# NusaCrowd: A Call for Open and Reproducible NLP Research
in Indonesian Languages

Samuel Cahyawijaya , Alham Fikri Aji11footnotemark: 1 , Holy
Lovenia11footnotemark: 1 ,
Genta Indra Winata11footnotemark: 1 , Bryan Wilie11footnotemark: 1 , Rahmad
Mahendra, Fajri Koto,
David Moeljadi, Karissa Vincentio, Ade Romadhony, Ayu Purwarianti
IndoNLP
Equal contribution.

###### Abstract

At the center of the underlying issues that halt Indonesian natural language
processing (NLP) research advancement, we find data scarcity. Resources in
Indonesian languages, especially the local ones, are extremely scarce and
underrepresented. Many Indonesian researchers refrain from publishing and/or
releasing their dataset. Furthermore, the few public datasets that we have are
scattered across different platforms, thus makes performing reproducible and
data-centric research in Indonesian NLP even more arduous. Rising to this
challenge, we initiate the first Indonesian NLP crowdsourcing effort,
NusaCrowd. NusaCrowd strives to provide the largest datasheet aggregation with
standardized data loading for NLP tasks in all Indonesian languages. By
enabling open and centralized access to Indonesian NLP resources, we hope
NusaCrowd can tackle the data scarcity problem hindering NLP progress in
Indonesia and bring NLP practitioners to move towards collaboration.

##  1 What is  NusaCrowd?

Natural language processing (NLP) resources in Indonesian languages,
especially the local language ones, are extremely scarce and underrepresented
in the research community. This introduces bottlenecks to Indonesian NLP
research, restraining it from opportunities and hindering its progress. In
response to this issue, several Indonesian and NLP communities have sourced
various types of datasets to also be available in Indonesian languages (Winata
et al., 2022; Koto et al., 2021a; Cahyawijaya et al., 2021; Koto et al.,
2021b; Mahendra et al., 2021; Aji et al., 2021; Wilie et al., 2020; Abdul-
Mageed et al., 2020; Ilmania et al., 2018; Pan et al., 2017). However, a
significant mass of the local resources is scattered across different
platforms (Etsa et al., 2018; Apriani et al., 2016; Dewi et al., 2020), and a
lack of access to public datasets still persists Aji et al. (2022).

Figure 1: Open access to the datasheets collected is provided through
NusaCatalogue, and the dataloader scripts to retrieve the resources are
implemented in NusaCrowd Data Hub.

To address this vital problem, inspired by other open collaboration projects
Orife et al. (2020); Nekoto et al. (2020); Dhole et al. (2021); Alyafeai et
al. (2021); McMillan-Major et al. (2022); Srivastava et al. (2022); Fries et
al. (2022); Gehrmann et al. (2022), we take a step and initiate NusaCrowd, a
joint movement to collect and centralize NLP datasets in Indonesian and
various Indonesia’s local languages, and engage the linguistics community in
collaboration.

Figure 2: The outline of public Indonesian NLP datasheet submission to
NusaCatalogue.

Powered by the collective effort of our contributors, NusaCrowd aims to
increase the accessibility of these datasets and promote reproducible research
on Indonesian languages through three fundamental facets: 1) Curated public
corpora datasheet sourcing111https://indonlp.github.io/nusa-catalogue/, 2)
Open-access centralized data hub222https://github.com/IndoNLP/nusa-crowd, and
3) Promoting private-to-public data access. We maintain the quality of the
contributions, both the consolidation efforts and the programmatic means, by
enforcing a quality control with a mix of automatic and manual evaluation
schemes. NusaCrowd is currently open for contribution, the movement is held
from 25 June 2022 to 18 November 2022. Let’s bring Indonesian NLP research one
step forward together.

##  2 Contributing in  NusaCrowd

Together, contributors in NusaCrowd drive Indonesian NLP forward by developing
a multi-faceted solution to improve data accessibility and research
reproducibility. To assist our widespread open collaboration and collective
progression, we formulate three main ways of contributing in NusaCrowd in the
following sections, each corresponds to a fundamental aspect in NusaCrowd’s
main objective.

###  2.1 Submit public  NLP datasheet

We encourage contributors to register the datasheets of public datasets on
NusaCatalogue333https://indonlp.github.io/nusa-catalogue/ by submitting them
through an online form at https://forms.gle/31dMGZik25DPFYFd6. NusaCatalogue
is a public datasheet catalogue website, inspired by Alyafeai et al. (2021),
in which we list all datasets collected in NusaCrowd. We build NusaCatalogue
to improve the discoverability of Indonesian NLP datasets and to assist users
in searching and locating Indonesian NLP datasets based on their metadata.

A datasheet is a dataset metadata which contains various information about the
dataset, including but not limited to: dataset name, original resource URL,
relevant publication, supported tasks, and dataset licence. The datasheet will
be reviewed and scored within a week or two. The contribution point
calculation for the public Indonesian NLP corpora datasheet is based on three
criteria: 1) whether the relevant dataset is previously public or not, 2)
dataset quality, and 3) language rarity. Details on the scoring mechanism will
be explained in §3.1.

Once the datasheet passes the review, we will notify the responsible
contributor and list the approved dataset’s datasheet on NusaCatalogue and
also reported on NusaCrowd data hub task
list444https://github.com/orgs/IndoNLP/projects/2 so its dataloader could be
implemented. The complete flow of how to submit the public Indonesian NLP
corpora datasheet is shown in Figure 2.

Figure 3: The outline of dataloader implementation for NusaCrowd data hub.

###  2.2 Implement dataloader(s) for  NusaCrowd data hub

A large-scale centralized data hub has to be equipped with the capability of a
simple and standardized programmatic data access that spans across diverse
resources, regardless of their separate hosting locations, distinct data
structures or formats, and different configurations. For this purpose,
building NusaCrowd data hub requires a few key elements: datasheet
documentation (§2.1), task schema standardization to support common NLP tasks,
and dataloader implementation. While the datasheets become the backbone of
NusaCrowd and standardized task schemas compose the skeleton, the dataloader
implementation is the heart of NusaCrowd data hub.

Each dataset requires a specific dataloader script tailored to its source task
type, structure, and configuration to enable easy loading and enforce
interoperability. To centralize all of the Indonesian NLP resources, NusaCrowd
needs a large number of proper dataloaders to be implemented. Therefore, we
invite all collaborators to contribute through creating these dataloaders via
NusaCrowd’s GitHub555https://github.com/IndoNLP/nusa-crowd.

Firstly, a contributor can view the task list in NusaCrowd Github project,
then choose the dataset they want to implement by assigning themself to the
related issue. Afterwards, the contributor can start setting up the
environment needed for development. To help with the dataloader
implementation, we provide a template script specifying all the parts the
contributor needs to complete, and task schemas for common NLP tasks, e.g.,
knowledge base, question answering, text classification, text-to-text, text
pairs, question answering, and more. We also equip NusaCrowd’s repository with
several working dataloader scripts that the contributor can check for
examples. To fill in the details of the dataset in the dataloader, such as its
source URL or its publication, the contributor can refer to the corresponding
datasheet recorded in NusaCatalogue.

The contributor can ensure that their dataloader is implemented correctly
through a manual inspection, a direct attempt of execution, and a unit test
provided in the repository. We also encourage the contributor to tidy up their
code accordingly with our formatter before they make a pull request to submit
their changes. The implemented dataloader then will go through a code review
process by NusaCrowd maintainers. If an adjustment in the code is required,
the maintainer will request some changes and provide their feedback on a
comment on the respective pull request, so the contributor will be able to
improve the dataloader accordingly. Once two maintainers give their approvals,
the dataloader will be merged to NusaCrowd repository. The flow overview of
the dataloader implementation is depicted in Figure 3. A comprehensive guide
for contributing through dataloader implementation can be accessed
here666https://github.com/IndoNLP/nusa-crowd/blob/master/DATALOADER.md.

###  2.3 Provide information on private  NLP datasets

Figure 4: The outline of private Indonesian NLP dataset listing in
NusaCatalogue.

Many studies in Indonesian NLP use private datasets, which in turn hampers the
research reproducibility. Therefore, the last method to contribute is to list
research papers of Indonesian NLP in which the data is not shared publicly. We
then contact the author to participate in open data access and ask for their
approval to include the dataset in NusaCrowd. Contribution points for the
authors that release their data to public will follow the scoring defined in
§3.3. The paper lister will also be awarded with a contribution point.

As far as we know, there is no data or analysis yet on why local researchers
prefer not to share their dataset publicly. Some reasons that we are aware of
include: 1) not accustomed to open data and open research, 2) restricted by
the university or funding policy, or 3) keep the data private as property. By
listing research with private datasets, we can additionally ask the authors
about their consideration to improve our understanding on this matter. Steps
to provide information on private Indonesian NLP datasets are illustrated in
Figure 4.

##  3 Contribution Point

To support fairness and transparency for all of our contributors, we establish
a scoring system of which co-authorship eligibility will be decided from. To
be eligible as a co-author in the upcoming NusaCrowd publication, a
contributor needs to earn at least 10 contribution points. The score is
aggregated from all the contributions made by the contributors. In order to
earn contribution points, we introduce three different methods to contribute
(for the method details, see §2): 1) submitting public Indonesian NLP corpora
datasheet, 2) implementing dataloader(s) for NusaCrowd data hub, and 3)
provide information on non-public Indonesian NLP datasets. The point for each
type of contribution is described in the following paragraphs.

Figure 5: A glance at the contribution matrix used for recapitulation. The
contributor list is clipped for simplicity.

###  3.1 Public  NLP corpora datasheet

A contributor can help to register public NLP corpora in NusaCrowd. For any
datasheet listed, the contributor is eligible for +2 contribution point as a
referrer. To support the development of local language datasets, we provide
additional contribution points according to the rarity of the dataset
language. Specifically, a contributor of any Sundanese (sun), Javanese (jav),
or Minangkabau (min) dataset, will receive +2 contribution points, while a
contributor of any other local language dataset will be granted +3
contribution points.

In addition, to encourage more diverse NLP corpora, we provide additional +2
contribution points for tasks that are considered rare. Based on our
observation, we find that the common NLP tasks in Indonesian languages
include: machine translation (MT), language modeling (LM), sentiment analysis
(SA), and named entity recognition (NER). All other NLP tasks are considered
rare and are eligible for the +2 contribution points. Lastly, we also notice
that publicly available Indonesian NLP corpora involving another modality
(e.g., speech or image) are very scarce, for instance: speech-to-text or
automatic speech recognition (ASR), text-to-speech (TTS) or speech synthesis,
image-to-text (e.g., image captioning), text-to-image (e.g., controllable
image generation), etc. To encourage more coverage over these data, we will
give additional +2 contribution points for the relevant datasheets submitted.

We understand that dataset quality can vary a lot. To support fairness in
scoring datasets with different qualities, for any dataset that does not pass
a certain minimum standard, 50% penalty will be applied. This penalty affects
any dataset that is collected with: 1) crawling without manual validation
process, 2) machine or heuristic-rule labelled dataset without manual
validation, and 3) machine-translated dataset without manual validation.

###  3.2 Implementing  dataloader

A contributor can help to implement a dataloader for any dataset listed on the
task list in NusaCrowd GitHub project (see §3.2). As a rule of thumb, one
dataloader implementation is generally worth 3 contributions points. However,
there are some exceptions where a dataloader can be worth more. The
contribution points will be counted once the respective pull request is merged
to master.

###  3.3 Finding and opening private  NLP datasets

Contributors can help to list research papers introducing non-public
Indonesian NLP dataset. For every private dataset listed, the corresponding
contributor will be eligible for +1 contribution point. While for the original
author of the dataset, if the author agrees to make the dataset publicly
available, the author will be eligible for +3 contribution points. Note that
we might request the author to clean up their data until it is proper for
public release (i.e. formatting, consistency, or additional filtering). In
addition to the contribution points obtained from publicly releasing the
dataset, the author is also eligible for additional points from the datasheet
listing, as mentioned in §3.1, when the datasheet is submitted to
NusaCatalogue.

###  3.4 Other ways to contribute in

Other than the previously mentioned contributions methods, we also open for
other forms of contribution, subject to NusaCrowd’s open discussion. To get
more details on the open discussion, please join our Slack and Whatsapp group
(see §7).

###  3.5 Contribution point recapitulation

The total contribution point for all contributors will be recapped every week
by a maintainer, and a contribution matrix will be published and updated on a
weekly basis. The contribution matrix and more detailed information to the
contribution point can be accessed on the following
link777https://docs.google.com/spreadsheets/d/e/2PACX-1vS3Kbi9s3o_V-
lyFRHeONOI7jFnMlUswqKj-D6cpgiSYOSxbijC4DIrjAstqxj-H-EI6I2lFhhyKe5s/pubhtml.
The final score will be recapped in November and the finalized contribution
matrix (see Figure 5 for an example) will be published along with the research
paper.

Figure 6: Timeline of the NusaCrowd movement. The datasheet collection and
dataloader implementation start from 25 June 2022 to 2 October 2022. Then
NusaCrowd will continue with experiments, paper writing, and point
recapitulation. The paper will be submitted to a top-level computational
linguistics conference, ACL 2023.

Figure 7: Join NusaCrowd’s Slack
(https://join.slack.com/t/nusacrowd/shared_invite/zt-1b61t06zn-
rf2rWw8WFZCpjVp3iLXf0g), Whatsapp group
(https://chat.whatsapp.com/Jn4nM6l3kSn3p4kJVESTwv), and Github
(https://github.com/IndoNLP/nusa-crowd).

##  4 Dataset Licensing and Ownership

NusaCrowd does not make a clone or copy the submitted dataset. The owner and
copyright holder will remain to the original data owner. All data access
policy will follow the original data licence without any modification from
NusaCrowd. NusaCrowd simply downloads and reads the file from the original
publicly available data source location when creating the dataloader, and, in
addition, the NusaCatalogue datasheet also directly points to the original
data site and publication.

##  5 Timeline

The current NusaCrowd movement is started from 25 June 2022 and will be closed
on 18 November 2022. The registration of datasheets and dataloaders will be
completed on 2 October 2022. From October onwards, we will focus more on
preparing extension and set of experiments to show the benefit of having
NusaCrowd platform. In addition, the contribution point for each contributors
(see Figure 5 for an example) and the research paper will also be finalized by
early November, followed by the final submission of the paper to the
Association of Computational Linguistics (ACL) 2023 conference. The detailed
phase for the paper development is shown in Figure 6.

##  6 Summary

NLP resources in Indonesian languages, especially the local ones, are
extremely low-resource and underrepresented in the research community. There
are multitudes factors causing this limitation. Here we solve this problem by
initiating the largest Indonesian NLP crowd sourcing efforts, NusaCrowd. In
the spirit of fairness, openness, and transparency; NusaCrowd comes with
various ways to contribute towards openness and standardization in Indonesian
NLP, while at the same time, introducing a scoring mechanism that provides
equal chance for all contributors to show the best out of their contributions.
We hope that, NusaCrowd can bring a new perspective to all Indonesian NLP
practitioners to focus more on openness and collaboration through code and
data sharing, complete documentation, and community efforts.

##  7 Call for participation

We invite all Indonesian NLP enthusiasts to participate in NusaCrowd. For any
inquiry and further information, contributors can join our community channel
on Slack or Whatsapp Group (see Figure 7). Let’s work together to advance the
progress of Indonesian language NLP!

## Acknowledgements

Thank you for all initiators, without whom NusaCrowd initiative would never be
possible, and salute to all contributors for the amazing efforts in NusaCrowd.

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wedge with opening angle $\Theta$, found from the correlation coefficient by
$\cos\Theta=-\delta$. The exact target window is colored in blue, and the
sector defining the easier integration subregion for the target in yellow. The
two components of the shadow window are found by reflecting the exact target
window across the lines $\mathcal{D}_{L}$ and $\mathcal{D}_{R}$ and are
colored in orange, with the easier bounding shadow integration region, which
necessarily covers these areas, in green.

The corners $\tilde{c}$ and $\tilde{d}$ of the target window lie on line
$\mathcal{D}_{R}$, and $\tilde{b}$ and $\tilde{d}$ on line $\mathcal{D}_{L}$.
Thus we need only reflect $\tilde{a}$ and $\tilde{b}$ about $\mathcal{D}_{R}$,
and $\tilde{a}$ and $\tilde{c}$ about $\mathcal{D}_{L}$. The coordinates of
the points reflected about $\mathcal{D}_{R}$ are

$\displaystyle\\{\tilde{a}_{R},$
$\displaystyle\,\tilde{b}_{R}\\}=\sqrt{2Dt}\,\times$
$\displaystyle\Bigg{\\{}\frac{\lambda(\beta-\alpha)}{\cosh(2\ln\lambda)}\hat{e}_{-}+\left(\frac{\beta-\alpha}{\lambda\cosh(2\ln\lambda)}+\alpha\lambda\right)\hat{e}_{+}\leavevmode\nobreak\
,$
$\displaystyle\qquad\left(\frac{\lambda(\beta-2\alpha)}{\cosh(2\ln\lambda)}+\frac{\alpha-\beta}{\lambda}\right)\hat{e}_{-}$
$\displaystyle\qquad\quad+\left(\frac{\beta-2\alpha}{\lambda\cosh(2\ln\lambda)}+\lambda(\alpha+\beta)\right)\hat{e}_{+}\Bigg{\\}},$
(80)

with similar forms for $\tilde{a}_{L}$ and $\tilde{c}_{L}$. The four-sided
figures described by the exact shadow window are evidently complicated. As
with the target window, we bound the area using a sector which scales in the
same way, however in this case an upper bound is required. The upper limit
$\rho^{\mathrm{sw}}_{+}$ is the radial coordinate of points $\tilde{c}_{L}$
and $\tilde{b}_{R}$, and the lower limit $\rho^{\mathrm{sw}}_{-}$ is that
shared by the corners $\tilde{b}$ and $\tilde{c}$. The angular half-width is
the maximum of the angular half-widths of points $\tilde{c}$ and
$\tilde{a}_{R}$; this depends on the specific value of $\Theta$. Again we find
integration limits
$\rho^{\mathrm{sw}}_{\pm}=C^{\mathrm{sw}}_{\pm}(\alpha,\beta,\delta)\sqrt{Dt}$,
and $\psi^{\mathrm{sw}}=\psi^{\mathrm{sw}}(\alpha,\beta,\delta)$.

Based on the previous calculation,
$p^{2\mathrm{d}}_{\mathrm{sw}}(\alpha,\beta,\delta)=\frac{4\sin(\nu\psi^{\mathrm{sw}})}{\nu\Gamma(\frac{\nu}{2})}\mathcal{I}^{\mathrm{sw}}(\alpha,\beta,\delta)$
and the corrected fraction is

$\displaystyle
p^{{2\mathrm{d}},\mathrm{corr}}_{\mathrm{w}}(\alpha,\beta,\delta)$
$\displaystyle=p^{2\mathrm{d}}_{\mathrm{w}}(\alpha,\beta,\delta)-p^{2\mathrm{d}}_{\mathrm{sw}}(\alpha,\beta,\delta)$
(81)
$\displaystyle=\frac{4}{\nu\Gamma(\frac{\nu}{2})}\big{(}\sin(\nu\psi)\mathcal{I}-\sin(\nu\psi^{\mathrm{sw}})\mathcal{I}^{\mathrm{sw}}\big{)}\leavevmode\nobreak\
.$ (82)

By working explicitly through the algebra one can verify that
$p^{{2\mathrm{d}},\mathrm{corr}}_{\mathrm{w}}$ is positive for all values of
$\delta\in[0,1)$, e.g., for the choice $\alpha=1$, $\beta=2$.

Now, taking $t=L/2$, for any such $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ we
can construct a RW which satisfies Condition 1 for end-to-end decimation in
the quantum chain as
$\mathfrak{m}=\mathfrak{m}_{1}\oplus\mathfrak{I}\mathfrak{m}_{2}$. Therefore a
lower bound on the simultaneous end-to-end decimation probability is given by
$p_{\mathrm{e}}\geq[p^{{2\mathrm{d}},\mathrm{corr}}_{\mathrm{w}}S(L/2)]^{2}\sim
L^{-\pi/\Theta}$. In combination with the upper bound, this shows that the
power law exponent controlling end-to-end decimation probability (and
consequently $\eta^{\mathrm{e}}_{z}$) varies continuously with $\delta$ as

$\eta^{\mathrm{e}}_{z}=\pi/\arccos(-\delta)\leavevmode\nobreak\ .$ (83)

####  V.4.2 Bulk decimation probability in two locally correlated Majorana
chains

Once again we can extend the result for end-to-end decimation
$p_{\mathrm{e}}$—requiring that both Majorana chains satisfy Condition 1—to
the bulk likelihood $p_{\mathrm{b}}(r)$ (for two fixed spins separated by $r$)
by considering also Condition 2. We first write a lower bound on the bulk pair
decimation probability by identifying exterior RWs which are guaranteed to
satisfy Condition 2 when properly adjoined to an interior RW of the type used
for the lower bound on $p_{\mathrm{e}}$ in the previous section. Specifically,
we restrict to exterior RWs with endpoints at time $t\equiv r$ (for
concreteness, but any constant multiple of $r$ would do as well) within a
particular sector (specified below) in the solution geometry. In the physical
geometry, $\Delta(r)=\beta\sqrt{2Dr}$ is an upper bound on the total deviation
of each of the 1d RWs $\mathfrak{m}_{x}$ and $\mathfrak{m}_{y}$ described by
the 2d interior RW $\mathfrak{m}_{\mathrm{int}}$.

One way to guarantee the bulk decimation is to require that each of the
physical 1d RWs described by each of the exterior 2d RWs
$\mathfrak{m}_{\mathrm{ext,L}}$ and $\mathfrak{m}_{\mathrm{ext,R}}$ survive,
and exceed $\Delta(r)$ at $t=r$. A point $(\rho,\phi)$ in the solution
geometry corresponds to

$x=\frac{\rho\sin(\Theta-\phi)}{\sqrt{\sin\Theta}}\leavevmode\nobreak\
,\leavevmode\nobreak\ y=\frac{\rho\sin(\phi)}{\sqrt{\sin\Theta}}$ (84)

in the physical geometry. Employing angular integration limits
$\phi\in(\theta-\psi,\theta+\psi)$, where $\psi$ can be chosen to be the same
value used for $\mathfrak{m}_{\mathrm{int}}$, sufficient radial limits for our
purposes are
$\rho^{\mathrm{ext}}_{-}=\Delta(r)\sqrt{\sin\Theta}/\sin(\theta-\psi)$ and
$\rho^{\mathrm{ext}}_{+}\to\infty$ (noticing that
$\sin(\theta-\psi)\leq\sin(\theta+\psi)$ for all $\psi\in[0,\theta]$). From
the calculation of the previous section there is a constant probability
$\kappa(\alpha,\beta,\delta)$ that any surviving RW lies in a window bounded
by $\rho\in[\rho^{\mathrm{ext}}_{-},\rho^{\mathrm{ext}}_{+}]$ and
$\phi\in[\theta-\psi,\theta+\psi]$ at $t=r$. Such a RW has deviation at least
$\Delta(r)$ in the physical $x$ and $y$ coordinates and thus as either
$\mathfrak{m}_{\mathrm{ext,L}}$ or $\mathfrak{m}_{\mathrm{ext,R}}$ is suitable
for satisfying Condition 2 for bulk decimation when properly adjoined to
$\mathfrak{m}_{\mathrm{int}}$ as constructed previously; thus
$p_{\mathrm{b}}(r)\geq p_{\mathrm{e}}(r)[\kappa S(t=r)]^{2}\sim
r^{-2\pi/\Theta}$.

Similar to the case of a single Majorana chain, for an upper bound we make use
of the probability $\omega(\rho,\phi;A)$ of a RW with spatial coordinates
$(\rho^{\prime},\phi^{\prime})$ reaching radius $\rho^{\prime}=A$ in the wedge
given a starting point $(\rho,\phi)$. This probability follows Laplace’s
equation $\nabla^{2}\omega=0$, now with boundary conditions
$\omega(\rho,\phi=0)=\omega(\rho,\phi=\Theta)=0$, $\omega(\rho=A,\phi)=1$.
Assuming a separable solution $\omega(\rho,\phi)=R(\rho)T(\phi)$, we find that
for the angular coordinate the solutions are $T_{n}(\phi)=\sin(n\nu\phi)$,
$n=1,2,3,\ldots{}$, where as before $\nu=\pi/\Theta$. For the radial
coordinate

$\rho^{2}\frac{\partial^{2}R}{\partial\rho^{2}}+\rho\frac{\partial
R}{\partial\rho}-(n\nu)^{2}R=0\leavevmode\nobreak\ ,$ (85)

which has solutions of the form $R_{n}(\rho)=\rho^{\pm n\nu}$. Determining the
constants from the boundary conditions,

$\omega(\rho,\phi;A)=\sum_{\begin{subarray}{c}n=1\\\ n\leavevmode\nobreak\
\text{odd}\end{subarray}}^{\infty}\frac{4}{n\pi}\left(\frac{\rho}{A}\right)^{n\nu}\sin(n\nu\phi).$
(86)

Along the relevant line $\phi=\theta$, the probability simplifies to

$\omega(\rho,\phi=\theta;A)=\frac{4}{\pi}\arctan\left[\left(\frac{\rho}{A}\right)^{\nu}\right]\leq\frac{4}{\pi}\left(\frac{\rho}{A}\right)^{\nu}\leavevmode\nobreak\
.$ (87)

In order to write an upper bound on the bulk decimation probability, we
consider a full RW satisfying both Conditions assembled from an
$\mathfrak{m}_{\mathrm{int}}=\mathfrak{m}_{1}\oplus\mathfrak{I}\mathfrak{m}_{2}$,
where each of $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ must survive until
$t\equiv r/2$, along with exterior RWs $\mathfrak{m}_{\mathrm{ext,L}}$ and
$\mathfrak{m}_{\mathrm{ext,R}}$ which must reach a particular radial
coordinate (determined from $\mathfrak{m}_{\mathrm{int}}$ as specified below)
without being absorbed. Suppose that $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$
terminate at coordinates $(\rho_{1},\phi_{1})$ and $(\rho_{2},\phi_{2})$,
which define the deviations of the physical RWs $(\Delta_{x,1},\Delta_{y,1})$,
and $(\Delta_{x,2},\Delta_{y,2})$ according to Eq. (84). The full deviation of
the interior walk $\mathfrak{m}_{\mathrm{int}}$ in the physical coordinates is
$(\Delta_{x},\Delta_{y})=(\Delta_{x,1}+\Delta_{x,2},\Delta_{y,1}+\Delta_{y,2})$
and the physical 1d RWs described by $\mathfrak{m}_{\mathrm{ext,L}}$ and
$\mathfrak{m}_{\mathrm{ext,R}}$ must exceed the corresponding $\Delta_{x}$ or
$\Delta_{y}$ before being absorbed. For this to be the case it is necessary,
but not sufficient, that the 2d exterior RWs each survive in the wedge until
reaching radial coordinate
$A\equiv\sqrt{\sin\Theta}\min(\Delta_{x},\Delta_{y})$ in the solution
geometry. Defining for $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ similar
$A_{1}\equiv\sqrt{\sin\Theta}\min(\Delta_{x,1},\Delta_{y,1})$ and
$A_{2}\equiv\sqrt{\sin\Theta}\min(\Delta_{x,2},\Delta_{y,2})$, we note that
$A\geq A_{1},A_{2}$. The probability of finding two such
$\mathfrak{m}_{\mathrm{ext,L}}$ and $\mathfrak{m}_{\mathrm{ext,R}}$ given the
terminating locations of $\mathfrak{m}_{1}$ and $\mathfrak{m}_{2}$ is

$\displaystyle p(\mathrm{ext}|\rho_{1},\phi_{1},\rho_{2},\phi_{2})$
$\displaystyle\leq\omega(\rho_{0},\theta;A)^{2}$ (88)
$\displaystyle\leq\left(\frac{4}{\pi}\right)^{2}\left(\frac{\rho_{0}}{A}\right)^{2\nu}$
(89)
$\displaystyle\leq\left(\frac{4}{\pi}\right)^{2}\left(\frac{\rho_{0}}{A_{1}}\right)^{\nu}\left(\frac{\rho_{0}}{A_{2}}\right)^{\nu}.$
(90)

Then, integrating over the distribution of the interior half-chain
coordinates,

$\displaystyle p_{\mathrm{b}}(r)$
$\displaystyle=\int\rho_{1}\,d\rho_{1}\,d\phi_{1}\,G(\rho_{1},\phi_{1},r/2;\rho_{0},\theta)\times$
$\displaystyle\qquad\int\rho_{2}\,d\rho_{2}\,d\phi_{2}\,G(\rho_{2},\phi_{2},r/2;\rho_{0},\theta)\times$
$\displaystyle\qquad\qquad
p(\mathrm{ext}|\rho_{1},\phi_{1},\rho_{2},\phi_{2})$ (91)
$\displaystyle\leq\Bigg{[}\frac{8}{\pi}\int_{0}^{\infty}\rho_{1}\,d\rho_{1}\int_{0}^{\theta}d\phi_{1}\times$
$\displaystyle\qquad\qquad
G(\rho_{1},\phi_{1},r/2;\rho_{0},\theta)\left(\frac{\rho_{0}}{\rho_{1}\sin\phi_{1}}\right)^{\nu}\Bigg{]}^{2}$
(92)
$\displaystyle=\left(\frac{16\mathcal{I}_{\phi}}{\pi^{2}\Gamma(\nu)}\right)^{2}\left(\frac{\rho_{0}^{2}}{2Dr}\right)^{2\nu}.$
(93)

We restrict to the right half-wedge, as the integrand is symmetric about
$\phi=\theta$. The angular integral is

$\displaystyle\mathcal{I}_{\phi}=\int_{0}^{\theta}d\phi_{1}\frac{\sin(\nu\phi_{1})}{(\sin\phi_{1})^{\nu}}\leavevmode\nobreak\
,$ (94)

which converges for $\Theta>\pi/2$, equivalently $\delta>0$. (The exponent we
are bounding is known at $\delta=0$, and follows from the result of Sec.
V.2.2.)

Combining the upper and lower bounds, we prove that $p_{\mathrm{b}}(r)\sim
r^{-2\nu}$, and the bulk correlations exponent for two locally correlated
Majorana chains with parameter $\delta$ is

$\eta_{z}=2\pi/\arccos(-\delta)\leavevmode\nobreak\ .$ (95)

###  V.5 Numerical SDRG study

The final results of this section, Eqs. (83) and (95), are in qualitative
agreement with the quantum simulations of Sec. IV.2 for relatively short
Majorana chains, and are consistent with previously-known results at the
points $\delta=0,1$, where the locally-correlated model describes the random
uncorrelated XY and perfectly correlated XX IRFPs. For further verification we
implement the SDRG update Eq. (19) directly for two Majorana chains with
locally-correlated terms, and are able to access larger system sizes. This
also allows us to study the bulk $C^{\perp}(r)$ power laws, which are not
analytically tractable in the mapping to RWs used in the preceding
subsections.

The numerically extracted exponents are shown in Fig. 19. The bare correlation
coefficient $\delta$ may become slightly renormalized from the lattice scale
definition in Eqs. (36)–(37) compared to the meaning in the continuum 2d RW
treatment in Sec. V.4, but these simulations are in good agreement with the
analytic forms for $\eta_{z}^{\mathrm{e}}(\delta)$ and $\eta_{z}(\delta)$ we
obtained. While we have precise analytical knowledge only of the critical
exponents $\eta_{z}^{\mathrm{e}}$ and $\eta_{z}$, we observe that
$\eta_{\perp}$ also varies continuously. In contrast,
$\eta_{\perp}^{\mathrm{e}}=1$ for any value of $\delta$, by the argument
presented in Sec. II.4.

Figure 19:  Numerical SDRG data are shown for two locally-correlated Majorana
chains, with the end-to-end and bulk decimation probability
exponents—equivalent to $\eta^{\mathrm{e}}_{z}$ and $\eta_{z}$, respectively,
in the quantum model—compared to the analytic forms Eqs. (83) and (95) (dashed
lines). Also shown are critical exponents $\eta_{\perp}^{\mathrm{e}}$ and
$\eta_{\perp}$ measured in the numerical SDRG, as well as red stars indicating
known values of bulk correlations exponents at $\delta=0$ and $1$, and yellow
diamonds indicating known values of end-to-end correlations exponents. The
end-to-end correlations data were taken from 1 000 000 disorder realizations
each for system sizes up to $N=128$, and the bulk correlations data were taken
from 100 000 disorder realizations at system size $N=256$, utilizing the
middle half of each of the two Majorana chains.

##  VI Fixed points for the interacting model

In Sec. V we performed a study of the behavior of critical exponents under a
varying degree of local correlations in a random free-fermion model. Despite
the lack of tractable SDRG flow equations, we showed that the local
correlation controlled by $\delta$ is a marginal perturbation which tunes
along a line of IRFPs. In the present section we advance the perspective that
this line of non-interacting fixed points in fact also controls the long-
distance behavior of the interacting model for small $J^{z}$ strength below
the transition to the $z$-AFM phase.

To do so requires a study of the SDRG at intermediate stages, taking into
account more general terms produced by the interactions. Equation (29)
describes the result of an initial decimation, but eventually descendant terms
will be frequent and must also be taken into account. We change our
conventions here from those of Sec. II.3.2 for convenience: namely, we denote
the Majorana chains by $\mathrm{I},\mathrm{II}$ rather than
$\mathcal{X},\mathcal{Y}$; and by a gauge transformation (described at the end
of Sec. II.3.1) we set the signs of $h^{\mathrm{I}}_{n},h^{\mathrm{II}}_{n}>0$
for all $n=1,\ldots,N-1$, and $K_{n}\equiv K_{n,n}<0$.

In order to capture the effect of iterated decimations, we observe that in Eq.
(29) descendants of the form
$K_{n,m}(i\gamma^{\mathrm{I}}_{n}\gamma^{\mathrm{I}}_{n+1})(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})$
are produced, which generalize the $K_{n}$ of Eq. (22). We enlarge the space
of couplings to include all such terms, with initial distribution $K_{n,m}=0$,
$n\neq m$. If the average $K\equiv|\langle K_{n,m}\rangle|$ can be considered
to be a small parameter (for weak interactions $K<|\langle h_{n}\rangle|$),
the higher-fermion term in Eq. (29) appears at order $O(K^{2})$ and can thus
be neglected. We will demonstrate that the space of couplings including all
$K_{n,m}$ is closed under RG flow up to $O(K)$, and that the structure of the
signs is preserved. Furthermore, we will show that the strength of the $K$
terms decreases in some sense relative to the $h$ terms, suggesting that
interactions are irrelevant, at least in the neighborhood of the free-fermion
fixed point.

Following the approach of Sec. II.3, denote the largest term as
$H_{0}=ih^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k+1}$
and associate with the eigenstates of this term a complex fermion
$f^{\dagger}_{0}=\frac{1}{2}(\gamma^{\mathrm{I}}_{k}+i\gamma^{\mathrm{I}}_{k+1})$
with projectors $\pi^{+}=f_{0}f^{\dagger}_{0}$ and
$\pi^{-}=f^{\dagger}_{0}f_{0}$ into the even- and odd-parity sectors, or the
high- and low-energy eigenstates, respectively, of $H_{0}$. The off-diagonal
terms in the Schrieffer-Wolff treatment share exactly one Majorana operator
with $H_{0}$:

$\displaystyle V_{\mathrm{od}}$
$\displaystyle=ih^{\mathrm{I}}_{k-1}\gamma^{\mathrm{I}}_{k-1}\gamma^{\mathrm{I}}_{k}+ih^{\mathrm{I}}_{k+1}\gamma^{\mathrm{I}}_{k+1}\gamma^{\mathrm{I}}_{k+2}$
(96)
$\displaystyle\quad+\sum_{m=1}^{N-1}\big{(}K_{k-1,m}(i\gamma^{\mathrm{I}}_{k-1}\gamma^{\mathrm{I}}_{k})(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})$
$\displaystyle\qquad\qquad+K_{k+1,m}(i\gamma^{\mathrm{I}}_{k+1}\gamma^{\mathrm{I}}_{k+2})(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\big{)}\leavevmode\nobreak\
.$ (97)

Separating $V_{\mathrm{od}}$ into symmetry sectors, we find that

$\displaystyle\pi^{+}H\pi^{-}$
$\displaystyle=\left[\left(h^{\mathrm{I}}_{k-1}+\sum_{m}K_{k-1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\right)i\gamma^{\mathrm{I}}_{k-1}+\left(h^{\mathrm{I}}_{k+1}+\sum_{m}K_{k+1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\right)\gamma^{\mathrm{I}}_{k+2}\right]f_{0}$
(98)
$\displaystyle\equiv(ih^{\mathrm{I},\text{int}}_{k-1}\gamma^{\mathrm{I}}_{k-1}+h^{\mathrm{I},\text{int}}_{k+1}\gamma^{\mathrm{I}}_{k+2})f_{0}\leavevmode\nobreak\
,$ (99) $\displaystyle\pi^{-}H\pi^{+}$
$\displaystyle=\left[\left(h^{\mathrm{I}}_{k-1}+\sum_{m}K_{k-1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\right)i\gamma^{\mathrm{I}}_{k-1}-\left(h^{\mathrm{I}}_{k+1}+\sum_{m}K_{k+1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\right)\gamma^{\mathrm{I}}_{k+2}\right]f^{\dagger}_{0}$
(100)
$\displaystyle\equiv(ih^{\mathrm{I},\text{int}}_{k-1}\gamma^{\mathrm{I}}_{k-1}-h^{\mathrm{I},\text{int}}_{k+1}\gamma^{\mathrm{I}}_{k+2})f^{\dagger}_{0}\leavevmode\nobreak\
.$ (101)

We make use of the “interacting couplings” notation used also in Sec. II.3.2
to connect with the non-interacting case, but here it is not evident that
these couplings—which are really operators—all commute. Nevertheless, a
suitably generalized version of Eq. (19) implements the Schrieffer–Wolff
transformation:

$\displaystyle H^{\prime}$
$\displaystyle=H_{0}+V_{\mathrm{d}}+\frac{(h^{\mathrm{I},\text{int}}_{k-1})^{2}+(h^{\mathrm{I},\text{int}}_{k+1})^{2}}{2h^{\mathrm{I}}_{k}}(i\gamma^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k+1})+\frac{h^{\mathrm{I},\text{int}}_{k-1}h^{\mathrm{I},\text{int}}_{k+1}+h^{\mathrm{I},\text{int}}_{k+1}h^{\mathrm{I},\text{int}}_{k-1}}{2h^{\mathrm{I}}_{k}}(i\gamma^{\mathrm{I}}_{k-1}\gamma^{\mathrm{I}}_{k+2})$
(102)
$\displaystyle=H_{0}+V_{\mathrm{d}}+(i\gamma^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k+1})\left[\frac{(h^{\mathrm{I}}_{k-1})^{2}+(h^{\mathrm{I}}_{k+1})^{2}}{2h^{\mathrm{I}}_{k}}+\frac{h^{\mathrm{I}}_{k-1}}{h^{\mathrm{I}}_{k}}\sum_{m}K_{k-1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})+\frac{h^{\mathrm{I}}_{k+1}}{h^{\mathrm{I}}_{k}}\sum_{m}K_{k+1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\right]$
$\displaystyle\quad+(i\gamma^{\mathrm{I}}_{k-1}\gamma^{\mathrm{I}}_{k+2})\left[\frac{h^{\mathrm{I}}_{k-1}h^{\mathrm{I}}_{k+1}}{h^{\mathrm{I}}_{k}}+\frac{h^{\mathrm{I}}_{k-1}}{h^{\mathrm{I}}_{k}}\sum_{m}K_{k+1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})+\frac{h^{\mathrm{I}}_{k+1}}{h^{\mathrm{I}}_{k}}\sum_{m}K_{k-1,m}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})\right]+O(K^{2})\leavevmode\nobreak\
.$ (103)

The effective terms in the first line of Eq. (103) (and the first term of the
second line) are $h$-type, with positive coefficients in the low-energy sector
of $H_{0}$ where $\langle
i\gamma^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k+1}\rangle=-1$. Conversely, the
remaining terms in the second line are $K$-type (recalling that
$\gamma^{\mathrm{I}}_{k-1}$ and $\gamma^{\mathrm{I}}_{k+2}$ become adjacent
after the decimation of $\gamma^{\mathrm{I}}_{k}$ and
$\gamma^{\mathrm{I}}_{k+1}$), and have coefficients with negative signs. One
sees that the signs of the initial distributions, namely
$h_{n}^{\mathrm{I},\mathrm{II}}>0$ and $K_{n,m}<0$, are maintained during the
RG flow, and it is evident from Eq. (103) that these types of terms are closed
under the SDRG up to $O(K)$.

As a measure of the evolution of the relative strength of $K$ terms to $h$
terms under this RG step, we compare the renormalized
$K^{\mathrm{eff}}_{k-1,m}$ to the geometric mean of the proximate $h$ terms
$h^{\mathrm{I},\mathrm{eff}}_{k-1}$ and $h^{\mathrm{II}}_{m}$:

$\frac{K^{\mathrm{eff}}_{k-1,m}}{\sqrt{h^{\mathrm{I},\mathrm{eff}}_{k-1}h^{\mathrm{II}}_{m}}}=\sqrt{\frac{h^{\mathrm{I}}_{k-1}}{h^{\mathrm{I}}_{k}}}\frac{K_{k+1,m}}{\sqrt{h^{\mathrm{I}}_{k+1}h^{\mathrm{II}}_{m}}}+\sqrt{\frac{h^{\mathrm{I}}_{k+1}}{h^{\mathrm{I}}_{k}}}\frac{K_{k-1,m}}{\sqrt{h^{\mathrm{I}}_{k-1}h^{\mathrm{II}}_{m}}}$
(104)

We see that if such ratios are small to begin with, i.e.,
$K_{k+1,m}/\sqrt{h^{\mathrm{I}}_{k+1}h^{\mathrm{II}}_{m}},K_{k-1,m}/\sqrt{h^{\mathrm{I}}_{k-1}h^{\mathrm{II}}_{m}}\ll
1$ before the decimation, they will likely become even smaller under the RG
flow if the disorder in the Majorana hoppings is strong, so that
$h_{k-1}^{\mathrm{I}},h_{k+1}^{\mathrm{I}}\ll h_{k}^{\mathrm{I}}$. This
suggests that if the $h$ terms are dominant initially, they will be even more
so during the SDRG and will asymptotically constitute the entirety of the
decimations.

The diagonal terms which contain both decimated Majoranas are

$\sum_{m}K_{k,m}(i\gamma_{k}^{\mathrm{I}}\gamma_{k+1}^{\mathrm{I}})(i\gamma_{m}^{\mathrm{II}}\gamma_{m+1}^{\mathrm{II}})\leavevmode\nobreak\
.$ (105)

Upon decimation, setting $\langle
i\gamma^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k+1}\rangle=-1$ in the ground
state gives $O(K)$ contributions to the Majorana hoppings in the other chain,
$h_{m}^{\mathrm{II},\text{eff}}=h_{m}^{\mathrm{II}}-K_{k,m}$. Given the
opposite signs of the $h$ and $K$ couplings, this increases the overall
strength of the remaining Majorana hoppings. This is the local SDRG analog of
the “mean field” of Eqs. (34) and (35) where the $J^{z}$ interactions
renormalize the $J^{x}$ and $J^{y}$ couplings by strengthening and correlating
them, as was already noted in Sec. II.3.2 and discussed in Sec. IV. Here we
note that including these renormalizations of the $h$ couplings only improves
our arguments for the persistence of the dominance of these couplings over the
$K$ couplings.

The terms omitted from Eq. (103) at $O(K^{2})$ are the following:

$\displaystyle\frac{1}{2h^{\mathrm{I}}_{k}}$
$\displaystyle(i\gamma^{\mathrm{I}}_{k}\gamma^{\mathrm{I}}_{k+1})\left[\sum_{m}(K_{k-1,m}^{2}+K_{k+1,m}^{2})+\sum_{m,l\neq
m,m\pm
1}(K_{k-1,m}K_{k-1,l}+K_{k+1,m}K_{k+1,l})(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})(i\gamma^{\mathrm{II}}_{l}\gamma^{\mathrm{II}}_{l+1})\right]$
$\displaystyle+\frac{1}{h^{\mathrm{I}}_{k}}(i\gamma^{\mathrm{I}}_{k-1}\gamma^{\mathrm{I}}_{k+2})\left[\sum_{m}K_{k-1,m}K_{k+1,m}+\sum_{m,l\neq
m,m\pm
1}K_{k-1,m}K_{k+1,l}(i\gamma^{\mathrm{II}}_{m}\gamma^{\mathrm{II}}_{m+1})(i\gamma^{\mathrm{II}}_{l}\gamma^{\mathrm{II}}_{l+1})\right]\leavevmode\nobreak\
.$ (106)

The first terms in each line are corrections to the ground-state energy and
the strength of the renormalized bond coupling on chain $\mathrm{I}$ [which
again preserves the sign structure and strengthens this hopping compared to
the leading contribution in Eq. (103)]. Along with these, four-fermion terms
within chain $\mathrm{II}$ and six-fermion inter-chain terms appear at
$O(K^{2})$. The former are expected to be ultimately irrelevant, based on
previous studies of a single Majorana chain realized in the quantum Ising
model Fisher (1994). However these four-fermion terms and the six-fermion
terms will produce yet more complicated descendants in subsequent RG steps,
and there will also be “degradation” processes leading to fewer-fermion terms,
including renormalization of the two-fermion terms, similar to the discussion
after Eq. (105) Monthus (2018). In this case we must rely on the perturbative
argument to justify dropping them, viewing them as irrelevant other than
feeding into strictly marginal correlations among the effective Majorana
hoppings in the two chains.

Together with the understanding of the locally correlated XY model in the
previous section, this leads us to propose the following picture for the
critical XYZ chain along the line separating the $x$-AFM and $y$-AFM phases.
For small $\tilde{J}^{z}$, this critical line is actually controlled by the
line of free Majorana fixed points with locally correlated hoppings
characterized in Sec. V. The effect of the interactions $J^{z}$ in the
original model with no correlations among the couplings ($\delta=0$) is to
develop such correlations among the renormalized $J^{x}$ and $J^{y}$ couplings
under RG while the $J^{z}$ couplings flow to zero. The ultimate degree of such
correlations (i.e., the fully renormalized parameter $\delta_{\text{eff}}$)
then determines the long-distance power laws in the average spin correlation
functions. We further conjecture that this persists for all
$\tilde{J}^{z}<\tilde{J}^{z}_{\text{crit}}=1$ below the transition to the
$z$-AFM phase. While we do not have perturbative control close to this
transition, any alternative would require yet another transition below
$\tilde{J}^{z}_{\text{crit}}$ which we did not observe and consider to be less
natural. Note that in this scenario the transition to the $z$-AFM phase is
controlled by a different _non-free-fermion_ fixed point, and we do not have
access to this $S_{3}$-symmetric fixed point in the present study. We will
further discuss the above conjecture, its corollaries and possible tests, as
well as open questions in the concluding section.

##  VII Discussion

In this paper, motivated by the observations of Slagle  _et al._ (2016), we
have performed a study of the low-energy properties of the random XYZ model
using unbiased numerics. We focus on the line separating the $x$-AFM and
$y$-AFM phases, which exhibits statistical symmetry between $J^{x}$ and
$J^{y}$ couplings. At all points allowing comparison our results are in
general agreement with the previous findings of Ref. Slagle  _et al._ (2016)
which used SBRG and presumed critical MBL physics at arbitrary energy density.
Our results strongly suggest that—regardless of the behavior of highly excited
states—there is quantum critical behavior in the ground state and the critical
line is described by IRFPs with continuously varying critical exponents in the
disorder-averaged correlation functions. Perhaps surprisingly, a Hartree–Fock
mean-field theory treating the $J^{z}$ interaction terms as perturbations
around the random XY (free-fermion) fixed point yielded results that are
qualitatively rather consistent with the full interacting model at small to
moderate $J^{z}$ couplings, including continuously varying power laws. This is
in contrast to the clean case, where the mean field model is not qualitatively
accurate due to divergences in the perturbation theory Giamarchi (2004).

The locally correlated XY effective model, introduced with the idea of
distilling the essential feature of the mean field theory, again exhibited
continuously varying critical exponents, which we were able to establish
numerically in larger sizes than for the XYZ chain. Because of the particular
free-fermion form of this effective model, we were able to treat it in the
SDRG using the random walk formulation in two dimensions. By making use of a
connection between survival probability and the structure of decimation in the
RG, we showed analytically that critical exponents for end-to-end and bulk
$C^{z}$ spin correlations vary continuously as the coupling correlation
parameter $\delta$ is tuned, and we also observed varying exponents in the
bulk $C^{\perp}$ correlations by running the SDRG numerically. This result
singles out and proves one of the scenarios of Fisher (1994) that random
anisotropy is strictly marginal along the critical line connecting the random
XX and random XY fixed points; that is, there is a line of fixed points
connecting the XX and XY IRFPs as sketched in Fig. 1.

Motivated by the successful understanding of the locally correlated XY model,
we revisited the SDRG for the full interacting XYZ chain in the regime of
small interactions and proposed a scenario where these interactions are
irrelevant, but during the initial flows they generate effective correlations
between the local $J^{x}$ and $J^{y}$ couplings (i.e., Majorana hopping
amplitudes on the two chains). Such flows are sketched in Fig. 2. These local
correlations in the free-fermion couplings then lead to non-universal power
laws in the average spin correlations: this is our story for the continuously
varying criticality in the XYZ spin chain.

We note that continuously varying critical exponents were previously observed
in IRFPs associated with correlated disorder by Rieger and Iglói (1999),
however in a qualitatively different setting than ours. Specifically,
disordered fixed points perturbed by the introduction of long-range
correlations $\sim r^{-a}$ to the disorder in the random transverse-field
Ising chain exhibit critical indices varying continuously with $a$ for $a<1$.
Their setting has only one Majorana chain and the correlated disorder is
within the chain. Also, in their case the $\psi$ exponent varies continuously,
which reflects a different character of the corresponding “random walker”
imprinted by the long-range correlations in the disorder.

Non-universal exponents at IRFPs were also observed in cases with very broad
(singular) distributions of random couplings Karevski  _et al._ (2001);
Krishna and Bhatt (2020). This again occurs already in a single chain and has
varying exponent $\psi$, and the variation can be traced directly to the
singularity in the probability distribution of the microscopic couplings,
while the exponents are universal for non-singular probability distributions.

The XYZ chain studied here is different from the above examples with varying
exponents in that there are no long-range correlations or singular
distributions input into the microscopic disorder. In this way the
continuously varying exponents are intrinsic to this system rather than
imprinted extrinsically. What is important in the XYZ chain is that we have
two simultaneously critical Majorana chains whose couplings become locally
correlated. This insight may be useful when looking for other IRFPs with
intrinsic continuously varying critical indices.

We conclude by returning to the discussion of the proposed scenario for the
fully interacting XYZ chain. This scenario is based on the conjecture that the
four-fermion and higher terms are irrelevant other than feeding into
correlations between the Majorana hoppings. While this is plausibly justified
for small interactions in Sec. VI, we have not fully proved it and the status
for intermediate interactions is less certain. In this respect, it would be
useful to carry out a systematic numerical SDRG study of the fully interacting
problem (e.g., using the scheme of Monthus (2018)) keeping track of all
generated interactions as well as allowing decimations of the interaction
terms when they happen to be the strongest. If our scenario is correct, we
should see the interaction terms progressively decreasing relative to the
Majorana hoppings. One should be able to perform such a study also directly in
the spin variables using the SBRG approach of Slagle  _et al._ (2016)
projected onto the ground state branch, e.g., as used in Ref. Duque  _et al._
(2021) in a different problem. Employing the insights gained here, it should
be helpful to interpret various Pauli string terms generated under the SBRG as
either Majorana hoppings or specific multi-fermion interactions. The SBRG can
also be indispensable for studying the putative $S_{3}$-symmetric fixed point
describing the transition to the $z$-AFM phase, as a possible new IRFP that is
not tractable with available analytical tools.

Thinking about a broader phase diagram, our work suggests that it could be
fruitful to add another parameter “axis” and study the XYZ chain with locally
correlated $J^{x}$ and $J^{y}$ couplings in the bare model (analogous to
parameter $\delta$ in the correlated XY model), in addition to the
interactions $J^{z}$. Figure 2 shows this parameter space, and constitutes a
mild abuse inasmuch as it serves as both a phase diagram and a picture of RG
flows, the latter of which occur in space not captured by just the two
parameters. In the space shown, the bare $\delta=0$ corresponds to the present
XYZ chain, with the transition from the critical phase to the $z$-AFM phase at
the $S_{3}$ symmetric point, marked XYZC in Fig. 2. On the other hand,
$\delta=1$ corresponds to the XXZ chain studied in the original work by Fisher
(1994). For $\tilde{J}^{z}$ below some threshold value, the XXZ spin chain is
critical and controlled by the free-fermion XX point, while for larger
$\tilde{J}^{z}$ it undergoes a transition to the $z$-AFM phase. Fisher
concluded that this transition is controlled by the so-called XXZC fixed point
which is essentially random singlet–like, also marked in Fig. 2. An
interesting question is the nature of the transition to the $z$-AFM phase
driven by the $\tilde{J}^{z}$ coupling as we vary the disorder correlation
parameter from $\delta=1$ (XXZC fixed point) to the statistically isotropic
XYZC fixed point. This line is marked with a question mark in Fig. 2, and one
possibility is that it is also described by a line of fixed points, but we
cannot at present exclude other scenarios. We leave these questions for future
investigations, noting that the possibility of novel IRFPs is quite
tantalizing and worth further exploration.

###### Acknowledgements.

We acknowledge helpful discussions with Jason Alicea, Matteo Ippoliti, Cheng-
Ju Lin, Sanjay Moudgalya, Gil Refael, Kevin Slagle, and Christopher White. We
are also grateful for earlier collaboration with Thomas Vidick on the RRG
which led us to look for new applications of this method. O.M. is also
grateful for previous collaborations with Kedar Damle, David Huse, and Daniel
Fisher on the IRFPs which provided important background for this project. This
work was supported by National Science Foundation through grant DMR-2001186.
Part of this work was performed at Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-1607611.

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