# THE SECOND MULTI-CHANNEL MULTI-PARTY MEETING TRANSCRIPTION CHALLENGE (M2MeT
2.0): A BENCHMARK FOR SPEAKER-ATTRIBUTED ASR

###### Abstract

With the success of the first Multi-channel Multi-party Meeting Transcription
challenge (M2MeT), the second M2MeT challenge (M2MeT 2.0) held in ASRU2023
particularly aims to tackle the complex task of _speaker-attributed ASR (SA-
ASR)_ , which directly addresses the practical and challenging problem of “who
spoke what at when” at typical meeting scenario. We particularly established
two sub-tracks. 1) The fixed training condition sub-track, where the training
data is constrained to predetermined datasets, but participants can use any
open-source pre-trained model. 2) The open training condition sub-track, which
allows for the use of all available data and models. In addition, we release a
new 10-hour test set for challenge ranking. This paper provides an overview of
the dataset, track settings, results, and analysis of submitted systems, as a
benchmark to show the current state of speaker-attributed ASR.

Index Terms—  M2MeT 2.0, Alimeeting, Meeting Transcription, Multi-speaker ASR,
Speaker-attributed ASR

##  1 INTRODUCTION

Despite years of research, meeting transcription accuracy still faces
significant challenges, including but not limited to overlapping speech, an
unknown number of speakers, far-field attenuated speech signals, noise,
reverberation, and other factors that can degrade transcription performance.
The ICASSP2022 Multi-Channel Multi-Party Meeting Transcription (M2MeT)
challenge [1, 2] has played a crucial role in the development of Mandarin
meeting transcription technology by addressing the challenge of speech overlap
in actual meetings. The challenge consists of two distinct tasks: speaker
diarization and multi-speaker automatic speech recognition (ASR). The former
involves identifying _who spoke when_ in the meeting, while the latter aims to
transcribe speech from multiple speakers. In the second M2MeT challenge (M2MeT
2.0), these two tasks are merged into a single speaker-attributed task.

The M2MeT 2.0 challenge presents two key differences from its predecessor, the
first M2MeT. Firstly, the evaluation metric in the first M2MeT challenge was
speaker-independent, meaning that transcription could be determined, but the
corresponding speaker was not identified. To overcome this limitation and
advance current multi-talker ASR systems, the M2MeT 2.0 challenge introduces
the _speaker-attributed ASR (SA-ASR)_ task. This task not only transcribes the
speech but also assigns speaker labels to each transcription. Specifically, we
introduce the concatenated minimum permutation character error rate (cpCER)
metric to evaluate the performance of the submitted systems. The cpCER is
proposed for Mandarin particularly, which is defined similarly to the
concatenated minimum permutation word error rate (cpWER) [3]. Secondly, unlike
other similar challenge such as Computational Hearing in Multisource
Environments (CHIME) [3], Multimodal Information Based Speech Processing
(MISP) [4], and M2MeT challenges, the M2MeT 2.0 challenge offers participants
the freedom to utilize any open-source pre-trained model, which is typically
prohibited in these challenges. This flexibility aims to explore the feasible
industrial application of academic research proposed in previous studies,
utilizing various open-source pre-trained models trained on a large amount of
data for the SA-ASR task.

The M2MeT 2.0 challenge consists of two sub-tracks: 1) The fixed training
condition track, is designed to enable reproducible research in this field by
providing a fixed set of training data, open-source pre-trained models, and
evaluation criteria. 2) The open training condition track, aims to benchmark
state-of-the-art performance in speaker-attributed ASR by allowing
participants to use their own data and training techniques.

##  2 RALATED WORKS

The SA-ASR task [5] involves identifying multiple speakers and transcribing
overlapped speech within a single session. One common approach to address this
cocktail party challenge is to use speaker diarization to identify the active
regions of different speakers. Then, a single-talker ASR system with a speaker
separation module can be used to transcribe speech from the known active
regions. Alternatively, an end-to-end multi-talker ASR system can be used to
transcribe speech and assign speaker labels simultaneously based on
corresponding speaker information provided by the diarization system.

Speaker diarization techniques that follow conventional clustering-based
approaches usually include two main steps: speaker embedding extraction and
clustering. These approaches begin by transforming the input audio stream into
a speaker-specific representation, followed by a clustering process like
Variational Bayesian HMM clustering (VBx) [6] that groups the regions of each
speaker into separate clusters. Clustering-based methods typically assign a
single speaker label to each frame, making it challenging for them to handle
speech overlap. With the rapid development of deep learning, End-to-End
speaker diarization methods like end-to-end neural diarization (EEND) [7] are
proposed, leveraging a single neural network to replace the modular cluster-
based system. Inspired by the target speaker extraction [8, 9, 10, 11], target
speaker voice activity detection (TS-VAD) [12, 13, 14] has been proposed,
which can estimate the activity level of each speaker in the presence of
overlapping speech, providing a promising solution for speaker diarization.

Single-talker ASR has been extensively investigated in recent years, and
various architectures, such as Conformer [15], Branchformer [16], and
Paraformer [17], have been proposed to push the limit of the accuracy of ASR.

Meanwhile, there has been a growing interest in multi-talker ASR, which is
designed to transcribe speech containing several speakers. One recent
approach, called Speaker-Attributed Transformer(SA-Transformer) [18],
generates token-level speaker labels during the decoding phase and can
directly produce speaker-attributed transcription only using the clustered
speaker profile. Moreover, TS-ASR [19] is also a promising approach that
utilizes speaker embedding for target speaker extraction (TSE) and achieves
good performance when TSE and ASR are jointly trained. By extracting the
speaker embedding, TS-ASR can identify the target speaker’s voice and enhance
the accuracy of the overall ASR system.

The field of rich transcription with speech overlap has undergone extensive
research, with advancements facilitated by numerous challenges and open-source
datasets [4, 20, 21, 22]. Table 1 outlines the primary datasets utilized in
this scenario.

WSJ0-2mix [23] and Libri2Mix [24] datasets involve mixing pairs of utterances
from different speakers at random SNRs, making them primarily used for speech
separation tasks where there is full overlap. On the other hand, AMI [25],
LibriCSS [26], and CHIME-6 [3] datasets are recorded in real rooms. However,
the AMI dataset’s fixed speaker count of four and poor recording quality limit
its practical applications. LibriCSS, similar to Libri2Mix, utilizes the
Librispeech corpus to produce speech mixes. However, due to the fixed
intonation and pace of reading in Librispeech, there remains a disparity
between LibriCSS and real meeting scenarios. CHIME-6 dataset is designed for
conference or indoor conversation transcription tasks and accounts for
overlapped speech.

Although these datasets significantly contribute to the progress of
transcription overlapping speech, they are limited to English. The language
barrier poses a challenge in achieving comparable results for non-English
languages, such as Mandarin. To overcome this challenge, the AISHELL-4 [27]
and AliMeeting [1] datasets have been developed specifically for Mandarin
meeting transcription. AISHELL-4 has a lower overlap ratio, while AliMeeting
contains intense discussions. Additionally, AliMeeting records the near-field
signal of each participant using a headset microphone, ensuring that only the
participant’s speech is transcribed.

Table 1: Datasets available in the literature in multi-talker speech transcription (OR: overlap rate) Dataset | Hours | #SPK | Devices | OR (%)
---|---|---|---|---
WSJ0-2mix [23] | 43 | 129 | Simu | Full
Libri2Mix [24] | 292 | 1252 | Simu | Full
AMI [25] | 100 | 190 |  Headset mic, 8-ch mic array | <10
LibriCSS [26] | 10 | 40 | 7-ch mic array | 0-40
CHIME-6 [3] | 35 | 26 | 4-ch mic array | 40
AISHELL-4 [27] | 120 | 61 | 8-ch mic array | 19
AliMeeting [1] | 129 | 481 |  Headset mic, 8-ch mic array | 42

##  3 DATASET AND TRACKS

AliMeeting [1, 2], AISHELL-4 [28], and CN-Celeb [29] corpus are adopted as our
training data, which is the same as the first M2MeT challenge. AliMeeting
dataset is a collection of multi-talker conversation records in a meeting
setting, comprising a total of 118.75 hours of speech data. The dataset is
split into 104.75 hours for training (_Train_), 4 hours for evaluation
(_Eval_), and 10 hours for testing (_Test_). The AliMeeting corpus includes
both far-field overlapped audios and corresponding near-field audios, which
exclusively record and transcribe single-speaker speech. To evaluate the
submitted systems, an additional 10 hours of audio data, called _Test-2023_ ,
is incorporated specifically for testing purposes. _Test-2023_ comprises 10
sessions that were recorded in 5 different rooms and include 58 speakers. It
is crucial to highlight that none of the speakers in _Test-2023_ overlap with
those in the AliMeeting corpus.

Fig. 1:  The main difference between the speaker-attributed ASR task (M2MeT
2.0) and the multi-speaker ASR task (M2MeT).

This challenge introduces the speaker-attributed ASR task which poses a unique
challenge of transcribing speech from multiple speakers and assigning a
speaker label to the transcription simultaneously. Figure 1 illustrates the
difference between the speaker-attributed ASR task and the multi-speaker ASR
task. The speaker-attribution ASR task groups transcriptions from the same
speaker together, while the multi-speaker ASR task combines overlapping
sentences spoken by different speakers. Sub-track 1 restricts participants
from using constrained datasets, while sub-track 2 allows the use of any
dataset, including private ones. Any open-source pre-trained models are
available for both of these two sub-tracks. The accuracy of a speaker-
attributed ASR system is evaluated using the cpCER. The calculation of cpCER
involves three steps. The first step is to concatenate the reference and
hypothesis transcriptions from each speaker in chronological order. This
produces a single transcription within one session. Next, the character error
rate (CER) is calculated between the concatenated reference and hypothesis
transcriptions, and this process is repeated for all possible speaker
permutations. Finally, the permutation with the lowest CER is selected as the
cpCER for that session.

To provide a clear and concise description of the cpCER computation, we
illustrate it in Algorithm 1. To run the algorithm, it is necessary to provide
both the ground truth and hypothesis transcriptions for a given session, which
are arranged in chronological order. In cases where the length of $Y$ and $H$
is not equal, we employ padding with blank transcriptions to ensure that both
sets have the same length.

1

input : Ground truth: $\left\\{Y_{1},Y_{2},...,Y_{S}\right\\}$, hypothesis of
different speakers: $\left\\{H_{1},H_{2},...,H_{\hat{S}}\right\\}$. $S$ is the
oracle speaker number and $\hat{S}$ is the predicted speaker number

output : The $cpCER$ of given session

2 $Y\leftarrow\left\\{Y_{1},Y_{2},...,Y_{S}\right\\}$;

3 $H\leftarrow\left\\{H_{1},H_{2},...,H_{\hat{S}}\right\\}$;

4 $mindistance\leftarrow INFINITY$;

5 foreach  _permutation of $H$_ do

6 $distance\leftarrow 0$;

7 for  _$i\leftarrow 1$ to $\max(S,\hat{S})$_ do

8 $distance\leftarrow distance+editdistance(H\left[i\right],Y\left[i\right])$;

9

10 end for

11 if  _$distance <mindistance$_ then

12 $mindistance\leftarrow distance$;

13

14 end if

15

16 end foreach

17$totaltoken\leftarrow\sum_{i=0}^{S}length(Y\left[i\right])$;

$cpCER\leftarrow mindistance/totaltoken\times 100\%$

Algorithm 1 Computation of cpCER

##  4 SYSTEM DESCRIPTION

###  4.1 Baseline system

We release an E2E SA-Transformer baseline built on FunASR [30] toolkit for
easy and reproducible research. The model architecture is illustrated in
Figure 2. It comprises an ASR block and a speaker block to carry out ASR and
token-level speaker identification. The ASR block is represented as

$\displaystyle H^{asr}$ $\displaystyle=\text{AsrEncoder}(X),$ (1)
$\displaystyle o_{n}$
$\displaystyle=\text{AsrDecoder}(y_{\left[1:n-1\right]},H^{asr},\bar{d}_{n}).\vspace{-0.1cm}$
(2)

In the ASR block, the AsrEncoder converts the given acoustic feature $X$ into
a series of hidden embeddings $H^{asr}$. The AsrDecoder then produces the
output distribution $o_{n}$ step-by-step. To generate each token, the
AsrDecoder takes in the history token $y_{\left[1:n-1\right]}$, the hidden
embeddings $H^{asr}$, and the weighted speaker profile $\bar{d}_{n}$. Compared
to the other encoder-decoder-based ASR models, our model differs in its use of
the weighted speaker profile $\bar{d}_{n}$, computed in the speaker block.
This profile is employed to bias the transcription towards a specific speaker,
which enhances the model’s ability to identify individual speakers within a
session. The posterior probability of token $i$ at the $n$-th decoding step is
formulated as

$\displaystyle
Pr(y_{n}=i|y_{\left[1:n-1\right]},s_{\left[1:n\right]},X,D)=o_{n,i}.\vspace{-0.1cm}$
(3)

On the other hand, the speaker block is denoted as

$\displaystyle H^{spk}$ $\displaystyle=\text{SpeakerEncoder}(x),$ (4)
$\displaystyle q_{n}$
$\displaystyle=\text{SpeakerDecoder}(y_{\left[1:n-1\right]},H^{spk},H^{asr}),$
(5) $\displaystyle\beta_{n,k}$
$\displaystyle=\frac{\exp(\cos(q_{n},d_{k}))}{{\textstyle\sum_{j}^{K}}\exp(\cos(q_{n},d_{j}))},$
(6) $\displaystyle\bar{d}_{n}$
$\displaystyle=\sum_{k=1}^{K}\beta_{n,k}d_{k}.\vspace{-0.1cm}$ (7)

The speaker encoder takes the input acoustic feature $X$ and produces the
speaker embedding $H^{spk}$, which has the same shape as the ASR embedding
$H^{asr}$ and represents the speaker’s unique characteristics. During each
decoding step $n$, the SpeakerDecoder uses $y_{1:n-1}$, $H^{spk}$, and
$H^{asr}$ to generate a speaker query $q_{n}$. Based on this query, a cosine
distance-based attention weight $\beta_{n,k}$ is calculated for every profile
$d_{k}$ in $D$. These weights can be viewed as posterior probabilities for the
$n$-th token being attributed to the $k$-th speaker, taking into account all
previous estimations, as well as relevant information from the input $X$ and
the speaker profiles in $D$. The $\beta_{n,k}$ can be represented as

$\displaystyle
Pr(s_{n}=k|y_{\left[1:n-1\right]},s_{\left[1:n-1\right]},X,D)=\beta_{n,k}.$
(8)

To extract speaker embeddings and initialize the SpeakerEncoder, we leveraged
a pre-trained x-vector extractor from ModelScope trained on CN-Celeb. To train
our E2E SA-ASR system, we employed a two-stage training strategy. In the first
stage of training the E2E SA-ASR system, we trained a standard conformer for
the ASR task, which was utilized in the second stage to initialize the ASR
block. In this second stage, both the ASR and speaker losses were incorporated
to fine-tune the model. By using Eqs. 3 and 8, the joint posterior of token
$Y$ and speaker $S$, optimized to be maximized during training, is represented
as

$\displaystyle
Pr(Y,S|X,D)=\prod_{n=1}^{N}Pr(y_{n}|y_{\left[1:n-1\right]},s_{\left[1:n\right]},X,D)$
(9) $\displaystyle\times
Pr(s_{n}|y_{\left[1:n-1\right]},s_{\left[1:n-1\right]},X,D).$

During the training phase, speaker embeddings were extracted from audio solely
containing one speaker to generate speaker profiles, utilizing the oracle time
stamp. However, during the decoding phase, when the oracle speaker label was
absent, we turned to spectral clustering for providing the speaker profile.

Fig. 2:  The SA-Transformer [18] baseline system uses in M2MeT 2.0 challenge.

###  4.2 Official modular system

In addition to the E2E SA-Transformer baseline, we also develop an official
modular system based on pre-trained models that serves as a strong
alternative. With the goal of encouraging research and development in the
field of speaker-attributed ASR, we are dedicated to releasing this system in
the near future. The modular baseline is illustrated in Figure 3. Our Front-
End process leverages two methods to enhance the input audio: weighted
prediction error (WPE), a well-known dereverberation technique, and guided
source separation (GSS), which utilizes prior segmentation information to
assist the separation process. To obtain speaker-labeled segments for GSS, we
employ a speaker diarization module. The input long-form audio is first
processed by a VBx diarization system to obtain an initial diarization output.
Using this output, an x-vector extractor generates a speaker profile for each
session from non-overlapping speech segments. The speaker profiles are then
utilized by a pre-trained speaker overlap-aware neural diarization (SOND) [31]
model to generate the final diarization output for GSS. Following GSS, we
segment the audio based on the diarization output and feed it into a
Paraformer for single-talker ASR with speaker labels.

Fig. 3:  The official modular system basing on open-source pre-trained models.

##  5 EXPERIMENTAL SETUP

In this section, we describe the settings of the SA-Transformer baseline and
the modular system based on the pre-train models.

###  5.1 SA-Transformer baseline

The ASR Encoder is composed of 12 conformer layers, each equipped with 4-head
multi-head attention (MHA). The MHA and feed-forward network (FFN) dimensions
are set to 256 and 2048, respectively. The ASR Decoder is made up of 6
transformer layers. The Speaker Encoder is initialized with a pre-trained
ResNet x-vector extractor, but with an additional linear layer that transforms
the embedding dimension into 256. The Speaker Decoder comprises 3 transformer
layers. The speaker loss weight is set to 0.5, while the CTC loss weight is
set to 0.3.

###  5.2 Official modular system

We have implemented VBx diarization in our modular speaker-attributed system,
following the workflow from the first M2MeT challenge baseline. To extract
x-vectors, we use the same extractor as in SA-Transformer. When generating
speaker profiles, we discard segments shorter than 0.5 seconds. Our system
utilizes the pre-trained SOND and Paraformer models, which are both open-
sourced and can be accessed on ModelScope. The train set of AliMeeting is
utilized to fine-tune the pre-trained paraformer model for 50 epochs with a
learning rate of 0.00005.

##  6 RESULTS AND ANLYSIS

Table 2: Top 5 ranking teams in terms of cpCER and their major techniques.
Team code O represents the official modular system and Team code B represents
the SA-Transformer baseline system

Team Code | Tracks  | Data  | Spk-Emb Extractor  | Diarization  | Front-End  | ASR  | Data Augmentation  | Post-prosessing  | cpCER(%)
---|---|---|---|---|---|---|---|---|---
|  Sub-track 1 |  Sub-track 2 |  Constraint Data |  WenetSpeech |  ResNet-34 |  ResNet-101 |  CAM++ |  TS-VAD |  SOND |  Spectral Clustering |  VBx Clustering |  WPE |  CMGAN |  BeamformIt |  GSS |  U2++ |  Paraformer |  MFCCA |  SOT |  SA-Transformer |  Speed Perturb. |  Noise & Reverb. |  Pitch Shifting |  Data Simulation |  Audio Codec |  SpecAugment |  Diarization System Fusion |  ASR System Fusion |  2-Speaker sessions (6) |  3-Speaker sessions (6) |  4-Speaker sessions (8) |  Average
O  | $\checkmark$ |  $\checkmark$ | $\checkmark$ |  | $\checkmark$ | $\checkmark$ |  |  | $\checkmark$ |  |  $\checkmark$ | $\checkmark$ |  |  |  $\checkmark$ |  | $\checkmark$ |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  | 9.01 | 7.34 | 10.31 | 8.84
X27  | $\checkmark$ |  $\checkmark$ | $\checkmark$ |  | $\checkmark$ |  |  | $\checkmark$ |  | $\checkmark$ |  | $\checkmark$ |  | $\checkmark$ |  | $\checkmark$ |  |  |  |  | $\checkmark$ |  |  | $\checkmark$ |  |  $\checkmark$ | $\checkmark$ |  $\checkmark$ | 11.81 | 9.58 | 12.52 | 11.27
M42  | $\checkmark$ |  $\checkmark$ | $\checkmark$ |  |  |  |  $\checkmark$ | $\checkmark$ |  | $\checkmark$ |  |  | $\checkmark$ |  |  | $\checkmark$ | $\checkmark$ |  |  |  | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$ |  $\checkmark$ |  |  $\checkmark$ | 18.34 | 16.99 | 20.86 | 18.64
C17  |  |  $\checkmark$ | $\checkmark$ |  $\checkmark$ |  | $\checkmark$ |  |  |  |  |  $\checkmark$ |  |  |  |  | $\checkmark$ |  |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  | 27.01 | 22.98 | 17.7 | 22.83
V29  |  |  $\checkmark$ | $\checkmark$ |  | $\checkmark$ |  |  |  |  | $\checkmark$ |  |  |  |  |  |  | $\checkmark$ |  |  |  |  |  |  |  |  |  $\checkmark$ |  |  | 20.44 | 18.25 | 33.05 | 23.52
C31  | $\checkmark$ |  $\checkmark$ | $\checkmark$ |  | $\checkmark$ |  |  |  | $\checkmark$ |  |  |  |  |  |  |  |  | $\checkmark$ | $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  $\checkmark$ |  |  | 21.72 | 19.01 | 34.97 | 24.82
B  | $\checkmark$ |  $\checkmark$ | $\checkmark$ |  | $\checkmark$ |  |  |  |  | $\checkmark$ |  |  |  |  |  |  |  |  | $\checkmark$ |  $\checkmark$ |  |  |  |  |  |  $\checkmark$ |  |  | 35.95 | 40.49 | 49.4 | 41.55

In this section, we provide a comprehensive analysis of various systems used
in this challenge. The major techniques utilization and evaluation results of
each team is shown in Table 2. A total of 30 teams registered for this
challenge and 8 of them submitted their results. Almost all teams use
constraint data to build their system except C17, so they submit the same
result to the two sub-tracks. While we initially released an end-to-end SA-ASR
baseline, most participants opted for a modular system approach due to the
long training time and limited performance with restricted data for end-to-end
approaches. Moreover, with the availability of open-source pre-trained models,
developing a modular system is more straightforward and yields satisfactory
results. Consequently, we examine different modules, data augmentation
techniques, and post-processing methods to ascertain their effectiveness in
improving the overall system performance.

###  6.1 Speaker modules

The speaker modules play a crucial role in the speaker-attributed ASR system
by providing time-stamp and speaker profiles for subsequent ASR module. The
input audio is typically divided into shorter segments, and then a speaker
embedding extractor, such as ResNet [32], or CAM++ [33], is employed to encode
the audio into an embedding that captures the speaker characteristics. The
speaker embedding is used to identify the speaker and corresponding speech
segment throughout the ASR process.

The performance of the diarization model is typically measured by the
diarization error rate (DER), which is impacted by both speaker activation
region prediction and speaker identification accuracy. However, in the context
of speaker-attribute ASR tasks, the speaker activation region is not always
necessary. As a result, we present the results of speaker counting in Table 3.
Teams X27 and M42 both employ TS-VAD with different speaker embedding
extractors. By utilizing the clustering result as initialization to further
enhance diarization accuracy, team X27 achieves DER of 3.64% on the _Test_
set. Team C17 and V29 opted for spectral clustering and VBx clustering,
respectively. Due to the low overlap ratio in the _Test-2023_ set, cluster-
based diarization methods can also perform well on speaker counting. Team C31
and the official modular system use the SOND model and achieved an accuracy of
80% and 100% in speaker counting, respectively. The main difference is that
the official modular system uses the VBx to split out the single-talker speech
part which can produce more accurate speaker profiles. Benefiting from the
accuracy of speaker counting, the official modular system achieves DER of
1.51% on the _Test-2023_ set.

Table 3: The speaker counting accuracy (%) on 20 sessions in _Test-2023_ set. $e_{spk}$ and $o_{spk}$ represent the estimated speaker number and oracle speaker number, respectively Team code |  $e_{spk}$<$o_{spk}$ |  $e_{spk}$=$o_{spk}$ |  $e_{spk}$>$o_{spk}$
---|---|---|---
Official | 0 | 100 | 0
X27 | 0 | 90 | 10
M42 | 0 | 100 | 0
C17 | 10 | 90 | 0
V29 | 0 | 95 | 5
C31 | 15 | 80 | 5
Baseline | 10 | 50 | 40

###  6.2 Front-End

Out of all the participating teams, only the top 2 teams use the Front-End
process. The winning team, X27, implements the weighted prediction error (WPE)
and weighted delay-and-sum acoustic beamforming (BeamformIt) techniques, which
have proven effective in recognizing far-field audio. Team M42 utilizes a
Conformer-based Metric GAN (CMGAN) [34] to separate multi-talker segments into
single-talker segments and compared the effectiveness of separate training and
joint training strategies. Joint training of the speech separation module and
ASR module results in a significant 4.2% reduction in the character error rate
(cpCER) on the _Test-2023_ setting (26.60% $\to$ 22.40%). The official modular
system adopts the WPE and GSS. GSS is proven to be an effective method in
various challenges when the time stamp can be easily obtained. In the official
modular system, it results in a 1.80% cpCER reduction compared to the process
of WPE and BeamformIt (11.98% $\to$ 10.18%).

###  6.3 ASR module

Multiple ASR models are explored by the participants, including U2++ [35],
Paraformer [17], MFCCA [36], and SA-Transformer [18]. In their experiment,
Team M42 utilizes the U2++ and Paraformer models and assessed their
performance by comparing the cpCER obtained after joint training with the
speaker separation module. The results indicate that Paraformer outperforms
U2++ with a lower cpCER of 20.67% as compared to 22.40%. Notably, the top four
teams processed the audio into single talker segments, which could be
effectively handled using a pre-trained standard ASR model.

Team X27 successfully transforms a single-talker ASR model into a target
speaker model [37] by incorporating the target speaker’s embedding into a
fully connected layer and injecting it with an element-wise product between
the encoder layers. This approach enabled the model to accurately recognize
the target speaker’s speech with high precision. Team C31 implements the
frame-level diarization with SOT (FD-SOT) [19] multi-talker ASR system to
mitigate the issue of overlap in the ASR module. This multi-talker ASR system
combines diarization and ASR results by adhering to the first-in first-out
property of SOT. By splitting the ASR result using speaker change symbols and
assigning speaker labels using diarization results, FD-SOT ensures one-to-one
correspondence. Furthermore, the MFCCA multi-channel solution is adopted as
their ASR model, which effectively leverages fine-grained channel-wise
information at each time step. It should be noted that MFCCA achieves state-
of-the-art performance levels with limited data from AliMeeting. However, the
conversion of multi-talker ASR transcripts to speaker-attributed ASR
transcripts remains a challenging task. The official modular system adopts the
Paraformer as its ASR model. We use the AliMeeting dataset to fine-tune the
pre-trained model and achieve the cpCER of 8.84% while before fine-tune, the
cpCER is 10.18%.

###  6.4 Data augmentation and post-processing

Data augmentation methods are only adopted by the top 2 teams. One such team,
X27, uses the non-overlapping segments in Aishell-4 and CN-Celeb as simulation
data for training their TS-VAD model. They have also applied speed
perturbation to the training data of their TS-ASR model. Furthermore, Dover-
lap is employed to merge the diarization result from different channels and
ROVER is also employed as the final system fusion method, which results in a
0.24% cpCER reduction on the Test set, reducing it from 17.08% to 16.84%. Team
M42 effectively employs a wide range of data augmentation methods, such as
speed perturbation, adding noise and reverb, pitch shifting, data simulation,
audio codec, and SpecAugment [38]. They have also developed U2++ and
Paraformer based speaker-attributed systems, and utilize a system fusion
method that depends on whether the audio has undergone speech separation. They
discover that the conformer model performs better on audio that goes through
speech separation while the paraformer model is more effective on audio
without speech separation. Therefore, the processed audio is taken as the
input of U2++, while the Paraformer takes other inputs to produce the fused
hypothesis. This approach results in 2.03% cpCER reduction on the _Test-2023_
set.

##  7 Conclusion

This paper provides an overview of the outcomes of the second multi-channel
multi-party meeting transcription challenge, with a focus on the techniques
used by the top-ranking teams. Given the limited training time and data,
leveraging open-source pre-trained models to construct a modular system is an
effective approach for tackling this challenge. For speaker modules, TS-VAD
and SOND are potent methods, but accurate speaker profiles are necessary for
optimal performance. Front-End process methods such as WPE, beeamformIt,
CMGAN, and GSS are beneficial for transcribing far-field data, although CMGAN
may have a negative effect on ASR accuracy for non-overlapped speech. In the
ASR module, SOT-based methods underperform due to limited training data, while
single-talker ASR models trained on a large amount of data perform well when
speech is well-preprocessed in a modular system. Data simulation is less
important than in the first M2MeT challenge, as pre-trained ASR models provide
sufficient initialization for fine-tuning with small data amounts. The best-
performing system achieves 8.84% cpCER given the limited training data.

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# Effects of exchange distortions in the magnetic Kagome lattice

A. A. Coker  Department of Physics, University of North Florida, Jacksonville,
FL 32224, USA  A. Saxena  Theoretical Division and Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
J.T. Haraldsen  Department of Physics, University of North Florida,
Jacksonville, FL 32224, USA

###### Abstract

This study examines the effect of distorted triangular magnetic interactions
in the Kagome lattice. Using a Holstein-Primakoff expansion, we determine the
analytical solutions for classical energies and the spin-wave modes for
various magnetic configurations. By understanding the magnetic phase diagram,
we characterize the changes in the spin waves and examine the spin distortions
of the ferromagnetic (FM), Antiferrimagnetic (AfM), and 120∘ phases that are
produced by variable exchange interactions and lead to various non-collinear
phases, which provides a deeper understanding of the magnetic fingerprints of
these configurations for experimental characterization and identification.

##  I Introduction

Complex magnetic lattice configurations have gained attention both
theoretically and experimentally during the last decade as part of the search
for topological and non-collinear magnetic orders, flat-band and Kitaev
interactions, and the elusive quantum spin liquid state[1, 2, 3, 4, 5, 6, 7,
8, 9, 10, 11, 12]. These magnetic systems are so interesting because of their
potential in technology for sustainable, energy-efficient memory devices and
computational power[13].

One such lattice that has increased in popularity is the Kagome lattice, which
is described as a two-dimensional triangular three-sublattice structure and
the foundational structure for the three-dimensional pyrochlore lattice[14,
15, 16, 17]. This extra sublattice helps distinguish the Kagome from the two-
sublattice honeycomb lattice[18]. As shown in Fig. 1(a), the Kagome lattice
consists of a lattice of coupled trimers, which introduces multiple inversion
points. Theoretical studies on materials with Kagome lattices have qualified
their relevance to these technologies and have further probed for numerous
magnetic and electronic properties[19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31]. Experimental studies have offered a different perspective on many of
the same properties[32, 33, 34, 35, 36, 37, 38, 32, 39].

Previously, Boyko et al. examined the spin-wave dynamics of the Kagome lattice
for three different magnetic configurations [out-of-plane ferromagnetic (FM),
out-of-plane antiferrimagnetic (AfM), and 120∘ phase] and various isotropic
nearest, next-nearest, and next-next-nearest neighbor interactions[40]. In
this work, it was shown that, to first order, the FM phase produced three
modes, wherein two modes were dispersive, long-range order modes, and one was
a non-dispersive, cluster-like flat band. These types of cluster modes are not
unheard of as they have been observed in structures like the pyrochlore
lattice[16]. Furthermore, Boyko et al. also revealed that, to first order, the
120∘ phase mimics the antiferromagnetic (AFM) honeycomb lattice due to the net
in and out spin configurations[40]. However, unlike the AFM honeycomb lattice,
the Kagome lattice can break this degeneracy with second- and third-order
interactions.

Figure 1: The atoms in a Kagome lattice are colored by spin orientation. The
various spin configurations are the ferromagnetic (FM - all spin angles in the
same direction) (a), antiferrimagnetic (AfM - spin angles are anti-aligned)
(b), and the $120^{\circ}$ phase, where each spin is rotated 120 degrees away
from its neighbors’ (c). The $120^{\circ}$ phase makes up triangular “in” and
“out” atomic spin groups (e), which further illustrate its AFM nature (d).

When considering the collinear Heisenberg model, exchange competition can
produce frustration in the system and require an axial anisotropy to stabilize
due to underlying non-collinear states[40, 6, 12]. Frustrated states can also
come from the interactions between orbitals that typically result in FM or AFM
orders but may become complicated by the competition between interactions as
well as any induced crystal-field anisotropies. The frustration in the
triangular interactions can lead to the need for more complex interactions to
describe the excitations of the system, which is a fairly standard approach as
any deviation of a known model indicates the need for new and exciting
physics. In the Kagome lattice, this complexity tends to lead researchers into
the realm of more exotic interactions like Kitaev model and spin liquids[1, 3,
2].

Figure 2: (a) The triangular subgeometry on which our calculations are
focused. The exchange interactions $(J_{ij})$, which may not all be equal
between all atom pairs. (b) A top-down view of the FM spin waves in a Kagome
lattice with symmetry points M, K and $\Gamma$ overlaid.

Another avenue that can alleviate these frustrations is the presence of some
underlying, possibly non-collinear, magnetic ground state that is not
considered[5]. Given the large number of magnetic configurations for this
structure, many groups are forced to examine numerical methods to interpret
and understand neutron scattering experiments. While these numerical methods
provide critical information for the characterization and identification of
magnetic systems, there is little understanding of how the system’s
interactions compete to produce the given state, which can hinder further
development of magnetic materials. By examining analytical solutions and the
evolution of complexity for simple models, one can gain insight into the
effects of exchange interactions on various configurations of spins in the
Kagome lattice. These insights can aid experimentalists and theorists (or
modelers) in the identification of different magnetic orders by simple
comparison, which can, in turn, allow for the tunability of structures.

In this study, we look to understand many of the Kagome lattice’s magnetic
configurations by first producing analytical solutions for the spin waves of
in-plane and out-of-plane ferromagnetic (FM) and antiferrimagnetic (AfM)
arrangements along with the in-plane 120∘ phase, then numerically
characterizing various spin states with a distorting nearest neighbor
interaction. The term antiferrimagnetic denotes a mixed composition of AFM and
FM aligned spins, which result in a structure that still produces a net
magnetic moment (shown in Fig. 1(b)). Using a Heisenberg spin-spin exchange
Hamiltonian with on-site anisotropy, we determine the energy phase diagrams
for this distorted system as well as the spin-wave dynamics. Furthermore, we
look beyond the 120∘ phase and push the analytical limit by generalizing the
in-plane magnetism to a 120∘ \+ $d\theta$ phase, where $d\theta$ goes from
-120∘ (FM phase) to 60∘ (AfM phase).

These calculations allow for a detailed understanding of the changes in the
spin-wave dynamics expected for various configurations of the magnetic Kagome
lattice, which is useful for experimental identification and interpretation.
Furthermore, deviation from these spin-wave dynamics provides evidence for
non-standard interactions like those determined by the Kitaev model or other
non-Heisenberg models.

Figure 3: The spin angles for the three-atom structure are represented as
deviations from a default state both in and out of the xy-plane. (a) The
azimuthal spin angles $\theta$, illustrated by the red arrows on each atom,
are represented as deviations from the angles $\theta_{FM}$ (gray arrows) that
reflect the FM state. As the significance of these angles is their relation to
one another rather than absolute orientation, the angle $\theta_{A}$ was set
arbitrarily to 0. Deviation angles $q$ and $r$ are taken as the difference
between angle $\theta$ and $\theta_{FM}$. (b) The in-plane spin angles $\phi$,
illustrated by the red arrows on each atom, are represented as deviations from
the angles $\phi_{120}$ (gray arrows) which make up the 120∘ phase. The angle
$\phi_{A}$ is set as $\pi/2$. Deviation angles $g$ and $h$ are taken as the
difference between angle $\phi$ and $\phi_{120}$ Figure 4: (a) Phase diagram
for spin waves out of the plane for all values of $J$ and $\mathcal{D}$. (b)
and (c) are the in-plane diagrams for all values of $\mathcal{D}$, where (b)
is for positive $J$ values and (c) for negative values. The phase borders were
determined by setting the simplified classical energies for each pair of
phases equal to each other. The configuration for each region was determined
by testing which phase offered the lowest energy for the $\alpha$ and $\beta$
values encompassed by that region. There may be other spin configurations to
consider in thoroughly characterizing these phases. However, these would
require a considerable amount of extra parameterization. To remain within two
dimensions and make as few assumptions as possible, we consider only the most
basic configurations for this portion of the analysis.

##  II Spin-exchange Hamiltonian

To gain a complete understanding of how local interaction changes can affect
the spin state of the Kagome lattice, a Heisenberg spin-spin Hamiltonian is
used to model first-order interactions with variable exchange values (shown in
Fig. 2(a)). From this Hamiltonian, the energy eigenstates and spin-wave
dynamics are examined within the analytical limit and expanded numerically to
explore the spin evolution of the magnetic properties, where the spin-spin
exchange Hamiltonian with $z$-axis anisotropy is given as

$H=-\frac{1}{2}\sum_{i\neq
j}J_{ij}\bar{S}_{i}\cdot\bar{S}_{j}-D\sum_{i}\bar{S}_{iz}^{2}.$ (1)

Here, $J_{i,j}$ is the exchange interaction between the spin sites and $D$ is
the anisotropy energy[6, 12]. Since the Kagome lattice can produce numerous
collinear and non-collinear phases, we must be able to consider the azimuthal
and polar angles of the site spins. Therefore, the Hamiltonian must be shifted
to study non-collinear spin configurations. As described in Ref. [6], the spin
rotation is performed using an Euler rotation matrix $U$ upon the Hamiltonian,
which is dependent on spherical coordinates $\theta$ (azimuthal) and $\phi$
(polar) between the two spins [6, 5]. Applying this rotation to the
Hamiltonian gives

$H=\displaystyle-\frac{1}{2}\sum_{i\neq
j}J_{ij}\bar{S}_{i}\cdot\underline{U}_{i}\underline{U}_{j}^{-1}\bar{S}_{j}-D\sum_{i}\underline{U}_{i}^{-1}\bar{S}_{iz}^{2}.$
(2)

Figure 5: $\alpha$ as a function of $q$ and $g$ such that classical energy is
minimized for the out-of-plane and in-plane spin configurations, respectively,
where $J$ is positive. Negative $J$ values are characterized by a reflection
of this graph about the $\alpha$ axis.

Through a $(1/S)$ Holstein-Primakoff expansion of this Hamiltonian, the system
breaks into various orders

$H=E_{0}+H_{1}+H_{2}+\cdot\cdot\cdot.$ (3)

Here, $E_{0}$ is the classical energy, which can be used to determine the
system’s overall ground state for a given spin configuration. The $H_{1}$ term
is the vacuum contribution to the spin dynamics, which vanishes in a stable
system. $H_{2}$ produces the first-order contributions to the spin-wave
dynamics within the quadratic limit. Higher order terms can also be
determined. However, these terms produce quantum fluctuations which can be
ignored for large $S$[6].

Figure 6: The spin-wave dynamics for the simplest spin configurations, where
$J=J^{\prime}=J^{\prime\prime}$ ($\alpha=\beta=\mathcal{J}$) and no magnetic
field is present. (a), (d), (g), and (j) draw out the spin waves through the
path in $k$-space which passes through symmetry points $\Gamma$, M, K, then
back to $\Gamma$ for the out-of-plane FM, out-of-plane AfM, in-plane FM, and
120∘ configurations, respectively. (b) and (c) are the out-of-plane FM
($\mathcal{J}=1$) spin waves with anisotropies $\mathcal{D}$ = 0 and
$\mathcal{D}$ = 1, respectively. (e) and (f) show a similar evolution in the
AfM configuration ($\mathcal{J}=-1$). However, since the equilateral spin
waves are not stable without anisotropy, the anisotropies here are
$\mathcal{D}$ = 1 (e) and $\mathcal{D}$ = 2 (f). (h) and (i) are the in-plane
FM ($\mathcal{J}=1$) spin waves with anisotropies $\mathcal{D}$ = 0 and
$\mathcal{D}$ = -1, respectively. (k) and (l) are spin waves for the 120∘
($\mathcal{J}=-1$) configuration. Although stable with none (k), (l) reflects
a large anisotropy value of -3 to illustrate the full transformation that this
configuration’s spin waves undergo as this parameter is amplified. Color
scales applied to the spin waves are grouped by configuration/column. All 3D
figures include a projection of the central energy level of the FM
configuration spin waves onto the $k$-plane to visualize the symmetry points.
The heat map projection’s colors are scaled relative to that of that single
energy level’s minima and maxima, as in figure 2(b)

The Kagome system can produce many spin configurations. The most well-known
are collinear FM and AfM systems. The spins in two configurations can be
either in-plane or out-of-plane and make distinct changes in the spin
dynamics. Outside of the collinear systems is the non-collinear 120∘ phase
[shown in Figs. 1(c)-(e)], which is typically an in-plane rotation of spin
produced through AFM frustration.

First, we examine the classical energy for this system and then move on to the
spin-dynamics. Within these sections, we will investigate the out-of-plane and
in-plane structures for the FM and AfM phases. With those established, we will
then enable a magnetic distortion of the exchange interactions and simulate
the evolution of the spin dynamics of both collinear and non-collinear phases
with various parameters.

##  III Classical Energy

Before determining the spin-wave dynamics for specific configurations, the
classical energy of the system is used to determine the ground-state spin
configuration within the 3 sublattice (3-SL) Kagome system.

Figure 7: Variously distorted isosceles spin waves for the out-of-plane
configurations where, in all cases, $\mathcal{J}=1$. The first row shows the
spin-wave energy values through the path in $k$-space that passes through
symmetry points $\Gamma,M,K$, and back to $\Gamma$. These 2-D representations
exclude half of the $\alpha$ values shown in the 3D graphs due to a large
amount of overlap rendering the paths unreadable. The $\alpha$ value for each
row of graphs is given by its central-column triangle diagram. The coloring of
atoms indicates the AfM configuration. While the FM state would be more
accurately by one uniform color, the dichromatic coloring of the AfM phase was
chosen to represent either state here. Unstable spin waves were included to
illustrate the results of the exchange interaction manipulations that oppose
the intuited signs. For both configurations, the evolution of these graphs
with anisotropy was included in its second column. All 3D FM graphs (first two
columns) are color-grouped together and all 3D AfM graphs are similarly
grouped. Figure 8: Variously distorted isosceles spin waves for the in-plane
FM and AfM configurations, each of which has $\mathcal{J}=1$. The first row
shows the spin-wave energy values through the path in k-space that passes
through symmetry points $\Gamma$, M, K, then back to $\Gamma$. These each
exclude half of the $\alpha$ values shown in the 3D graphs, as a large amount
of overlap with the other values made them too difficult to read. The $\alpha$
value for each row of graphs is given by the diagram in the central column.
For both configurations, the evolution of the spin waves with anisotropy was
included in the second column for each. FM graphs are color-grouped together
and all AfM graphs are similarly grouped.

The classical energy of each magnetic configuration is given by

$\begin{array}[]{ll}E_{0}&\displaystyle=-\frac{1}{2}\sum_{i,j}J_{ij}S^{2}\Big{(}\sin(\theta_{i})\sin(\theta_{j})\cos(\phi_{j}-\phi_{i})\\\
&+\cos(\theta_{i})\cos(\theta_{j})\Big{)}\displaystyle-
DS^{2}\sum_{i}\cos^{2}(\theta_{i}),\end{array}$ (4)

where the various spin angles are illustrated in Fig. 3[6]. Within the
different spin configurations, we will specialize angles for the purpose of
achieving analytical and understandable solutions.

###  III.1 Out-of-plane spin configurations

Figure 9: The FM spin waves configuration with scalene exchange interaction
proportions. The $\alpha$ and $\beta$ values for each are separated by a comma
as $\alpha$, $\beta$ in the corresponding triangle diagrams. All graphs are
grouped together in a single color group.

The out-of-plane spin configurations have collinear spins with $\theta$ = 0 or
$\pi$. Here, the classical energy can be written as

$\begin{array}[]{ll}\displaystyle\frac{E_{0}}{|J|S^{2}}=&-\frac{2}{3}\Big{(}\alpha\cos(\theta_{A})\cos(\theta_{B})+\beta\cos(\theta_{A})\cos(\theta_{C})\\\
&\displaystyle+\mathcal{J}\cos(\theta_{B})\cos(\theta_{C})\Big{)}-\mathcal{D},\end{array}$
(5)

where $\mathcal{J}\equiv J/|J|=\pm 1$ and determines the general exchange
interaction of the system. Additionally, $\mathcal{D}\equiv D/|J|$, $\alpha$
$\equiv$ $\frac{J^{\prime}}{|J|}$ and $\beta$ $\equiv$
$\frac{J^{\prime\prime}}{|J|}$. By pulling out an overall $J$ and looking at
the ratio of exchange interactions, we reduce the number of variables and
produce an overall scaling factor that helps determine the energy scale of a
material system.

For ease of calculation, angles $\theta_{A}$, $\theta_{B}$, and $\theta_{C}$
are replaced with the values 0, 0 + $q$, and 0 + $r$, respectively, where $q$
and $r$ are deviation angles from the FM state, as illustrated in Fig. 3(a).
The energy becomes

$\displaystyle\frac{E_{0}}{|J|S^{2}}=-\frac{2}{3}\Big{(}\alpha\cos(q)+\beta\cos(r)+\mathcal{J}\cos(q)\cos(r)\Big{)}-\mathcal{D}.$
(6)

Assuming no magnetic field nor anisotropy, the energy function was inspected
for minima and maxima. The values for $\alpha$ and $\beta$ that minimize
$E_{0}$ out the plane belong to the sinusoids $\alpha=-\mathcal{J}\cos(r)$ and
$\beta=-\mathcal{J}\cos(q)$. It is important to remember that this
relationship is only accurate for $\theta$ values that are multiples of $\pi$,
as the consideration of any other value reinstates the necessity for the first
term of Eq. 4, leading these relationships to oscillate discretely between
$-1$ and 1 as $(-1)^{n+1}\mathcal{J}$ where $n$ is $q/\pi$ for $\alpha$ and
$r/\pi$ for $\beta$.

The case where $\beta$ and $\alpha$ are equal to each other, but not
necessarily to $J$, is termed the “isosceles” case. While $r$ and $q$ are not
mathematically required to be equivalent when $\alpha$ and $\beta$ are equal,
it is likely that the angles will be equal through symmetry. The minimizing
formula is then $\alpha=-\mathcal{J}\cos(q)$, which is shown in Fig. 5. In
order to minimize $E_{0}$, $J^{\prime}$ may never exceed $J$ in magnitude.
Assuming a positive $J$ value, the maximum value $J^{\prime}=J$ requires the
AfM configuration ($q$ = $\pi$) to minimize energy and the minimum value
($J^{\prime}=-J$) requires the FM ($q$ = 0) configuration for minimization.

Figure 10: The spin waves for the AfM configuration with scalene exchange
interaction proportions. The $\alpha$ and $\beta$ values for each case are
separated by a comma as $\alpha,\beta$ in the corresponding triangle diagrams.
A single color group encompasses all 3D graphs shown.

The simplest case, where all exchange interactions are equal
($\beta=\alpha=\mathcal{J}$), is termed the “equilateral” case. In this state,
the value for $g$ that minimizes $E_{0}/|J|S^{2}$ is $\pi$, which represents
the AfM configuration. $E_{0}/|J|S^{2}$ is maximized by $g=0$, indicating the
FM configuration. These configurations are therefore the focus of our out-of-
plane analyses.

The classical energy for the out-of-plane FM configuration ($\theta=0$) is
reduced to

$\displaystyle\frac{E_{0,FM}}{|J|S^{2}}=-\frac{2}{3}(\alpha+\beta+\mathcal{J})-\displaystyle\mathcal{D}$
(7)

and the AfM energy to

$\displaystyle\frac{E_{0,AfM}}{|J|S^{2}}=-\frac{2}{3}(-\alpha-\beta+\mathcal{J})-\mathcal{D}.$
(8)

These energies were used to generate the phase diagram in Fig. 4(a).

###  III.2 In-plane spin configurations

For in-plane configurations ($\theta=\pi/2$), the classical energy becomes

$\begin{array}[]{ll}\displaystyle\frac{E_{0}}{|J|S^{2}}=&\displaystyle-\frac{2}{3}\Big{[}\alpha\cos(\phi_{A}-\phi_{B})+\beta\cos(\phi_{A}-\phi_{C})\\\
&+\mathcal{J}\cos(\phi_{B}-\phi_{C})\Big{]}.\end{array}$ (9)

To reduce the number of variables, the energy is rearranged in terms of
deviation from the $120^{\circ}$ phase such that $\phi_{A}$, $\phi_{B}$, and
$\phi_{C}$ become $(1/2)\pi$, $(7/6)\pi+g$, and $(11/6)\pi–h$, respectively,
as shown in Fig. 3(b). Assuming all variables are real valued, the energy
becomes

$\begin{array}[]{ll}\displaystyle\frac{E_{0}}{|J|S^{2}}=&\displaystyle\frac{2}{3}\Big{[}\textstyle\alpha\sin\Big{(}\frac{\pi}{6}+g\Big{)}+\beta\sin\Big{(}\frac{\pi}{6}+h\Big{)}\\\
&+\cos\Big{(}\textstyle\frac{\pi}{3}+g+h\Big{)}\Big{]}.\end{array}$ (10)

It is important to note that since we are using the isotropic Heisenberg
model, all spins in this configuration can be rotated in the plane by any
phase factor of $\phi^{\prime}$ as it is energetically degenerate. We choose
this particular orientation to simplify the expressions. Using this system,
the relationships between spin angle and exchange interaction strength which
minimize the classical energy are determined analytically as:

$\displaystyle\alpha=\mathcal{J}\frac{\sin\Big{(}\frac{\pi}{3}+g+h\Big{)}}{\cos\Big{(}\frac{\pi}{6}+g\Big{)}}~{}{\rm
and}~{}\displaystyle\beta=\mathcal{J}\frac{\sin\Big{(}\frac{\pi}{3}+g+h\Big{)}}{\cos\Big{(}\frac{\pi}{6}+h\Big{)}}.$
(11)

Unlike the out-of-plane case, these relationships hold true for any value of g
and h.

Figure 11: Isosceles distortions of spin waves for the 120∘ configuration. The
first row shows the spin-wave energy values through the path in $k$-space that
passes through symmetry points $\Gamma$, M, K, then back to $\Gamma$. The
$\alpha$ value for each row of 3D graphs is given by the diagram in the left-
hand column. Note that there are no stable spin waves for this configuration
without anisotropy, aside from the equilateral ($\alpha=-1$) mode. This being
the case, multiple magnitudes of anisotropy were included to better illustrate
the stable possibilities for the 120∘ phase. Because no positive-$\alpha$
state could be stabilized with anisotropy lesser in magnitude than $3|J|$,
only negative values were included here. A single color group encompasses all
3D graphs in this figure.

In the isosceles case, the minimizing relationship becomes

$\displaystyle\alpha=\mathcal{J}\frac{\sin\Big{(}\frac{\pi}{3}+2g\Big{)}}{\cos\Big{(}\frac{\pi}{6}+g\Big{)}}.$
(12)

Equation (12), illustrated in Fig. 5, shows that to minimize $E_{0}$, the
absolute value of exchange interaction $J^{\prime}$ may never exceed twice the
value of $J$. It may also be observed that when $J$ is positive, the maximum
value $J^{\prime}=J$ requires the AfM configuration ($g$ = $\frac{\pi}{3}$) to
minimize classical energy and the minimum value $J^{\prime}$ = $-J$ requires
the FM configuration ($g$ = $-\frac{2\pi}{3}$).

Examination of the equilateral case showed that the minima and maxima of
$E_{0}/|J|S^{2}$, regardless of the sign of $J$, are produced by the 120∘ and
FM configurations, respectively. The 120∘ phase has the lowest energy of all
equilateral structures in and out of the plane and is determined as the ground
state.

The AfM configuration features as a local minimum for the classical energy,
yet proved too unstable for spin-wave examination in the equilateral case. It
is analyzed, however, with further distortions.

The FM, AfM, and 120∘ phases will therefore be the initial subjects for in-
plane analyses. In the plane, the ferromagnetic energy is simplified to

$\displaystyle\frac{E_{0,FM}}{|J|S^{2}}=-\frac{2}{3}(\alpha+\beta+\mathcal{J}),$
(13)

the 120∘ configuration energy to

$\displaystyle\frac{E_{0,120}}{|J|S^{2}}=\frac{1}{3}(\alpha+\beta+\mathcal{J}),$
(14)

and the AfM energy to

$\displaystyle\frac{E_{0,AfM}}{|J|S^{2}}=\frac{1}{3}(-\alpha-\beta+\mathcal{J}).$
(15)

Overall, the classical energies allow for the general understanding of where
these few configurations are stable with respect to each other. It is
important to note that other magnetic structures, especially canted non-
collinear or $>$3 SL magnetic configurations could also exist. However, there
are too many to meaningfully characterize all possibilities here. Therefore,
it is essential to look at the spin dynamics to gain insight into the spin
configurations’ stability. If a system is a stable ground state according to
the classical considerations but unstable from the standpoint of the spin
dynamics, then this is an indication of the presence of a canted non-collinear
state. Therefore, the next step is to evaluate these configurations for
dynamic stability.

##  IV Spin-wave Dynamics

Exploring the spin-wave dynamics for the five most interesting spin
configurations determined by the classical energy, we first generate solutions
to the simplest case where all exchange interactions are equal, no magnetic
field is applied, the spin angles are held constant, the physical distance
between each atom remains static, and the only variation is anisotropy. Beyond
this, we produce spin-wave solutions for the sublattice where exchange
interactions are no longer equal, first exploring only one distortion (the
isosceles case), then examining the case where no two interactions are equal,
termed the “scalene” case. Considering purely in-plane spin configurations,
the latter two cases are further probed by varying the spin angles. The varied
spin angles considered belong to the classical energy minimizing relationships
discussed in the previous section.

There are a few formatting rules applied throughout the figures in this next
section. For every 3-D spin-wave graph, a color scale is applied which spans
all colors from purple to red, where purple reflects the minimum $\omega/|J|S$
value 0 and red indicates the maximum $\omega/|J|S$ value between all graphs
in the group of graphs to which that color scale is applied. For example, in
Fig. 6, the $120^{\circ}$ graphs (k) and (l) are grouped, and the color scale
for both spans from purple at 0 to red at 4.5, where 4.5 is the highest
$\omega/|J|S$ value between both graphs. The color grouping used between
graphs is specified in each figure caption.

To contextualize the spin waves’ form with respect to the reciprocal lattice,
all 3D spin-wave graphs include a projection of the equilateral FM
configuration’s central energy level (pictured in Fig. 6(a)) onto the
$k$-plane. The projection is a heat map whose colors are scaled relative to
that single energy level’s minima and maxima, as in Fig. 2(b).

The 2D spin-wave graphs in the isosceles figures include only three $\alpha$
values, as opposed to the six considered in the 3D graphs, because the
inclusion of all six $\alpha$ values created massive overlap in the lines
defining the energies, rendering the graphs unreadable. Any destabilizing
$\alpha$ values were thus excluded from these representations to offer greater
clarity.

All spin-wave figures after the equilateral figure include small triangular
diagrams. These are included to visualize the proportions of the exchange
interaction strengths between atoms. Red lines indicate negative valued
interactions and black positive. The atoms are colored to reflect the spin
configurations as in Fig. 1, save those in the isosceles graphs: as the FM and
AfM configurations share triangle diagrams, the diagrams were colored to
reflect the AfM mode.

Figure 12: Isosceles distortions of in-plane spin waves according to the
relationship defined in Eq. 12. Aside from the two graphs whose $\alpha$ value
approaches 2, all spin waves pictured have anisotropy $\mathcal{D}$ = 1. Only
negative $\alpha$ values are depicted because when both the $\alpha$ and $J$
signs are reversed for any given combination, they produce the same spin
waves, with the exception of the addition of anisotropy. Additionally, the
spin rotating angles are provided in Table 1, and two groups are defined for
clarity in color scaling, where those without anisotropy are grouped together
and those with are grouped separately.

###  IV.1 Out-of-plane Configurations

####  IV.1.1 Ferromagnetic Phase

Since all spins in the FM phase point in the same direction, it is expected
this phase is stable when each of the exchange interactions between these
aligned sites ($\mathcal{J}$, $\alpha$, and $\beta$) are positive. We examine
the effects of interaction competition on the FM phase to understand when it
becomes unstable from a spin-wave standpoint.

Figure 6(b) and (c) show how the equilateral ($\alpha$ = $\beta$ =
$\mathcal{J}$ = +1) out-of-plane FM spin waves evolve with anisotropy. The
anisotropy present in Fig. 6(c) is equal to the exchange interaction
($\mathcal{D}$ = +1). It is observed that, without anisotropy, the spin waves
are identical to the FM configuration pointing in the plane Fig. 6(h). As
expected, easy-axis anisotropy present in the out-of-plane configuration
simply adds uniformly across the $k$-plane to all energy states. The spin-wave
energies $\omega_{i}$ are represented analytically as

$\begin{array}[]{ll}\omega_{0}/|J|S=&2\mathcal{D}+6\mathcal{J}\\\
\omega_{\pm}/|J|S=&2\mathcal{D}+3\mathcal{J}\pm\sqrt{8\mathcal{J}^{2}\cos(k_{x})^{2}\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})^{2}+8\mathcal{J}^{2}\cos(\frac{k_{x}}{2}\pm\frac{\sqrt{3}k_{y}}{2})\cos(k_{x})\sin(k_{x})\sin(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})+\mathcal{J}^{2}}.\\\
\end{array}$ (16)

Moving to the first distortion of exchange interactions, the spin-wave
dynamics of various isosceles states ($\alpha$ = $\beta$) are explored for the
FM configuration in Fig. 7, which illustrates the transformation of the spin
waves as the value $\alpha$ changes. With an $\alpha$ value of 2, the
interaction between atoms B and C has half the magnitude of the other two.
This distortion is reflected in the dispersion shown in the $k_{y}$ direction
when inspecting the spin-wave diagram. A similar, perpendicular phenomenon can
be seen in the $\alpha=\frac{1}{2}$ case. The inclusion of anisotropy energy,
predictably, has a similar additive effect to that observed in the equilateral
spin waves. As the spins are uni-directional in a ferromagnet, it is expected
that the exchange interactions are all positive. This instability is confirmed
visually by inspecting the negative $\alpha$-valued graphs, except for the
$\alpha$ = -2 spin waves, which seem to become stable due to a symmetry effect
in the interactions. Larger negative $\alpha$ values produce equally stable-
looking waves. It becomes apparent, however, that this state is not truly
stable when considering its behavior with anisotropy. Notice that the graph in
the second column, which reflects $\mathcal{D}=+1$, has lower overall energy
than its no-anisotropy counterpart, which is indicative of an unstable system.
This is in agreement with the classical energy, which indicates that this
state is only metastable. A similar effect in the AfM configuration is
realized in the next analysis. While calculable, the analytical representation
of the isosceles spin waves is too large to be presented here and cannot be
used to provide insight.

Figure 9 shows the scalene spin waves with all permutations of a 1-2-3
proportionality between $\alpha$, $\beta$, and $J$, given a static FM spin
configuration. It is observed that, as in the isosceles case, both the in-
plane and out-of-plane FM configurations produce identical spin waves
throughout the variation of exchange interaction, so long as anisotropy is not
considered. However, their changes with respect to anisotropy differ. For
simplicity, anisotropy was excluded from the scalene discussion. These
visualizations illustrate the dispersion that occurs along the direction of
the most robust exchange interaction.

####  IV.1.2 Antiferrimagnetic Phase

In this configuration, sites B and C in Fig. 2(a) are aligned and site A
points in the opposite direction. It is thus expected that this configuration
will be stabilized by a positive exchange interaction $J$ and negative
interactions $J^{\prime}$ and $J^{\prime\prime}$. However, with frustration in
this system, the boundaries of stability are of interest, as with the FM
phase.

The out-of-plane, equilateral AfM energies are unstable without anisotropy.
The anisotropies $\mathcal{D}$ present in Fig. 6 (e) and (f) are valued at +1
and +2, respectively. As discussed prior, it is expected that the AfM
configuration contains one positive exchange interaction (between the two
same-spin atoms) and two negative interactions. However, as all exchange
interactions are strictly equal in the equilateral case, all exchange
interactions were assigned a negative value to reflect the net negativity of
the three interactions together. Therefore, this state has
$\mathcal{J}=\alpha=\beta=-1$. The analytical representation of these energies
is too large for inclusion.

Figure 7 includes the isosceles ($\alpha$ = $\beta$) out-of-plane AfM spin
waves. The expected stabilizing state is $\alpha$ = -1, while $J$ is positive,
as the exchange interactions $\alpha$ and $\beta$ are equal and belong to the
opposite-spin atom pairs. The spin waves at this $\alpha$ value reflect this
expectation in their visually apparent stability. Interestingly, it is shown
that various $\alpha$ values that stabilize this otherwise unstable
configuration. Similar to the FM phase, the AfM system has an unexpected
metastable state where all three exchange interactions are positive with an
$\alpha$ value of +2. Again, we see an overall lowering of energy when
anisotropy is included, revealing the volatile nature of this arrangement.

The scalene spin waves with all permutations of a 1-2-3 proportionality
between $\alpha$, $\beta$, and $J$, for an AfM spin configuration, are shown
in Fig. 10. As in the isosceles case, the in-plane and out-of-plane
configurations, with no anisotropy, produce identical spin waves throughout
the exchange interaction variation. For the sake of simplicity, anisotropy-
inclusive spin waves are excluded from the narrative. As with the isosceles
case, dispersion in the spin waves occurs along the direction of the strongest
exchange interaction.

Isosceles spins
---
$\mathcal{J}$ |  $\alpha$ |  $\beta$ |  $\phi_{A}(^{\circ}$) |  $\phi_{B}(^{\circ}$) |  $\phi_{C}(^{\circ}$)
-1 |  -2 |  -2 |  90 |  270.0 |  270.0
-1 |  -3/2 |  -3/2 |  90 |  228.6 |  311.4
-1 |  -1 |  -1 |  90 |  210.0 |  330.0
-1 |  -3/4 |  -3/4 |  90 |  202.0 |  338.0
-1 |  -1/2 |  -1/2 |  90 |  194.5 |  345.5
1 |  -2 |  -2 |  90 |  90.00 |  90.00
1 |  -3/2 |  -3/2 |  90 |  131.4 |  48.60
1 |  -1 |  -1 |  90 |  150.0 |  30.00
1 |  -3/4 |  -3/4 |  90 |  158.0 |  22.00
1 |  -1/2 |  -1/2 |  90 |  165.5 |  14.50
Table 1: The spin angles calculated for $\alpha$ values by the relationship in
Eq. 12

###  IV.2 In-plane Configurations

####  IV.2.1 Ferromagnetic Phase

Figure 6 (h) and (i) show the evolution of the equilateral in-plane FM spin
waves with the presence of anisotropy. Easy-plane anisotropy added to the in-
plane configuration stretches the spin waves as interactions strain the easy
orientation. For this configuration, the spin-wave energies can be represented
analytically and shown to be

$\begin{array}[]{ll}\omega_{0}/|J|S=&2\sqrt{9\mathcal{J}^{2}-3\mathcal{D}\mathcal{J}}\\\
\omega_{\pm}/|J|S=&\biggl{(}8\mathcal{J}^{2}\sin(k_{x})\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\sin(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\cos(k_{x})+8\mathcal{J}^{2}(\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2}))^{2}(\cos(k_{x}))^{2}-6\mathcal{D}\mathcal{J}+10\mathcal{J}^{2}\\\
&\pm
2\mathcal{J}\sqrt{8(\mathcal{D}-3\mathcal{J})^{2}(\sin(k_{x})\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\sin(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\cos(k_{x})+\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})^{2}(\cos(k_{x}))^{2}+1)}\biggr{)}^{\frac{1}{2}}.\\\
\end{array}$ (17)

Various isosceles FM spin waves are illustrated in Fig. 8. It is shown that,
as in the equilateral case, the spin waves are identical to the out-of-plane
configuration where anisotropy is not considered. Otherwise, the inclusion of
anisotropy energy has the same stretching effect as that in the equilateral
spin waves. Like the out-of-plane FM configuration, the analytical
representation of the energies that describe an isosceles in-plane FM state
were too large to include here.

As no anisotropy was considered for the scalene cases, Fig. 9 represents both
in- and out-of-plane configurations and is not discussed redundantly.

####  IV.2.2 Antiferrimagnetic Phase

Even with the presence of anisotropy five times the magnitude of the exchange
interaction, the equilateral spin waves for the in-plane AfM configuration
were unstable and therefore excluded from figure 6.

Figure 8 illustrates isosceles in-plane AfM spin waves. As in the out-of-plane
case, there are various $\alpha$ values which stabilize this unstable
configuration. Without anisotropy, these spin waves are identical to the out-
of-plane spin waves. Yet, when present, a stretching of energy is observed as
a result of anisotropy.

Scalene spinwaves for the in-plane AfM configuration with $\mathcal{D}=0$ are
shown in figure 10 and are identical to those of their out-of-pane
counterparts.

Figure 13: Scalene distortions of in-plane spin waves according to the
relationships in Eq. 11. All spin waves shown have anisotropy $\mathcal{D}$ =
1. The $\alpha$ and $\beta$ values for each graph are presented within the
triangle diagrams, respectively. Additionally, the spin rotating angles are
provided in Table 2. A single color group encompasses all graphs.

####  IV.2.3 120∘ Phase

The ground state of the Kagome lattice behaves peculiarly in contrast to the
behaviors of the FM and AfM configurations. The no-anisotropy spin waves in
Fig. 6 (k) show a degeneracy of energy levels for the 120∘ phase. This
degeneracy is lost, however, as anisotropy increases. As anisotropy energy
increases to outweigh the exchange interactions, Dirac nodes form in the
system. While lower values of anisotropy maintained stable spin waves, the
relatively large amount of $\mathcal{D}$ = -3 is included for the purpose of
illustrating this phenomenon. Lesser anisotropies are included in the 2-D spin
waves graph (j) to convey its interesting path to this point. Here, the
analytical spin-wave energies can be shown as

$\begin{array}[]{ll}\omega_{0}/|J|S=&\sqrt{6\mathcal{J}\mathcal{D}}\\\
\omega_{\pm}/|J|S=&\biggl{(}-4\mathcal{J}^{2}\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})^{2}\cos(k_{x})^{2}-4\mathcal{J}^{2}\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\sin(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\cos(k_{x})\sin(k_{x})+3\mathcal{D}\mathcal{J}+4\mathcal{J}^{2}\\\
&\pm\mathcal{J}\sqrt{(8\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})^{2}\cos(k_{x})^{2}+8\cos(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\sin(\frac{k_{x}}{2}-\frac{\sqrt{3}k_{y}}{2})\cos(k_{x})\sin(k_{x})+1)\mathcal{D}^{2}}\biggr{)}^{\frac{1}{2}}.\\\
\end{array}$ (18)

The distortion of exchange energies did not provide any additional stability
to the 120∘ configuration. In fact, any distortion destabilized the system and
required anisotropy on the order of the exchange interaction for stability.
Even anisotropy, however, could not stabilize any state whose $\alpha$ value
had a sign opposite to that of $J$. This instability is predicted by the state
being a particular case of energy minimization related to the $\alpha$ and
$\beta$ values.

Because isosceles distorting exchange interactions for the 120∘ phase offered
no stability, scalene spin wave analyses for this configuration were omitted
entirely.

Scalene spins
---
$\mathcal{J}$ |  $\alpha$ |  $\beta$ |  $\phi_{A}(^{\circ}$) |  $\phi_{B}(^{\circ}$) |  $\phi_{C}(^{\circ}$)
-1 |  -5/4 |  -3/2 |  90 |  217.1 |  311.6
-1 |  -3/4 |  -5/4 |  90 |  175.1 |  306.7
-1 |  -1/2 |  -3/4 |  90 |  152.7 |  306.3
-1 |  -3/4 |  -1/2 |  90 |  233.7 |  27.27
-1 |  -5/4 |  -3/4 |  90 |  233.3 |  4.940
-1 |  -3/2 |  -5/4 |  90 |  228.4 |  322.9
Table 2: The spin angles calculated for various combinations of $\alpha$ and
$\beta$ determined by the relationships in Eq. 11

###  IV.3 Spin Angle Distortions

The 120∘ phase showed to be quite a unique configuration. While it is the
ground state when all exchange interactions are equal, any distortion of the
exchange parameters destabilize the phase, unlike the other configurations
which could maintain stability through some distortion. In light of this, we
looked for stability in new states whose spin angles and exchange interactions
together minimize the classical energy.

Before considering all distorting parameters, we investigate the isosceles
($\alpha$ = $\beta$) states under the assumption of the simplest case, where
rotation angles $g$ and $h$ are also equivalent, as described in Eq. 12. It
can be inferred from this minimizing relationship, illustrated in Fig. 5, that
there are two angle $g$ solutions for any given $\alpha$. However, these two
values produce equivalent spin waves as the two $g$ values produce the same
two values for spin angles for atoms B and C, but the angles assigned to each
site are reversed. Spinwaves for some interesting $\alpha$ values are
illustrated in Fig. 12. As nearly every result was unstable, an anisotropy of
$\mathcal{D}=-1$ was applied to all states except those two which have an
$\alpha$ value approaching 2. These do not include anisotropy.

Examining further distortion, figure 13 shows scalene spin waves with various
values of $\alpha$ and $\beta$, given a negative $J$, and with spin angles
defined by the classical energy minimizing relationships in Eq. 11. As the
scalene spin waves were also unstable without anisotropy, the solutions
pictured reflect a value $\mathcal{D}=-1$. These visualizations illustrate the
dispersion that occurs along the direction of the strongest exchange
interaction. The fact that these polar spin angle solutions which minimize the
classical energy rely on anisotropy for stability may indicate that the ground
state configurations for these distorted exchange interaction states are
canted out of the lattice plane to some degree.

Overall, these results indicate that purely in-plane magnetic configurations
which deviate from the FM, AfM, and 120∘ states require anisotropy to
stabilize within the Kagome lattice. We expect that there will be a reduction
in the required anisotropy as the spin angles are allowed to cant in any out-
of-plane direction (not necessarily along the $z$-axis) and form distinct non-
collinear phases. Further study into the realm of these out-of-plane canted
phases needs to be done. However, the results here are important for
understanding the evolution of the magnetic Kagome lattice’s spin waves with
variable exchange interactions.

##  V Discussion and Conclusion

The quest for an understanding of quantum spin states, especially the quantum
spin liquid, has led to a wealth of experimental realizations and studies on
the structural, magnetic, and thermodynamics of Kagome systems[41, 42, 43, 44,
45, 46] as well as distorted Kagome systems[47, 48, 49, 50]. While many
studies focus on the interpretation of either one material or even one
magnetic configuration, the ability to discern the various magnetic
interactions with and without distortions has been a challenge that leads most
to using numerical approaches in modeling experimental data. However,
examining these systems within an analytical limit for the symmetric systems
and evolving out into the distorted systems allows for one to gain a deeper
understanding of the effects of the interactions and how they distort the
magnetic systems.

In this paper, we examine the effects of variable first-order magnetic
distortions on the spin-wave dynamics of the Kagome lattice. Using an
isotropic spin-spin exchange Hamiltonian with $z$-axis anisotropy, we
determine the phase diagrams for various out-of-plane and in-plane spin
configurations and then examine how the spin waves are affected by exchange
interaction distortions.

By analyzing the spin waves with static spin angles and varying exchange
interactions, we gain novel insights on the effect this distortion alone has
on the magnetic identity of this sublattice. A dispersive effect on the spin
waves in accordance with exchange interaction proportionality is illustrated
in the isosceles and scalene cases for all five of the most straightforward
configurations, and some unexpectedly metastable states fell out of unlikely
exchange interaction states for the FM and AfM cases.

Although it is the ground state classically, the special-case nature of the
120∘ phase is underlined by its inability to retain stability as its exchange
interactions are distorted. This led to our employing the energy-minimizing
relationships for spin angles in seeking stable spin waves for these changing
$\alpha$ values, which pointed to even further spin distortions for stability.
The next natural step in this vein would be to numerically minimize the
classical energy with full freedom of spin angles, allowing for continuity in
both the polar and azimuthal angles. Limiting to purely in-plane or out-of-
plane angles offered analytical insights that are invaluable in characterizing
magnetic relationships. However, to more deeply describe the most natural
behaviors of atoms in this lattice, the allowance of canted configurations is
a necessary consideration.

Overall, this paper aims to provide insight into the spin dynamics of the
Kagome lattice to help in the characterization of its non-collinear phases.
Therefore, we show how the distortion of the exchange parameters and spin
angles affect the dynamics, which provides useful information for the
characterization of material systems, especially when investigating these
phases using inelastic neutron scattering.

## Acknowledgements

A.A.C. and J.T.H acknowledge support from the Institute for Materials Science
at Los Alamos National Laboratory. The work at Los Alamos National Laboratory
was carried out under the auspices of the U.S. DOE and NNSA under Contract No.
DEAC52-06NA25396 and supported by U.S. DOE (A.S.).

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