# Are we describing the same sound? An analysis of word embedding spaces of
expressive piano performance

Silvan David Peter<EMAIL_ADDRESS>0009-0000-8328-291X Johannes Kepler
UniversityLinzAustria ,  Shreyan Chowdhury<EMAIL_ADDRESS>0000-0003-1171-4036 Johannes Kepler UniversityLinzAustria ,  Carlos Eduardo
Cancino-Chacón<EMAIL_ADDRESS>0000-0001-5770-7005
Johannes Kepler UniversityLinzAustria and  Gerhard Widmer
<EMAIL_ADDRESS>0000-0003-3531-1282 Johannes Kepler
UniversityLinzAustria

(2023)

###### Abstract.

Semantic embeddings play a crucial role in natural language-based information
retrieval. Embedding models represent words and contexts as vectors whose
spatial configuration is derived from the distribution of words in large text
corpora. While such representations are generally very powerful, they might
fail to account for fine-grained domain-specific nuances. In this article, we
investigate this uncertainty for the domain of characterizations of expressive
piano performance. Using a music research dataset of free text performance
characterizations and a follow-up study sorting the annotations into clusters,
we derive a ground truth for a domain-specific semantic similarity structure.
We test five embedding models and their similarity structure for
correspondence with the ground truth. We further assess the effects of
contextualizing prompts, hubness reduction, cross-modal similarity, and
k-means clustering. The quality of embedding models shows great variability
with respect to this task; more general models perform better than domain-
adapted ones and the best model configurations reach human-level agreement.

Semantic Similarity, Embeddings, Evaluation, Music Performance

††journalyear: 2023††copyright: rightsretained††conference: Forum for
Information Retrieval Evaluation; December 15–18, 2023; Panjim,
India††booktitle: Forum for Information Retrieval Evaluation (FIRE 2023),
December 15–18, 2023, Panjim, India††doi: 10.1145/3632754.3632759††isbn:
979-8-4007-1632-4/23/12††ccs: Information systems Test collections††ccs:
Information systems Similarity measures††ccs: Information systems Top-k
retrieval in databases††ccs: Applied computing Performing arts

##  1\. Introduction

Semantic embeddings are a foundational concept in Natural Language Processing
(NLP). NLP embedding models map words and their contexts to high-dimensional
vector spaces while encoding as much of the semantic information as possible.
Computers process such numerical data more readily than text, and vector
spaces enable simple yet powerful similarity representations. Semantic
embeddings enable a wide a variety of downstream tasks, from retrieval to
classification to context-enhanced few shot learning. State-of-the-art (SOTA)
embedding models are trained on vast datasets of natural language covering
many domains and disciplines, underpinned by a general distributional
hypothesis — that words with similar meanings occur in similar contexts.

This hypothesis and its corresponding inductive bias lead to one of the major
open challenges related to semantic embeddings: whether context dependence,
polysemy, and fine-grained domain-specific idiosyncrasies are adequately
represented by general purpose semantic embeddings. Specialized domains of
language use such as talk of specific arts might exhibit incommensurable
associations and similarites, e.g., a bright piano sound evokes different
meanings than a bright student. While specific domains and their possibly
nuanced differentiations do occur in large datasets, they only occur in their
specific context, where the distributional hypothesis runs counter to
different encoding. In other words, if certain emotionally dissimilar
adjectives (happy, sad) only occur in similar contexts, they end up close in
the embedding space despite opposing meanings.

Figure 1. Multidimensional Scaling (MDS) of the term data based the equally
weighted pile group one, pile group two, and performance similarities. The
legend on the left lists all piles of both groups, first the group number,
then the names the musicians assigned them. Each term of the 150 in our ground
truth data is shown in the scatter plot to the right and colored by the two
piles it was sorted into, one for group one (large dots), one for group two
(small dots). The musicians did not rate any similarities between piles, the
color progressions for the piles do not encode closeness.  Figure 2. Box plots
of distributions of pairwise similarities in various embedding spaces. The
main embedding models tested are labelled as EWE, CLAP, ADA, GTE, and BGE (Li
et al., 2023; OpenAI, 2022; BAAI, 2023; Elizalde et al., 2023; Agrawal et al.,
2018). The three leftmost distributions relate to cross-modal audio and text
embeddings as discussed in Section 4.2 and the distributions labeled with
”context” are addressed in Section 4.4.

In this article, we address the question whether general semantic embeddings
can recover similarity relations in a specific domain of language use. In
particular, we are interested in adjectival spaces used in the
characterization of expressive performance of Western classical solo piano
music. The underlying motivation is the possibility of expressivity- or
emotion-based music retrieval using intuitive verbal queries, which would be a
valuable and sought-after service in the digital music world.

Characterizations of expressive performance get at the finest details of
performance technique, expression, timbre, emotions, metaphors, and
associations. Note that it’s crucially not the piece that is being described
by such characterizations, but its expressive interpretation and rendition.

Characterizations of expressive performance are both highly specific as well
as very important to domain experts such as performers, teachers, and
committed listeners. In fact, a crucial skill for aspiring performers consists
in developing a sensibility as well as a language for performance nuances.
Likewise, discriminating classical music lovers are highly sensitive to
interpretation differences and can be very articulate in describing aspects of
a performance that they don’t like.

We use a dataset of terms used for the characterization of expressive
performance in a large scale listening study. The dataset is annotated with
similarity clusters of 150 terms created by two groups of domain experts in
Western classical music performance. This data gives us an ecologically valid
adjectival space of domain specific terms along with expert-annotated
similarity annotations — an ideal experimental reference for general embedding
spaces.

In this article, we take this reference similarity space of 150 terms related
to the domain of expressive performance characterization, and compare it with
embeddings for these terms derived from five embedding models by means of
precision at k metrics. Furthermore, we present experiments investigating
several factors affecting the embedding spaces: adding context to terms,
reducing hubness in the embedding spaces, comparing audio to text embeddings,
and a comparison of different clusterings in the embedding spaces.

##  2\. Related Work

The literature on semantic embeddings is vast and multifaceted, with many
tasks and benchmarks making use of suitable language representations. For a
recent overview and online benchmarking results, we refer to the Massive Text
Embedding Benchmark (MTEB) (Muennighoff et al., 2022, 2023).

In our work, we investigate five models which we introduce in the following.
These models cannot represent the whole of the state of the art, however, we
do think they represent interesting models for our purpose. We use three
models among the top performers in the MTEB. First, the general text
embeddings (GTE) model ”gte-large” developed by the Alibaba DAMO Academy (Li
et al., 2023). This model is trained contrastively in both an unsupervised and
a supervised fine-tuning fashion and as of September 2023 leads the MTEB
leaderboard for semantic text similarity (STS) tasks. Second, we use the BAAI
General Embeddings (BGE) model ”bge-large-en” (BAAI, 2023). This model
currently tops the overview MTEB leaderboard with minimal background
information on the type of model and training available. Thirdly, we use
OpenAI’s general purpose embedding model ”text-embedding-ada-002” offered at
their API since December 2022 (OpenAI, 2022; Brown et al., 2020). Not many
architectural details about this model are known, however, it performs in the
top 20 models both in the overview as well as for STS tasks as of September
2023. Furthermore, it’s likely one of the most widely used models in
commercial applications.

We extend the model list with two specialized models: a pure word embedding
model for emotion-enriched word embeddings (EWE) (Agrawal et al., 2018),
trained to mitigate the inductive bias that emotion term embeddings are liable
to be influenced by, and Microsoft’s cross-modal text-audio embedding model
(CLAP) (Elizalde et al., 2023), trained to embed both text and audio excerpts
in the same space for cross-modal retrieval. We assume these models to be
better suited to the language in the domains of audio and emotion description,
respectively, both of which overlap with expressive performance
characterization.

The last year has seen several publications investigating retrieval of
perceptual structure information from large language models (LLM), and in
particular using the model chatGPT. Most closely related to our proposal are
ratings of timbral similarity (Siedenburg and Saitis, 2023), music similarity
(Flexer, 2023) , and general sensory judgment dimensions (Marjieh et al.,
2023), all of which find evidence for the recovery of human annotations by
chatGPT, albeit not at human level.

Our reference data is based on free text descriptions of expressive piano
performance. People involved with music take pleasure in talking about music.
This is no different for expressive performance of Western classical solo
piano. In doing so they develop a rich vocabulary that relates to different
aspects of performance such as evaluative/axiological terminology, emotions,
metaphors, playing technique, or timbre descriptors. In music research, these
aspects are often addressed separately, and with a generally reductionist
approach. That is, researchers are interested in the identification of
underlying factos, perceptual categories, and their relations to acoustic
features (Palmer, 1997; Lerch et al., 2020). This is somewhat opposed to our
approach, where no reduction of the semantic space is pursued. Nevertheless,
we want to briefly outline two relevant areas of inquiry: emotions and timbre.

Figure 3. Left plot: aP@k for k $\in\\{1,...,49\\}$ for several embeddings
models against the ground truth similarities. Right plot: aP@k of 45
performance embeddings represented as CLAP audio embeddings and as mean CLAP
text embeddings of terms (with and without context prompts).

There exists a substantial literature investigating the emotional language
related to music (Gabrielsson, 2003; Eerola and Vuoskoski, 2012). Among
prominent models are categorical models such as the Geneva Emotional Music
Scale (GEMS) (Zentner et al., 2008) and the dimensional valence and arousal
model (Russell, 1980). Crucial questions relate to the question whether
musical emotions are perceived or induced (Song et al., 2016) or the
paradoxical enjoyment of negative emotions such a sadness (Eerola et al.,
2018). Metaphors are another common linguistic device in the characterization
of music. We refer to Schaerlaken et al. (Schaerlaeken et al., 2019) who
identified five main factor in their analysis of metaphorical attributed
(GEMMES) and connected their perception to the GEMS (Schaerlaeken et al.,
2022).

Timbre is crucial topic in performance research and music psychology
(Krumhansl, 1989; Saitis et al., 2020; Esling et al., 2018). Timbre is
increasingly conceptualized as controlled expressive performance attribute,
i.e., as something that performers can influence with playing techniques and
gestures. Most relevant to our work are several piano timbre description
experiments by Bernays et al. (Bernays and Traube, 2013, 2014, 2010, 2011)
which brought five to eleven categorical terms to the fore. For our
experiments, we rely on as many terms with associated similarity annotations
as possible without reduction to principal factors or dimensions, which we
find in the con espressione dataset and its pile sorting extension, detailed
in Section 3.1.

Figure 4. Left plot: relative change in aP@k brought about by the inclusion of
contextualizing prompts. Right plot: relative change in aP@k due to hubness
reduction at neighborhoods of size eight.

##  3\. Methods

For our experiments we require an embedding space of words (for simplicity:
adjectives) with associated ground truth similarity ratings as a type of
ground truth reference data. We also need several SOTA embedding models and
the metrics by which we can meaningfully compare the resulting similarity
structures. In this section, we introduce these components.

###  3.1. Con Espressione Data

The Con Espressione Dataset (CED) collected descriptions of piano performances
through an online questionnaire (Cancino-Chacón et al., 2020, 2023).
Participants listened to 45 different performances by famous pianists of nine
excerpts of Western classical piano pieces. The participants were shown the
prompt: ’Please think of words (if possible, adjectives) that best describe
the character of each performance to you.’ and a text field allowed for free
text answers (as many words as they liked, in a language of their choice).
They were further instructed to concentrate on the performance aspects and not
on the piece of music itself. The CED characterizations contains 3,166 terms,
of which 1,415 are unique. Consequently, the CED consists of very loosely
structured text data for which relational semantic ground truth, i.e. which
answers refer to the same aspect or idea, is largely missing.

To mitigate this, a follow-up experiment was designed (Cancino-Chacón et al.,
2021). In two separate sessions, groups of professional
musicians/musicologists sorted the 150 most often occurring terms in the CED
into piles. These piles should cluster terms that describe a common expressive
character. The type of similarity and number of piles were left open. The two
groups of four professional musicians each sorted the terms into 25 and 19
piles, respectively, and gave a name to each pile, in the form of an adjective
that best summarizes the common meaning of the words associated with the pile.
111An interactive interface for the exploration of piles, terms, and
performances is available at: https://cpjku.github.io/expressivity/

###  3.2. Embeddings and Similarities

We use the same 150 terms as our adjectival space. To derive ground truth
similarities between the terms we use both term co-occurrence in piles (from
both groups) and co-occurrence for performance descriptions. For each of the
pile groups, we create a similarity matrix where pairwise similarities of
terms within a pile are set to one, outside the pile to zero. Term
similarities are also computed based on the CED directly, where two different
terms that occur in the characterization of the same performance are assumed
more similar than if they are used for different performance. We again compute
a similarity matrix where co-occurrence of two terms in the same performance
description sets the similarity to one, else to zero. We weigh each source of
similarity annotations the same, that is, the similarities from pile group
one, pile group two, and performance description are summed up and finally
normalized. Figure 1 illustrates the resulting similarity structure. The
legend on the left lists all piles of both groups (with the names the
musicians gave them) the scatter plot on the right shows a low-dimensional
approximation of the term similarities.

We compare our ground truth similarities with similarities for the same
adjectival space, i.e., the same 150 terms, embedded using five different
embedding models: ADA (embedding dimensionality 1536), CLAP (1024), EWE (300),
GTE (1024), and BGE (1024), detailed in Section 2.

###  3.3. Metrics

We assess the similarities using several metrics. All similarities are cosine
similarities in the embedding spaces:

(1) $CosineSimilarity(x,y)=\frac{x\cdot y}{|x||y|}$

for $x$ and $y$ term embeddings in the same space.

Our main evaluation metric is the average precision at k (aP@k) for k nearest
neighbors. For each term x in our adjectival space S, we compute two
neighborhoods of size k, one according to our ground truth embedding space U
(knn(x,U)), and one according to a test embedding space V (knn(x,V)). We then
compute the precision of retrieval of the U-neighborhood by the
V-neighborhood:

(2) $aP@k(U,V)=\frac{1}{|S|}\sum_{x\in S}\frac{|knn(x,U)\cap knn(x,V)|}{k}$

where knn(x,U) denotes the set of k nearest neighbors of x according to the
embedding space U. We choose aP@k over the more common Spearman correlation of
embedding similarities for STS tasks (Muennighoff et al., 2022), for its
capacity to naturally differentiate each level of neighborhood size k and its
rank-agnostic behavior within the neighborhoods, which corresponds to the
ground truth pile sortings.

In Section 4.5, we compare the agreements between the pile groups (ground
truth sets of clusters) with k-means clusterings in the embedding spaces. For
this purpose, we compute the overlap coefficient between sets of clusters
according to different similarities. In particular, we compute the average
maximal overlap between two different sets of clusters $\mathbb{C}_{1}$ and
$\mathbb{C}_{2}$ as:

(3)
$AvMaxOverlap(C_{1},C_{2})=\frac{1}{|\mathbb{C}_{1}|}\sum_{C_{1}\in\mathbb{C}_{1}}\max_{C_{2}\in\mathbb{C}_{2}}\frac{|C_{1}\cap
C_{2}|}{min(|C_{1}|,|C_{2}|)}$

Note that this metric is not symmetrical in $\mathbb{C}_{1}$ and
$\mathbb{C}_{2}$.

##  4\. Experiments and Results

This section presents experiments addressing structural correspondences
between similarity spaces. After a main experiment in Section 4.1, we
investigate more fine-grained issues regarding the correspondence of text and
audio embeddings in Section 4.2 the effects of hubness in the embedding spaces
in Section 4.3, the effects of contextualized terms in Section 4.4, and the
overlap in clusterings in Section 4.5.

###  4.1. Similarity Structure Correspondence

To what extent do the models’ embedding spaces correspond to the similarity
relations derived from expert annotations? We answer this question using aP@k
values. However, we first make a few general observations about the
distributions of similarities without reference to structural considerations.

Figure 2 shows box plots of the distributions of pairwise similarities in
several spaces under scrutiny. Note that for all language models (labelled as
EWE, CLAP, ADA, GTE, and BGE) the embeddings are very similar, with values
rarely falling below 0.7 and mean values above 0.8. This illustrates the
inductive bias of the embeddings as these terms are likely to occur in similar
and sometimes very particular domains.

Figure 3 (left) shows the performance of each embedding in terms of its
approximation of the neighborhood structure of a reference embedding. For
neighborhood sizes k between one and 49, the aP@k of each embedding space is
computed against our ground truth data. The models are are clearly ranked for
all k, from top to bottom: ADA, GTE, BGE, CLAP, and EWE. A yellow diagonal
denotes a random baseline. With the exception of EWE are all models
significantly better than random. The upper bound of the 95% confidence
interval (not shown) of the random baseline is approximately 0.02 above the
baseline itself at k=1 and gets closer to the baseline as k increases.

A blue line indicates the only values that are not compared against the ground
truth. Instead it compares similarities only based on pile group 1 (co-
occuring terms in a pile have similarity one, else zero) with similarities
only based on pile group 2. This is added as an indication of where human
agreement about similarity might fall, albeit with some caveats: The values
are only really meaningful for k in the approximate size of piles (roughly
5-10 terms/pile). For smaller k, the piles do not encode greater or lesser
similarity of terms within a pile, all values are one and an arbitrary
ordering was created by addition of some minimal noise. For larger k, the
piles do not encode greater or lesser similarity for terms beyond the pile,
all values are zero and again minimal noise was added for an arbitrary
ordering. It’s also not possible to compare these similarities against the
ground truth since they were part of the creation of the ground truth which
distorts the result unfairly. The blue line segment for k = 5 - 10 does give
an indication that none of the models reach the agreement of two groups of
expert annotators.

###  4.2. Audio vs. Text Embeddings

The CLAP model is trained to minimize distances between corresponding text and
audio embeddings which makes it possible to compare against audio embeddings
of the 45 performances that are characterized by the 150 terms in the con
espressione data. The three leftmost distributions in Figure 2 are related to
this approach: ”audio - audio” denotes the pairwise similarities between audio
embeddings, ”s terms - audio” shows cross-modal similarities between 150
individual terms and 45 performances, and ”p terms - audio” shows cross-modal
similarities between 45 performances averaged from their corresponding term
embeddings (the terms used to describe the performance in the CED) and 45
performance recording embeddings. Figure 3 (right) shows the latter in an aP@k
plot; the 45 performances are embedded in both in the text spaces of two
different text models (see Section 4.4 for a discussion on the ”context”
model) and in the audio space. The values are notably lower, for several k the
values are not significantly better than the random baseline, indicating that
either the audio or the text space do not have the granularity to represent
these minute differences. We conjecture that the audio model is the more
likely source of misalignment. After all, all performance recordings can
reasonably be classified as ”classical solo piano” which is closer to the
level of precision to be expected from an unspecific audio representation
model trained on a variety of (non-)musical audio (Elizalde et al., 2023).

###  4.3. The Effect of Hubness

Model | nbhd | Skewness |  | Robinhood |
---|---|---|---|---|---
ADA | 4 | 0.99 | 0.88 | 0.25 | 0.21
CLAP | 4 | 1.09 | 0.52 | 0.22 | 0.19
GTE | 4 | 1.11 | 0.91 | 0.24 | 0.20
BGE | 4 | 1.76 | 1.14 | 0.32 | 0.29
RB | 4 | 0.54 | 0.40 | 0.17 | 0.13
ADA | 8 | 1.12 | 0.58 | 0.25 | 0.21
CLAP | 8 | 1.19 | 0.54 | 0.25 | 0.23
GTE | 8 | 1.13 | 0.44 | 0.24 | 0.21
BGE | 8 | 1.97 | 1.63 | 0.39 | 0.36
RB | 8 | 0.37 | 0.17 | 0.13 | 0.11
ADA | 16 | 0.81 | 0.32 | 0.26 | 0.24
CLAP | 16 | 1.21 | 0.70 | 0.25 | 0.23
GTE | 16 | 0.67 | 0.24 | 0.25 | 0.24
BGE | 16 | 1.83 | 1.61 | 0.43 | 0.42
RB | 16 | 0.25 | 0.04 | 0.10 | 0.08
Table 1. Results of hubness measurement and reduction. Values for each model
(RB random baseline) are presented for three different neighborhood sizes
(nbhd). All skewness and robinhood values are doubled, left original, right
after hubness reduction.

The existence of hubs has repeatedly been found a source of distorted
similarity structures, especially in high-dimensional spaces (Feldbauer and
Flexer, 2019). Hubs are points that appear too often in k nearest
neighborhoods of other points and as such are liable to influence the aP@k. In
this experiment we address the influence of hubs for four models.

We first measure hubness as both skewness of the k-occurrence histogram
(higher skewness indicates more hubness) and as robinhood index (which
indicates the percentage of slots in nearest neighbor lists would need to be
redistributed for equal distribution). We carry this computation out for three
neighborhood sizes (4,8, and 16) and compute hubness reduction using an
approximate mutual proximity method (Schnitzer et al., 2012). For algorithmic
details regarding hubness measurement and reduction we refer the reader to
Feldbauer et al. (Feldbauer et al., 2018).

Table 1 shows the results of hubness reduction. For each metric (skewness,
robinhood) we note two values for each setting: before (left) and after
(right) hubness reduction. Several models show significant hubness (skewness ¿
1.0, robinhood ¿ 0.25), with BGE being a negative outlier, and they almost
universally benefit from reduction.

What does this mean for similarity structure recovery against the ground
truth? Figure 4 (right) shows the relative change in aP@k for 4 models after
hubness reduction at neighborhood size eight. This neighborhood was chosen for
being crucial against ground truth based on piles, which on average have
approximately this size. Notably, hubness reduction leads to a decrease for k
less than the set neighborhood. For values around eight, hubness reduction
universally leads to an increase in performance of about 20%, enough to boost
the highest performing models an the league of the expert agreement baseline
in this crucial area (see Figure 3 on the left, see section 4.1).

Figure 5. Visualization of the convex hull of terms of each pile as embedded
in the ground truth data. MDS dimension reduction for illustrative purposes
only, 8+ dimensions are required to represent the space with minimal loss of
information (¡10% reduction in aP@k against original data). The pile centers
are shown as average term embedding positions and annotated with the pile
names given by the musicians.

###  4.4. The Effect of Context

SOTA text embedding models are often capable of encoding term contexts into
their representations, i.e., terms can be augmented with suitable prompts to
specify the context of use. Such contextualizing information ideally enables
the model to learn domain-specific similarities and relations that might not
be apparent in other situations, such as metaphorical usage of terms related
to movement, weight, and flow, or words borrowed from other sensory modalities
like sweet, rough, and warm, which are common in our ground truth data. In
this experiment, we augment the 150 terms with a common context prompt. In the
original Con Espressione game, listeners were asked to ‘please think of words
(if possible, adjectives) that best describe the character of each performance
to you.’. We translate this to the prompt: ’I listen to a solo piano
performance of a classical piece of music and I’d describe the character of
the performance as TERM’ and recompute all embeddings for four test models.

Figure 2 shows the distributions of pairwise similarities for context prompts.
For all tested models, the prompts led to higher pairwise similarities. To
test the similarity structure of these context embeddings, we again compute
the relative change in aP@k for four models after adding contexts. Figure 4
(left) shows this relative change, i.e., the aP@k of the embeddings with
context divided by the aP@k of those without. Not all models react positively
context information, GTE and CLAP stay largely the same or get worse. The
other two models see performance increases of 20 % and more, which in the case
of ADA pushes the model higher than the expert reference (see Figure 3, see
section 4.1).

###  4.5. Clustering and Piles

The aP@k compares neighborhoods of the same size, however, the experts’ groups
of piles are not homogeneous in size and number. On the other hand, the piles
do provide a complete clustering of the adjectival space which can be compared
against automatic clusterings (see Figure 5 for an illustration of the
clustering provided by pile group two in a the ground truth space). In this
last experiment, we compute k-means clusterings in several embedding spaces
and compare them against the two groups of piles by means of average maximal
overlap coefficients.

Table 2 shows the results. All k-means clusterings are computed with k=22, the
average of the pile group sizes ($|P1|=25,|P2|=19$). Average maximal overlap
coefficients are not symmetrical, hence we report two values per setup. The
top row reports two values for reference: the average maximal overlap of pile
group one with pile group two (”P1 in P2”), and vice versa (”P2 in P1”).
K-means clusterings are computed for six models, four embedding models, the
ground truth model, and a random baseline consisting of a 100-dimensional
Gaussian. The two columns below ”Overlap v P1” average maximal overlap
coefficients for group one in k-means clusters (left) and k-means clusters in
group one (right). The next two columns marked ”Overlap v P2” report the same
values for group two.

Note that smaller sets of clusters generally reach a higher average maximal
overlap due to larger clusters (P2 in P1 ¿ P1 in P2, $|P1|=25$, $|P2|=19$).
None of the embedding models reach the agreement between the two groups of
piles, however, they clearly outperform the random baseline. The ground truth
reaches higher overlap values with both groups than in between the groups,
which is to be expected, as it was derived from the performance annotations
and the two groups of piles.

Ref | P1 in P2: | 0.62 | P2 in P1: | 0.65
---|---|---|---|---
Model | Overlap v P1 | Overlap v P2
ADA | 0.55 | 0.58 | 0.48 | 0.43
CLAP | 0.52 | 0.51 | 0.47 | 0.46
GTE | 0.59 | 0.59 | 0.53 | 0.52
BGE | 0.51 | 0.53 | 0.53 | 0.48
GT | 0.66 | 0.70 | 0.66 | 0.70
RB | 0.39 | 0.41 | 0.40 | 0.39
Table 2. Overlap and distance ratios.

##  5\. Discussion and Conclusion

Specialized datasets created by domain experts like the CED and its sorted
pile groups usually serve a primary reductionist research purpose: the
identification and description of dimensions, categories, and relations in
perceptual-linguistic spaces. However, they also allow for a quantitative
glimpse into the similarity structures of these spaces: similarity structures
which are hypothesized to be recovered by SOTA term embedding models. In a
series of experiments, we address this hypothesis for a presumably highly
specialized type of adjectival space, that of characterizations of expressive
performance of Western classical solo piano works.

Our results show that domain-specific semantic similarity structures are
indeed represented in the embedding spaces — to a degree. The tested models
span the full range from near the random baseline to near human agreement.
General-purpose models perform better than domain-adapted ones, running
counter to our initial assumption.

In our experiments, cross-modal audio embeddings of the performance recordings
fail to exhibit the same similarity structure as text embeddings. Hubness
reduction helps the similarity correspondence universally, albeit only for a
specific and small segment of neighborhoods. The inclusion of contextualizing
prompts affects the models differently, with the best models receiving a clear
boost in quality. Overlap statistics show that all embedding space clusters
show less correspondence with the expert sortings than those show among
themselves.

The groups of piles are used as a reference for correspondence in similarity
structures throughout as they represent a rough estimate of inter-rater
agreement to be expected. They are however the largest source of uncertainty.
We do not know how likely similar groups of piles are or have other means of
assessing of inter-rater agreement. Neither do we have direct similarity
ratings or know whether performance expressivity-specific similarities are
indeed notably different from general similarity. Research into dimensional
and categorical structure on music perception as discussed in section 2 may
ground the pile group similarities, however, for the number of terms used or
even the free-text CED annotations more research is required to illuminate the
robustness of this data.

To conclude, general state-of-the-art text embedding models can show
correspondence with expert annotated perceptual-linguistic similarities that
reach the experts’ inter-rater agreement while other — even plausibly better
suited domain models — fail at this task. Future research includes the
investigation of the robustness of the annotation data as well as the
extension of this approach to other domains where fine-grained and possibly
idiosyncratic adjectival spaces are used.

##  6\. Reproducibility

Our data and code is available at:

https://github.com/CPJKU/performance_embeddings_fire23

###### Acknowledgements.

This work is supported by the European Research Council (ERC) under the EU’s
Horizon 2020 research & innovation programme, grant agreement No. 101019375
(“Whither Music?”), and the Federal State of Upper Austria (LIT AI Lab).

## References

  * (1)
  * Agrawal et al. (2018) Ameeta Agrawal, Aijun An, and Manos Papagelis. 2018.  Learning emotion-enriched word representations. In _Proceedings of the 27th international conference on computational linguistics_. 950–961.
  * BAAI (2023) BAAI. 2023.  BGE repository.  https://github.com/FlagOpen/FlagEmbedding
  * Bernays and Traube (2010) Michel Bernays and Caroline Traube. 2010.  Expression of piano timbre: gestural control, perception and verbalization. In _Proceedings of CIM09: The 5th Conference on Interdisciplinary Musicology_.
  * Bernays and Traube (2011) Michel Bernays and Caroline Traube. 2011.  Verbal expression of piano timbre: Multidimensional semantic space of adjectival descriptors. In _Proceedings of the international symposium on performance science (ISPS2011)_. European Association of Conservatoires (AEC) Utrecht, Netherlands, 299–304.
  * Bernays and Traube (2013) Michel Bernays and Caroline Traube. 2013.  Expressive production of piano timbre: touch and playing techniques for timbre control in piano performance. In _Proceedings of the 10th Sound and Music Computing Conference (SMC2013)_. KTH Royal Institute of Technology Stockholm, Sweden, 341–346.
  * Bernays and Traube (2014) Michel Bernays and Caroline Traube. 2014.  Investigating pianists’ individuality in the performance of five timbral nuances through patterns of articulation, touch, dynamics, and pedaling.  _Frontiers in Psychology_ 5 (2014), 157.
  * Brown et al. (2020) Tom Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared D Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020\.  Language models are few-shot learners.  _Advances in neural information processing systems_ 33 (2020), 1877–1901.
  * Cancino-Chacón et al. (2021) Carlos Cancino-Chacón, Silvan Peter, Shreyan Chowdhury, Anna Aljanaki, and Gerhard Widmer. 2021\.  Sorting Musical Expression: Characterization of Descriptions of Expressive Piano Performances. In _Extended Abstract in 16th International Conference on Music Perception and Cognition ICMPC 2021) and 11th Triennial Conference of ESCOM (ICMPC-ESCOM 2021)._
  * Cancino-Chacón et al. (2023) Carlos Cancino-Chacón, Silvan Peter, Shreyan Chowdhury, Anna Aljanaki, and Gerhard Widmer. 2023\.  Con Espressione Dataset.  https://zenodo.org/record/3968828
  * Cancino-Chacón et al. (2020) Carlos Cancino-Chacón, Silvan David Peter, Shreyan Chowdhury, Anna Aljanaki, and Gerhard Widmer. 2020\.  On the Characterization of Expressive Performance in Classical Music: First Results of the Con Espressione Game. In _Proceedings of the 21st International Society for Music Information Retrieval Conference, ISMIR 2020_. Online.
  * Eerola and Vuoskoski (2012) Tuomas Eerola and Jonna K Vuoskoski. 2012.  A review of music and emotion studies: Approaches, emotion models, and stimuli.  _Music Perception: An Interdisciplinary Journal_ 30, 3 (2012), 307–340.
  * Eerola et al. (2018) Tuomas Eerola, Jonna K Vuoskoski, Henna-Riikka Peltola, Vesa Putkinen, and Katharina Schäfer. 2018\.  An integrative review of the enjoyment of sadness associated with music.  _Physics of Life Reviews_ 25 (2018), 100–121.
  * Elizalde et al. (2023) Benjamin Elizalde, Soham Deshmukh, Mahmoud Al Ismail, and Huaming Wang. 2023.  Clap learning audio concepts from natural language supervision. In _ICASSP 2023-2023 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)_. IEEE, 1–5.
  * Esling et al. (2018) Philippe Esling, Axel Chemla-Romeu-Santos, and Adrien Bitton. 2018\.  Generative timbre spaces: regularizing variational auto-encoders with perceptual metrics.  arXiv:1805.08501 [cs.SD]
  * Feldbauer and Flexer (2019) Roman Feldbauer and Arthur Flexer. 2019.  A comprehensive empirical comparison of hubness reduction in high-dimensional spaces.  _Knowledge and Information Systems_ 59, 1 (2019), 137–166.
  * Feldbauer et al. (2018) Roman Feldbauer, Maximilian Leodolter, Claudia Plant, and Arthur Flexer. 2018.  Fast approximate hubness reduction for large high-dimensional data. In _2018 IEEE International Conference on Big Knowledge (ICBK)_. IEEE, 358–367.
  * Flexer (2023) Arthur Flexer. 2023\.  Can ChatGPT be useful for distant reading of music similarity?. In _Proceedings of the 2nd Workshop on Human-Centric Music Information Retrieval (HCMIR), 2023_.
  * Gabrielsson (2003) Alf Gabrielsson. 2003\.  Music performance research at the millennium.  _Psychology of music_ 31, 3 (2003), 221–272.
  * Krumhansl (1989) Carol L Krumhansl. 1989\.  Why is musical timbre so hard to understand.  _Structure and perception of electroacoustic sound and music_ 9 (1989), 43–53.
  * Lerch et al. (2020) Alexander Lerch, Claire Arthur, Ashis Pati, and Siddharth Gururani. 2020.  An Interdisciplinary Review of Music Performance Analysis.  _Transactions of the International Society for Music Information Retrieval_ (Nov 2020).  https://doi.org/10.5334/tismir.53
  * Li et al. (2023) Zehan Li, Xin Zhang, Yanzhao Zhang, Dingkun Long, Pengjun Xie, and Meishan Zhang. 2023\.  Towards General Text Embeddings with Multi-stage Contrastive Learning.  arXiv:2308.03281 [cs.CL]
  * Marjieh et al. (2023) Raja Marjieh, Ilia Sucholutsky, Pol van Rijn, Nori Jacoby, and Thomas L. Griffiths. 2023\.  Large language models predict human sensory judgments across six modalities.  arXiv:2302.01308 [cs.CL]
  * Muennighoff et al. (2022) Niklas Muennighoff, Nouamane Tazi, Loïc Magne, and Nils Reimers. 2022.  MTEB: Massive Text Embedding Benchmark.  (2022).  arXiv:2210.07316 [cs.CL]
  * Muennighoff et al. (2023) Niklas Muennighoff, Nouamane Tazi, Loïc Magne, and Nils Reimers. 2023.  MTEB Leaderboard.  https://huggingface.co/spaces/mteb/leaderboard
  * OpenAI (2022) OpenAI. 2022.  New and improved embedding model.  https://openai.com/blog/new-and-improved-embedding-model
  * Palmer (1997) Caroline Palmer. 1997\.  Music performance.  _Annual review of psychology_ 48, 1 (1997), 115–138.
  * Russell (1980) James A Russell. 1980\.  A Circumplex Model of Affect.  _Journal of Personality and Social Psychology_ 39, 6 (1980), 1161–1178.
  * Saitis et al. (2020) Charalampos Saitis, Stefan Weinzierl, Katharina von Kriegstein, Sølvi Ystad, and Christine Cuskley. 2020\.  Timbre semantics through the lens of crossmodal correspondences: A new way of asking old questions.  _Acoustical Science and Technology_ 41, 1 (2020), 365–368.
  * Schaerlaeken et al. (2022) Simon Schaerlaeken, Donald Glowinski, and Didier Grandjean. 2022\.  Linking musical metaphors and emotions evoked by the sound of classical music.  _Psychology of Music_ 50, 1 (2022), 245–264.
  * Schaerlaeken et al. (2019) Simon Schaerlaeken, Donald Glowinski, Marc-André Rappaz, and Didier Grandjean. 2019.  “Hearing music as…”: Metaphors evoked by the sound of classical music.  _Psychomusicology: Music, Mind, and Brain_ 29, 2-3 (2019), 100\.
  * Schnitzer et al. (2012) Dominik Schnitzer, Arthur Flexer, Markus Schedl, and Gerhard Widmer. 2012.  Local and global scaling reduce hubs in space.  _Journal of Machine Learning Research_ 13, 10 (2012).
  * Siedenburg and Saitis (2023) Kai Siedenburg and Charalampos Saitis. 2023.  The language of sounds unheard: Exploring musical timbre semantics of large language models.  arXiv:2304.07830 [cs.CL]
  * Song et al. (2016) Yading Song, Simon Dixon, Marcus T Pearce, and Andrea R Halpern. 2016\.  Perceived and induced emotion responses to popular music: Categorical and dimensional models.  _Music Perception: An Interdisciplinary Journal_ 33, 4 (2016), 472–492.
  * Zentner et al. (2008) Marcel Zentner, Didier Grandjean, and Klaus R Scherer. 2008\.  Emotions evoked by the sound of music: characterization, classification, and measurement.  _Emotion_ 8, 4 (2008), 494.


# Spin superfluidity in noncollinear antiferromagnets

Bo Li  Department of Physics and Astronomy and Nebraska Center for Materials
and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588, USA  Alexey
A. Kovalev  Department of Physics and Astronomy and Nebraska Center for
Materials and Nanoscience, University of Nebraska, Lincoln, Nebraska 68588,
USA

###### Abstract

We explore the spin superfluid transport in exchange interaction dominated
three-sublattice antiferromagnets. The system in the long-wavelength regime is
described by an $SO(3)$ invariant field theory. Additional corrections from
Dzyaloshinskii-Moriya interactions or anisotropies can break the symmetry;
however, the system still approximately holds a $U(1)$-rotation symmetry.
Thus, the power-law spatial decay signature of spin superfluidity is
identified in a nonlocal-measurement setup where the spin injection is
described by the generalized spin-mixing conductance. We suggest iron
jarosites as promising material candidates for realizing our proposal.

Spintronics has been extremely successful in combining advanced theoretical
concepts with practical applications and experiment [1]. Materials with
antiferromagnetic ordering are a focus of active research in spintronics due
to many desirable properties such as spin dynamics in terahertz range [2], the
absence of stray fields, and insensitivity to the presence of magnetic fields
[3]. In addition, antiferromagnetic insulators are characterized by long spin
diffusion length associated with transport of magnons, making them
particularly suitable for spintronics applications such as low dissipation
electronic devices [4].

Magnetic insulators can also transport spins in a regime in which the
transport can be describe as spin superfluidity [5; 6]. In easy-plane magnets,
the spin is then transported over large distances by the coherent order
parameter precession [7; 8; 9]. The power-law decay of spin current can enable
spin transport over longer distances compared to the diffusive regime [10; 7;
11; 8; 12; 9; 13; 14; 15]. Nevertheless, in ferromagnets the dipole
interaction can limit the range of spin superfluid transport [9]. On the other
hand, collinear antiferromagnetic insulators could provide a viable platform
for realizing the spin superfluidity [8; 16; 17] as demonstrated in
experiments on Cr2O3 [14] and antiferromagnetic $\nu=0$ quantum Hall state of
graphene [15].

Noncollinear antiferromagnets (nAFM) are yet another viable platform for
realizing spin flows [18; 19; 20]. The noncollinear conducting magnets can
exhibit a multitude of phenomena associated with topology of electronic bands
[21], e.g., Mn3X (X = Ge, Sn, Ga, Ir, Rh, or Pt) magnets exhibit the anomalous
[22] and spin [23] Hall responses. Various magnon-mediated responses relying
on magnon spin-momentum locking, topology of magnonic bands, and coupling to
phonons have been studied theoretically, promising observation of spin related
phenomena in insulating antiferromagnets [24; 25; 26; 27; 28; 29; 30; 31; 32].

In this work, we analytically study viability of spin superfluid transport in
insulating nAFM. In general, the U(1) symmetry of magnetic ordering can be
hampered by various anisotropies. The highly symmetric hexagonal environment
considered in this work can be beneficial for realizing spin superfluid
transport. Hexagonal nAFM can exhibit relevant phenomena, e.g., the appearance
of domain walls [33; 34] and Goldstone modes [35]. Furthermore, spin
superfluid transport has been studied numerically in a triangular nAFM [36].
In this work, we offer analytical results with a detailed discussion of the
generalized spin-mixing conductance and spin current injection into nAFM. We
identify the power-law decay feature of the spin superfluid transport in a
nonlocal experimental setup. Our simple results can help in designing and
interpreting experiments on spin superfluidity in nAFM.

Long-wavelength Hamiltonian— In nAFM, the exchange interaction is often
dominant, which approximately endows the system with an $SO(3)$ symmetry given
that all other interactions, e.g., anisotropy, DMI, are very weak. We start
with constructing a long-wavelength $SO(3)$ field theory to describe the nAFMs
and regard other weak terms as additional perturbations. In a two-dimensional
nAFM with three sublattices (e.g., kagome, triangular), the exchange
interactions favor fully compensated spin configurations, which in the
presence of other interactions may acquire a very small net magnetization.
Therefore, we parametrize the spins $\bm{S}_{i}$ of length $S$ in each
triangular plaquette as [37]

$\displaystyle\bm{S}_{i}=S\hat{R}(\bm{n}_{i}+\bm{L})/(1+2\bm{L}\cdot\bm{n}_{i}+L^{2})^{1/2},$
(1)

where $\bm{n}_{i}$ ($i=1,2,3$) sets a reference ordered state allowed by
exchange interactions with

$\displaystyle\bm{n}_{1}=(0,1,0),\quad\bm{n}_{2}=(-\frac{\sqrt{3}}{2},-\frac{1}{2},0),$
$\displaystyle\bm{n}_{3}=(\frac{\sqrt{3}}{2},-\frac{1}{2},0);$ (2)

$\hat{R}\in SO(3)$ is a rotation matrix which generates degenerate states by
acting on the reference state; $\bm{L}$ describes small deviation from the
compensated spin structure with the magnitude $L\ll 1$. $\hat{R}$ and $\bm{L}$
together generate all possible spin configurations on three sublattices. To
the leading order in $\bm{L}$,
$\bm{S}_{i}=S\hat{R}[\bm{n}_{i}+\bm{L}-\bm{n}_{i}(\bm{L}\cdot\bm{n}_{i})]$,
and the net angular momentum density,

$\displaystyle\bm{m}=\hbar/A_{\text{uc}}\sum_{i}\bm{S}_{i}=3s\hat{R}(\hat{T}\bm{L}),$
(3)

where $s=S\hbar/A_{\text{uc}}$,
$T_{ab}=\delta_{ab}-\frac{1}{3}\sum_{i}n_{i}^{a}n^{b}_{i}$, and
$A_{\text{uc}}$ is the area of a unit cell.

With the forgoing parametrization, the system is generally described by a
Lagrangian [37],

$\displaystyle\mathcal{L}=\bm{m}\cdot\bm{\Omega}_{t}-\frac{\bm{m}^{2}}{2\varrho}-\text{tr}[\hat{P}_{ij}\partial_{i}\hat{R}^{T}\partial_{j}\hat{R}]$
(4)

where
$(\Omega_{t})_{i}=-\frac{1}{2}R_{i\alpha}\epsilon_{\alpha\beta\gamma}(\hat{R}^{T}\partial_{t}\hat{R})_{\beta\gamma}$.
Here, the first term is derived from the spin kinetic energy; the second term
originates from the exchange interaction, e.g., for the nearest exchange $J$,
we obtain $\varrho\propto\hbar^{2}/(JA_{\text{uc}})$; the last term describes
the second-order gradient expansion of exchange coupling with tensor
$\hat{P}_{ij}$ encoding the exchange interactions and lattice geometry. From
the Euler-Lagrange equation [33], we obtain:

$\displaystyle\bm{m}=\varrho\bm{\Omega}_{t},$ (5)

from which the field $\bm{m}$ can be removed from Eq. (4), i.e.,

$\displaystyle\mathcal{L}=\frac{\varrho}{4}\text{tr}[\partial_{t}\hat{R}^{T}\partial_{t}\hat{R}]-\text{tr}[\hat{P}_{ij}\partial_{i}\hat{R}^{T}\partial_{j}\hat{R}].$
(6)

This is the so-called nonlinear $\sigma$ model [37; 38].

Figure 1: Noncollinear kagome (left) and triangular (right) antiferromagnets.
The red arrows indicate the spin directions of the reference state. The blue
arrows (left) indicate the out-of-plane and in-plane DMI vectors; the in-plane
DMI vectors are defined for anti-clockwise direction in each triangular
plaquette.

To determine the tensor $\hat{P}_{ij}$ for a hexagonal-symmetry lattice,
kagome or triangular, when exchange is the dominant interaction, we explore
the spin wave behavior by following Ref. [35]. We consider small fluctuations
upon the reference state by using
$\hat{R}=\exp[-i\bm{\theta}\cdot\hat{\bm{\mathcal{J}}}]$, where
$(\mathcal{J}_{i})_{jk}=-i\epsilon_{ijk}$ with $\epsilon_{ijk}$ being the
Levi-Civita tensor and a vector $\bm{\theta}$ describes the small deviation
with $|\bm{\theta}|\ll 1$. The spin-wave energy is obtained from the leading-
order expansion of the second term in Eq. (6),

$\displaystyle\mathcal{U}$
$\displaystyle=\text{tr}[\hat{P}_{ij}\hat{\mathcal{J}}_{k}\hat{\mathcal{J}}_{l}]\partial_{i}\theta_{k}\partial_{j}\theta_{l}$
(7)
$\displaystyle=\partial_{i}\theta_{k}\partial_{j}\theta_{k}\text{tr}[\hat{P}_{ij}]-P_{ij,kl}\partial_{i}\theta_{k}\partial_{j}\theta_{l}.$

Due to the highly symmetric hexagonal environment, $\theta_{z}$ acts as a
scalar and $\theta_{x},\theta_{y}$ act as two components of a vector under
symmetry transformations. Using the symmetry constraints, we recover the form
of $\hat{P}_{ij}$ tensor:

$\displaystyle
P_{ij,kl}=\eta\delta_{ij}\delta_{kl}+\lambda\delta_{ik}\delta_{jl}+\mu\delta_{il}\delta_{jk}\quad\text{for}\quad
k,l=1,2,$ $\displaystyle P_{ij,zz}=\kappa\delta_{ij}.$ (8)

where the arbitrary coefficients $\kappa$, $\eta$, $\lambda$, and $\mu$ scale
as the exchange strength $J$. For triangular and kagome lattices with the
nearest exchange interaction, their values are summarized in Table. 1.

From Eq. (6), the spin-wave Lagrangian reads
$\mathcal{L}=(\varrho/2)(\dot{\bm{\theta}})^{2}-\mathcal{U}$. By using Eq.
(Spin superfluidity in noncollinear antiferromagnets), we can obtain three
linearly dispersive Goldstone modes [35] $\omega_{i}=v_{i}k$ with
$v_{1}=\sqrt{2(2\eta+\lambda+\mu)/\varrho}$,
$v_{2}=\sqrt{2(\kappa+\eta+\lambda+\mu)/\varrho}$,
$v_{3}=\sqrt{2(\kappa+\eta)/\varrho}$. Here, $\omega_{1}$ corresponds to the
scalar mode $\theta_{z}$, and $\omega_{2,3}$ comes from the vector modes
$\theta_{x},\theta_{y}$ [35].

Lattice | $\varrho$ | $A_{\text{uc}}$ | $\kappa$ | $\eta$ | $\lambda$ | $\mu$
---|---|---|---|---|---|---
Triangular | $\frac{2\hbar^{2}}{9\sqrt{3}Ja^{2}}$ | $\frac{3\sqrt{3}a^{2}}{2}$ | 0 | $\frac{\sqrt{3}JS^{2}}{8}$ | 0 | 0
Kagome | $\frac{\hbar^{2}}{4\sqrt{3}Ja^{2}}$ | $2\sqrt{3}a^{2}$ | 0 | 0 | $\frac{\sqrt{3}JS^{2}}{16}$ | $\frac{\sqrt{3}JS^{2}}{16}$
Table 1: Parameters describing spin-wave excitations in triangular and kagome
lattices with only the nearest exchange interaction. Here, $a$ is the lattice
constant.

DMI and anisotropy— In nAFMs, the field theory used to describe exchange
interactions should be modified by the aforementioned weak interactions that
remove the $SO(3)$ symmetry and gap out the Goldstone modes. To take these
interactions into account in centrosymmetric crystals, we first consider their
microscopic expressions.

In a kagome lattice (see Fig. 1), we consider the DMI term, e.g. typical to
jarosites [39],

$\displaystyle H_{D}=\sum_{i,j}\bm{D}_{ij}\cdot(\bm{S}_{i}\times\bm{S}_{j}),$
(9)

where $\bm{D}_{ij}=D_{z}\hat{\bm{z}}+\bm{D}_{\parallel}$,
$\bm{D}_{\parallel}=D_{p}\hat{\bm{n}}_{ij}$ with
$\hat{\bm{n}}_{12}=(\frac{\sqrt{3}}{2},-\frac{1}{2},0)$,
$\hat{\bm{n}}_{23}=(0,1,0)$, and
$\hat{\bm{n}}_{31}=(-\frac{\sqrt{3}}{2},-\frac{1}{2},0)$. We first consider
DMI to the leading-order in spatial gradients and obtain the energy density,

$\displaystyle\mathcal{H}_{D}\approx-i\text{tr}[\hat{\bm{X}}\cdot(\hat{R}^{T}\hat{\bm{\mathcal{J}}}\hat{R})],$
(10)

where
$(\hat{X}_{k})_{ab}=\sum_{i,j=1}^{3}S^{2}\mathcal{D}^{k}_{ij}n_{i}^{a}n_{j}^{b}$
and $\mathcal{D}^{k}_{ij}=D^{k}_{ij}/A_{\text{uc}}$. By using Eq. (10) and the
representation of the rotation matrix,
$\hat{R}=\exp[-i\bm{\theta}\cdot\hat{\bm{\mathcal{J}}}]\exp[-i\phi\hat{\mathcal{J}}_{z}]$,
the leading correction of the DMI term is obtained by expanding Eq. (10) to
the lowest order of $\theta_{i}$ ($i=x,y,z$),

$\displaystyle\mathcal{\delta}\mathcal{U}=\frac{\Delta}{2}(\theta_{x}^{2}+\theta_{y}^{2}),$
(11)

where $\Delta=-3\sqrt{3}\mathcal{D}_{z}S^{2}>0$. The out-of-plane DMI
suppresses spin rotations other than those with respect to $z$-axis, reducing
the SO(3) symmetry to a $U(1)$ rotation symmetry. Furthermore, the DMI in Eq.
(9) constrains the ground state of the system and gaps out the Goldstone modes
$\omega_{i}=\sqrt{v_{i}^{2}k^{2}+\Delta/\varrho}$ for $i=2,3$, while the
$\omega_{1}$ mode is intact.

To capture a small gap in the $\omega_{1}$ mode, we expand Eq. (1) to the
first order in $\bm{L}$ and substitute it in Eq. (9). The contribution
proportional to $\bm{m}$ can be written in a compact form,

$\displaystyle\mathcal{H}_{D}^{(1)}=\bm{\mathcal{B}}\cdot\bm{m},$ (12)

where $\bm{\mathcal{B}}$ is a “magnetic field”:

$\displaystyle\mathcal{B}_{l}=-i\text{tr}[\hat{Z}_{kl}\hat{R}^{T}\hat{\mathcal{J}}_{k}\hat{R}],$
(13)

with
$(Z_{kl})_{ab}=(S/6\hbar)\sum_{i,j}\sum_{c}D_{ij}^{k}(n^{a}_{i}\delta^{bc}-n_{i}^{a}n_{j}^{b}n_{j}^{c}+\delta^{ac}n_{j}^{b}-n_{i}^{a}n_{j}^{b}n_{i}^{c})(\hat{T}^{-1}\hat{R}^{T})_{cl}$.
Combing this term with Eqs. (4) and (10) and eliminating $\bm{m}$, the
effective Lagrangian becomes:

$\displaystyle\mathcal{L}=\frac{\varrho}{2}(\bm{\Omega}_{t}-\bm{\mathcal{B}})^{2}-\mathcal{U}[\hat{R}],$
(14)

where
$\mathcal{U}[\hat{R}]=\text{tr}[\hat{P}_{ij}\partial_{i}\hat{R}^{T}\partial_{j}\hat{R}]+\mathcal{H}_{D}$.
The $\bm{\mathcal{B}}$ term breaks the rotation symmetry and gaps out the
$\omega_{1}$ mode.

In a triangular lattice (see Fig. 1), the intrinsic DMI is forbidden by the
lattice symmetry, while the ground state can be stabilized by the energy
density,

$\displaystyle\mathcal{H}_{A}$
$\displaystyle=\sum_{i=1,2,3}-\mathcal{K}(\hat{\bm{n}}_{i}\cdot\bm{S}_{i})^{2}+\mathcal{K}_{z}(\hat{\bm{z}}\cdot\bm{S}_{i})^{2},$
(15)

where $\mathcal{K}=K/A_{\text{uc}}$, $\mathcal{K}_{z}=K_{z}/A_{\text{uc}}$,
with $K,K_{z}$ being the easy-axis and easy-plane anisotropy constants,
respectively. By substituting $\bm{S}_{i}\approx\hat{R}\bm{n}_{i}$ in Eq (15),
the anisotropy term gives a correction,

$\displaystyle\delta\mathcal{U}=\frac{\Delta}{2}(\theta_{x}^{2}+\theta_{y}^{2})+3\mathcal{K}S^{2}\theta_{z}^{2},$
(16)

where $\Delta=3(\mathcal{K}_{z}+\mathcal{K})S^{2}$. When
$\mathcal{K}\ll\mathcal{K}_{z}$, we can approximately neglect the easy-axis
term, and thus the system approximately respects $U(1)$ symmetry. The
Goldstone modes, $\omega_{2,3}$, acquire a gap, $\sqrt{\Delta/\varrho}$. A
small gap in the $\omega_{1}$ mode is described by $\mathcal{K}$.

Spin superfluidity— In the following, we focus on the spin transport
facilitated by approximate $U(1)$ symmetry. We assume that the system is
driven by a weak perturbation when compared to the gap of the $\omega_{2}$ and
$\omega_{3}$ modes, while large enough to overcome the barrier corresponding
to the gap of the $\omega_{1}$ mode. By adding $\mathcal{H}_{A}$ or
$\mathcal{H}_{D}$ to the Lagrangian (6) and neglecting the hard modes, the
Lagrangian of the soft mode becomes:

$\displaystyle\mathcal{L}=\frac{\varrho}{2}\dot{\phi}^{2}-\frac{\mathcal{A}}{2}(\nabla\phi)^{2},$
(17)

where $\mathcal{A}=2(2\eta+\lambda+\mu)$. On the other hand, the third
component of Eq. (5) is reduced to $m_{z}\approx\varrho\partial_{t}\phi$.
Therefore, we arrive at a continuity equation,

$\displaystyle\partial_{t}m_{z}-\mathcal{A}\nabla^{2}\phi=0,$ (18)

where a spin current density with polarization along $z$-axis can be
identified as

$\displaystyle\bm{j}_{s}=-\mathcal{A}\bm{\nabla}\phi.$ (19)

Figure 2: A nonlocal measurement setup containing normal metal/nAFM/normal
heterostructure. A charge current in the left layer generates a spin
accumulation $\bm{\mu}_{s}$ via spin Hall effect, which injects a spin current
into nAFM layer. The spin current mediated by the collective modes in the
middle layer passes the second interface by virtue of spin pumping effect. The
pumped spin current is measured in the right layer via the inverse spin Hall
effect.

The spin superfluidity in nAFM will be affected by dissipation effects. Within
the Lagrangian formalism, we can add dissipation using the Rayleigh
dissipation function [40]:

$\displaystyle\mathcal{R}=\frac{1}{2}Q_{ij}\dot{\bm{S}}_{i}\cdot\dot{\bm{S}}_{j},$
(20)

where $\hat{Q}$ is a symmetric matrix with non-negative eigenvalues [41; 42].
From symmetry considerations applied to three equivalent sublattices, we
obtain that $Q_{ij}=r_{1}$ for $i=j$ and $Q_{ij}=r_{2}$ for $i\neq j$ where
$r_{1}$ and $r_{2}$ are real parameters. The Rayleigh function can be
approximately written as
$\mathcal{R}=\text{tr}[\hat{\mathcal{Q}}\partial_{t}\hat{R}^{T}\partial_{t}\hat{R}]$
with $\mathcal{Q}_{ab}=\frac{1}{2}\sum_{ij}Q_{ij}n_{i}^{a}n_{j}^{b}$ to
describe dissipation of the soft mode, i.e.,

$\displaystyle\mathcal{R}=\alpha s\dot{\phi}^{2}/2,$ (21)

where $\alpha=3(r_{1}+2r_{2})/s$ is a dimensionless dissipation parameter.

To activate spin dynamics in a magnetic insulator, a spin Hall current can be
induced in a neighbouring normal metal layer (see Fig. 2). A build-up of spin
accumulation in a normal metal will then lead to injection of spin current
into the magnetic insulator layer. The boundary condition on the interface can
be derived via the magnetoelectronic circuit theory [43; 44; 45; 46; 47]. When
exchange interactions dominate, the spin injection and pumping together give
the total spin current across the interface (see details in Supplemental
Material):

$\displaystyle\bm{I}_{s}=\frac{1}{4\pi}\hat{\mathcal{G}}^{m}\cdot(\bm{\mu}_{s}-\hbar\bm{\omega}).$
(22)

where $\bm{\omega}$ is the instantaneous angular velocity for slow dynamics of
the order parameter and $\hat{\mathcal{G}}^{m}$ is the generalized spin-mixing
conductance tensor [47]. Under assumption of $C_{3}$ symmetry (or axial
symmetry with respect to $\bm{l}$ axis), the tensor $\hat{\mathcal{G}}^{m}$
takes the following form:

$\displaystyle\hat{\mathcal{G}}^{m}(\bm{l})$ $\displaystyle=$ $\displaystyle
2\mathcal{G}^{\uparrow\downarrow}_{r}(\mathbbm{1}-\bm{l}\otimes\bm{l})+2\mathcal{G}^{\uparrow\downarrow}_{i}(\bm{l}\times)+2\mathcal{G}_{\parallel}\bm{l}\otimes\bm{l},$

where ${\cal
G}^{\uparrow\downarrow}_{r(i)}=\sum_{mn}\operatorname{Re}(\operatorname{Im})(\delta_{nm}-r_{mn}^{\uparrow\uparrow}r_{mn}^{\downarrow\downarrow*})$,
${\cal
G}_{\parallel}=\sum_{mn}(|r_{mn}^{\uparrow\downarrow}|^{2}+|r_{mn}^{\downarrow\uparrow}|^{2})$,
and $\bm{l}$ is the unit vector normal to the plane spanned by three-
sublattice spins (cf. Eq. (59) in Ref. [47]). Here,
$r^{\sigma\sigma^{\prime}}_{mn}$ stands for reflection amplitudes for
electrons reflected from channel $n$ into channel $m$ in the normal metal and
$\bm{l}$ is the quantization axis for
$\sigma,\sigma^{\prime}=\uparrow,\downarrow$.

By writing the Euler-Lagrange equation with Rayleigh dissipation for Eqs. (17)
and (21), the dynamic equation for $\phi$ reads,

$\displaystyle\varrho\ddot{\phi}-\mathcal{A}\partial_{x}^{2}\phi+\alpha
s\dot{\phi}=0.$ (24)

We use a steady-state ansatz [7; 8], $\phi(x,t)=\varphi(x)+\Omega t$, where
$\Omega$ is a constant frequency. For almost in-plane spin order, the angular
velocity is $\bm{\omega}\approx\dot{\phi}\hat{z}$ and $\bm{l}=\hat{z}$. We
also assume spin accumulation along $z$-direction, $\bm{\mu}_{s}=\mu\hat{z}$.
Equations (19), (22), and (Spin superfluidity in noncollinear
antiferromagnets) then lead to the boundary conditions on the left ($x=0$) and
right ($x=L$) interfaces:

$\displaystyle-\mathcal{A}\partial_{x}\varphi(0)$
$\displaystyle=-\frac{g_{L}}{4\pi}(\hbar\Omega-\mu),$ (25a)
$\displaystyle-\mathcal{A}\partial_{x}\varphi(L)$
$\displaystyle=\frac{g_{R}}{4\pi}\hbar\Omega,$ (25b)

where $g_{a}=\mathcal{G}_{\parallel,a}/\mathcal{V}$ ($a=L,R$) with
$\mathcal{V}$ being the area of interface. Combining the boundary condition
and the steady-state ansatz, we obtain,

$\displaystyle\Omega=\frac{\mu g_{L}}{\hbar(g_{L}+g_{R}+g_{\alpha})},\quad
j^{s}_{R}=\frac{\mu}{4\pi}\frac{g_{L}g_{R}}{(g_{L}+g_{R}+g_{\alpha})},$ (26)

where $g_{\alpha}=4\pi\alpha sL/\hbar$.

Above approximations need to be revisited when weak in-plane DMI or easy-axis
anisotropy are present. We obtain a Lagrangian describing the soft mode:

$\displaystyle\mathcal{L}[\phi]=\frac{\varrho}{2}[(\partial_{t}\phi)^{2}-c_{s}^{2}(\nabla\phi)^{2}+m_{s}^{2}\cos
2\phi],$ (27)

where $c_{s}=\sqrt{\mathcal{A}/\varrho}$ and $m_{s}$ is the mass term due to
weak in-plane DMI or easy-axis anisotropy. To activate spin transport, the
gradient needs to overcome the barrier induced by the gap, i.e.,
$|\nabla\phi|\geq m_{s}/c_{s}$. The spiral-like phase supporting spin
superfluid will become energetically unstable when the field $\phi$ varies
faster than $1/\xi$ where $\xi=\sqrt{\varrho c_{s}^{2}/\Delta}$ is the
characteristic length associated with the gap of $\omega_{2}$ and $\omega_{3}$
modes, i.e., $|\nabla\phi|\leq\xi^{-1}$. Thus, the spin-superfluid transport
is only possible under the assumption,

$\displaystyle m_{s}\sqrt{\varrho/\Delta}\ll 1.$ (28)

We first discuss the kagome lattice nAFM with in-plane DMI in which case
$m_{s}=\sqrt{3/2}D_{p}S/\hbar$. As estimated in Table. 2, we find that
different iron jarosites fulfil criteria (28) very well and hence are very
promising for experimental realization of spin superfluidity. In a triangular
lattice, the easy-axis anisotropy hinders ideal spin superfluidity leading to
the last term in Eq. (27) with $m_{s}=3S\sqrt{KJ}/\hbar$. Equation (28) leads
to the condition $\sqrt{K/K_{z}}\ll 1$.

The spin-superfluid transport can be measured in a nonlocal setup in Fig. 2
[48]. The spin Hall current builds up an effective spin accumulation,
$\mu_{s}=(4\pi/g_{L})J_{\text{SH}}^{s}$, where
$J_{\text{SH}}^{s}=\vartheta_{\text{SH}}(\hbar/2e)J^{c}_{L}$ is the spin
current induced by the charge current $J^{c}_{L}$, and $\vartheta_{\text{SH}}$
is the spin Hall angle in the leads [49; 48; 50]. The spin current mediated by
the collective dynamics of nAFM passes across the second interface by virtue
of the spin pumping effect, and it is converted into a charge current in the
right lead, $J^{c}_{R}=(\vartheta_{\text{SH}}\sigma/d)(\hbar/2e)\Omega$, where
$\sigma$ and $d$ are, respectively, the conductivity and thickness of the
right metal layer. The nonlocal transport is characterized by a drag
coefficient,
$\mathcal{D}=J_{R}^{c}/J^{c}_{L}=\mathcal{D}_{0}/(1+L/L_{\alpha})$, where
$\mathcal{D}_{0}=\pi\vartheta_{\text{SH}}^{2}\sigma\hbar/(2e^{2}gd)$,
$g=g_{L}=g_{R}$, and $L_{\alpha}=\hbar g/(2\pi\alpha s)$. Assuming
$\vartheta_{\text{SH}}=0.1$, $\sigma=0.1\,\mu\Omega^{-1}\cdot\text{cm}^{-1}$,
$d=1\text{nm}$, $g\sim 10^{19}\text{m}^{-2}$, $\alpha=10^{-3}$,
$s\sim\hbar/a^{3}$, and a lattice constant $a\sim 1\text{nm}$, we obtain
$\mathcal{D}_{0}\sim 0.1$ and $L_{\alpha}\sim 1\mu\text{m}$. These results are
similar to collinear systems [7; 8] and show that the long crossover length
$L_{\alpha}$ can be used as a key signature of spin superfluidity.

Material |  $J$(meV) | $D_{p}/J$ | $D_{z}/J$ | $m_{s}\sqrt{\varrho/\Delta}$
---|---|---|---|---
KFe3(OH)6(SO4)2 | 3.18 | 0.062 | -0.062 | 0.088
AgFe3(OH)6(SO4)2 | 3.18 | 0.057 | -0.053 | 0.088
AgFe3(OD)6(SO4)2 | 3.18 | 0.075 | -0.053 | 0.115
Table 2: Relevant material parameters for iron jarosites taken from Refs. [51;
52].

Conclusions— We have used an $SO(3)$-invariant field theory to describe three-
sublattice antiferromagnets with hexagonal lattice in an exchange interaction
dominated limit. When weak interactions, such as DMI or anisotropy, are added,
the symmetry is approximately reduced to $U(1)$. We have shown that in this
limit, three-sublattice antiferromagnets can facilitate a spin superfluid
transport. Using generalized spin-mixing conductance, we have also described
the injection of spin current and its power-law decay in a nonlocal
experimental setup. Our results indicate that the magnitude of spin current is
constrained by parasitic DMI or anisotropies, which can help in finding
suitable materials. In particular, we estimate that iron jarosites can be
promising for realizing spin superfluidity in noncollinear antiferromagnets.
Noncollinear antiferromagnets hold promise for realizing spin flows with low
dissipation and the theoretical framework presented here can be useful for
exploring the interplay between transport phenomena [53; 54] and topological
defects, i.e., domain walls [33; 34], or skyrmions [55].

This work was supported by the U.S. Department of Energy, Office of Science,
Basic Energy Sciences, under Award No. DE-SC0021019.

## References

  * Tsymbal and Žutić [2019] E. Y. Tsymbal and I. Žutić, eds., _Spintronics Handbook: Spin Transport and Magnetism_ (CRC Press, Taylor & Francis Group, Boca Raton, 2019).
  * Olejník et al. [2018] K. Olejník, T. Seifert, Z. Kašpar, V. Novák, P. Wadley, R. P. Campion, M. Baumgartner, P. Gambardella, P. Němec, J. Wunderlich, et al., Sci. Adv. 4, eaar3566 (2018).
  * Baltz et al. [2018] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. Phys. 90, 015005 (2018).
  * Lebrun et al. [2018] R. Lebrun, A. Ross, S. A. Bender, A. Qaiumzadeh, L. Baldrati, J. Cramer, A. Brataas, R. A. Duine, and M. Kläui, Nature 561, 222 (2018).
  * Sonin [2010] E. Sonin, Adv. Phys. 59, 181 (2010).
  * Halperin and Hohenberg [1969] B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898 (1969).
  * Takei and Tserkovnyak [2014] S. Takei and Y. Tserkovnyak, Phys. Rev. Lett. 112, 227201 (2014).
  * Takei et al. [2014] S. Takei, B. I. Halperin, A. Yacoby, and Y. Tserkovnyak, Phys. Rev. B 90, 094408 (2014).
  * Skarsvåg et al. [2015] H. Skarsvåg, C. Holmqvist, and A. Brataas, Phys. Rev. Lett. 115, 237201 (2015).
  * König et al. [2001] J. König, M. C. Bønsager, and A. H. MacDonald, Phys. Rev. Lett. 87, 187202 (2001).
  * Sonin [2017] E. B. Sonin, Phys. Rev. B 95, 144432 (2017).
  * Flebus et al. [2016] B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A. Duine, Phys. Rev. Lett. 116, 117201 (2016).
  * Iacocca et al. [2017] E. Iacocca, T. J. Silva, and M. A. Hoefer, Phys. Rev. B 96, 134434 (2017).
  * Yuan et al. [2018] W. Yuan, Q. Zhu, T. Su, Y. Yao, W. Xing, Y. Chen, Y. Ma, X. Lin, J. Shi, R. Shindou, et al., Sci. Adv. 4, eaat1098 (2018).
  * Stepanov et al. [2018] P. Stepanov, S. Che, D. Shcherbakov, J. Yang, R. Chen, K. Thilahar, G. Voigt, M. W. Bockrath, D. Smirnov, K. Watanabe, et al., Nat. Phys. 14, 907 (2018).
  * Qaiumzadeh et al. [2017] A. Qaiumzadeh, H. Skarsvåg, C. Holmqvist, and A. Brataas, Phys. Rev. Lett. 118, 137201 (2017).
  * Takei et al. [2016] S. Takei, A. Yacoby, B. I. Halperin, and Y. Tserkovnyak, Phys. Rev. Lett. 116, 216801 (2016).
  * Mook et al. [2019a] A. Mook, R. R. Neumann, J. Henk, and I. Mertig, Phys. Rev. B 100, 100401 (2019a).
  * Flebus et al. [2019] B. Flebus, Y. Tserkovnyak, and G. A. Fiete, Phys. Rev. B 99, 224410 (2019).
  * Ma et al. [2020] B. Ma, B. Flebus, and G. A. Fiete, Phys. Rev. B 101, 035104 (2020).
  * Šmejkal et al. [2018] L. Šmejkal, Y. Mokrousov, B. Yan, and A. H. MacDonald, Nat. Phys. 14, 242 (2018).
  * Chen et al. [2014] H. Chen, Q. Niu, and A. H. MacDonald, Phys. Rev. Lett. 112, 017205 (2014).
  * Železný et al. [2017] J. Železný, Y. Zhang, C. Felser, and B. Yan, Phys. Rev. Lett. 119, 187204 (2017).
  * Owerre [2017] S. A. Owerre, Phys. Rev. B 95, 014422 (2017).
  * Laurell and Fiete [2018] P. Laurell and G. A. Fiete, Phys. Rev. B 98, 094419 (2018).
  * Mook et al. [2019b] A. Mook, J. Henk, and I. Mertig, Phys. Rev. B 99, 014427 (2019b).
  * Kim et al. [2019] K.-S. Kim, K. H. Lee, S. B. Chung, and J.-G. Park, Phys. Rev. B 100, 064412 (2019).
  * Okuma [2017] N. Okuma, Phys. Rev. Lett. 119, 107205 (2017).
  * Li et al. [2020a] B. Li, A. Mook, A. Raeliarijaona, and A. A. Kovalev, Phys. Rev. B 101, 024427 (2020a).
  * Li et al. [2020b] B. Li, S. Sandhoefner, and A. A. Kovalev, Phys. Rev. Research 2, 013079 (2020b).
  * Park et al. [2020] S. Park, N. Nagaosa, and B.-J. Yang, Nano Letters 20, 2741 (2020).
  * Go et al. [2019] G. Go, S. K. Kim, and K.-J. Lee, Phys. Rev. Lett. 123, 237207 (2019).
  * Ulloa and Nunez [2016] C. Ulloa and A. S. Nunez, Phys. Rev. B 93, 134429 (2016).
  * Yamane et al. [2019] Y. Yamane, O. Gomonay, and J. Sinova, Phys. Rev. B 100, 054415 (2019).
  * Dasgupta and Tchernyshyov [2020] S. Dasgupta and O. Tchernyshyov, arXiv:2004.02790 (2020).
  * Goli and Manchon [2020] V. M. L. D. P. Goli and A. Manchon, arXiv:2005.13481 (2020).
  * Dombre and Read [1989] T. Dombre and N. Read, Phys. Rev. B 39, 6797 (1989).
  * Azaria et al. [1992] P. Azaria, B. Delamotte, and D. Mouhanna, Phys. Rev. Lett. 68, 1762 (1992).
  * Elhajal et al. [2002] M. Elhajal, B. Canals, and C. Lacroix, Phys. Rev. B 66, 014422 (2002).
  * Gilbert [2004] T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).
  * Yuan et al. [2019] H. Y. Yuan, Q. Liu, K. Xia, Z. Yuan, and X. R. Wang, Europhys. Lett. 126, 67006 (2019).
  * Kamra et al. [2018] A. Kamra, R. E. Troncoso, W. Belzig, and A. Brataas, Phys. Rev. B 98, 184402 (2018).
  * Brataas et al. [2000] A. Brataas, Y. V. Nazarov, and G. E. W. Bauer, Phys. Rev. Lett. 84, 2481 (2000).
  * Brataas et al. [2006] A. Brataas, G. E. Bauer, and P. J. Kelly, Phys. Rep. 427, 157 (2006).
  * Belashchenko et al. [2016] K. D. Belashchenko, A. A. Kovalev, and M. van Schilfgaarde, Phys. Rev. Lett. 117, 207204 (2016).
  * Tserkovnyak and Ochoa [2017] Y. Tserkovnyak and H. Ochoa, Phys. Rev. B 96, 100402 (2017).
  * Flores et al. [2020] G. G. B. Flores, A. A. Kovalev, M. van Schilfgaarde, and K. D. Belashchenko, Phys. Rev. B 101, 224405 (2020).
  * Tserkovnyak and Bender [2014] Y. Tserkovnyak and S. A. Bender, Phys. Rev. B 90, 014428 (2014).
  * Tserkovnyak et al. [2005] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I. Halperin, Rev. Mod. Phys. 77, 1375 (2005).
  * Ochoa et al. [2018] H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev. B 98, 054424 (2018).
  * Matan et al. [2006] K. Matan, D. Grohol, D. G. Nocera, T. Yildirim, A. B. Harris, S. H. Lee, S. E. Nagler, and Y. S. Lee, Phys. Rev. Lett. 96, 247201 (2006).
  * Matan et al. [2011] K. Matan, B. M. Bartlett, J. S. Helton, V. Sikolenko, S. Mat’aš, K. Prokeš, Y. Chen, J. W. Lynn, D. Grohol, T. J. Sato, et al., Phys. Rev. B 83, 214406 (2011).
  * Kim et al. [2015] S. K. Kim, S. Takei, and Y. Tserkovnyak, Phys. Rev. B 92, 220409 (2015).
  * Zou et al. [2019] J. Zou, S. K. Kim, and Y. Tserkovnyak, Phys. Rev. B 99, 180402 (2019).
  * Rosales et al. [2015] H. D. Rosales, D. C. Cabra, and P. Pujol, Phys. Rev. B 92, 214439 (2015).
  * Tserkovnyak et al. [2002] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev. B 66, 224403 (2002).
  * Brouwer [1998] P. W. Brouwer, Phys. Rev. B 58, R10135 (1998).
  * Brataas et al. [2012] A. Brataas, Y. Tserkovnyak, G. Bauer, and P. Kelly, _Spin pumping and spin transfer_ (Oxford University Press, United Kingdom, 2012), pp. 87–135.

Supplemental Material

##  I DMI or anisotropy induced gap

In the main text, we have shown that the out-of-plane DMI or easy-plane
anisotropy can suppress spin rotations except for the one with respect to
$z$-direction, thus reduce the $SO(3)$ symmetry to a $U(1)$-rotation symmetry.
Here, we show that in the presence of a slowly varying $\phi$-angle spiral
background allowed by the $U(1)$-rotation symmetry, the local spectrum of hard
rotation modes with respect to $x$\- or $y$-directions are always separated
from the soft rotation by a constant gap. To capture this scenario, we use the
rotation matrix
$\hat{R}=\exp[-i\bm{\theta}\cdot\hat{\bm{\mathcal{J}}}]\exp[-i\phi\hat{\mathcal{J}}_{z}]$,
where the first component describes the spin waves ($|\bm{\theta}|\ll 1$), the
second component parametrizes the local background state. With this
representation, the local spin-wave Lagrangian for a system with out-of-plane
DMI or easy-plane anisotropy reads

$\displaystyle\mathcal{L}=\frac{\varrho}{2}\dot{\bm{\theta}}^{2}-\mathcal{U}[\phi]$
(29)

with

$\displaystyle\mathcal{U}[\phi]=$
$\displaystyle\frac{\mathcal{A}}{2}(\nabla\theta_{z})^{2}+\frac{\Delta}{2}(\theta_{x}^{2}+\theta_{y}^{2})$
(30)
$\displaystyle+(\mathcal{C}_{1}+\mathcal{C}_{2}\sin^{2}\phi)[(\partial_{x}\theta_{x})^{2}+(\partial_{y}\theta_{y})^{2}]$
$\displaystyle+(\mathcal{C}_{1}+\mathcal{C}_{2}\cos^{2}\phi)[(\partial_{x}\theta_{y})^{2}+(\partial_{y}\theta_{x})^{2}]$
$\displaystyle+C_{2}\sin(2\phi)[\partial_{y}\theta_{x}\partial_{y}\theta_{y}-\partial_{x}\theta_{x}\partial_{x}\theta_{y}]$
$\displaystyle+2(\mu\sin^{2}\phi-\lambda\cos^{2}\phi)\partial_{x}\theta_{x}\partial_{y}\theta_{y}$
$\displaystyle+2(\lambda\sin^{2}\phi-\mu\cos^{2}\phi)\partial_{y}\theta_{x}\partial_{x}\theta_{y}.$

Here, $\mathcal{C}_{1}=\kappa+\eta$, $\mathcal{C}_{2}=\lambda+\mu$, $\Delta$
is a mass term induced by the out-of-plane DMI or easy-plane anisotropy.
Specifically, $\Delta=-3\sqrt{3}\mathcal{D}_{z}S^{2}$ for the kagome nAFM with
an out-of-plane DMI and $\Delta=3\mathcal{K}_{z}S^{2}$ for the triangle nAFM
with an easy-plane anisotropy. By applying the Euler-Lagrange equation and
converting the equations to frequency-momentum space, we obtain

$\displaystyle\varrho\omega^{2}\theta_{z}=\mathcal{A}k^{2}\theta_{z},$
$\displaystyle\varrho\omega^{2}\left(\begin{array}[]{cc}\theta_{x}\\\
\theta_{y}\end{array}\right)=\left(\begin{array}[]{cc}\Delta+(2\mathcal{C}_{1}+\mathcal{C}_{2})k^{2}+\mathcal{C}_{2}\cos(2\phi)(k_{y}^{2}-k_{x}^{2})&\mathcal{C}_{2}\sin(2\phi)(k_{y}^{2}-k_{x}^{2})-2\mathcal{C}_{2}\cos(2\phi)k_{x}k_{y}\\\
\mathcal{C}_{2}\sin(2\phi)(k_{y}^{2}-k_{x}^{2})-2\mathcal{C}_{2}\cos(2\phi)k_{x}k_{y}&\Delta+(2\mathcal{C}_{1}+\mathcal{C}_{2})k^{2}-\mathcal{C}_{2}\cos(2\phi)(k_{y}^{2}-k_{x}^{2})\end{array}\right)\left(\begin{array}[]{cc}\theta_{x}\\\
\theta_{y}\end{array}\right).$ (37)

The spectrum is solved from equations above

$\displaystyle\omega_{1}=\sqrt{\frac{\mathcal{A}}{\varrho}}k,\qquad\omega_{2,3}=\Bigg{(}\frac{\Delta}{\varrho}+\frac{k^{2}}{\varrho}\big{[}2\mathcal{C}_{1}+\mathcal{C}_{2}\pm\mathcal{C}_{2}\sqrt{\cos^{2}(2\phi)\cos^{2}(2\xi_{\bm{k}})+\sin^{2}(2\phi+2\xi_{\bm{k}})}\big{]}\Bigg{)}^{1/2},$
(39)

where $(k_{x},k_{y})=k(\cos\xi_{\bm{k}},\sin\xi_{\bm{k}})$. We found that the
size of the gap induced by the mass term is independent of the local
background. In the limit $\phi\rightarrow 0$, the Lagrangian describes the
spin wave in the ground state and the spectrum recovers the three Goldstone
modes in the absence of DMI or anisotropy, i.e., $\Delta=0$. When the system
is driven by a weak perturbation with a characteristic energy scale below the
gap induced by the mass term, the dynamics and spatial variation of hard modes
are unactivated. In this regime, we can concentrate on the slow, long-
wavelength dynamics of the soft $\phi$-mode by dropping all variations of hard
modes, as described by Eq. (17).

##  II Boundary condition

According to the magnetoelectronic circuit theory, the general boundary
current across a magnet/normal metal interface on the metal side is [43; 44]

$\displaystyle\hat{i}_{N}=\frac{e}{h}\sum_{mn}\left[\hat{t}^{\prime}_{mn}\hat{f}^{M}(\hat{t}^{\prime}_{mn})^{\dagger}-\left(\delta_{mn}\hat{f}^{N}-\hat{r}_{mn}\hat{f}^{N}(\hat{r}_{mn})^{\dagger}\right)\right].$

Here, $\hat{f}^{M},\hat{f}^{N}$ are distribution function matrix for magnet
and normal metal, respectively, $\hat{t}^{\prime}_{mn}$ donates the spin-
dependent transmission amplitude for electrons transmitted from channel $n$ in
node magnet into channel $m$ in normal metal, $\hat{r}_{mn}$ stands for the
spin dependent reflection amplitude in the normal metal from channel $m$ to
channel $n$. In our discussion, the transmission matrix vanishes,
$\hat{t}^{\prime}_{mn}=0$, as we consider an insulating magnet. The current
matrix and distribution function matrix can be decomposed as below [49]

$\displaystyle\hat{i}_{N}=\frac{1}{2}(\hat{\sigma}_{0}i^{N}_{c}+\frac{e}{\hbar}\hat{\bm{\sigma}}\cdot\bm{i}^{N}_{s}),$
(42)
$\displaystyle\hat{f}^{a}=\hat{\sigma}_{0}f^{a}_{0}+\hat{\bm{\sigma}}\cdot\bm{f}^{a}_{s},\quad(a=M,N).$

Here, $i^{N}_{c}$ and $\bm{i}^{N}_{s}$ stand for the charge and spin current
at a given energy, respectively; $f_{0}^{a}$ reflects the charge accumulation,
$\bm{f}^{a}_{s}$ is a vector whose direction depends on the device
configuration and applied biases; $\sigma_{0}$ is the identity matrix and
$\bm{\sigma}=\\{\sigma_{x},\sigma_{y},\sigma_{z}\\}$ with $\sigma_{i}$ being
Pauli matrix.

With applying spin bias in the normal metal, a spin accumulation,
$\bm{\mu}_{s}$, can be built up near the interface providing the metal layer
is a poor spin sink. This spin accumulation can inject a spin current across
the normal metal/magnetic insulator interface. At a given energy $\epsilon$,
the spin current on the magnetic insulator side (opposite to the metal side)
is obtained from the Eq. (II) and Eq. (42) [45; 47]

$\displaystyle\bm{i}_{s}(\epsilon)=-\bm{i}^{N}_{s}(\epsilon)=\frac{1}{2\pi}\hat{\mathcal{G}}^{m}\cdot\bm{f}_{s}(\epsilon),$
(43)

where $\hat{\mathcal{G}}^{m}$ is a spin conductance tensor. The spin
accumulation vector and distribution function are connected as below

$\displaystyle\bm{\mu}_{s}=\int
d\epsilon\text{tr}[\hat{\bm{\sigma}}\hat{f}(\epsilon)]=2\int
d\epsilon\bm{f}_{s}(\epsilon).$ (44)

By integrating Eq. (43) over energy and using Eq. (44), the total spin current
across the interface is obtained

$\displaystyle\bm{I}_{s}=\frac{1}{4\pi}\hat{\mathcal{G}}^{m}\cdot\bm{\mu}_{s},$
(45)

In our discussion, we dropped the contribution form transmission across the
interface as we mentioned in the beginning. The tensor $\hat{\mathcal{G}}^{m}$
is solely expressed in terms of reflection matrix elements [45; 47]

$\displaystyle\mathcal{G}^{m}_{ij}=2\delta^{kl}_{ij}\sum_{mn}\mathcal{R}_{mn}^{kl}+\mathcal{R}_{mn}^{lk}+i\epsilon_{kl\nu}(\mathcal{R}_{mn}^{0\nu}-\mathcal{R}_{mn}^{\nu
0}),$ (46)

where

$\displaystyle\mathcal{R}_{mn}^{\mu\nu}=\frac{1}{4}\text{Tr}[(\hat{r}_{mn}\otimes\hat{r}^{\ast}_{mn})\cdot(\sigma^{\mu}\otimes\sigma^{\nu})],$
(47)

and $\delta^{kl}_{ij}=\delta_{ij}\delta_{kl}-\delta_{ik}\delta_{jl}$. The
explicit form of $\hat{\mathcal{G}}^{m}$ is

$\displaystyle\hat{\mathcal{G}}^{m}=\left(\begin{array}[]{c c
c}|r^{\downarrow\uparrow}-r^{\uparrow\downarrow}|^{2}+|r^{\downarrow\downarrow}-r^{\uparrow\uparrow}|^{2}&2\text{Im}(r^{\downarrow\uparrow\ast}r^{\uparrow\downarrow}+r^{\downarrow\downarrow}r^{\uparrow\uparrow\ast})&2\text{Re}(r^{\downarrow\downarrow\ast}r^{\uparrow\downarrow}-r^{\downarrow\uparrow\ast}r^{\uparrow\uparrow})\\\
2\text{Im}(r^{\downarrow\uparrow\ast}r^{\uparrow\downarrow}+r^{\downarrow\downarrow\ast}r^{\uparrow\uparrow})&|r^{\downarrow\uparrow}+r^{\uparrow\downarrow}|^{2}+|r^{\downarrow\downarrow}-r^{\uparrow\uparrow}|^{2}&2\text{Im}(r^{\downarrow\downarrow}r^{\uparrow\downarrow\ast}+r^{\downarrow\uparrow^{\ast}}r^{\uparrow\uparrow})\\\
2\text{Re}(r^{\downarrow\downarrow\ast}r^{\downarrow\uparrow}-r^{\uparrow\downarrow\ast}r^{\uparrow\uparrow})&2\text{Im}(r^{\downarrow\downarrow\ast}r^{\downarrow\uparrow}+r^{\downarrow\uparrow}r^{\uparrow\uparrow\ast})&2(|r^{\downarrow\uparrow}|^{2}+|r^{\uparrow\downarrow}|^{2})\end{array}\right).$
(51)

Assume the coupling between normal metal and magnet on the boundary is
dominated by exchange interaction, thus the boundary condition stays unchanged
upon a $SO(3)$ rotation in the spin space. Therefore, we are able to consider
a special case with all three-sublattice spins lying in the plane of the
interface and chose the direction normal to the interface as $z$-direction.
The system should respect a rotation symmetry with respect to $z$-axis.

The tensor in principle should be a function of magnetic spin orders on the
boundary, i.e.,
$\hat{\mathcal{G}}^{m}=\hat{\mathcal{G}}^{m}(\\{\bm{S}_{i}\\})$. For a given
symmetry $\hat{\mathcal{O}}$ of the system, the tensor respects the condition

$\displaystyle\hat{\mathcal{G}}^{m}(\\{\det[\hat{\mathcal{O}}]\hat{\mathcal{O}}\bm{S}_{i}\\})=\hat{\mathcal{O}}\hat{\mathcal{G}}^{m}(\\{\bm{S}_{i}\\})\hat{\mathcal{O}}^{-1}.$
(52)

When the symmetry operation is unitary ($\det[\hat{\mathcal{O}}]=1$), the
variation in spin space does not change the tensor form as a result of the
$SO(3)$ symmetry, i.e.,
$\hat{\mathcal{G}}^{m}(\\{\det[\hat{\mathcal{O}}]\hat{\mathcal{O}}\bm{S}_{i}\\})=\hat{\mathcal{G}}^{m}(\\{\bm{S}_{i}\\})$.
Therefore, the unitary symmetry constraint on the tensor reads

$\displaystyle\hat{\mathcal{G}}^{m}(\\{\bm{S}_{i}\\})=\hat{\mathcal{O}}\hat{\mathcal{G}}^{m}(\\{\bm{S}_{i}\\})\hat{\mathcal{O}}^{-1}.$
(53)

Applying the rotation symmetry with respect to $z$-direction produces

$\displaystyle\hat{\mathcal{G}}^{m}_{0}=\left(\begin{array}[]{ccc}x_{1}&x_{2}&0\\\
-x_{2}&x_{1}&0\\\ 0&0&x_{3}\end{array}\right),$ (57)

where $x_{1},x_{2},x_{3}$ are coefficients that will be determined later.
Since the system has a $SO(3)$ symmetry, we are allowed to rotate the
perpendicular direction of the spin-spanned plane (spin quantization axis)
from $z$-direction to arbitrary direction
$\bm{l}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$ by appling a
rotation matrix
$\hat{\mathcal{O}}(\bm{l})=\text{exp}[-i\hat{\mathcal{J}}_{z}\phi]\text{exp}[-i\hat{\mathcal{J}}_{y}\theta]$.
The spin conductance tensor under a representation with new quantization axis
becomes

$\displaystyle\hat{\mathcal{G}}^{m}_{0}\rightarrow\hat{\mathcal{G}}^{m}(\bm{l})=\hat{\mathcal{O}}(\bm{l})\hat{\mathcal{G}}^{m}_{0}\hat{\mathcal{O}}^{T}(\bm{l}),$
(58)

where $\hat{\mathcal{G}}^{m}(\bm{l})$ is explicitly written as

$\displaystyle\hat{\mathcal{G}}^{m}(\bm{l})$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{ccc}x_{1}&0&0\\\ 0&x_{1}&0\\\
0&0&x_{1}\end{array}\right)-x_{2}\left(\begin{array}[]{ccc}0&-l_{z}&l_{y}\\\
l_{z}&0&-l_{x}\\\ -l_{y}&l_{x}&0\end{array}\right)$ (69)
$\displaystyle+(x_{3}-x_{1})\left(\begin{array}[]{ccc}l_{x}^{2}&l_{x}l_{y}&l_{x}l_{z}\\\
l_{y}l_{x}&l^{2}_{y}&l_{y}l_{z}\\\
l_{z}l_{x}&l_{z}l_{y}&l_{z}^{2}\end{array}\right)$ $\displaystyle=$
$\displaystyle
x_{1}(\mathbbm{1}-\bm{l}\otimes\bm{l})-x_{2}(\bm{l}\times)+x_{3}\bm{l}\otimes\bm{l}.$
(70)

Now, we can fix the coefficient in Eq. (69) by comparing it with Eq. (51). In
the frame with $\bm{l}$ being the spin quantization axis, only the elements
fitting the form of Eq. (69) take finite values, all others are forced to
vanish by symmetry. Combing with the scattering matrix normalization relation,
$\sum_{mn}\hat{r}_{mn}\hat{r}_{mn}^{\dagger}=M\sigma_{0}$, the tensor takes
the form in the main text, i.e.,
$x_{1}=2\mathcal{G}^{\uparrow\downarrow}_{r}$,
$x_{2}=-2\mathcal{G}^{\uparrow\downarrow}_{i}$, and
$x_{3}=2\mathcal{G}_{\parallel}$.

Above, we discussed spin current across the boundary induced by spin
accumulation. However, when magnetic spins start to process, the dynamic
variation of spins will also generate a pumping current. In a steady state
with adjacent metal and magnet staying in a (dynamic) mutual equilibrium, if
one observes the system in a rotation frame stick-on the magnet, the net spin
current will vanish as spins in the magnet are static. This could be
understood as that the spin accumulation is canceled by a rotation induced
effective magnetic field $\bm{H}_{\text{eff}}=\hbar\bm{\omega}$ with
$\bm{\omega}$ being the vectorial angular velocity [49; 56]. This means that
the total current in the laboratory frame should take the form

$\displaystyle\bm{I}_{s}=\frac{1}{4\pi}\hat{\mathcal{G}}^{m}\cdot(\bm{\mu}_{s}-\hbar\bm{\omega}).$
(71)

In the trilayer structure discussed in the main text, the boundary current
density across two interfaces from left to right are

$\displaystyle\bm{j}_{s}|_{L}=-\frac{\hat{g}_{L}}{4\pi}\cdot(\hbar\bm{\omega}-\bm{\mu}_{s}),$
$\displaystyle\bm{j}_{s}|_{R}=\frac{\hat{g}_{R}}{4\pi}\cdot\hbar\bm{\omega},$
(72)

where we assumed that the spin accumulation in the right normal metal is zero.
Here,

$\displaystyle\hat{g}_{a}=\hat{\mathcal{G}}^{m}_{a}/\mathcal{V},\qquad a=L,R,$
(73)

with $\mathcal{V}$ being the area of the interface. When all spins lie in the
plane, both vectorial angular velocity and spin accumulation are pointing
along the out-of-plane direction, the boundary condition will be reduced to
Eq. (25).

##  III Spin pumping

We discussed the spin pumping effect by switching to a frame rotating with the
magnet in the last section. Here, we apply a parametric pumping theory [57] to
confirm this argument. Assuming the scattering matrix depends on time through
a set of real parameter $\\{X_{i}(t)\\}$, the current pumped to the normal
metal is [58]

$\displaystyle\hat{I}_{N}=e\frac{\partial\hat{n}}{\partial
X_{j}}\frac{dX_{j}}{dt}$ (74)

where

$\displaystyle\frac{\partial\hat{n}}{\partial X_{j}}=\frac{1}{4\pi
i}\sum_{mn}\frac{\partial\hat{r}_{mn}}{\partial
X_{j}}\hat{r}_{mn}^{\dagger}+\text{H.c.}$ (75)

Here, the scattering matrix is reduced to the reflection matrix in the normal
metal because the transmission amplitude from the magnet to the normal metal
is zero. The reflection matrix follows the rotation of order parameters by

$\displaystyle\hat{r}_{mn}(t)=U\hat{r}^{(0)}_{mn}U^{\dagger}$ (76)

where $r^{(0)}_{mn}$ is the reflection matrix for the initial state, $U$ is a
$SU(2)$ spin rotation matrix corresponding to the $SO(3)$ rotation of the
order parameters. We use the representation
$U=\exp[i\beta(\bm{\nu}\cdot\hat{\bm{\sigma}})/2]$ with $\bm{\nu}$ and $\beta$
being the instantaneous rotation axis and angle, respectively. At a given
moment, $\hat{I}_{N}=e(\partial\hat{n}/\partial\beta)\dot{\beta}$, namely the
rotation axis is instantaneously static. Therefore, we have

$\displaystyle\frac{\partial\hat{n}}{\partial\beta}=$
$\displaystyle\frac{1}{4\pi
i}\sum_{mn}\frac{\partial\hat{r}_{mn}}{\partial\beta}\hat{r}_{mn}^{\dagger}+\text{H.c.}$
$\displaystyle=$
$\displaystyle\frac{1}{4\pi}\sum_{mn}[(\bm{\nu}\cdot\hat{\bm{\sigma}})\delta_{mn}-\hat{r}_{mn}(\bm{\nu}\cdot\hat{\bm{\sigma}})\hat{r}^{\dagger}_{mn}],$
(77)

where we used $\partial_{\beta}U=i(\bm{\nu}\cdot\hat{\bm{\sigma}}/2)U$ and
$\sum_{mn}\hat{r}_{mn}\hat{r}_{mn}^{\dagger}=M\sigma_{0}$ with $M$ being the
number of channels in the normal metal. The current matrix can be decomposed
as
$\hat{I}=\frac{1}{2}(\hat{\sigma}_{0}I_{c}^{N}+\frac{e}{\hbar}\hat{\bm{\sigma}}\cdot\bm{I}^{N}_{s})$.
By plugging Eq. (III) into Eq. (74) and taking
$I_{s,i}=\frac{\hbar}{e}\text{Tr}[\hat{\sigma}_{i}\hat{I}_{N}]$, we obtain the
pumped spin current in the metal layer

$\displaystyle
I_{s,i}=\frac{\hbar}{4\pi}\sum_{mn}\big{(}2\delta_{ij}\delta_{mn}-\text{Tr}[\sigma_{i}\hat{r}_{mn}\hat{\sigma}_{j}\hat{r}^{\dagger}_{mn}]\big{)}\omega_{j},$
(78)

where $\omega_{j}=\dot{\beta}\nu_{j}$ is the instantaneous angular velocity.
It can be shown that the tensor in the equation above agrees with spin-mixing
conductance Eq. (51). By fitting the above result to Eq. (69), we can verify
that the spin pumping expression is consistent with Eq. (71).