0.006 | -0.001 | 0.000 | 0.000
0.003 | -0.004 | -0.004 | 0.000 | 0.000 | 0.000
-0.001 | 0.001 | 0.002 | -0.001 | 0.000 | 0.000
-0.013 | 0.014 | 0.013 | -0.001 | 0.000 | 0.000
0.008 | -0.008 | -0.008 | 0.000 | 0.000 | 0.000
0.006 | -0.006 | -0.006 | -0.002 | 0.000 | 0.000
-0.013 | 0.013 | 0.013 | 0.000 | 0.000 | 0.000
-0.009 | 0.009 | 0.009 | 0.000 | 0.000 | 0.000
-0.005 | 0.005 | 0.005 | 0.001 | 0.000 | 0.000
-0.007 | 0.007 | 0.007 | 0.001 | 0.000 | 0.000
0.001 | -0.001 | -0.001 | 0.000 | 0.000 | 0.001
0.009 | -0.009 | -0.008 | -0.001 | 0.000 | 0.000
-0.003 | 0.003 | 0.003 | 0.000 | 0.000 | 0.000
0.003 | -0.003 | -0.003 | 0.000 | 0.000 | 0.000
-0.014 | 0.014 | 0.014 | 0.001 | 0.000 | 0.001
0.005 | -0.006 | -0.005 | 0.000 | 0.000 | 0.001
-0.006 | 0.005 | 0.005 | 0.000 | 0.000 | 0.001
0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000
-0.003 | 0.003 | 0.003 | 0.000 | 0.000 | 0.000
-0.003 | 0.003 | 0.004 | 0.001 | 0.000 | 0.000
-0.009 | 0.009 | 0.009 | -0.001 | 0.000 | 0.000
-0.011 | 0.011 | 0.010 | 0.001 | 0.000 | 0.000
-0.001 | 0.001 | 0.001 | -0.001 | 0.000 | 0.000
-0.009 | 0.009 | 0.008 | -0.001 | 0.000 | -0.001
0.006 | -0.006 | -0.006 | -0.001 | 0.000 | 0.000
-0.002 | 0.002 | 0.002 | 0.002 | 0.000 | 0.000
0.002 | -0.002 | -0.002 | 0.000 | 0.000 | 0.000
0.005 | -0.005 | -0.005 | -0.002 | 0.000 | 0.000
-0.005 | 0.005 | 0.004 | 0.001 | 0.000 | 0.000
0.007 | -0.007 | -0.006 | 0.001 | 0.000 | 0.000
0.004 | -0.004 | -0.003 | 0.000 | 0.000 | 0.000
0.005 | -0.006 | -0.006 | 0.001 | 0.000 | 0.000
-0.005 | 0.006 | 0.005 | 0.001 | 0.000 | 0.001
0.007 | -0.008 | -0.008 | -0.001 | 0.000 | 0.000
0.005 | -0.005 | -0.004 | 0.000 | 0.000 | -0.001
0.002 | -0.002 | -0.002 | -0.001 | 0.000 | 0.000
0.003 | -0.003 | -0.002 | -0.001 | 0.000 | 0.001
0.007 | -0.007 | -0.007 | 0.001 | 0.000 | 0.000
Table C.3: TransE’s 50-dimensional embedding vectors for the different relations in the WN18RR dataset, rounded to the third decimal. This table shows that the vectors for all relations are close to the zero-vector. DRF = derivationally_related_form, IH = instance_hypernym, MM = member_meronym and SDT = synset_domain_topic of. DRF | has_part | hypernym | IH | MM | SDT
---|---|---|---|---|---
0.000 | -0.002 | -0.003 | -0.004 | -0.027 | -0.028
0.000 | 0.000 | 0.012 | 0.081 | -0.001 | 0.010
0.000 | -0.003 | 0.007 | 0.010 | -0.012 | 0.009
0.000 | -0.003 | 0.003 | -0.148 | -0.042 | 0.000
0.000 | 0.017 | 0.003 | 0.005 | 0.002 | 0.007
0.000 | -0.001 | 0.002 | 0.139 | -0.014 | 0.003
0.000 | -0.002 | -0.002 | -0.115 | 0.031 | -0.006
0.000 | -0.006 | 0.005 | 0.005 | -0.015 | 0.008
0.000 | -0.001 | 0.001 | 0.150 | -0.021 | 0.010
0.000 | -0.007 | -0.011 | 0.119 | 0.012 | 0.019
0.000 | -0.004 | 0.013 | 0.118 | -0.019 | 0.012
0.000 | -0.004 | -0.002 | -0.069 | 0.042 | -0.001
0.000 | -0.002 | 0.002 | -0.165 | -0.025 | 0.002
0.000 | 0.000 | -0.012 | -0.017 | 0.007 | -0.012
0.000 | -0.018 | -0.002 | -0.111 | 0.008 | 0.015
0.000 | 0.021 | 0.001 | 0.006 | 0.003 | -0.041
0.000 | 0.001 | -0.015 | 0.005 | 0.043 | -0.026
0.000 | -0.002 | 0.005 | -0.152 | -0.017 | -0.006
0.000 | 0.002 | 0.002 | -0.082 | -0.002 | -0.009
0.000 | -0.002 | 0.001 | 0.134 | -0.048 | 0.044
0.000 | -0.001 | 0.007 | 0.142 | -0.003 | 0.016
0.000 | -0.001 | -0.001 | -0.134 | -0.060 | -0.009
0.000 | 0.018 | -0.003 | -0.142 | -0.020 | -0.016
0.000 | 0.002 | -0.008 | -0.084 | 0.010 | -0.030
0.000 | -0.004 | 0.009 | -0.062 | -0.007 | -0.019
0.000 | 0.005 | 0.005 | 0.067 | 0.000 | 0.011
0.000 | -0.003 | 0.005 | 0.113 | -0.009 | 0.028
0.000 | 0.002 | -0.005 | 0.002 | -0.038 | -0.010
0.000 | 0.007 | 0.001 | -0.127 | 0.000 | -0.010
0.000 | -0.002 | 0.000 | 0.036 | -0.007 | 0.013
0.001 | -0.005 | 0.002 | 0.002 | 0.009 | -0.010
0.000 | -0.004 | 0.004 | -0.075 | -0.045 | 0.017
-0.001 | 0.016 | 0.001 | 0.110 | 0.018 | -0.024
0.000 | 0.009 | 0.000 | 0.156 | -0.002 | -0.020
0.001 | 0.000 | 0.007 | 0.023 | -0.043 | -0.001
0.000 | -0.002 | -0.003 | -0.103 | 0.001 | 0.000
0.000 | -0.008 | 0.007 | 0.167 | -0.005 | 0.019
0.000 | 0.000 | -0.017 | 0.021 | 0.012 | -0.020
0.000 | -0.007 | -0.003 | -0.001 | 0.025 | -0.002
0.000 | -0.008 | -0.006 | 0.029 | 0.047 | 0.006
-0.001 | 0.002 | 0.000 | -0.001 | 0.047 | -0.004
0.000 | -0.002 | 0.014 | 0.139 | -0.036 | 0.005
0.000 | 0.001 | 0.010 | 0.010 | -0.056 | 0.003
0.000 | -0.001 | 0.001 | -0.005 | 0.000 | 0.019
0.000 | -0.002 | -0.001 | -0.128 | 0.005 | 0.007
0.000 | -0.001 | 0.002 | 0.131 | 0.001 | -0.002
0.000 | -0.004 | 0.011 | 0.137 | -0.008 | 0.024
0.000 | 0.005 | -0.002 | -0.092 | -0.045 | -0.006
0.000 | 0.001 | 0.011 | -0.058 | -0.005 | -0.023
0.000 | 0.004 | -0.013 | -0.041 | 0.021 | -0.046

((a)) Original WN18RR KG

((b)) Extended WN18RR KG

((c)) Original family KG

((d)) Extended family KG

Figure C.1: Distribution of PCA confidence of mined rules by entity selection strategy. PCA confidence scores are calculated on the original KG and the extended KGs from which the rules are mined. Rules may count multiple times if they were mined from different extensions. The dashed line represents the median PCA confidence of the rules mined from the original KG. Strategy | Original | New | Missed
---|---|---|---
| Quantity | % of mined | Quantity | % of mined | Quantity | % of original
Random | 10 | 53% | 9 | 47% | 0 | 0%
Most frequent | 10 | 42% | 14 | 58% | 0 | 0%
Least frequent | 10 | 48% | 11 | 52% | 0 | 0%
Probabilistic | 10 | 45% | 12 | 55% | 0 | 0%
Table C.4: Count of rules mined or missed by entity selection strategy for the WN18RR KG. Strategy | Original | New | Missed
---|---|---|---
| Quantity | % of mined | Quantity | % of mined | Quantity | % of original
Random | 90 | 71% | 36 | 29% | 4 | 4%
Most frequent | 88 | 46% | 105 | 54% | 6 | 6%
Least frequent | 94 | 77% | 28 | 33% | 0 | 0%
Probabilistic | 91 | 73% | 33 | 27% | 3 | 3%
Table C.5: Count of rules mined or missed by entity selection strategy for the Family KG. Rank cutoff | Kept | New | Missed
---|---|---|---
| Quantity | % of mined | Quantity | % of mined | Quantity | % of original
1 | 10 | 50% | 10 | 50% | 0 | 0%
4 | 4 | 27% | 11 | 73% | 6 | 60%
7 | 3 | 20% | 12 | 80% | 7 | 70%
Table C.6: Count of rules mined or missed by rank cutoff value for the WN18RR KG (excluding TransE-only rules). Rank cutoff | Original | New | Missed
---|---|---|---
| Quantity | % of mined | Quantity | % of mined | Quantity | % of original
1 | 94 | 94% | 11 | 10% | 0 | 0%
4 | 90 | 51% | 85 | 49% | 4 | 4%
7 | 88 | 45% | 108 | 55% | 6 | 6%
Table C.7: Count of rules mined or missed by rank cutoff value for the Family
KG (excluding TransE-only rules).

((a)) Original WN18RR KG

((b)) Extended WN18RR KG

((c)) Original family KG

((d)) Extended family KG

Figure C.2: Distribution of PCA confidence of mined rules by rank cutoff values. PCA confidence scores are calculated on the original KG and the extended KGs from which the rules are mined. Rules may count multiple times if they were mined from different extensions. The dashed line represents the median PCA confidence of the rules mined from the original KG. Table C.8: Rules mined from the original family KG, with their corresponding PCA confidences and how many times they were mined from the 48 extensions. Note that all rules with the $relative$ predicate in the consequent have lower PCA confidence and were mined less frequently. Rule | Frequency | PCA Confidence
---|---|---
father(y, x) $\Rightarrow$ child(x, y) | 48 | 0.998071
spouse(y, x) $\Rightarrow$ spouse(x, y) | 48 | 0.992355
mother(y, x) $\Rightarrow$ child(x, y) | 48 | 0.991906
sibling(y, x) $\Rightarrow$ sibling(x, y) | 48 | 0.989634
father(z, y) $\wedge$ mother(z, x) $\Rightarrow$ spouse(x, y) | 48 | 0.967952
father(z, y) $\wedge$ sibling(z, x) $\Rightarrow$ father(x, y) | 48 | 0.961199
father(z, y) $\wedge$ sibling(x, z) $\Rightarrow$ father(x, y) | 48 | 0.961023
father(z, x) $\wedge$ sibling(y, z) $\Rightarrow$ child(x, y) | 48 | 0.954603
father(z, x) $\wedge$ sibling(z, y) $\Rightarrow$ child(x, y) | 48 | 0.954328
father(z, x) $\wedge$ mother(z, y) $\Rightarrow$ spouse(x, y) | 48 | 0.953170
mother(x, z) $\wedge$ spouse(z, y) $\Rightarrow$ father(x, y) | 48 | 0.872686
mother(x, z) $\wedge$ spouse(y, z) $\Rightarrow$ father(x, y) | 48 | 0.867655
child(x, z) $\wedge$ sibling(y, z) $\Rightarrow$ child(x, y) | 48 | 0.851012
child(x, z) $\wedge$ sibling(z, y) $\Rightarrow$ child(x, y) | 48 | 0.850519
relative(y, x) $\Rightarrow$ relative(x, y) | 48 | 0.846306
mother(y, z) $\wedge$ spouse(z, x) $\Rightarrow$ child(x, y) | 48 | 0.839947
mother(y, z) $\wedge$ spouse(x, z) $\Rightarrow$ child(x, y) | 48 | 0.838288
mother(z, y) $\wedge$ sibling(x, z) $\Rightarrow$ mother(x, y) | 48 | 0.787816
mother(z, y) $\wedge$ sibling(z, x) $\Rightarrow$ mother(x, y) | 48 | 0.787401
child(y, x) $\Rightarrow$ father(x, y) | 48 | 0.769746
child(z, x) $\wedge$ mother(y, z) $\Rightarrow$ sibling(x, y) | 48 | 0.709805
child(z, y) $\wedge$ mother(x, z) $\Rightarrow$ sibling(x, y) | 48 | 0.709139
father(x, z) $\wedge$ spouse(z, y) $\Rightarrow$ mother(x, y) | 48 | 0.708957
mother(x, z) $\wedge$ mother(y, z) $\Rightarrow$ sibling(x, y) | 48 | 0.707149
mother(z, x) $\wedge$ sibling(y, z) $\Rightarrow$ child(x, y) | 48 | 0.704016
mother(z, x) $\wedge$ sibling(z, y) $\Rightarrow$ child(x, y) | 48 | 0.703385
father(x, z) $\wedge$ spouse(y, z) $\Rightarrow$ mother(x, y) | 48 | 0.701812
child(z, y) $\wedge$ spouse(x, z) $\Rightarrow$ child(x, y) | 48 | 0.688723
child(z, y) $\wedge$ spouse(z, x) $\Rightarrow$ child(x, y) | 48 | 0.687653
child(z, x) $\wedge$ father(y, z) $\Rightarrow$ sibling(x, y) | 48 | 0.668446
child(z, x) $\wedge$ child(z, y) $\Rightarrow$ sibling(x, y) | 48 | 0.664745
child(z, y) $\wedge$ father(x, z) $\Rightarrow$ sibling(x, y) | 48 | 0.662206
father(x, z) $\wedge$ father(y, z) $\Rightarrow$ sibling(x, y) | 48 | 0.658865
sibling(x, z) $\wedge$ sibling(y, z) $\Rightarrow$ sibling(x, y) | 48 | 0.590059
father(y, z) $\wedge$ spouse(z, x) $\Rightarrow$ child(x, y) | 48 | 0.588533
father(y, z) $\wedge$ spouse(x, z) $\Rightarrow$ child(x, y) | 48 | 0.588257
sibling(z, y) $\wedge$ sibling(x, z) $\Rightarrow$ sibling(x, y) | 48 | 0.588151
sibling(z, x) $\wedge$ sibling(y, z) $\Rightarrow$ sibling(x, y) | 48 | 0.584857
child(y, z) $\wedge$ sibling(x, z) $\Rightarrow$ father(x, y) | 48 | 0.584069
child(y, z) $\wedge$ sibling(z, x) $\Rightarrow$ father(x, y) | 48 | 0.583590
sibling(z, x) $\wedge$ sibling(z, y) $\Rightarrow$ sibling(x, y) | 48 | 0.564657
child(y, x) $\Rightarrow$ mother(x, y) | 48 | 0.535350
relative(x, z) $\wedge$ sibling(y, z) $\Rightarrow$ relative(x, y) | 40 | 0.484528
relative(x, z) $\wedge$ sibling(z, y) $\Rightarrow$ relative(x, y) | 40 | 0.483459
child(y, z) $\wedge$ mother(z, x) $\Rightarrow$ spouse(x, y) | 48 | 0.480628
child(x, z) $\wedge$ mother(z, y) $\Rightarrow$ spouse(x, y) | 48 | 0.477785
child(x, z) $\wedge$ child(y, z) $\Rightarrow$ spouse(x, y) | 48 | 0.471973
child(x, z) $\wedge$ father(z, y) $\Rightarrow$ spouse(x, y) | 48 | 0.471666
child(y, z) $\wedge$ father(z, x) $\Rightarrow$ spouse(x, y) | 48 | 0.469716
child(y, z) $\wedge$ sibling(x, z) $\Rightarrow$ mother(x, y) | 48 | 0.422559
child(y, z) $\wedge$ sibling(z, x) $\Rightarrow$ mother(x, y) | 48 | 0.422363
child(z, x) $\wedge$ spouse(z, y) $\Rightarrow$ mother(x, y) | 48 | 0.386296
child(z, x) $\wedge$ spouse(y, z) $\Rightarrow$ mother(x, y) | 48 | 0.382227
father(y, z) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 37 | 0.378085
father(z, y) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 37 | 0.372340
child(z, y) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 37 | 0.364472
child(z, x) $\wedge$ spouse(y, z) $\Rightarrow$ father(x, y) | 48 | 0.351479
relative(z, x) $\wedge$ sibling(z, y) $\Rightarrow$ relative(x, y) | 40 | 0.351014
mother(y, z) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 36 | 0.350490
child(z, x) $\wedge$ spouse(z, y) $\Rightarrow$ father(x, y) | 48 | 0.350117
relative(z, x) $\wedge$ sibling(y, z) $\Rightarrow$ relative(x, y) | 40 | 0.348765
child(y, z) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 37 | 0.343180
relative(x, z) $\wedge$ spouse(z, y) $\Rightarrow$ relative(x, y) | 37 | 0.305043
relative(x, z) $\wedge$ spouse(y, z) $\Rightarrow$ relative(x, y) | 37 | 0.305043
father(z, y) $\wedge$ relative(z, x) $\Rightarrow$ relative(x, y) | 37 | 0.301061
father(y, z) $\wedge$ relative(z, x) $\Rightarrow$ relative(x, y) | 37 | 0.294304
mother(z, y) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 33 | 0.283544
child(z, y) $\wedge$ relative(z, x) $\Rightarrow$ relative(x, y) | 37 | 0.275454
child(y, z) $\wedge$ relative(z, x) $\Rightarrow$ relative(x, y) | 37 | 0.273356
mother(y, z) $\wedge$ relative(z, x) $\Rightarrow$ relative(x, y) | 33 | 0.241645
mother(z, y) $\wedge$ relative(z, x) $\Rightarrow$ relative(x, y) | 30 | 0.224490
relative(z, x) $\wedge$ spouse(y, z) $\Rightarrow$ relative(x, y) | 35 | 0.206044
relative(z, x) $\wedge$ spouse(z, y) $\Rightarrow$ relative(x, y) | 32 | 0.205761
father(z, x) $\wedge$ father(y, z) $\Rightarrow$ relative(x, y) | 30 | 0.196078
father(z, x) $\wedge$ relative(y, z) $\Rightarrow$ relative(x, y) | 33 | 0.179144
child(z, y) $\wedge$ father(z, x) $\Rightarrow$ relative(x, y) | 26 | 0.166667
child(x, z) $\wedge$ father(y, z) $\Rightarrow$ relative(x, y) | 27 | 0.153584
child(x, z) $\wedge$ relative(y, z) $\Rightarrow$ relative(x, y) | 32 | 0.148287
father(z, x) $\wedge$ relative(z, y) $\Rightarrow$ relative(x, y) | 30 | 0.140871
child(z, y) $\wedge$ child(x, z) $\Rightarrow$ relative(x, y) | 25 | 0.130759
relative(z, y) $\wedge$ sibling(x, z) $\Rightarrow$ relative(x, y) | 39 | 0.127978
relative(z, y) $\wedge$ sibling(z, x) $\Rightarrow$ relative(x, y) | 39 | 0.126881
father(z, x) $\wedge$ spouse(y, z) $\Rightarrow$ relative(x, y) | 21 | 0.124101
child(x, z) $\wedge$ relative(z, y) $\Rightarrow$ relative(x, y) | 31 | 0.123643
father(z, x) $\wedge$ spouse(z, y) $\Rightarrow$ relative(x, y) | 17 | 0.122083
relative(y, z) $\wedge$ sibling(z, x) $\Rightarrow$ relative(x, y) | 38 | 0.120448
relative(y, z) $\wedge$ sibling(x, z) $\Rightarrow$ relative(x, y) | 38 | 0.120360
father(z, y) $\wedge$ father(x, z) $\Rightarrow$ relative(x, y) | 16 | 0.108774
mother(x, z) $\wedge$ relative(y, z) $\Rightarrow$ relative(x, y) | 21 | 0.104492
child(x, z) $\wedge$ spouse(y, z) $\Rightarrow$ relative(x, y) | 13 | 0.102020
child(x, z) $\wedge$ spouse(z, y) $\Rightarrow$ relative(x, y) | 13 | 0.102020
mother(z, x) $\wedge$ relative(y, z) $\Rightarrow$ relative(x, y) | 18 | 0.101064
father(x, z) $\wedge$ sibling(z, y) $\Rightarrow$ relative(x, y) | 10 | 0.100971
relative(z, y) $\wedge$ relative(x, z) $\Rightarrow$ relative(x, y) | 29 | 0.100440


#  Floquet engineering of Mott insulators with strong spin-orbit coupling

Naoya Arakawa<EMAIL_ADDRESS>The Institute of Science and
Engineering, Chuo University, Bunkyo, Tokyo 112-8551, Japan  Kenji Yonemitsu
The Institute of Science and Engineering, Chuo University, Bunkyo, Tokyo
112-8551, Japan  Department of Physics, Chuo University, Bunkyo, Tokyo
112-8551, Japan

###### Abstract

We propose a method for controlling the exchange interactions of Mott
insulators with strong spin-orbit coupling. We consider a multiorbital system
with strong spin-orbit coupling and a circularly polarized light field and
derive its effective Hamiltonian in the strong-interaction limit. Applying
this theory to a minimal model of $\alpha$-RuCl3, we show that the magnitudes
and signs of three exchange interactions, $J$, $K$, and $\Gamma$, can be
changed simultaneously. Then, considering another case in which one of the
hopping integrals has a different value and the other parameters are the same
as those for $\alpha$-RuCl3, we show that the Heisenberg interaction $J$ can
be made much smaller than the anisotropic exchange interactions $K$ and
$\Gamma$.

Periodic driving enables us to control the magnetic properties of solids. The
solution to the Schrödinger equation for a periodically driven system
satisfies the Floquet theorem because of time periodicity of the driving field
Floquet1 ; Floquet2 . In particular, the time evolution in steps of the
driving period $T$ can be described by a time-independent Hamiltonian Kuwahara
. Since its parameters usually depend on the amplitude and frequency of the
driving field, the properties could be controlled by tuning the driving field;
such control is called Floquet engineering review1 ; review2 ; review3 . For
example, by applying $E(t)=E_{0}\cos\omega t$ to a single-orbital Mott
insulator and tuning $E_{0}$ and $\omega$, we can change the magnitude and
sign of the antiferromagnetic Heisenberg interaction between magnetic moments
Floquet-1orbMott1 ; Floquet-1orbMott2 . Moreover, for a multiorbital Mott
insulator without spin-orbit coupling (SOC), we can control the
antiferromagnetic and the ferromagnetic contributions to the Heisenberg
interaction via a time-periodic electric field Floquet-MultiMott1 ; Floquet-
MultiMott2 . Such control could be used to engineer the magnetic properties of
solids because the exchange interactions are key quantities describing
magnetization dynamics review-dynamics  and spintronics phenomena review-
spintronics .

Although the magnetic properties of solids are affected by the SOC, Floquet
engineering of Mott insulators with strong SOC is lacking. The magnetic
properties of Mott insulators with strong SOC are described by spin and
orbital coupled degrees of freedom StrongLS-Mott ; StrongLS-Mott-review . As a
result, not only the Heisenberg interaction but also the anisotropic exchange
interactions contribute to the effective Hamiltonian StrongLS-Superex1 ;
StrongLS-Superex2 ; StrongLS-Superex3 ; Rau-PRL ; NA . For example, the
effective Hamiltonians for $\alpha$-RuCl3 and the honeycomb iridates possess
the Heisenberg interaction, the Kitaev interaction, and the off-diagonal
symmetric exchange interaction Rau-PRL ; Rau-review ; Valenti-PRB ; Valenti-
review . Then the combinations of the Heisenberg interaction and the
anisotropic exchange interactions cause various types of magnetic order Rau-
PRL ; Valenti-review ; PD1 ; PD2 ; PD3 ; PD4 ; PD5 ; if the Kitaev interaction
is dominant, the spin-liquid states are stabilized Kitaev . Controlling the
exchange interactions via a time-periodic field may provide a new opportunity
to engineer their properties. Nevertheless, it is unclear how a time-periodic
field changes the exchange interactions of Mott insulators with strong SOC.

In this work, we study the exchange interactions of periodically driven Mott
insulators with strong SOC. We use a $t_{2g}$-orbital Hubbard model in the
presence of strong SOC and a circularly polarized light field on the honeycomb
lattice and derive its effective Hamiltonian in the strong-interaction limit.
Applying this theory to the case of $\alpha$-RuCl3, we show that the
magnitudes and signs of three exchange interactions can be changed. Then,
studying another case of our model, we show that the Heisenberg interaction
can be made much smaller than the anisotropic exchange interactions.

Setup. We consider a periodically driven multiorbital system described by

$\displaystyle H=H_{\textrm{KE}}+H_{\textrm{SOC}}+H_{\textrm{int}},$ (1)

where $H_{\textrm{KE}}$, $H_{\textrm{SOC}}$, and $H_{\textrm{int}}$ represent
the kinetic energy, the atomic SOC StrongLS-Mott-review , and the local
multiorbital Coulomb interactions Kanamori , respectively. The kinetic energy
is given by the hopping integrals of the $t_{2g}$-orbital electrons on the
honeycomb lattice (Fig. 1) in the presence of a circularly polarized light
field $\bm{E}(t)={}^{t}(E_{0}\cos\omega t\ -E_{0}\sin\omega t)$. The effects
of $\bm{E}(t)$ are treated as Peierls phase factors:

$\displaystyle
H_{\textrm{KE}}=\sum_{i,j}\sum_{a,b}\sum_{\sigma}t_{iajb}e^{-ie(\bm{R}_{i}-\bm{R}_{j})\cdot\bm{A}(t)}c_{ia\sigma}^{\dagger}c_{jb\sigma},$
(2)

where $\bm{A}(t)={}^{t}(-\frac{E_{0}}{\omega}\sin\omega t\
-\frac{E_{0}}{\omega}\cos\omega t)$; hereafter, we use $\hbar=1$. Then the
atomic SOC of $H_{\textrm{SOC}}$ is given by the $LS$ coupling for the
$t_{2g}$-orbital electrons StrongLS-Mott-review . The terms of
$H_{\textrm{int}}$ consist of the following interactions Kanamori :

$\displaystyle
H_{\textrm{int}}=\sum_{i}\Bigl{\\{}\sum_{a,b}c_{ia\uparrow}^{\dagger}c_{ia\downarrow}^{\dagger}[U\delta_{a,b}+J^{\prime}(1-\delta_{a,b})]c_{ib\downarrow}c_{ib\uparrow}$
$\displaystyle+\sum_{\begin{subarray}{c}a,b\\\
a>b\end{subarray}}\sum_{\sigma,\sigma^{\prime}}c_{ia\sigma}^{\dagger}c_{ib\sigma^{\prime}}^{\dagger}(U^{\prime}c_{ib\sigma^{\prime}}c_{ia\sigma}-J_{\textrm{H}}c_{ib\sigma}c_{ia\sigma^{\prime}})\Bigr{\\}}.$
(3)

As a specific example, we consider a minimal model of $\alpha$-RuCl3 Rau-PRL :
$t_{iajb}$’s in $H_{\textrm{KE}}$ can be parametrized by three hopping
integrals between nearest-neighbor sites on the honeycomb lattice (Fig. 1).
Namely, for the $Z$ bond, the finite $t_{iajb}$’s are given by
$t_{id_{yz}jd_{yz}}=t_{id_{zx}jd_{zx}}=t_{1}$,
$t_{id_{yz}jd_{zx}}=t_{id_{zx}jd_{yz}}=t_{2}$, and $t_{id_{xy}jd_{xy}}=t_{3}$;
for the $X$ or $Y$ bond, similar relations can be obtained from symmetry
arguments Rau-PRL . Then the low-energy properties can be described by the
$j_{\textrm{eff}}=1/2$ states Rau-PRL ; Jeff-RuCl3-1 ; Jeff-RuCl3-2 ,
$|+\rangle_{i}=\frac{1}{\sqrt{3}}(c_{id_{yz}\downarrow}^{\dagger}+ic_{id_{zx}\downarrow}^{\dagger}+c_{id_{xy}\uparrow}^{\dagger})|0\rangle$
and
$|-\rangle_{i}=\frac{1}{\sqrt{3}}(c_{id_{yz}\uparrow}^{\dagger}-ic_{id_{zx}\uparrow}^{\dagger}-c_{id_{xy}\downarrow}^{\dagger})|0\rangle$,
in which the spin and the orbital are entangled by strong SOC. Since such
entanglement is the key property of strong SOC StrongLS-Mott-review , this
model will be sufficient for analyzing essential physics in the presence of
strong SOC.

Figure 1:  Structure of the honeycomb lattice. $X$, $Y$, and $Z$ denote three
different nearest-neighbor bonds. Black and white circles represent the $A$
and $B$ sublattices, respectively.

Floquet theory of Mott insulators. We derive an effective Hamiltonian for a
periodically driven Mott insulator using the Floquet theory Floquet-MultiMott2
. To derive it, we consider the strong-interaction limit in which $t_{iajb}$’s
are much smaller than the energies of doubly occupied states, $U+2J^{\prime}$,
$U-J^{\prime}$, $U^{\prime}-J_{\textrm{H}}$, and $U^{\prime}+J_{\textrm{H}}$
NA ; Ishihara . In this limit, we can approximately express the solution to
the Schrödinger equation, $|\Psi\rangle_{t}$, as
$|\Psi\rangle_{t}\approx|\Psi_{0}\rangle_{t}+|\Psi_{1}\rangle_{t}$ Floquet-
MultiMott2 , where $|\Psi_{0}\rangle_{t}$ and $|\Psi_{1}\rangle_{t}$ denote
the states without and with, respectively, a doubly occupied site. As a
result, the Schrödinger equation reduces to a set of simultaneous equations:

$\displaystyle
i\partial_{t}|\Psi_{0}\rangle_{t}=\mathcal{P}_{0}H_{\textrm{KE}}|\Psi_{1}\rangle_{t}+H_{\textrm{SOC}}|\Psi_{0}\rangle_{t},$
(4) $\displaystyle
i\partial_{t}|\Psi_{1}\rangle_{t}=H_{\textrm{KE}}|\Psi_{0}\rangle_{t}+(\mathcal{P}_{1}H_{\textrm{KE}}\mathcal{P}_{1}+\tilde{H}_{\textrm{int}})|\Psi_{1}\rangle_{t},$
(5)

where $\tilde{H}_{\textrm{int}}=H_{\textrm{int}}+H_{\textrm{SOC}}$; and
$\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ denote the projections onto the
subspaces without and with, respectively, a doubly occupied site. Hereafter,
we concentrate on the high-frequency case in which $\omega$ is much larger
than $t_{iajb}$’s. In this case, we could replace the time-dependent operator
$\mathcal{P}_{1}H_{\textrm{KE}}\mathcal{P}_{1}$ in Eq. (5) by its time-
averaged one
$\bar{H}_{\textrm{KE}}=\sum_{i,j}\sum_{a,b}\sum_{\sigma}t_{iajb}\mathcal{J}_{0}(u_{ij})\mathcal{P}_{1}c_{ia\sigma}^{\dagger}c_{jb\sigma}\mathcal{P}_{1}$
Floquet-MultiMott2 , where $\mathcal{J}_{n}(u_{ij})$ is the $n$th Bessel
function of the first kind and $u_{ij}=\frac{eE_{0}}{\omega}\textrm{sgn}(i-j)$
remark1 ; the distance between nearest-neighbor sites is set to unity. By
using this replacement, we can solve Eq. (5); the result Supp  is

$\displaystyle|\Psi_{1}\rangle_{t}=\sum_{i,j,a,b,\sigma}\sum_{n=-\infty}^{\infty}\frac{t_{iajb}\tilde{\mathcal{J}}_{-n}(u_{ij})e^{-in\omega
t}}{n\omega-\bar{H}_{\textrm{KE}}-\tilde{H}_{\textrm{int}}}c_{ia\sigma}^{\dagger}c_{jb\sigma}|\Psi_{0}\rangle_{t},$
(6)

where
$\tilde{\mathcal{J}}_{-n}(u_{ij})=\mathcal{J}_{-n}(u_{ij})e^{-in\theta_{ij}}$,
and $\theta_{ij}=\theta_{ji}=\frac{\pi}{3}$, $\pi$, and $\frac{5\pi}{3}$ for
the $Y$, $Z$, and $X$ bonds, respectively. Furthermore, since
$H_{\textrm{int}}$ gives the largest contribution of the terms of
$\bar{H}_{\textrm{KE}}$ and
$\tilde{H}_{\textrm{int}}(=H_{\textrm{int}}+H_{\textrm{SOC}})$, we replace
$n\omega-\bar{H}_{\textrm{KE}}-\tilde{H}_{\textrm{int}}$ in Eq. (6) by
$n\omega-H_{\textrm{int}}$; this replacement may be sufficient if $\omega$ is
nonresonant, i.e., the denominator of Eq. (6) does not diverge. By using Eq.
(6) with this replacement and omitting the constant term (i.e.,
$H_{\textrm{SOC}}|\Psi_{0}\rangle_{t}$), we can rewrite Eq. (4) as

$\displaystyle
i\partial_{t}|\Psi_{0}\rangle_{t}=H_{\textrm{eff}}(t)|\Psi_{0}\rangle_{t},$
(7)

where

$\displaystyle H_{\textrm{eff}}(t)=$
$\displaystyle\sum_{i,j}\sum_{a,b,c,d}\sum_{\sigma,\sigma^{\prime}}\sum_{n,m=-\infty}^{\infty}t_{jcid}t_{iajb}\mathcal{P}_{0}c_{jc\sigma^{\prime}}^{\dagger}c_{id\sigma^{\prime}}$
$\displaystyle\times$
$\displaystyle\frac{\tilde{\mathcal{J}}_{m}(u_{ji})\tilde{\mathcal{J}}_{-n}(u_{ij})e^{i(m-n)\omega
t}}{n\omega-
H_{\textrm{int}}}c_{ia\sigma}^{\dagger}c_{jb\sigma}\mathcal{P}_{0}.$ (8)

The leading term of $H_{\textrm{eff}}(t)$ is given by the time-independent
Floquet Hamiltonian. Since $H_{\textrm{eff}}(t)$ is time periodic, it can be
expressed as the Fourier series $H_{\textrm{eff}}(t)=\sum_{l}e^{il\omega
t}H_{l}$. Furthermore, by using a high-frequency expansion of the Floquet
theory review1 ; review2 ; review3 , $H_{\textrm{eff}}(t)$ can be written in
the form $H_{\textrm{eff}}(t)=H_{0}+O(\omega^{-1})$. Therefore, the time-
averaged $H_{\textrm{eff}}(t)$, $\bar{H}_{\textrm{eff}}$, gives the leading
term of Eq. (8); $\bar{H}_{\textrm{eff}}$ is given by

$\displaystyle\bar{H}_{\textrm{eff}}=$
$\displaystyle\sum_{i,j}\sum_{a,b,c,d}\sum_{\sigma,\sigma^{\prime}}\sum_{n=-\infty}^{\infty}t_{jcid}t_{iajb}\mathcal{P}_{0}c_{jc\sigma^{\prime}}^{\dagger}c_{id\sigma^{\prime}}\frac{\mathcal{J}_{n}(u_{ij})^{2}}{n\omega-
H_{\textrm{int}}}$ $\displaystyle\times
c_{ia\sigma}^{\dagger}c_{jb\sigma}\mathcal{P}_{0}.$ (9)

Figure 2:  The $|u_{ij}|(=|\frac{eE_{0}}{\omega}|)$ dependences of $J$, $K$,
and $\Gamma$ in (a) and (b) the case of $\alpha$-RuCl3 and (c) and (d) another
case.

Application to periodically driven $\alpha$-RuCl3. Applying the above theory
to the minimal model of $\alpha$-RuCl3, we derive its Floquet Hamiltonian.
This derivation can be performed in a way similar to the derivation in the
absence of a driving field. Here we describe the main points of the derivation
(for the details, see the Supplemental Material Supp ). To derive the
expression of $\bar{H}_{\textrm{eff}}$ for the minimal model of
$\alpha$-RuCl3, we suppose that in the subspace of $|\Psi_{0}\rangle_{t}$ a
single hole occupies $j_{\textrm{eff}}=1/2$ state per site. We also rewrite
$H_{\textrm{int}}$ using the irreducible representations of doubly occupied
states NA :
$H_{\textrm{int}}=\sum_{i}\sum_{\Gamma,g_{\Gamma}}U_{\Gamma}|i;\Gamma,g_{\Gamma}\rangle\langle
i;\Gamma,g_{\Gamma}|$, where $U_{A_{1}}=U+2J^{\prime}$, $U_{E}=U-J^{\prime}$,
$U_{T_{1}}=U^{\prime}-J_{\textrm{H}}$, and
$U_{T_{2}}=U^{\prime}+J_{\textrm{H}}$; $|i;\Gamma,g_{\Gamma}\rangle$’s are
expressed in the Supplemental Material Supp . Then, by calculating the
contributions of possible hopping processes to $\bar{H}_{\textrm{eff}}$, we
obtain Supp

$\displaystyle\bar{H}_{\textrm{eff}}=\sum_{\langle
i,j\rangle}[J\bm{S}_{i}\cdot\bm{S}_{j}+KS_{i}^{\gamma}S_{j}^{\gamma}+\Gamma(S_{i}^{\alpha}S_{j}^{\beta}+S_{i}^{\beta}S_{j}^{\alpha})],$
(10)

where

$\displaystyle J=$
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{4\mathcal{J}_{n}(u_{ij})^{2}}{27}\Bigl{\\{}\frac{(2t_{1}+t_{3})^{2}}{U+2J^{\prime}-n\omega}+\frac{6t_{2}^{2}}{U^{\prime}+J_{\textrm{H}}-n\omega}$
$\displaystyle\ \ \ \ \
+\frac{2[(t_{1}-t_{3})^{2}-3t_{2}^{2}]}{U-J^{\prime}-n\omega}+\frac{6t_{1}(t_{1}+2t_{3})}{U^{\prime}-J_{\textrm{H}}-n\omega}\Bigr{\\}},$
(11) $\displaystyle K=$
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{4\mathcal{J}_{n}(u_{ij})^{2}}{9}\Bigl{\\{}\frac{4t_{2}^{2}}{U-J^{\prime}-n\omega}-\frac{[(t_{1}-t_{3})^{2}+t_{2}^{2}]}{U^{\prime}+J_{\textrm{H}}-n\omega}$
$\displaystyle\ \ \ \ \
+\frac{[(t_{1}-t_{3})^{2}-3t_{2}^{2}]}{U^{\prime}-J_{\textrm{H}}-n\omega}\Bigr{\\}},$
(12) $\displaystyle\Gamma=$
$\displaystyle\sum_{n=-\infty}^{\infty}\frac{16\mathcal{J}_{n}(u_{ij})^{2}t_{2}(t_{1}-t_{3})J_{\textrm{H}}}{9(U^{\prime}-J_{\textrm{H}}-n\omega)(U^{\prime}+J_{\textrm{H}}-n\omega)},$
(13)

and $(\alpha,\beta,\gamma)=(x,y,z)$, $(y,z,x)$, and $(z,x,y)$ for the $Z$,
$X$, and $Y$ bonds, respectively; $J$, $K$, and $\Gamma$ are the Heisenberg
interaction, the Kitaev interaction, and the off-diagonal symmetric exchange
interaction, respectively. [These expressions hold also for
$\bm{E}(t)={}^{t}(E_{0}\cos\omega t\ E_{0}\sin\omega t)$.]

We now show how $J$, $K$, and $\Gamma$ vary with $\omega$ and $u_{ij}$. To do
it, we numerically evaluate Eqs. (11)–(13). We set $t_{1}=47$ meV, $t_{2}=160$
meV, $t_{3}=-129$ meV remark2 , $J^{\prime}=J_{\textrm{H}}$,
$U^{\prime}=U-2J_{\textrm{H}}$, $U=3$ eV, and $J_{\textrm{H}}=0.5$ eV; we
replace $\sum_{n=-\infty}^{\infty}$’s by
$\sum_{n=-n_{\textrm{max}}}^{n_{\textrm{max}}}$’s and set
$n_{\textrm{max}}=500$. Figures 2(a) and 2(b) show the $|u_{ij}|$ dependences
of $J$, $K$, and $\Gamma$ at $\omega=3$ and $2.2$ eV. We see that by changing
$|u_{ij}|$, the magnitudes of $J$, $K$, and $\Gamma$ can be changed at
$\omega=3$ and $2.2$ eV; and that at $\omega=2.2$ eV it is possible to change
not only their magnitudes but also their signs.

For a deeper understanding of the above results, we perform some analyses of
Eqs. (11)–(13). Since $J^{\prime}=J_{\textrm{H}}$ and
$U^{\prime}=U-2J_{\textrm{H}}$, $J$, $K$, and $\Gamma$ can be rewritten as
follows: $J=J_{1}+J_{2}+J_{3}$, where
$J_{1}=\sum_{n}\frac{4\mathcal{J}_{n}(u_{ij})^{2}(2t_{1}+t_{3})^{2}}{27(U+2J_{\textrm{H}}-n\omega)}$,
$J_{2}=\sum_{n}\frac{8\mathcal{J}_{n}(u_{ij})^{2}(t_{1}-t_{3})^{2}}{27(U-J_{\textrm{H}}-n\omega)}$,
and
$J_{3}=\sum_{n}\frac{8\mathcal{J}_{n}(u_{ij})^{2}t_{1}(t_{1}+2t_{3})}{9(U-3J_{\textrm{H}}-n\omega)}$.
$K=K_{1}+K_{2}$, where
$K_{1}=\sum_{n}\frac{4\mathcal{J}_{n}(u_{ij})^{2}[3t_{2}^{2}-(t_{1}-t_{3})^{2}]}{9(U-J_{\textrm{H}}-n\omega)}$and
$K_{2}=\sum_{n}\frac{4\mathcal{J}_{n}(u_{ij})^{2}[(t_{1}-t_{3})^{2}-3t_{2}^{2}]}{9(U-3J_{\textrm{H}}-n\omega)}$.
$\Gamma=\Gamma_{1}+\Gamma_{2}$, where
$\Gamma_{1}=\sum_{n}\frac{8\mathcal{J}_{n}(u_{ij})^{2}t_{2}(t_{1}-t_{3})}{9(U-3J_{\textrm{H}}-n\omega)}$
and
$\Gamma_{2}=\sum_{n}\frac{8\mathcal{J}_{n}(u_{ij})^{2}t_{2}(t_{3}-t_{1})}{9(U-J_{\textrm{H}}-n\omega)}$.
For the hopping parameters of $\alpha$-RuCl3, $J_{1}$ is much smaller than
$J_{2}$ and $J_{3}$; as a result, $J\approx J_{2}+J_{3}$. This is the origin
of the in-phase $|u_{ij}|$ dependences of $J$, $K$, and $\Gamma$ [Figs. 2(a)
and 2(b)]. Then we can understand the sign changes of $J$, $K$, and $\Gamma$
at $|u_{ij}|\sim 0.4$, $3.5$ [Fig. 2(b)] by estimating the dominant
contributions. We make the estimate of $J$ because the sign changes of $K$ and
$\Gamma$ can be understood similarly. For $\omega=2.2$ eV, the dominant
contributions are given by

$\displaystyle
J\approx(J_{2}^{0}+J_{3}^{0})\mathcal{J}_{0}(u_{ij})^{2}+(J_{2}^{0}c_{2}-J_{3}^{0}c_{3})\mathcal{J}_{1}(u_{ij})^{2},$
(14)

where $J_{2}^{0}=\frac{8(t_{1}-t_{3})^{2}}{27(U-J_{\textrm{H}})}$,
$J_{3}^{0}=\frac{8t_{1}(t_{1}+2t_{3})}{9(U-3J_{\textrm{H}})}$,
$c_{2}=\frac{U-J_{\textrm{H}}}{\delta\omega_{2}}$,
$c_{3}=\frac{U-3J_{\textrm{H}}}{\delta\omega_{3}}$, and
$\omega=U-3J_{\textrm{H}}+\delta\omega_{3}=U-J_{\textrm{H}}-\delta\omega_{2}$
(i.e., $\delta\omega_{2}=0.3$ eV and $\delta\omega_{3}=0.7$ eV). At
$|u_{ij}|=0$, $J$ is ferromagnetic, i.e., negative, because $J_{2}^{0}$ and
$J_{3}^{0}$ satisfy $J_{2}^{0}>0$, $J_{3}^{0}<0$, and $J_{2}^{0}+J_{3}^{0}<0$.
As $|u_{ij}|$ increases, the term including $\mathcal{J}_{1}(u_{ij})^{2}$ in
Eq. (14), the positive-sign contribution, becomes considerable and causes a
sign change of $J$. With further increases in $|u_{ij}|$,
$\mathcal{J}_{1}(u_{ij})^{2}$ approaches zero, and the sign of $J$ changes
again.

Application to another case. We consider another case and study the effects of
the driving field on the exchange interactions. In this case, we set
$t_{3}=129$ meV and use the same values of the other parameters as those used
in the case of $\alpha$-RuCl3; in a set of these values, $J_{1}$ is comparable
to $J_{2}$ and $J_{3}$. Although it may be difficult to change the value of
$t_{3}$ in $\alpha$-RuCl3, we study this case to clarify how the driving field
changes $J$ in the presence of non-negligible $J_{1}$. Figures 2(c) and 2(d)
show the $|u_{ij}|$ dependences of $J$, $K$, and $\Gamma$ in this additional
case. We see that the $|u_{ij}|$ dependence of $J$ differs from that of $K$ or
$\Gamma$. In particular, $J$ can be very small in magnitude, while $K$ and
$\Gamma$ are finite [see, for example, their values at $|u_{ij}|=1.6$ in Fig.
2 (d)].

Discussion. We comment on the validity of our theory. First, the hopping
integrals of our model are simplified compared with those obtained in the
first-principles calculations Valenti-PRB . However, since the leading terms
are $t_{2}$ and $t_{3}$ Valenti-PRB , our model may be appropriate for a
minimal model of $\alpha$-RuCl3. Then the obtained $|u_{ij}|$ dependences of
$J$, $K$, and $\Gamma$ might be affected by the doublon-holon hoppings
described by $\bar{H}_{\textrm{KE}}$. Nevertheless, we believe our results
remain qualitatively unchanged. This is because the previous studies Floquet-
MultiMott1 ; Floquet-MultiMott2  show that in the frequency range where the
correction due to $\mathcal{J}_{1}(u_{ij})^{2}$ is important and the
corrections due to $\mathcal{J}_{n}(u_{ij})^{2}$’s for $n\geq 2$ are less
important (the range of $U-2J_{\textrm{H}}<\omega<U$ in Ref. Floquet-
MultiMott1, ), the effects of the driving field on the exchange interactions
remain qualitatively unchanged even if the doublon-holon hoppings are taken
into account.

We also remark on heating effects. The periodically driven system eventually
approaches an infinite-temperature state Heat-InfT1 ; Heat-InfT2 . However, at
intermediate times $t\lesssim\tau$ remark3 , it can be approximately described
by the Floquet Hamiltonian as long as $\omega$ is nonresonant Floquet-
MultiMott2  and much larger than the exchange interactions Kuwahara ; Heat-
Floq1 ; Heat-Floq2 ; Heat-Floq3 ; Heat-Floq4 . Since these conditions hold in
our study, the properties similar to our results could be realized
experimentally.

We now discuss the implications of our results. First, our results in the case
of $\alpha$-RuCl3 indicate that by tuning $\omega$ and changing $E_{0}$, one
can change the magnitudes and signs of the three exchange interactions of
periodically driven $\alpha$-RuCl3. In particular, by using this method, the
Kitaev interaction can be made ferromagnetic (negative) or antiferromagnetic
(positive). Since its sign drastically affects the magnetic properties of
materials with strong SOC Valenti-review ; Vojta-review , our results will
provide an opportunity for connecting the ferromagnetic and the
antiferromagnetic Kitaev physics. Such control of the exchange interactions
could be achieved by pump-probe measurements. Then our results in another case
suggest that if the contribution from the doubly occupied state with $A_{1}$
symmetry is non-negligible, it is possible to make the Kitaev interaction much
larger in magnitude than the Heisenberg interaction. Therefore the
periodically driven Mott insulator with strong SOC and hopping integrals that
lead to such a contribution may be suitable for realizing the Kitaev model
Kitaev  and the spin liquid.

Conclusions. ]We have studied the exchange interactions of the Mott insulators
with a circularly polarized light field and strong SOC in two cases. In the
case of $\alpha$-RuCl3, we have shown that $J$, $K$, and $\Gamma$ have the
similar $|u_{ij}|$ dependences, and that their magnitudes and signs can be
changed by tuning $\omega$ and varying $E_{0}$. These properties can be
utilized for changing the exchange interactions of $\alpha$-RuCl3 and
controlling its magnetic properties. In another case, we have shown that the
$|u_{ij}|$ dependence of $J$ differs from those of $K$ and $\Gamma$, and that
$J$ can be made much smaller than $K$ and $\Gamma$ by tuning $|u_{ij}|$. The
latter property suggests a new possibility of realizing the Kitaev spin
liquid. Our results will provide the first step towards controlling the
exchange interactions and magnetic properties of periodically driven Mott
insulators with strong SOC.

###### Acknowledgements.

This work was supported by JST CREST Grant No. JPMJCR1901, JSPS KAKENHI Grants
No. JP19K14664 and No. JP16K05459, and MEXT Q-LEAP Grant No. JP-MXS0118067426.

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