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# GRI: Graph-based Relative Isomorphism of Word Embedding Spaces

Muhammad Asif Ali,1 Yan Hu,1 Jianbin Qin,2 Di Wang1
1 King Abdullah University of Science and Technology, KSA
2 Shenzhen University, China
{muhammadasif.ali, yan.hu<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

Automated construction of bilingual dictionaries using monolingual embedding
spaces is a core challenge in machine translation. The end performance of
these dictionaries relies upon the geometric similarity of individual spaces,
i.e., their degree of isomorphism. Existing attempts aimed at controlling the
relative isomorphism of different spaces fail to incorporate the impact of
semantically related words in the training objective. To address this, we
propose GRI that combines the distributional training objectives with
attentive graph convolutions to unanimously consider the impact of
semantically similar words required to define/compute the relative isomorphism
of multiple spaces. Experimental evaluation shows that GRI outperforms the
existing research by improving the average P$@$1 by a relative score of up to
63.6%. We release the codes for GRI at https://github.com/asif6827/GRI.

##  1 Introduction

Bilingual Lexical Induction (BLI) aims at the construction of lexical
dictionaries using different mono-lingual word embeddings. Automated
construction of bilingual dictionaries plays a significant role, especially
for resource-constrained languages where hand-crafted dictionaries are almost
non-existent. It is also a key tool to bootstrap the performance of many down-
streaming applications, e.g., cross-lingual information retrieval (Artetxe et
al., 2018a), neural machine translation Lample et al. (2018).

The most prevalent way for the construction of cross-lingual embeddings is to
map the mono-lingual embeddings in a shared space using linear and/or non-
linear transformations, also known as mapping-based methods Conneau et al.
(2017); Joulin et al. (2018); Patra et al. (2019). A core limitation of the
mapping-based methods is their reliance on the approximate isomorphism
assumption, i.e., the underlying monolingual embedding spaces are
geometrically similar. This severely limits the applicability of the mapping-
based methods to closely related languages and similar data domains. This
isomorphism assumption does not hold, especially in case of domain-mismatch
and for languages exhibiting different characteristics Conneau et al. (2017);
Søgaard et al. (2018); Glavas et al. (2019); Patra et al. (2019). Other
dominant factors identified in the literature that limit the end performance
of BLI systems include: (i) linguistic differences (ii) algorithmic mismatch,
(iii) variation in data size, (iv) parameterization etc. Similar to the
supervised models, the unsupervised variants of BLI are also unable to cater
to the above-mentioned challenges Kim et al. (2020); Marie and Fujita (2020).

Figure 1: Semantically related tokens for English and Ukrainian languages.
These words though lexically varying carry the same semantics and their impact
should be unanimously considered.

Instead of relying on embedding spaces trained completely independent of each
other, in the recent past there have been a shift in explicitly using the
isomorphism measures alongside distributional training objective Marchisio et
al. (2022). In order to control the relative isomorphism of monolingual
embedding spaces, these models use existing bilingual dictionaries as training
seeds. However, one core limitation of these models is their inability to
incorporate the impact of semantically relevant tokens into the training
objective. This severely deteriorates the relative isomorphism of the
resultant cross-lingual embedding space.

This phenomenon is illustrated in Figure 1 for English and Ukrainian
languages. For example, for the English language, we often use terms
{_“terrible", “horrible"_} within the same context without a significant
change in the meaning of the sentence. For these terms, corresponding terms in
the Ukrainian language {“страшни́й”, “жахливо”} may also be used
interchangeably without a significant change in the context. Likewise, for the
bottom row in Figure 1, the words {_“good", “great", “excellent"_} are
semantically related words in the English language, with {“відмінно”,
“чудово”, “добре”} as corresponding semantically related words in the
Ukrainian language.

To address these challenges, in this paper we propose a novel framework named:
Graph-based Relative Isomorphism (GRI). GRI uses attentive graph convolutions
to pay attention to semantically related tokens, followed by using isomorphism
metrics to inject this information into the model training. Later, it combines
the isomorphism loss with the distributional training objective to train the
complete model.

We argue GRI offers a better alternative for BLI, as it allows injecting
information about the semantic variations of tokens in the training objective,
which is a more natural setting in order to control the relative isomorphism
of linguistic data. An immediate benefit of the proposed model is obvious in
the domain-mismatch settings, where attentive graph convolutions mechanism by
GRI offer the ability to unanimously analyze and/or model similar concepts
represented by lexically varying terms across different corpora. This is also
evident by a relatively stable performance of GRI for both domain-sharing and
domain-mismatch settings (Section 6.1). We summarize the core contributions of
this paper as follows:

  1. 1.

We propose GRI that combines isomorphism loss functions (guided by graph
convolutions) along with the distributional training objective for BLI.

  2. 2.

We propose attentive graph convolutions for GRI in order to control the
relative isomorphism by sharing information across semantically related
tokens.

  3. 3.

We illustrate the effectiveness of GRI via comprehensive experimentation. For
benchmark data sets, GRI outperforms the existing state of the art by
approximately 63.6% for average P$@$1.

##  2 Related Work

Due to limited space, we primarily categorize the related work on relative
isomorphism of cross-lingual embeddings into: (i) mapping to shared space, and
(ii) joint training.

##### Mapping to shared space.

These models aim to find a linear and/or non-linear transformation for pre-
trained word embeddings in order to map them to a shared space. These models
rely on the assumption that the embedding models share similar structure
across different languages Mikolov et al. (2013a), which allows them to
independently train embeddings for different languages and learn mapping
functions to align them in a shared space. Supervised variants in this regard
use existing bilingual resources, such as parallel dictionaries Xing et al.
(2015); Joulin et al. (2018); Jawanpuria et al. (2019). The unsupervised
variants use distributional matching Zhang et al. (2017); Conneau et al.
(2017); Artetxe et al. (2018b); Zhou et al. (2019). These models have also
been applied to the contextualized embeddings Aldarmaki and Diab (2019);
Schuster et al. (2019).

##### Joint Training

These models put additional constraints on model learning, i.e., a hard or
soft cross-lingual constraints in addition to the monolingual training
objectives. Similar to the mapping-based models, early works in this domain
include the supervised variants relying on bilingual dictionaries (Ammar et
al., 2016; Luong et al., 2015; Gouws et al., 2015). Recently, the unsupervised
approaches have gained attention because of their ease of implementation. For
instance, Lample et al. (2018) analyzed the performance for concatenated
monolingual corpora with shared vocabulary without any additional cross-
lingual resources. Results show that this setting outperforms many carefully
crafted alignment based strategies for unsupervised machine translation. Other
unsupervised approaches with good results on benchmark data sets include zero-
shot cross-lingual transfer by Artetxe and Schwenk (2019) and cross-lingual
pre-training by Lample and Conneau (2019). Marchisio et al. (2022) proposed
IsoVec that introduces multiple different losses along with the skip-gram loss
function to control the relative isomorphism of mono-lingual spaces.

A major limitation of these methods is their inability to incorporate the
lexico-semantic variations of word pairs across different languages in the
model training, which severely limits the end performance of these models.

##  3 Background

In this section, we discuss the notation and mathematical background of the
tools and techniques used in this paper.

###  3.1 Notation

Throughout this paper, we use $\mathbf{U}\in\mathbf{R}^{p\times d}$ and
$\mathbf{V}\in\mathbf{R}^{q\times d}$ to represent the embeddings of the
source and target languages. We assume the availability of seeds pairs for
both source and target languages, represented by:
$\\{(u_{0},v_{0}),(u_{1},v_{1}),...(u_{s},v_{s})\\}$.

###  3.2 VecMap toolkit

For mapping across different embedding spaces, we use vecmap
toolkit111https://github.com/artetxem/vecmap. We follow Zhang et al. (2019) to
pre-process the embeddings, i.e., the embeddings are unit-normed, mean-
centered and unit-normed again. For bilingual induction, we follow the steps
outlined by (Artetxe et al., 2018a), i.e., whitening each space, and solving
Procrustes. This is followed by re-weighting, de-whitening, and mapping of
translation pairs via nearest-neighbor retrieval. For details, refer to the
original work by Artetxe et al. (2018a).

##  4 Proposed Approach

###  4.1 Problem Definition

In this paper, we address a core challenge in controlling the relative
isomorphism for cross-lingual data sets, i.e., incorporate the impact of
semantically coherent words for BLI.

###  4.2 Overview

We propose Graph-based Relative Isomorphism GRI, shown in Figure 2, that aims
to learn distributional information in the source embedding space
$\mathbf{U}$, in such a way: (i) $\mathbf{U}$ is geometrically similar to the
target embedding space $\mathbf{V}$ to the best possible extent, (ii)
$\mathbf{U}$ captures information about the semantically related terms in
$\mathbf{V}$. In order to capture the distributional information GRI uses
skip-gram with negative sampling. In order to control the geometry and
isomorphism of embedding space $\mathbf{U}$ relative to space $\mathbf{V}$,
GRI uses attentive graph convolutions. Finally, it uses multiple different
isomorphism metrics along with the skip-gram loss function for model training.

We claim the proposed model provides the provision to perform BLI in a
performance-enhanced fashion by using attentive graph convolutions for
effective propagation of semantic relatedness of tokens across different
languages.

Figure 2: Proposed framework for Graph-based Relative Isomorphism(GRI). It
combines attentive graph convolutions with the skip-gram to control the
relative isomorphism for source $\mathbf{U}$ and target $\mathbf{V}$
embeddings.

###  4.3 GRI

In order to learn the distributional embeddings for the source language that
are geometrically similar to the target embeddings GRI incorporates attentive
graph convolutions along with the distributional training objective. GRI
relies on the assumption that each language possesses multiple variations of
semantically similar tokens that may be used interchangeably. And, in order to
effectively model the relative isomorphism for the multi-lingual data sets
this phenomenon needs to be captured explicitly.

The proposed model (GRI) is based on the assumption that sharing information
among semantically related words is inevitable in order to control the
relative isomorphism of the embedding spaces. From the linguistic perspective,
there are an arbitrary number of words semantically related to a given word,
which makes graphs a natural choice to unanimously consider the impact of
these words in the end model. We explain the individual components of GRI in
the following sub-sections:

####  4.3.1 Distributional Representation

In order to learn the distributional representations GRI uses skip-gram with
negative sampling. For training the skip-gram model, we follow the same
settings as outlined by Mikolov et al. (2013b), i.e., embed a word close to
its contextual neighbors and far from a set of words randomly chosen from the
vocabulary. The formulation of the skip-gram loss function is illustrated in
Equation 1.

$\displaystyle\vspace{-2.7ex}\mathcal{L}_{SG}=\log\sigma({u^{{}^{\prime}}_{c_{O}}}^{\mathsf{T}}u_{c_{I}})+$
(1) $\displaystyle\sum_{i}^{k}\mathbf{E}_{c_{i}\sim
P_{n}(c)}\big{[}\log\sigma(-{u^{{}^{\prime}}_{c_{i}}}^{\mathsf{T}}u_{c_{I}})\big{]}\vspace{-2.7ex}$

Here $u_{c_{O}}$ and $u_{c_{I}}$ correspond to the output and input vector
representation of the word $c$. $u^{{}^{\prime}}_{c_{i}}$ the embedding
vectors for noise terms. $P_{n}(c)$ corresponds to the noise distribution, $k$
is the number of noisy samples. We use $k=10$ in our case.

####  4.3.2 Capturing Semantics

In order to control the relative isomorphism across the source and target
embeddings GRI uses attentive graph convolutions under transductive settings
in order to share information among semantically related words. The graph
construction is summarized in Algorithm 1, and explained as follows:

##### Graph Construction.

Inputs for the graph construction include: (i) the supervision seed pairs for
the target language, (ii) existing pre-trained word2vec
embeddings222https://code.google.com/archive/p/word2vec/, trained using
Google-News Corpus of 100 billion words. Mikolov et al. (2013b). The graph
construction process proceeds as follows:

Firstly, we organize the target words into all possible pairs, i.e.,
combinations of two words at a time. For each word pair, we compute a score
(cosine similarity) of their corresponding embedding vectors. The word pairs
with scores higher than a threshold ($thr$) are stored as the probable
semantically related terms ($\text{Pairs}_{prob}$), illustrated in lines
(2-6). We observed that using a significantly higher value for $thr$ is
beneficial, because: (i) it helps in capturing the most confident word pairs
thus overcoming noise, and (ii) only a few word pairs end up in
$\text{Pairs}_{prob}$ which makes it computationally efficient. Finally, for
all word pairs in $\text{Pairs}_{prob}$, we formulate edges to construct the
graph G. For each word pair, we use the cosine score of corresponding
embedding vectors as the attention weight.

Algorithm 1 Graph Construction

Input: Embedding (EMB);
D${}_{tar}=\text{Target}(D_{tr+dev+tst})$
Output: Graph: G

1:$\text{Pairs}_{prob}$ $\leftarrow$ $\mathbf{0}$;
$\text{G}\leftarrow\emptyset$

2:for $(w_{1},w_{2})\leftarrow\text{Pairs}(\text{D}_{tar})$ do

3: $y^{*}$ = $\text{score}_{\text{EMB}}(w_{1},w_{2})$

4: if $y^{*}\geq{thr}$ then

5: $\text{Pairs}_{prob}\leftarrow\text{Pairs}_{prob}\cup(w_{1},w_{2})$

6: end if

7:end for

8:for $pair\in\text{Pairs}_{prob}$ do

9: G $\leftarrow$ $\text{G}\cup\\{edge(pair)$}

10:end for

11:return G

##### Attentive Graph Convolutions.

Depending upon the value of $thr$, graph G surrounds each word by a set of
highly confident semantically related words (including their lexical
variations). The degree of similarity is controlled by the cosine similarity
of embedding vectors. Later, for each word, we aggregate the information
contained in the neighbors to come up with a new representation of the word
that accommodates information from semantically related neighbors.

Note, in our setting, unlike the existing work by Kipf and Welling (2016), we
propose attentive graph convolutions with pair-wise distributional similarity
scores as the hard attention weights. The attention weights are not updated
during the model training. Specifically, we use the following layer-wise
propagation mechanism:

$\vspace{-1.7ex}L^{(i+1)}=\rho(\tilde{\Gamma}L^{(i)}W_{i})$ (2)

where $\tilde{\Gamma}=\bar{D}^{-1/2}(\Gamma+I)\bar{D}^{-1/2}$ is the
normalized symmetric matrix, $\bar{D}$ is the degree matrix of $\Gamma$,
$\Gamma$ is the weighted adjacency matrix learned from graph G with pair-wise
scores as the attention weights, $L^{(i)}$ is the input from previous layer,
with $L^{(0)}=\UU\in\mathbf{R}^{p\times{d}}$ as the input matrix corresponding
to the embeddings of the source terms, $W_{i}$ is the learn-able weight
matrix, $\rho$ is the non-linear activation function.

Note, the end goal of the attentive convolutions is to analyze each word in G
in relation to the weighted combination of its semantic neighbors. For this,
we surround each word (node) with a group of semantically related words (nodes
in the graph) and perform weighted aggregation to re-compute the
representation of the word. We also allow self-connections for each word,
i.e., adding an identity matrix $I$ to $\Gamma$. This will enforce
"semantically related words" to get similar representations. We observed, that
for our problem settings, this attentive graph convolution framework
outperforms the basic settings with equal contribution from all neighbors Kipf
and Welling (2016).

For GRI, we use a two-layered network to learn the final embeddings of each
word ${\UU_{m}}\in\mathbf{R}^{p\times{d}}$ as follows:

$\vspace{-1.7ex}\UU_{m}=\tilde{(\Gamma)}(ReLU(\tilde{(\Gamma)}\UU
W_{0}))W_{1}$ (3)

###  4.4 Isomorphism Loss functions

In order to train the GRI, we experiment with multiple different isomorphism
loss functions on top of the attentive graph convolution network. Details
about each loss function are as follows:

##### L2 loss.

We use L2-norm averaged over the number of data samples.

$\vspace{-1.7ex}\mathcal{L}_{2}=\frac{1}{N}{||\mathbf{U}_{m}-\mathbf{V}||_{2}}$
(4)

##### Orthognal Procrustus loss.

Orthogonal Procrustes loss aims to find a linear transformation $W$ to solve:

$\mathcal{L}_{proc}=\operatorname*{arg\,min}_{\WW\in\mathbf{R}^{d\times
d},\WW^{T}\WW=I}\frac{1}{N}{||\mathbf{U}_{m}\mathbf{W}-\mathbf{V}||}_{2}$ (5)

Schönemann (1966) proposed a solution $\mathbf{W}=\QQ\PP^{T}$, where
$\PP\Sigma\QQ^{T}$ is the singular value decomposition of $\VV^{T}\UU_{m}$.

##### Procrustus loss with initialization.

It follows the same process as that of the Procrustus loss with the exception
that we initialize the embeddings for the source words with the embedding
vectors of their translation vectors corresponding to the target words. The
end goal of this setting is to analyze the ability of the GRI to propagate the
knowledge of the initialized embeddings during the model training. We also
allow updating the initialized word embeddings during model training. We
denote this loss by $\mathcal{L}_{proc_{init}}$. We use the symbol
$\mathcal{L}_{ISO}$ to represent the different variations of isomorphism
losses, i.e., $\mathcal{L}_{2}$, $\mathcal{L}_{proc}$ and
$\mathcal{L}_{proc_{init}}$.

###  4.5 The Complete Model

Finally, we combine the distributional training objective with the isomorphism
loss function to compute complete loss of GRI, as follows:

$\vspace{-0.7ex}\mathcal{L}_{GRI}=\alpha\mathcal{L}_{SG}+(1-\alpha)\mathcal{L}_{ISO}$
(6)

where $\alpha$ is the parameter used to control the contribution
$\mathcal{L}_{SG}$ and $\mathcal{L}_{ISO}$ respectively.

##  5 Experiments and Results

###  5.1 Datasets

In order to set up a unanimous platform for comparative analysis, we use the
data settings used by Marchisio et al. (2022). We use the first 1 million
lines from newscrawl-2020 data set for English (“en"), Bengali (“bn") and
Tamil (“ta") and the entire of newscrawl-2020 data for Ukrainian (“uk") to
train word embeddings. We used Moses scripts for data pre-processing333Moses
script. For evaluation, we used publically available train, dev, and test
splits provided by MUSE (Conneau et al., 2017). Out of approx 8000-word pairs
for each language, we used word pairs 0-5000, 5001-6500, and 6501-8000 as
train, test and dev set respectively. The train set is used for model
training, dev set is used for parameter tuning, and final results are reported
using the test set. All these data splits are non-overlapping.

###  5.2 Baseline Models

For comparative evaluation, we use independently trained distributional
embeddings for the source and target languages as a baseline model. Amongst
the existing research, we compare GRI against the prevalent state-of-the-art
work on BLI, i.e., IsoVec by Marchisio et al. (2022). IsoVec uses the skip-
gram training objective along with isomorphism training objectives. Note,
Marchisio et al. (2022) used exactly the same data settings as that of our
proposed model (i.e., GRI), so for performance comparison, we use the numbers
reported in the original paper.

###  5.3 Experimental Settings

For model training, we use Adam optimizer (Kingma and Ba, 2014) with learning
rate = 0.001; $\alpha$ in Equation 6 is set to 0.7; the value of $thr$ in
algorithm 1 is set to 0.5. For experiments, we use embeddings learnt for
English language as the target embeddings, and embeddings for other languages,
i.e., “ta", “uk", and “bn", as the source embeddings. For mapping across
different spaces, we use Vecmap toolkit with process-flow explained in Section
3.2. We use average P@1 as the evaluation metric. We report mean $(\mu)$ and
standard deviation $(\sigma)$ of the results over 5 runs of the experiment.
All experiments are performed using Intel Core-i9-10900X CPU, and Nvidia
3090Ti GPU.

###  5.4 Main Results

Methodology | bn | uk | ta
---|---|---|---
Baseline | 13.1 ($\pm$ 0.51) | 13.9 ($\pm$ 0.45) | 10.8 ($\pm$ 0.42)
IsoVec (L2) | 16.3 ($\pm$ 0.4) | 16.5 ($\pm$ 0.4) | 11.1 ($\pm$ 0.5)
IsoVec (Proc-L2) | 16.6 ($\pm$ 0.7) | 16.0 ($\pm$ 0.8) | 10.7 ($\pm$ 0.3)
IsoVec (Proc-L2-Init) |  16.9 ($\pm$ 0.2) |  17.1 ($\pm$ 0.6) |  11.8 ($\pm$ 0.3)
GRI ($\mathcal{L}_{2}$) | 17.28 ($\pm$ 0.02) | 18.75 ($\pm$ 0.41) | 13.47 ($\pm$ 0.04)
GRI ($\mathcal{L}_{{proc}_{init}}$) | 19.83 ($\pm$ 0.05) | 21.37 ($\pm$ 0.08) | 15.27 ($\pm$ 0.01)
GRI ($\mathcal{L}_{proc}$) |  20.52 ($\pm$ 0.02) |  27.97 ($\pm$ 2.63) |  18.97 ($\pm$ 0.2)

Table 1: GRI results for the proposed model. We compare results with IsoVec
(Marchisio et al., 2022).

The results for the proposed model (GRI) compared with the baseline models are
shown in Table 1. We boldface the overall best-performing scores with the
previous state-of-the-art underlined.

These results show the GRI has a relatively stable performance (with low
variance), it consistently outperforms the baseline and previous state-of-the-
art scores by a significant margin. For “bn", “uk", and “ta", GRI outperforms
the IsoVec (Marchisio et al., 2022) by 21.4%, 63.6% and 60.7% respectively.
Especially noteworthy is the performance improvement gained by GRI for the
Ukrainian language. We attribute this performance improvement to the fact that
the semantic relatedness of the words corresponding to the Ukrainian embedding
space is relatively better compared to other languages.

The performance comparison of different isomorphism loss functions shows that
$\mathcal{L}_{proc}$ consistently outperforms the $\mathcal{L}_{proc_{init}}$
and $\mathcal{L}_{2}$ across all data sets. A relatively low performance of
$\mathcal{L}_{proc_{init}}$ compared to the $\mathcal{L}_{proc}$ may be
attributed to the fact that randomly initialized embeddings are a better
choice compared to the initialization from the seed pairs. The initialization
from the seed pairs may not be helpful for the model training to improve the
performance at later stages.

Overall results show the significance of using attentive graph convolutions in
controlling the relative geometry of source language for BLI. Especially, the
ability of the attentive convolutions to accumulate the contribution of
semantically related terms plays a vital role in controlling the relative
geometry of the source embeddings relative to the target embeddings, as is
evident from the results in Table 1.

##  6 Discussion

In this sub-section, we perform a detailed analysis of the performance of GRI.
Specifically, we analyze: (i) Domain mis-match settings (ii) Impact of
attentive convolutions, (iii) Isometric metrics, and (iv) Error cases.

###  6.1 Domain mismatch

Domain-mismatch has been identified as one of the core limitations of existing
BLI methods. These methods fail badly in inferring bilingual information for
embeddings trained on data sets from different domains Søgaard et al. (2018);
Marchisio et al. (2020).

We claim that incorporating lexical variations for semantically related tokens
makes GRI robust to the domain mismatch settings. In order to validate these
claims for GRI, we re-run the experiments using target embeddings trained on
33.8 million lines of web-crawl data from the English Common Crawl data. The
embeddings for the source languages (“bn", “uk" and “ta") are trained using
the newscrawl-2020 data. The results for the domain-mismatch experiments for
different isomorphism loss functions are reported in Table 2.

These results are compared against the baseline distributional embeddings and
best-performing scores of the existing work, i.e., IsoVec by Marchisio et al.
(2022). Note, for the domain mismatch experiments, we use exactly same data
settings as that of Marchisio et al. (2022), so we report exactly the same
numbers as reported in original paper. Comparing the results of our model
against the IsoVec, the GRI improves the performance by 27.74%, 53.12% and
74.22% for the “bn", “uk" and “ta" languages respectively.

Methodology | bn | uk | ta
---|---|---|---
Baseline | 9.7 ($\pm$ 0.72) | 10.2 ($\pm$ 0.43) | 7.5 ($\pm$ 0.39)
IsoVec (Proc-L2-Init) | 15.5 ($\pm$ 0.7) | 17.3 ($\pm$ 0.4) | 10.9 ($\pm$ 0.5)
GRI ($\mathcal{L}_{2}$) | 13.97 ($\pm$ 0.02) | 17.32 ($\pm$ 0.32) | 11.93 ($\pm$ 0.01)
GRI ($\mathcal{L}_{{proc}_{init}}$) | 19.75 ($\pm$ 0.01) | 21.32 ($\pm$ 0.10) | 17.12 ($\pm$ 0.59)
GRI ($\mathcal{L}_{proc}$) |  19.80 ($\pm$ 0.50) |  26.49 ($\pm$ 0.50) |  18.99 ($\pm$ 0.20)

Table 2: GRI results for domain mis-match experiments compared with the
baseline models, IsoVec Marchisio et al. (2022).

Comparing these results against the main experiments reported in Table 1, we
can see the GRI yields a stable performance for both domain-shared as well as
domain mismatch settings. These results show that the attentive graph
convolutions indeed allow information sharing across semantically related
tokens along with their lexical variations that is in turn helpful in
controlling the relative isomorphism of the embedding spaces.

Comparing the results for different loss functions, we can see that similar to
the main experiments the performance of the model for the Procrustes loss
$(\mathcal{L}_{proc})$ is relatively higher than the $\mathcal{L}_{2}$ and
$\mathcal{L}_{proc_{init}}$.

Methodology | bn | uk | ta
---|---|---|---
GRI w/o G-Conv ($\mathcal{L}_{2}$) | 16.10 ($\pm$ 0.35) | 16.35 ($\pm$ 0.30) | 11.25 ($\pm$ 0.45)
GRI w/o G-Conv ($\mathcal{L}_{proc_{init}}$) | 16.75 ($\pm$ 0.20) | 16.98 ($\pm$ 0.30) | 11.70 ($\pm$ 0.25)
GRI w/o G-Conv ($\mathcal{L}_{proc}$) | 16.50 ($\pm$ 0.5) | 16.10 ($\pm$ 0.70) | 10.65 ($\pm$ 0.20)
GRI ($\mathcal{L}_{2}$) | 17.28 ($\pm$ 0.02) | 18.75 ($\pm$ 0.41) | 13.47 ($\pm$ 0.04)
GRI ($\mathcal{L}_{{proc}_{init}}$) | 19.83 ($\pm$ 0.05) | 21.37 ($\pm$ 0.08) | 15.27 ($\pm$ 0.01)
GRI ($\mathcal{L}_{proc}$) |  20.52 ($\pm$ 0.02) |  27.97 ($\pm$ 2.63) |  18.97 ($\pm$ 0.20)

Table 3: Analyzing the impact of attentive graph convolutions for GRI.

###  6.2 Impact of attentive convolutions

In this sub-section, we analyze in detail the performance improvement of GRI
attributable to the attentive graph convolutions. For this, we primarily
analyze the performance improvement of GRI with and without attentive graph
convolutions. The results of these experiments are reported in Table 3. These
results show the significance of attentive graph convolutions that help in
improving the performance across all three languages. The improvement in
performance for the “bn", “uk" and “ta" languages is 24.36%, 64.72% and 62.13%
respectively.

To gain further insight, we also analyzed the output of the model with and
without graph convolutions in order to dig out which class of translation
instances were correctly translated especially due to the attentive
convolutions part of GRI. We run this analysis only for the Ukrainian language
because: GRI yields a higher score for the Ukrainian language compared to
other languages. All the analyses were performed under the direct supervision
of a linguistic expert.

Detailed analyses show that a major portion (approx 51%) of the pairs
corrected especially by the graph convolutions belong to the nouns, with 21%
verbs and 20% adjectives. The rest 7% are assigned to other classes. This
analysis shows that the phenomenon of lexical variation is dominant among
nouns that results in better performance of GRI compared to the baseline
models.

###  6.3 Isometric metrics

We also correlate the results of GRI with different widely used isomorphism
metrics. Specifically, we use two metrics, namely: (a) Pearson’s correlation,
and (b) Eigenvector similarity. Details about these metrics and the
corresponding experimental setting are as follows:

| Pearson Correlation ($\uparrow$) | Eigenvector Similarity($\downarrow$)
---|---|---
Methodology | bn | uk | ta | bn | uk | ta
GRI ($\mathcal{L}_{2}$) | 0.47 | 0.36 | 0.42 | 35.55 | 30.64 | 69.72
GRI ($\mathcal{L}_{proc_{init}}$) | 0.47 | 0.36 | 0.43 | 31.23 | 10.92 | 45.56
GRI ($\mathcal{L}_{proc}$) | 0.49 | 0.37 | 0.44 | 32.16 | 29.53 | 62.81

Table 4: Analysis of different isometry metrics for GRI.

##### Pearson’s Correlation.

We compute Pearson’s correlation between the cosine similarities of the seed
translation pairs as an indicator of the relative isomorphism of corresponding
spaces. We expect our P@1 results to correlate positively ($\uparrow$) with
Pearson’s correlation.

We compute Pearson’s correlation over first 1000 translation seed pairs.
Corresponding results are shown in the first half of Table 4. We boldface the
best scores. These results show that for all languages, Pearson’s correlation
for the model GRI ($\mathcal{L}_{proc}$) is slightly higher compared to other
models. Although these results are aligned with our findings in Table 4,
however, one noteworthy observation is that Pearson’s correlation is not a
true indicator of the relative performance improvement across different
isomorphism losses.

##### Eigenvector Similarity.

In order to compute the eigenvector similarity of two spaces, we compute the
Laplacian spectra of the corresponding k-nearest neighbor graphs. This setting
is similar to Søgaard et al. (2018), and is summarized as follows. For seed
pairs construct unweighted graphs followed by computing the graph Laplacians.
Later, compute the eigenvalues of the graph Laplacians and retain the first
$k$ eigenvalues summing to less than 90% of the total sum of eigenvalues.
Finally, we compute the eigenvector similarity as the sum of squared
differences between partial spectra. The graphs with similar eigenvalue
spectra are supposed to have similar structures (a measure of relative
isomorphism).

We expect our eigenvector similarity results to correlate negatively
($\downarrow$) with P@1. The experimental results are shown in the right half
of Table 4, with the best scores boldfaced. These results show that the
eigenvector similarity scores for the model GRI ($\mathcal{L}_{proc_{init}}$)
are better than the other two models. This is in contrast to our findings in
Table 1, where GRI ($\mathcal{L}_{proc}$) shows relatively better performance.

Generally speaking, the results of the isometric metrics do not truly
correlate with the P@1. These findings are aligned with earlier studies by
Marchisio et al. (2022) that also emphasized the need for better metrics to
compute the relative isomorphism of the embedding spaces.

###  6.4 Error Analyses

We also analyze the error cases of GRI in order to know the limitations of the
model and room for future improvement. Note, similar to section 6.2, we only
perform the error analyses for the Ukrainian language and Procrustes loss
($\mathcal{L}_{proc}$). All experiments were performed with the help of
linguistic experts. We separately analyze the errors for the variants of GRI
with and without attentive graph convolutions (i.e., GRI; GRI w/o G-Conv) in
order to quantify the reduction in error attributable to the attentive graph
convolutions.

| Type-a | Type-b
---|---|---
GRI w/o G-Conv($\mathcal{L}_{proc}$) | 21.3% | 6.5%
GRI ($\mathcal{L}_{proc}$) | 50.2% | 16.6%

Table 5: Classification of Error Types

In order to better understand the errors from semantic perspective, we
categorize the errors into the following types:

##### Type-a:

The predicted target word for P@1 is semantically close to the true target
word.

##### Type-b:

The predicted target word is a k-nearest neighbor of the true word for k=5.

We limit the error cases to only the above-mentioned simple types because
these types give a rough picture of the relative isomorphism of the different
spaces from the semantic perspective. The percentage error counts for both
models are shown in Table 5. For the model GRI w/o
G-Conv($\mathcal{L}_{proc}$), 21.3% errors fall in error Type-a, and 6.5%
errors belong to error Type-b. For the model GRI($\mathcal{L}_{proc}$), 50.2%
errors fall in Type-a, and 16.6% errors belong to Type-b. As expected the
variant of GRI with graph convolutions shows a higher percentage for both
categories, i.e., Type-a and Type-b.

GRI ($\mathcal{L}_{proc}$) | GRI w/o G-Conv($\mathcal{L}_{proc}$)
---|---
source | target | target${}^{{}^{\prime}}$ | source | target | target${}^{{}^{\prime}}$
пошта | mail | mailing |  зникли | gone | shattered
спільний | shared | sharing |  олія | oil | 60g
вікно | window | windows |  банки | cans | merchants
внз | college | teaching |  ніс | nose | rubbing
йшов | walked | went |  заміна | replacing | overpriced
підручників | manuals | templates |  вулкан | volcano | 100mph
реформа | reform | reforms |  ріст | growth | decline

Table 6: Example error cases for the Ukrainian vs English language for the
models: GRI ($\mathcal{L}_{proc}$); GRI w/o G-Conv($\mathcal{L}_{proc}$). For
each model, the first column (source) corresponds to the Ukrainian words, the
second column (target) represents the true target word, third column
(target${}^{{}^{\prime}}$) represents the model predictions for target words.

These numbers clearly indicate that the attentive graph convolutions were not
only able to correct a major portion of errors made by (GRI w/o G-Conv), but
also the errors made by the model GRI are either highly semantically related
to the target words or a nearest neighbor of the target word.

In order to gain further insight, we manually look at the error cases. For
both models, a few examples are shown in Table 6. The majority of the
predictions made by GRI are indeed correct and closely related to the true
target words. For example, it predicts {“mailing", “sharing", “windows"} in
place of {“mail", “shared", “window"} respectively. These results clearly
indicate that the current performance of GRI is under-reported and there is a
need for better quantification measures (other than P@1) in order to compute
and/or report the true potential of the model.

Overall error analyses show the significance of the using attentive graph
convolutions to incorporate the lexical variations of semantically related
tokens in order to control the relative isomorphism and perform BLI in
performance-enhanced way.

##  7 Conclusion and Future Directions

In this paper, we propose Graph-based Relative Isomorphism (GRI) to
incorporate the impact of lexical variations of semantically related tokens in
order to control the relative isomorphism of cross-lingual embeddings. GRI
uses multiple different isomorphism losses (guided by the attentive graph
convolutions) along with the distributional loss to perform BLI in a
performance-enhanced fashion. Experimental evaluation shows that GRI
outperforms the existing research on BLI by a significant margin. Some
probable future directions include: (i) extending the concepts learned in this
research to contextualized embeddings, and (ii) augmenting the GRI to focus
more on preserving lexico-semantic relations.

## Limitations

Some of the core limitations of the proposed approach are as follows: (i)
current formulation of GRI is not defined and/or implemented for deep
contextualized embeddings which are more prevalent and a better alternate to
the distributional embeddings, (ii) existing limitations of the distributional
embeddings are inherited in the model, which limits the end-performance of
GRI. For example, as pointed out by Ali et al. (2019) the distributional
embedding space tends to inter-mix different lexico-semantic relations, and
yield a poor performance on a specific task. This phenomenon has a direct
impact on GRI especially on controlling the relative isomorphism of highly
interlinked relation pairs, e.g., Antonyms vs Synonyms.

##### Acknowledgements.

Di Wang, Yan Hu and Muhammad Asif Ali are supported in part by the baseline
funding BAS/1/1689-01-01, funding from the CRG grand URF/1/4663-01-01,
FCC/1/1976-49-01 from CBRC and funding from the AI Initiative REI/1/4811-10-01
of King Abdullah University of Science and Technology (KAUST). Di Wang is also
supported by the funding of the SDAIA-KAUST Center of Excellence in Data
Science and Artificial Intelligence (SDAIA-KAUST AI).

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XeLatex=

# Propagating charge carrier plasmon in Sr2RuO4

Martin Knupfer  Leibniz Institute for Solid State and Materials Research
Dresden, Helmholtzstr. 20, D-01069 Dresden, Germany  Fabian Jerzembeck  Max
Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187
Dresden, Germany  Naoki Kikugawa  National Institute for Materials Science,
Tsukuba 305-0003, Japan  Friedrich Roth  Institute of Experimental Physics, TU
Bergakademie Freiberg, Leipziger Str. 23, 09599 Freiberg, Germany  Center for
Efficient High Temperature Processes and Materials Conversion, TU Bergakademie
Freiberg, 09599 Freiberg, Germany  Jörg Fink  Leibniz Institute for Solid
State and Materials Research Dresden, Helmholtzstr. 20, D-01069 Dresden,
Germany  Institut für Festkörperphysik, Technische Universität Dresden,
D-01062 Dresden, Germany

###### Abstract

We report on studies of charge carrier plasmon excitations in Sr2RuO4 by
transmission Electron Energy-Loss Spectroscopy. In particular, we present
results on the plasmon dispersion and its width as a function of momentum
transfer. The dispersion can be qualitatively explained in the framework of
RPA calculations, using an unrenormalized tight-binding band structure. The
constant long-wavelength width of the plasmon indicates, that it is caused by
a decay into inter-band transition and not by quantum critical fluctuations.
The results from these studies on a prototypical “bad“ metal system show that
the long-wavelength plasmon excitations near 1 eV are caused by resilient
quasiparticles and are not influenced by correlation effects.

### Introduction.

Unconventional superconductors are one of the most active research fields in
solid state physics. The reason for this is not only the observed high
superconducting transition temperature. Very often, these materials are termed
“bad“ metals because of their strange normal state properties. The latter are
caused by strong electron-electron interactions leading to correlation effects
connected with quantum critical behavior. The renormalizations of single-
particle excitations are described by the complex self-energy $\Sigma$, the
real and the imaginary part of which are related to the mass renormalization
$m^{*}/m$ and the scattering rate $\Gamma$, respectively.

One of the most studied “bad“ metals are the doped cuprates. There are
numerous studies of single-hole excitations in these materials by ARPES
providing detailed knowledge of the self-energy yielding a mass
renormalization of the conduction band of about two [1]. Furthermore, there
are studies of the spin excitations compiled in Ref. [2]. On the other hand,
there are only a few studies on momentum dependent charge degrees of freedom.
There is an ongoing discussion, how the renormalization of the single-particle
excitations, i.e., the mass renormalization and the finite scattering rate,
influences the existence, the dispersion, and the lifetime broadening of two-
particle charge excitations (plasmons) in the cuprates.

Several theoretical studies [3, 4, 5] have predicted, that spin excitations
strongly influence the charge excitations. On the other hand, the observation
of a continuum instead of the high-energy plasmon was discussed for quantum
critical systems based on holographic theories [6], which predicted, different
from the classical Landau damping [7], a strong “over-damping“ even for the
long-wavelength plasmons. These theoretical investigations seem to be
supported by Electron Energy-Loss Spectroscopy (EELS) in reflection (R-EELS)
studies of cuprates [8, 9, 10] in which not well-defined propagating plasmons
or even only a continuum of excitations were detected. On the other hand,
there are EELS studies in transmission (T-EELS) [11, 12, 13, 14, 15, 16] on
cuprates [17, 18, 19, 20, 21, 22] where the dispersion of well defined
plasmons could be explained in terms of collective excitations within an
unrenormalized band structure.

To better understand the problem, in the present paper, we extend the studies
to Sr2RuO4, a “bad“ metal with an even higher average effective mass of about
four, using T-EELS. Sr2RuO4 is a prototype of unconventional superconductors
[23, 24]. The superconducting order parameter is still a matter of controversy
[25]. The strongly correlated normal state electronic structure has been
studied by optical spectroscopy [26, 27, 28, 29, 30], by R-EELS [10], by
tunneling spectroscopy [31], and by ARPES [32, 33, 34, 35, 36], in some cases
supported by DFT or combined DFT plus dynamical mean field theory (DFT+DMFT)
calculations [37, 38, 36]. The latter two experimental techniques probe
essentially one-particle (electron or hole) properties whereas the first two
methods probe two-particle excitations (hole plus electrons).

Using slices of the Sr2RuO4 crystal cut perpendicular to the ${\bf c}$ axis
together with an orientation of the ${\bf c}$ axis parallel to the electron
beam, in the present T-EELS measurements, the momentum can be varied within
the (${\bf a,b}$) plane. The momentum parallel to the beam is negligible.
Optical spectroscopy measuring excitations with zero momentum transfer [26,
27, 29, 30] found for a photon polarization perpendicular to the ${\bf
c}$-axis a well pronounced plasmon near 1.5 eV with a plasmon width of
$\approx$ 1 eV. For a polarization parallel to the ${\bf c}$-axis, a low
energy plasmon near 0.01 eV was detected [26, 27, 28]. Different from this
result, R-EELS study detected near 1 eV a continuum and at low energy a
dispersing acoustic plasmon [10], which was ascribed to a “Demon“ plasmon on
the basis of interacting excitations of heavy and light electrons.

The existence of a high-energy and a low-energy plasmon was already predicted
from a calculation of the high in-plane and the low out-of-plane Fermi
velocities on the basis of band structure calculations [39].

The main result of the present study is the observation of a well-defined
propagating plasmon which in the non-local long-wavelength limit can be
qualitatively explained by RPA calculations based on an unrenormalized DFT
band structure. This indicates that the low-momentum dispersion is not
influenced by the local on-site Coulomb interaction. The results also
demonstrate that single hole excitations (measured by ARPES) are much stronger
screened than two-particle electron-hole excitations (measured e.g. by
T-EELS).

### Experimental.

Sr2RuO4 crystals were grown using the traveling floating-zone method [40]. The
superconducting transition temperature of the sample was $T_{c}=$1.5 K. For
the EELS measurements, thin films with a thickness of $\approx$ 100 nm were
cut perpendicular to the crystal ${\bf c}$-axis from these single crystals
using an ultramicrotome equipped with a diamond knife. The films were then put
onto standard transmission electron microscopy grids and transferred into the
spectrometer. The measurements were carried out at a sample temperature of
$T=$ 20 K with a dedicated transmission electron energy-loss spectrometer [15,
16] employing a primary electron energy of 172 keV. The energy and momentum
resolution was set to $\Delta E$ = 80 meV and $\Delta q$ = 0.035 Å-1,
respectively. In all measurements, the momentum transfer was parallel to the
(${\bf a,b}$) plane. Before measuring the loss-function, the thin films have
been characterized by in-situ electron diffraction, in order to orient the
crystallographic axis with respect to the transferred momentum.

### Calculations.

For the calculation of the complex dielectric function we used the Ehrenreich-
Cohen expression [41, 42]:

$\epsilon(\omega,{\bf
q})=\epsilon_{\infty}-\frac{1}{q^{2}}\chi_{0}(\omega,{\bf q})$ (1)

with

$\chi_{0}(\omega,{\bf
q})\propto\frac{1}{q^{2}}M^{2}\int_{-\infty}^{\infty}\frac{F_{k}\Delta
E}{\omega^{2}-\Delta E^{2}+i\Gamma_{P}\omega}d{\bf k}.$ (2)

Here $\Delta E=E_{\bf k+q}-E_{\bf k}$ where $E_{\bf k}$ describes the band
structure, $F_{\bf k}$ is the Fermi function, $\Gamma_{P}$ is the lifetime
broadening, and M is the matrix element. Using the complex dielectric function
we then calculate the loss function $\Im[-1/\epsilon(\omega,{\bf q})]$. The
maximum of the loss function as a function of momentum q which occurs at
$\Re[\epsilon(\omega,{\bf q})]=0$ yields the plasmon dispersion. In the
calculation we use a constant matrix element, the absolute value being
determined by the condition that the plasmon energy for $q=0$ agrees with that
derived from optical spectroscopy. The small momentum dependence of the matrix
element is supported by the observation that for momentum $q\leq$0.5 Å-1 the
longitudinal f-sum rule [7] is reduced by less than $20~{}\%$. Similar
calculations were performed for poly-acetylene [43], La2CuO4 [19], optimally
doped Bi2Sr2CaCu2O8 [18, 20], and for the ladder compound CaxSr14-xCu24O41
[22]. Interestingly, in all cases the low-energy dispersion could be well
described using a simple unrenormalized tight-binding band structure.

In Sr2RuO4 the conduction band is composed of 3 Ru 4$d$ $t_{2g}$ bands, the
narrow quasi-1D $\alpha$ and $\beta$ bands, and the wider quasi-2D $\gamma$
band, having, without taking into account spin-orbit interaction,
predominantly $xz$, $yz$, and $xy$ orbital character, respectively [33, 36].
For the calculation of the plasmon dispersion we used an unrenormalized tight-
binding band structure [38]. We obtain a minimum of the difference between
calculated and experimental plasmon dispersion by a shift of 0.2 eV of the
band structure to lower binding energies. For calculations using Eq. 1 the
neglect of the spin-orbit coupling is reasonable because the spin-orbit
coupling is considerably smaller than the plasmon energy [44, 36]. For the
calculations we used the background dielectric function
$\epsilon_{\infty}=$2.3 from optical spectroscopy [29, 30].

Using RPA for the homogenous electron gas, the free electron long-wavelength
plasmon dispersion coefficient $\alpha$ (see Eq. 3) is proportional to
$v_{42}\propto<v_{F}^{4}>/<v_{F}^{2}>\approx<v_{F}^{2}>$ [20, 42]. To control
the anisotropy and orbital character of the calculated dispersions we have
calculated $v_{42}$ for the bands for ${\bf q}$ parallel to the [110] and
[100] direction.

### Results.

Figure 1:  Electron energy-loss spectra multiplied by $q^{2}$ ($q^{2}I$) of
Sr2RuO4 in the energy range between 0 to 4 eV for momentum transfer ${\bf q}$
between 0.15 and 0.5 Å-1. Upper panel: ${\bf q}$ parallel to [100] direction,
lower panel: ${\bf q}$ parallel to [110] direction. Solid lines: guide to the
eyes.

In Fig. 1 we present the experimental loss spectra of Sr2RuO4 for momentum
transfer parallel to the [110] and the [100] direction. The strongest
excitations occur between 1.5 and 1.8 eV, which are ascribed to plasmons of
the charge carriers. At low energies, the intensity is dominated by the quasi-
elastic peak. At higher energy near 3.3 eV, a small peak possibly due to
interband transitions is visible at higher momentum transfer.

We derive a positive plasmon dispersion by an evaluation of the maxima of the
loss data. In the long-wavelength limit the experimental and the calculated
dispersion can be well described by

$\omega(q)=\omega_{p}+\frac{\hbar^{2}}{m}\alpha q^{2}.$ (3)

$\omega_{p}$ is equal to 1.49 eV which roughly agrees with the in-layer
plasmon energy $\omega_{p}=$ 1.57 eV derived from optical spectroscopy [26,
29, 30].

Figure 2:  (Color online) Experimental data of the plasmon dispersion.
Circles: data for ${\bf q}$ parallel to [100]. Squares: data for ${\bf q}$
parallel to [110]. Dotted line: least squares fit to the experimental data.
Thick solid lines: results from the calculations using an effective mass of
one. All black data are for ${\bf q}$ parallel to [100]. The grey (red) data
are for ${\bf q}$ parallel to [110]. Dashed lines: calculations using an
effective mass of four.

The dispersions of the plasmon along the two directions together with a fit
for $q\leq$ 0.3 Å-1 is depicted in Fig. 2. The derived dispersion coefficients
are $\alpha^{exp}_{110}=$0.22 and $\alpha^{exp}_{100}=$0.30.

In this figure we have also added the calculated plasmon dispersion along the
[110] and the [100] direction. The calculated dispersion coefficients for
$q\leq$ 0.3 Å-1 are $\alpha^{cal}_{110}=$0.25 and $\alpha^{cal}_{100}=$0.21.

Figure 3:  (Color online) Experimental data of the plasmon width as a function
of momentum. Black circles: data for ${\bf q}$ parallel to the [100]
direction. Grey (red) squares: data for ${\bf q}$ parallel to the [110]
direction. Dashed line: guide to the eyes.

In Fig. 3 we present the plasmon width (FWHM), derived from a fit of a
Gaussian plus background to the loss data shown in Fig. 1. Along the two
directions, within error bars the width is constant and equal to 0.7 eV up to
$q=$ 0.4 Å-1. Above $q\approx$ 0.4 Å-1 an increase of the plasmon width is
detected.

### Discussion.

We have interpreted the principle excitation of the loss spectra near 1.5 eV
by a charge carrier plasmon. Its finite width of 0.7 eV is smaller than the
plasmon energy of 1.5 eV, indicating a quasi-particle excitation. The finite
dispersion signals a propagating plasmon. We emphasize that the width is
almost constant in the studied low momentum range. This means that it is not
related to electron-electron interaction which would lead to a quadratic
increase in momentum transfer [45]. The fact that the width is smaller than
the energy signals that the excitation is not overdamped due to fluctuations
in a quantum critical system, in contrast to theoretical predictions [6].
Rather, it is caused, as in most metallic systems, studied by T-EELS, by a
decay into interband transitions [46, 47, 48]. These interbands are caused by
a back-folding of bands from the second to the first Brillouin zone by a
finite pseudo-potential.

The observation of a continuum near 1.5 eV and an acoustic plasmon at low
energies in a recent R-EELS study [10] can be possibly explained by a large
momentum transfer perpendicular to the layers leading to out-of phase
collective excitations in the layer system. This view is supported by numerous
theoretical studies [49, 50, 51, 52, 53, 54] and by several recent Resonant
Inelastic X-ray Scattering (RIXS) experiments on layered n-type [55, 56, 57]
and p-type doped cuprates [58, 59]. In these experiments acoustic plasmons
with an out-of-plane dispersion have been detected in the low-($\omega,q$)
range with a very small width of $\approx 0.1$ eV. The interpretation of the
acoustic plasmons in terms of an out-of-phase excitation in layered compounds
is at variance with the assignment to a “Demon“ plasmon [10].

Looking at Fig. 2, the calculated plasmon dispersions, using an effective mass
of one, are close to the experimental ones. On the other hand, the small
experimental anisotropy is opposite to the calculated one. The difference
between the experimental and the calculated dispersion may be explained by the
distortion of the bands by spin-orbit interaction [36], not taken into account
in our RPA calculations. An analysis of average Fermi velocities yields some
information on the contribution of the 3 conduction bands to the plasmon
dispersion: along ${\bf q}$ || to the [110] directions, the dispersion is
predominantly determined by the 2D $\gamma$ band and to a lesser extent by the
1D $\alpha$ and $\beta$ bands. Along the [100] direction, both the $\gamma$
band and the $\alpha$ band contribute to the momentum dependence.

We emphasize, that we obtain a qualitative agreement between the experimental
and the calculated dispersion, although for this highly correlated system, we
have used in the calculation an unrenormalized tight-binding band structure.
The situation is similar to that in cuprates (see above) where the plasmon
dispersion can be well described by RPA using an unrenormalized tight-binding
band structure[18, 20]. In the cuprates, where the band structure is
renormalized by a factor of two, in the simplest approximation when the
dispersion coefficient is proportional to the square of the Fermi velocity,
one may expect a reduction of the plasmon dispersion due to flattening of the
bands by a factor of 4. However, in Sr2RuO4 the average mass renormalization
of the charge carrier is near 4.5 [36], which would lead in the above
mentioned approximation to a reduction of the plasmon dispersion by a factor
of about 20. Actually, our RPA calculation using an effective mass $m^{*}/m=$
4 yields a negative plasmon dispersion (see Fig. 2). As explained in previous
calculations [18, 20] negative plasmon dispersion may occur because the
dispersion coefficient is composed out of a positive and a negative
contribution. A large reduction of the plasmon dispersion is not compatible
with our experimental results. Thus our present results are a clear proof that
correlation effects do not influence the long-wavelength dispersion of the
plasmon in systems which are correlated by the on-site Coulomb interaction in
transition metal ions.

The weak influence of correlation effects on the plasmon dispersion can be
rationalized by the following consideration. The dispersion is a measure of
the compressibility of the electron liquid [7, 42]. In the long-wavelength
limit, the charge oscillations due to the plasmon excitation have a wavelength
that is much larger than the distance between the next nearest Ru atoms. On
the other hand, the mass enhancement in the correlated materials is
predominantly caused by the hopping between neighboring transition metal atoms
which is needed to overcome the on-site Coulomb interaction. In the long-
wavelength limit of the plasmon excitations, the occupation number on the
transition metal atoms is hardly changed and therefore the plasmon dispersion
is not influenced by correlation effects. Moreover, at high energies, the mass
enhancement is expected to be reduced because the charge carriers are no more
coupled to spin excitations [3, 4, 5], which have an energy of the order of
100 meV. Furthermore, one can also understand the difference between EELS and
ARPES studies. In the latter, the mass of a single hole is enhanced by an
Auger-like valence band electron-hole excitation, leading in the final state
to a two holes in the occupied part of the band and therefore is influenced by
the on-site Coulomb interaction.

Another indication for the non-sensitivity of the plasmon excitations to
correlation effects is the plasmon width, being in the long-wavelength limit
nearly independent of the momentum transfer (see Fig. 3). This demonstrates,
that in Sr2RuO4 the width is not caused by a lifetime broadening of the
valence band due to a finite imaginary part of the self-energy, observed by
ARPES. At $q=$0 the plasmon width must be rigorously zero in an free-electron
gas [13]. In the case of Al a quadratic momentum dependence of the width due
to intra-band particle-hole excitations was predicted by theoretical
calculations [45], which is much too small compared to experimental data of
Al. Later, theoretical calculations [46, 60] and experiments [61, 62, 63]
showed, that the plasmon width is predominantly caused by a decay of the
plasmon into inter-band transitions related to back-folded bands from the
second to the first Brillouin zone by a finite pseudo-potential. This could
lead to a constant or a negative $q$-dependence of the plasmon width observed
for K0.3MoO3 [63] or Li [61], respectively. Thus the constant width at low $q$
in Sr2RuO4 is probably also caused by a decay into inter-band transitions and
not by quantum critical fluctuations.

Moreover, there is also a more quantitative explanation of the differences of
the imprint of correlation effects in ARPES and EELS studies, based on
dynamical mean-field calculation theory [64, 29]. These calculations predict
for energies below about 0.1 eV a Fermi liquid behavior with a mass
enhancement of about four and above that energy a strongly reduced mass
enhancement. This leads for the spectral function at low energies ( 0.2 <
$\omega$ > -0.5 eV) to a strongly renormalized dispersion and at higher
energies to an unrenormalized dispersion of resilient quasiparticles which
determine the “bad“ metal properties. The observation of a plasmon dispersion,
related to high-energy excitations, which are not influenced by correlation
effects, is a clear indication of such resilient quasiparticles.

A comparison of the calculated susceptibility (not shown) and the plasmon
dispersion shows that independent of the momentum direction, the plasmon
merges into the single particle excitations near $q=$ 0.4 Å-1. At this
momentum also an additional broadening of the experimental plasmon width is
observed (see Fig. 3). This behavior is very similar to that observed in
nearly-free electron metals [48].

There is a certain flattening of the experimental dispersion at higher
momentum transfer. As it also appears in the RPA calculations, we conclude
that it is not caused by correlation induced local field corrections. Rather
we infer that it is caused by the finite band width of the conduction bands.
To obtain information on the dynamic local field correction experiments at
higher momentum transfer are required.

In summary, we have observed in Sr2RuO4 a propagating plasmon, which is at low
momentum and at high energies determined by resilient quasiparticles which are
not renormalized. Extending future EELS measurement to higher momentum and
into the low-energy range using an improved energy resolution will provide
valuable information on the renormalized part of the spectral function and on
the momentum dependence of correlation effects.

##  I  ACKNOWLEDGMENTS

We thank Peter Abbamonte, Bernd Büchner, Luis Craco, Stefan-Ludwig Drechsler,
Axel Lubk, and Andrew P. Mackenzie for helpful discussions, and Marco Naumann
for technical assistance. This work is supported by a KAKENHI Grants-in-Aids
for Scientific Research (Grant Nos. 18K04715, 21H01033, and 22K19093), and
Core-to-Core Program (No. JPJSCCA20170002) from the Japan Society for the
Promotion of Science (JSPS) and by a JST-Mirai Program (Grant No. JPMJMI18A3).

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