# Toward Blockchain-based Fashion Wearables in the Metaverse: the Case of
Decentraland

Amaury Trujillo  IIT-CNR
Pisa, Italy
<EMAIL_ADDRESS>Clara Bacciu  IIT-CNR
Pisa, Italy
<EMAIL_ADDRESS>
###### Abstract

Among the earliest projects to combine the Metaverse and non-fungible tokens
(NFTs) we find Decentraland, a blockchain-based virtual world that touts
itself as the first to be owned by its users. In particular, the platform’s
virtual wearables (which allow avatar appearance customization) have attracted
much attention from users, content creators, and the fashion industry. In this
work, we present the first study to quantitatively characterize Decentraland’s
wearables, their publication, minting, and sales on the platform’s
marketplace. Our results indicate that wearables are mostly given away to
promote and increase engagement on other cryptoasset or Metaverse projects,
and only a small fraction is sold on the platform’s marketplace, where the
price is mainly driven by the preset wearable’s rarity. Hence, platforms that
offer virtual wearable NFTs should pay particular attention to the economics
around this kind of assets beyond their mere sale.

###### Index Terms:

metaverse, virtual fashion, non-fungible tokens

##  I Introduction

The idea of the Metaverse as a fully immersive future iteration of the
Internet has fueled the imagination of many tech enthusiasts for a few decades
now, with the latest advancements in extended reality (XR) having further
ignited public interest. The fashion industry is no exception; in recent years
there has been an increasing participation in Metaverse-related initiatives,
ranging from independent designers to famous commodity and luxury brands [1].
After all, avatar customization is one of the main enjoyments of users and one
of the main factors of purchase intent in virtual worlds [2], with the
economic potential being highly enticing to the fashion industry [1]. Also
recently, non-fungible tokens (NFTs) —units of cryptographic data that
represent a unique asset via blockchain technology— are being used to certify
ownership of digital objects such as virtual garments.

Arguably, the first and most influential Metaverse-related project that
utilized NFTs is Decentraland, a virtual world based on the Ethereum
blockchain in which users can create, experience, and monetize their content
and assets (including wearables for avatars) via the in-platform
cryptocurrency, called MANA. The platform is managed through a Decentralized
Autonomous Organization (DAO) —a blockchain-based system that enables self
coordination and governance [3]— based on MANA ownership. In addition, users
can also “buy” several kinds of NFTs, namely parcels of land; groups of
parcels called estates; unique names; as well as wearables and the recently
introduced emotes, items to customize the appearance and animation sequences
of avatars, respectively.

Decentraland started in 2015 and launched in February 2020 to a moderate
success. In 2021, there was a surge in the public interest in NFTs caused by
several high-profile sales [4], giving rise to a sudden increase in the value
of MANA. A few months later, “Facebook Inc.” rebranded itself into “Meta
Platforms”, drawing even more attention to the Metaverse in general, and to
Decentraland in particular. In view of this newfound popularity, Decentraland
was host to the first Metaverse Fashion Week (MVFW) in March 2022, which
involved a wide variety of brands and several in-world fashion events.
However, the 2022 cryptowinter (i.e., a remarkable devaluation of
cryptoassets) greatly affected Decentraland. Such a volatility, together with
many uncertainties surrounding NFTs, brought into discussion the approach used
by Decentraland for its economy and the ownership of the in-platform digital
assets, particularly wearables, with potential ramifications for other
Metaverse-related projects.

Herein, we thus characterize the publication, minting, and marketplace sales
of Decentraland’s wearables. Our work’s main contribution is —to the best of
our knowledge— the first quantitative characterization in literature of
blockchain-based digital wearable assets on a virtual world.

##  II Background and Related Work

By Metaverse, we refer to a shared universe of interoperable virtual worlds,
with these in turn being “[s]hared, simulated spaces which are inhabited and
shaped by their inhabitants who are represented as avatars” [5]. We are still
in an early development phase of the Metaverse, but a plethora of independent
virtual worlds have already been created in the last decades, such as Second
Life, or massively multiplayer online (MMO) games such as World of Warcraft,
Minecraft and Fortnite. Nevertheless, these have been subject to much
criticism in terms of the governance and _ownership_ of in-world assets: a new
form of virtual feudalism in Second Life [6], conflicts of agency and
ownership in World of Warcraft between publisher and gamers [7], and an
overall separation of content and platform ownership in virtual worlds to the
detriment of users [8]. As a response, a wave of blockchain virtual
experiences with decentralized governance and assets emerged, of which
Decentraland is a prominent example.

Still, most previous research on Decentraland either focus on landholding
assets [9, 10], or take the platform’s tokens as a whole within a larger NFT
ecosystem [11]. As far as we know, Decentraland’s wearables have not been yet
studied in literature. Fashion within virtual worlds, on the other hand, has
been studied for several years now, mainly on the still on-going Second Life
(released in 2003), which popularized the idea of customizable virtual worlds
and avatars [12], but who many argue failed to meet the expectations of the
general public [13]. Nonetheless, within Second Life there is still a strong
economy based on avatar customization and many users have found a new passion
(and in some cases a source of income) by becoming in-world “fashion
designers” of avatar clothing and accessories[14]. In the last few years,
however, the fashion industry has begun to focus more on highly popular MMO
games —mainly through branded _skins_ , i.e., in-game character cosmetic
options— with a growing interest on more recent blockchain-based virtual
worlds [13]. For instance, many important luxury brands have started
experimenting with NFT collectibles [15], e.g., participating in initiatives
such as the MVFW or also creating Decentraland weareables [1].

These wearables consist of items in the form of clothing, accessories, and
partial or complete body features that change the appearance of avatars. Each
wearable item belongs to a collection and can be _minted_ into unique NFTs,
i.e. items themselves cannot be bought or sold, only the tokens minted from
them (see Figure 1). First, collection creators must pay a fee so that it can
be vetted by a curation committee (elected by the project’s DAO). If approved,
the collection items are available for minting. Afterward, creators can either
mint tokens and transfer them to any account in the blockchain network, or put
their items on sale for minting on the official marketplace. Other users can
obtain wearable tokens in three main ways: 1) by participating in an in-world
event (e.g., a fashion show) that airdrops (gives away) tokens; 2) by claiming
tokens airdropped by an external project (e.g., to reward its stakeholders);
and 3) by buying tokens within or without the official marketplace. Regardless
of the minting mechanism, the wearable NFTs are property of the holder; hence,
it is also possible to use other trading platforms for secondary sales, such
as OpenSea, the largest NFT marketplace. Herein we mainly focus on primary
sales on Decentraland’s marketplace.

Figure 1: After a Decentraland wearable collection is approved by a curation
committee, each distinct item in it becomes available for _minting_ into NFTs
up to the item’s rarity limit chosen by the collection creator.

Currently, there are two wearable versions in Decentraland’s marketplace: v1
was implemented on the Ethereum network, with the first collection being
published in October 2019. However, many concerns emerged within the community
around Ethereum’s scalability and volatile fees to process transactions.
Hence, v2 wearables were developed for the Polygon network, a blockchain
platform and an Ethereum-compatible sidechain that allows a more efficient
processing of transactions [16]. The last v1 collection was published in March
2021; the first v2 collection was published in June 2021, with v1 being
deprecated but remaining usable in-world and available on NFT marketplaces.
Therefore, users must be aware of the network in which they carry out
secondary sales.

##  III Methods

We collected various wearable data, from first record available up to December
31 2022, via Decentraland’s query endpoints on TheGraph —an indexing protocol
and service for querying blockchain networks. These endpoints are used across
the different applications within Decentraland and their implementations are
open source.111https://github.com/decentraland In detail, we retrieved
metadata regarding wearable collections (name, status, creator, blockchain
network) and items (name, category, rarity, price), the 3D asset metadata for
v2 wearables (number of triangles, meshes, textures), in addition to all v2
items’ mints on Polygon, as well as the primary and secondary NFT sales
(buyer, seller, kind, price) within Decentraland. To compare the monetary
value of wearable items and sales over time, we first converted the price in
MANA of a given item or sale to USD, based on data from Yahoo! Finance, and we
then adjusted for inflation on a monthly basis to the USD value in December
2022, based on the Consumer Price Index (CPI) by the Bureau of Labor
Statistics of the United States. Henceforth, all monetary value will be
expressed in CPI-adjusted USD. To present results, we use the median
($\tilde{x}$) and median absolute deviation (mad) to describe distributions,
and multinomial $\chi^{2}$ tests to compare proportions of wearable kinds
between groups. We used the R statistical software and its ecosystem for all
of the analyses.

##  IV Published Wearables

For v1, there were only 41 collections (representing 436 items), whereas at
the cutoff date there were 4,110 v2 collections (8,324 items) submitted, with
most (88%) being approved and published (3,756 collections with 7,323 items).
Concerning the 1,266 distinct v2 creator accounts, more than half (53.9%) had
only one collection; the most prolific creator by far had 79 collections with
616 items, mostly used as rewards by Decentral Games, a company whose main
product is a poker game within Decentraland. All v1 collections consist of two
items or more, with 19.5% having more than ten; whereas the majority of v2
collections (68.8%) consist of a single item. The monthly publication of
approved v2 items, (see Figure 2) follows the rise and decline in MANA, with
the most prolific month being March 2022, around the MVFW.

Figure 2: Monthly publication count of approved v2 items. The month with the
most publications corresponds to the Metaverse Fashion Week (MVFW).

The median asking price of v2 wearables at the cutoff date was 30.64 USD (mad
= 45.42). Interestingly, 987 items (13.4% of the total published) have a zero
price, but are not necessarily listed for sale, as these are usually used for
airdrops or claimed as rewards. Table I presents an overview of v2 items by
_rarity_ , which is set by their creators and limits the number of token
supply that can be minted. The share of items (from the grand total) by rarity
and version follows a bell-shaped distribution, with legendary being the most
popular in both versions. The relative minted supply (i.e. the proportion of
tokens already minted) by rarity has instead a linear-log correlation, with
the lower the limit the more supply minted.

TABLE I: Overview of published v2 wearable items by rarity Rarity | Limit | Item share | Median price | Minted supply
---|---|---|---|---
unique | 1 | 4.2% | $2003.04 | 0.879
mythic | 10 | 7.6% | $156.70 | 0.683
legendary | 100 | 32.3% | $41.46 | 0.452
epic | 1000 | 31.5% | $28.20 | 0.224
rare | 5000 | 11.7% | $11.69 | 0.179
uncommon | 10000 | 7.4% | $7.30 | 0.157
common | 100000 | 5.3% | $3.03 | 0.036

Wearable 3D models can be adapted to different body shapes: female, male, or
both (unisex). Between v1 and v2 there was a significant change in the share
of items by body shape ($\chi^{2}_{2}$ = 86.9; $p$ $<$ .001), with unisex
increasing by 12.7 percent points to 92.3%, whereas female and male decreased
to a respective share of 4.1% and 3.6%. Other 3D model attributes refer to
resources for its rendering, such as the number of triangles and textures,
which –––according to the guidelines— should be limited by wearable
_category_. These are: upper body, lower body, feet, accessory (i.e., earring,
hat, helmet, etc.), head (i.e., facial hair, eyebrows, etc.), and skin (which
changes the appearance of the whole avatar). For instance, no more than 5k
triangles and 5 textures for skin; no more than 1.5k triangles for all other
wearables except accessory (500 triangles) with at most 2 textures. However,
we found that these values are exceeded in 41% of the items for triangles and
15% for textures. By far the most occurring item categories are upper body and
accessory, with 36.6% and 36%, respectively. Based on Freeman’s theta measure
of association for ordinal and multinomial variables, rarity and category are
very weakly associated ($\theta$ = 0.06). Still, and as can be seen in the
item share by these two variables in Figure 3, skin has proportionally more
unique items (11%) compared to the mean of the other categories (4.3%).

Figure 3: Item share of published v2 wearables by category and rarity.

##  V Token Mints

The maximum token supply of v1 wearables was 246.2k, of which 70.9k tokens
(28.8%) have been minted. However, given the deprecation of v1, their minting
has practically halted. The maximum v2 token supply was 50.87M, of which only
3.56M tokens (7%) have been minted. There were 1,225 v2 creators (96% of the
total) that published 6,9k items (94.3%) minted at least once, or, stated
differently, 50 creators (4%) had 423 items (5.7%) never minted as of the cut-
off date. Around 66.6% of v2 creators with mints were involved on a primary
sale on the marketplace, in which 8,953 buyers minted the tokens. Of the mints
not associated with a primary sale, 682.3k mints (18.9% of the grand total)
were transfers from the creators to themselves, and the rest (80.8% of the
grand total) were transfers to other beneficiaries. Overall, there were 564.3k
distinct beneficiaries of 3.56M minted tokens, with 354.7k beneficiaries
(62.8%) having received only one token, and 35.85k (6.4%) ten or more.

##  VI Marketplace Sales

Concerning the overall sales within the marketplace, v1 had 9.2K sales with a
median sale price of 3.1 USD, and v2 had 166.9K sales with a median sale of
2.2 USD, with only 341 v1 sales being made since the introduction of v2. The
number of v2 primary sales was 120.7k (72.4% of the total), representing just
3.4% of v2 mints; the median primary sale value was 1.56 USD (mad = 2.32).
Despite both sales volume and value increasing at the end of 2021 and reaching
their peak in early 2022, the median value of primary sales decreased
remarkably during the second half of 2021 (see Figure 4), most likely due to
the highly increased supply of tokens. As for secondary sales, 9.7% were bids
(offers made for a token not listed for sale) and the rest were listings (of
tokens publicly listed for sale), with an overall median value of 3.03 USD
(mad = 3.91).

Figure 4: Monthly v2 primary sales by volume, value, and median price.

The number of items with at least one primary sale was 3,332 (45.5% of the
total). We modeled the item median price of these sales by first selecting
features intrinsic to the item (i.e., rarity, category, single or multiple
item collection, 3D model complexity), and then using a forward stepwise
regression linear model based on the Akaike information criterion (AIC). Table
II reports the predictor coefficients of the final model, with $R^{2}_{adj}$ =
0.29, F(4, 3327) = 341.3, $p$<.001. Of these predictors, and based on a simple
elasticity model, the most important is the item’s rarity limit ($R^{2}_{adj}$
= 0.24). As expected, the rarer the item, the more expensive it is. However,
this predictor alone systematically underestimates the price of unique items,
as can be seen in Figure 5, hence the dummy predictor for this rarity in the
final linear model. Based on this model, the elasticity of the price with
respect to the rarity limit is –0.389, and being unique is associated with a
+56.9% price change. On the contrary, belonging to a single-item collection is
associated with a –32.5% change. Interestingly, the number of textures (and
not triangles) is the most relevant 3D model metric, with an associated +5.7%
price change by texture.

TABLE II: Linear model of the item median price for primary sales | $\mathbf{log_{10}(\textbf{}{price})}$
---|---
Coefficient | Estimate | Std. Error | $p$
(Intercept) | 2.137 | 0.049 | <.001
$log_{10}$(rarity limit) | –0.389 | 0.012 | <.001
is single-item collection | –0.325 | 0.024 | <.001
is unique | 0.569 | 0.119 | <.001
textures | 0.057 | 0.013 | <.001
Figure 5: Beeswarm plots of the median item sale price by rarity for primary
sales (3.3k items). The black line depicts a simple elasticity linear model.

In regards to secondary sales, there were 3,269 distinct items, of which 1,508
(46%) did not have any primary sale within the marketplace, i.e., the
respective tokens were minted by other means and then sold on the marketplace.
Of the tokens minted via a primary sale, only 3.4% were the object of a
secondary sale. The tokens with multiple sales were 7,375, with 88.3% having
just two sales. Of the 840 tokens with three of more sales, 73 (8.6%) have at
least one round trip sale, i.e., an account eventually re-buys the token it
previously sold. This is indicative of trade washing, a manipulative practice
that affects most NFT markets in varying degrees [17]. The most blatant
instance concerns the most sold token, with only two accounts trading it 15
times in just 18 days.

##  VII Discussion

###  VII-A Change and standardization

The change from the short-lived v1 to v2 wearables was substantial in terms of
publication, implementation, and economics. The initial version was, in part,
victim to the sudden increase in interest on NFTs and on Ethereum in
particular, which put into the limelight the network’s transaction fees
volatility, scalability issues. Hence, such a speedy transition was
inevitable, but it was not without hiccups. For instance, changes in rendering
of materials entailed a cumbersome procedure of 3D model rejection and re-
submission, which puts in doubt the long-term functioning of in-world assets.
These issues are also a manifestation of the current frenzy around blockchain
and the Metaverse. Both are nowadays frequently associated with the so-called
Web3, an envisioned decentralized iteration of the Web based on blockchain,
and as it happened with the Web, to achieve such a vision it is necessary to
first agree on common standards on a global scale.

Recently, several companies (including Meta and Microsoft) have founded the
Metaverse Standards Forum, a platform for collaboration between standards
organizations and businesses to promote an interoperable and open Metaverse.
The blockchain-based community reacted with the Open Alliance for the
Metaverse (OMA3), with the intended goal of ensuring that virtual land,
digital assets, ideas, and services follow the principles of Web3.
Decentraland is among the founding members of OMA3, which has now joined the
Metaverse Standards Forum to work on 3D asset interoperability, digital asset
management, privacy, cybersecurity and identity. To add to the confusion, the
Open Metaverse Foundation (by the Linux Foundation) has been recently
launched. At the moment of writing all of these organizations had existed for
less than a year; it remains to be seen if their missions are indeed reached.

###  VII-B Wearables and the monetization of promotion

Surprisingly, only a mere share of mints (3.4%) is done via primary sales on
the platform marketplace, and the volume of secondary sales on it is much
lower than we had expected. Moreover, at the moment of writing (early 2023),
there were only 3.2k tokens of v2 wearables listed as “buy now” (i.e., listed
at a set price by the owner) on OpenSea, the most common external marketplace
for Decentraland assets. From the minting data alone, it seems that the
majority of tokens are given away as part of airdrops or awards to entice
potential or existing users. Upon manual verification, this is certainly the
case with the most prolific creators such as Decentral Games.

Further, for the most popular or complex items, it seems that the main scope
is not earn money from the sales per se, but to promote other products and
services. For instance, Figure 6 depicts: a) the top most minted item which
was part of a give away to promote its creator; b) the item with the most
complex 3D geometry used to promote Sophia The Robot; and c) the item with the
highest number of primary sales, at zero price plus network fees. Indeed, at
the moment of writing MVFW 2023 hast just taken place, a testament of the in-
world promotion appeal for the fashion industry. Concerning desirability of
wearables, based on our linear model the most important factor is the item’s
rarity, especially if unique, with other significant factors being the size of
the collection and the number of textures of the 3D model. It should be noted,
however, that the term _rarity_ as used in Decentraland wearables differs
considerably from the same term used on most NFT projects, in which it denotes
the relative frequency of a token with a given set of characteristics. In the
latter case, _rarity_ is also the most important factor for sale price, but
with different and varying statistical distributions per project [18].

Figure 6: Selection of outlier items: a) CCC - Hat - Gray, the most minted
with 99,961 mints; b) Sophia 42 - #2, the most complex 3D model with 8,657
triangles, 7 textures, and 6 meshes; c) The Royal Family Crown, with the most
primary sales being sold 7,781 times at zero price plus network fees.

###  VII-C Study limitations

One of the main limitations of our work is the focus on Decentraland’s
marketplace: a creator that mints a token for themselves could put it for sale
(within or without Decentraland’s marketplace), transfer it to other accounts
(e.g, as part of an airdrop or claimed awards), or simply keep it
indefinitely. At the same time, Decentraland’s marketplace is the entrance
point and main setting for trading wearables on the platform, particularly for
v2 primary sales, and it is the most common data source regarding Decentraland
used in previous literature [11, 10, 9]. Another limitation is the lack of
data and analysis regarding the effective use of wearables within
Decentraland’s virtual world. For instance, it is plausible that many users
own wearables that they have not worn in-world and perhaps will never do or
will do for a very brief period of time. However, the on-chain data concerns
only ownership of wearables, which is an interesting aspect by itself, but it
is a faulty proxy of actual fashion as seen in-world. In addition, despite the
popularity and influence of Decentraland, it would be interesting to see if
similar wearable dynamics are present on similar blockchain-based virtual
worlds.

##  VIII Conclusion

Decentraland’s blockchain-based approach to virtual wearables has piqued the
attention of digital fashion designers and consumers. Although items can be
minted and sold within the platform’s marketplace, our study shows that most
are distributed by other means, mainly with the scope of promoting other
projects and initiatives. For those sold within, the price is unsurprisingly
primarily driven by their predefined rarity, with collection cardinality and
number of textures also being important. However, being among the first in
this domain has also brought problems for the platform regarding price
volatility and long-term robustness of its technical implementations. We thus
hope that our work inspires others working on bringing the Metaverse into a
vogue reality.

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# Anisotropic topological superconductivity in Josephson junctions

Barış Pekerten  Joseph D. Pakizer  Benjamin Hawn  Alex Matos-Abiague
Department of Physics and Astronomy, Wayne State University, Detroit, MI
48201, USA

###### Abstract

We investigate the effects of magnetic and crystalline anisotropies on the
topological superconducting state of planar Josephson junctions (JJs). In
junctions where only Rashba spin-orbit coupling (SOC) is present, the
topological phase diagram is insensitive to the supercurrent direction, but
exhibits a strong dependence on the magnetic field orientation. However, when
both Rashba and Dresselhaus SOCs coexist, the topological phase diagram
strongly depends on both the magnetic field and junction crystallographic
orientations. We examine the impact of the magnetic and crystalline anisotropy
on the current-phase relation (CPR), energy spectrum, and topological gap of
phase-biased JJs, where the junction is connected in a loop and the
superconducting phase difference is fixed by a loop-threading magnetic flux.
The anisotropic CPR can be used to extract the ground-sate phase (i.e. the
superconducting phase difference that minimizes the system free energy)
behavior in phase-unbiased JJs with no magnetic flux. Under appropriate
conditions, phase-unbiased JJs can self-tune into or out of the topological
superconducting state by rotating the in-plane magnetic field. The magnetic
field orientations at which topological transitions occur strongly depend on
both the junction crystallographic orientation and the relative strength
between Rashba and Dresselhaus SOCs. We find that for an optimal practical
application, in which the junction exhibits topological superconductivity with
a sizable topological gap, a careful balancing of the magnetic field
direction, the junction crystallographic orientation, and the relative
strengths of the Rashba and Dresselhaus SOCs is required. We discuss the
considerations that must be undertaken to achieve this balancing for various
junction types and parameters.

##  I Introduction

Majorana bound states (MBSs) are zero-energy quasiparticle excitations
predicted to arise in topological superconductors (TSs). [1, 2, 3, 4, 5, 6]
Due to their non-Abelian exchange statistics they can be utilized as qubits
for fault-tolerant quantum computing, with quantum gates realized through
braiding operations. [7, 8, 9, 10] Driven by this technological impetus,
proposals to achieve topological superconductivity have included, among
others, 1D systems such as magnetic chains on s-wave superconductors, [11, 12,
13, 14, 15, 16] semiconductor nanowires with large spin-orbit coupling (SOC)
proximitized by s-wave superconductors, [17, 18, 19, 20, 21, 22, 23, 24, 25,
26, 27] and proximitized systems exposed to magnetic textures. [28, 29, 30,
31, 32, 33, 34, 35, 36] Because of their experimental feasibility, planar
Josephson junctions (JJs) have also been considered as an alternative
promising platform for creating and manipulating MBSs. [37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60,
61, 62] Moreover, the superconducting phase difference across JJs provides an
additional control knob that can enhance the parameter space leading to
topological superconductivity. However, tuning a planar JJ to a topologically
nontrivial state does not necessarily guarantee the existence of a sizable
topological gap, which is a practical requirement for the stability of MBSs
and braiding operations. [63, 64] Indeed, as shown later in this work (see
also Ref. 65), even when the parameter space for topological superconductivity
is relatively large, the topological gap may be sizable only over reduced
subregions.

Josephson junctions with noncentrosymmetric superconductors (particularly
$d$-wave superconductors) have been predicted to exhibit anisotropic effects.
[66, 67, 68] In this work, we consider the effects of magnetic and crystalline
anisotropies on the topological superconducting state in planar JJs formed in
a semiconducting two-dimensional electron gas proximitized by $s$-wave
superconductors. The interrelation between the Zeeman interaction and the
Rashba SOC emerging from the lack of structure inversion symmetry [69] in
proximitized planar JJs gives rise to a strong dependence of the system
properties on the magnetic field direction. Furthermore, in junctions where
both Rashba and Dresselhaus SOCs are relevant, not only the magnetic field
direction, but also the junction crystallographic orientation can strongly
affect the topological superconducting state. [65] The Dresselhaus SOC
originates from the bulk inversion asymmetry, [70] which can be particularly
large in some zinc blende semiconductors (e.g., InSb) suitable for building
superconductor/semiconductor proximitized JJs. [71] Here we investigate the
impact of SOC-induced anisotropies of topological phase transitions on the
topological gap, topological charge, energy spectrum, ground-state phase,
current phase relation, and critical currents in planar JJs.

Figure 1: [Color online] (a) A JJ composed of a non-centrosymmetric
semiconductor 2DEG in contact to two superconducting (S) leads. The $\hat{x}$
and $\hat{y}$ axes define the coordinate system in the junction reference
frame. The Rashba SOC strength can be controlled by using a gate on the top of
the normal region [42, 72]. The current flow is perpendicular to the junction.
(b) $\varphi_{B}$ and $\theta_{c}$ characterize the orientation of the in-
plane magnetic field ($\mathbf{B}$) and the junction reference frame,
respectively, with respect to the semiconductor [100] crystallographic axis.

##  II Theoretical Model

###  II.1 General considerations

We consider JJs composed of superconducting (S) and normal (N) regions (see
Fig. 1). The S regions are formed in a semiconducting 2DEG proximitized by a
superconducting (e.g., Al or Nb) covering. Excitations in the JJ are described
by the Bogoliubov-de Gennes (BdG) Hamiltonian,

$H=H_{0}\tau_{z}-\frac{g^{\ast}\mu_{B}}{2}\mathbf{B}\cdot\bm{\Sigma}+\Delta(x)\tau_{+}+\Delta^{\ast}(x)\tau_{-}\;,$
(1)

where

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\frac{\mathbf{p}^{2}}{2m^{\ast}}+V(x)-(\mu_{S}-\varepsilon)+\frac{\alpha}{\hbar}\left(p_{y}\sigma_{x}-p_{x}\sigma_{y}\right)$
$\displaystyle+$
$\displaystyle\frac{\beta}{\hbar}[(p_{x}\sigma_{x}-p_{y}\sigma_{y})\cos
2\theta_{c}-(p_{x}\sigma_{y}+p_{y}\sigma_{x})\sin 2\theta_{c}]\;,$

is the single-particle Hamiltonian of the 2DEG in the absence of a magnetic
field. In the equations above, $\mathbf{p}$ is the momentum, $m^{\ast}$ the
electron effective mass, $\alpha$ and $\beta$ are, respectively, the Rashba
and Dresselhaus SOC strengths, and $\theta_{c}$ characterizes the direction of
the current ($x$ axis) with respect to the [100] crystallographic direction of
the semiconductor [Fig. 1(b)]. The crystallographic orientation of the
junction is determined by the angle $\theta_{c}+\pi/2$. The length of the
junction is $L$ and the widths of the S and N regions are $W_{S}$ and $W_{N}$,
respectively (see Fig. 1). The gate-voltage-induced difference between the
chemical potentials in the N ($\mu_{N}$) and S ($\mu_{S}$) regions is
described by $V(x)=(\mu_{S}-\mu_{N})\,\Theta(W_{N}/2-|x|)$ and
$\sigma_{x,y,z}$ and $\tau_{x,y,z}$, with $\tau_{\pm}=(\tau_{x}\pm
i\tau_{y})/2$, represent Pauli and Nambu matrices, respectively. The chemical
potentials are measured with respect to the minimum of the single-particle
energies, $\varepsilon=m^{\ast}\lambda^{2}(1+\left|\sin
2\theta_{c}\right|)/2\hbar^{2}$, where we have used the SOC parametrization,

$\alpha=\lambda\cos\theta_{so}\;,\;\;\beta=\lambda\sin\theta_{so}\;,\;\;\lambda=\sqrt{\alpha^{2}+\beta^{2}}.$
(3)

Here $\lambda$ represents the overall strength of the combined Rashba +
Dresselhaus SOCs, while the spin-orbit angle,

$\theta_{so}={\rm arccot}(\alpha/\beta),$ (4)

characterizes the relative strength between Rashba and Dresselhaus SOC.

The second contribution in Eq. (1), with the Dirac spin matrices
$\bm{\Sigma}$, represents the Zeeman splitting due to an applied magnetic
field,

$\mathbf{B}=|\mathbf{B}|\begin{pmatrix}\cos\varphi_{B}\\\ \sin\varphi_{B}\\\
0\end{pmatrix}.$ (5)

The angle $\varphi_{B}$ characterizes the direction of the magnetic field with
respect to the current flow ($x$ axis), as shown in Fig. 1(b). The spatial
dependence of the superconducting gap is $\Delta(x)=\Delta e^{-i\;{\rm
sgn}(x)\phi/2}\,\Theta(|x|-W_{N}/2)$, where $\phi$ is the phase difference
across the JJ.

The temperature and magnetic field dependence of the superconducting gap is
taken into account by using the BCS relation,

$\Delta(T,B)\approx\Delta(T,0)\sqrt{1-\left[\frac{B}{B_{c}(T)}\right]^{2}},$
(6)

where $\Delta(T,0)\approx\Delta_{0}\tanh[1.74\sqrt{T_{c}/T-1}]$,
$\Delta_{0}=1.74$ $k_{B}T_{c}$, $k_{B}$ is the Boltzmann constant, and $T_{c}$
is the superconductor critical temperature. The temperature dependence of the
critical magnetic field can be approximated as,
$B_{c}(T)=B_{c}(1-T^{2}/T_{c}^{2})$, where $B_{c}$ is the critical magnetic
field at zero temperature.

Although we consider ballistic junctions throughout this work, we expect the
predicted anisotropic effects to qualitatively hold in the presence of weak
disorder. Junctions with low disorder are usually preferred because the
presence of disorder typically reduces the parameter space supporting
topological superconductivity, although in some cases weak disorder can
increase the robustness of the topological superconducting state [73, 74, 75,
76].

###  II.2 Topological gap and topological charge

Topological superconductivity (TS) is a superconducting phase featuring a pair
of degenerate zero-energy states, called Majorana bound states (MBSs), which
are isolated from the rest of the excitation spectrum by an energy gap, called
the topological gap ($\Delta_{top}$). In the TS state, the topological gap
cannot be destroyed by smooth local perturbations, providing protection for
the MBSs. However, the information stored in the MBSs can be damaged if the
perturbation energy becomes comparable or larger than the topological gap. A
large topological gap is therefore desirable for the practical use of fault-
tolerant qubits encoded in MBSs.

The topological gap can be estimated by imposing translational invariance
along the junction direction (the $y$ direction in our case). In such a
system, the momentum component $p_{y}$ is a conserved quantity and can be
substituted by $\hbar k_{y}$. Then $\Delta_{\rm top}$ is obtained as the
eigenenergy closest to zero,

$\Delta_{\rm top}=\min_{k_{y}}|E(k_{y})|.$ (7)

only when the system is in the TS state. When the system is in the trivial
state, the quantity defined in Eq. (7) simply denotes the lowest positive-
energy Andreev state.

The size of the topological gap strongly depends on the interrelation between
the spin-orbit angle ($\theta_{so}$), the junction crystallographic
orientation ($\theta_{c}$) and the in-plane magnetic field orientation
($\varphi_{B}$). [48, 65] In symmetric JJs (i.e., with identical left and
right S coverings) the optimal topological gap is achieved when the system is
in the TS state and the condition, [65]

$\tan{\varphi_{B}}=\cot\theta_{so}\;\sec{2\theta_{c}}-\tan{2\theta_{c}},$ (8)

is fulfilled.

This work focuses on the behavior of the current across the JJ (see Fig. 1a).
Hence, we use the current direction as the axis with respect to which the
magnetic field orientation, $\varphi_{B}$, is defined (see Fig. 1b) [77].

In the presence of Rashba and Dresselhauss SOC the topological gap exhibits
strong magnetic and crystalline anisotropies. [65] Therefore the fulfillment
of Eq. (8) is a vital prerequisite for the optimization of $\Delta_{top}$.
However, the relation in Eq. (8) alone is not sufficient for inducing TS.
Therefore, in a practical situation, one will need to first arrange the
experimental setup in a way that Eq. (8) is fulfilled, and then tune other
system parameters (e.g., chemical potential, magnetic field amplitude) to
drive the system into the TS state.

The TS state in symmetric JJs typically belongs to the D class. However, the
BDI class can emerge for some specific junction crystallographic orientations
and magnetic field directions (see Table I in Ref. 65). Therefore, the
transition between the trivial and TS states typically occur when the $Z_{2}$
topological index (also called topological charge, $Q$) associated with the
symmetry class D changes sign. According to the bulk-boundary correspondence,
we can obtain the phase diagram of a junction with finite length by computing
the topological charge of the translational-invariant version of the junction,

$Q={\rm sgn}\left[\frac{{\rm Pf}\\{H(k_{y}=\pi)\sigma_{y}\tau_{y}\\}}{{\rm
Pf}\\{H(k_{y}=0)\sigma_{y}\tau_{y}\\}}\right],$ (9)

where ${\rm Pf}\\{...\\}$ denotes the Pfaffian [78, 79, 80, 81]. The
topological charge determines whether the system is in the trivial ($Q=1$) or
topological ($Q=-1$) phase. [82, 83, 73, 74, 75, 84]

###  II.3 Current-phase relation and critical current in Josephson junctions

The supercurrent across the JJ can be obtained from the energy spectrum of the
BdG Hamiltonian given in Eq. (1). Indeed, the eigenenergies ($E_{n}$) can be
used to compute the free energy of the junction,

$F=-\sum_{E_{n}>0}\left[E_{n}+2k_{B}T\ln\left(1+e^{-\beta
E_{n}}\right)\right]$ (10)

with $\beta=1/k_{B}T$. The current-phase relation (CPR) is then obtained as

$I(\phi)=\frac{2e}{\hbar}\frac{dF}{d\phi}=I_{0}(\phi)+\Delta I(\phi,T).$ (11)

where the zero-temperature contribution is given by

$I_{0}(\phi)=-\frac{2e}{\hbar}\sum_{E_{n}>0}\frac{dE_{n}}{d\phi},$ (12)

while the temperature-dependent correction reads,

$\Delta I(\phi,T)=\frac{4e}{\hbar}\sum_{E_{n}>0}\left[\frac{1}{1+e^{\beta
E_{n}}}\right]\frac{dE_{n}}{d\phi}.$ (13)

In the low-temperature limit ($\beta E_{n}\gg 1$), Eqs. (11)–(13) yield

$\displaystyle I(\phi)$ $\displaystyle\approx$ $\displaystyle
I_{0}(\phi)+\frac{4e}{\hbar}\sum_{E_{n}>0}e^{-\beta
E_{n}}\frac{dE_{n}}{d\phi}$ (14) $\displaystyle=$ $\displaystyle
I_{0}(\phi)-\frac{4e}{\hbar\beta}\frac{d}{d\phi}\left(\sum_{E_{n}>0}\
e^{-\beta E_{n}}\right),$

while in the high-temperature regime ($\beta E_{n}\ll 1$),

$I(\phi)\approx-\frac{e\beta}{\hbar}\sum_{E_{n}>0}E_{n}\frac{dE_{n}}{d\phi}=-\frac{e\beta}{2\hbar}\frac{d}{d\phi}\left(\sum_{E_{n}>0}E_{n}^{2}\right).$
(15)

In the phase-biased case, the JJ is connected to a closed loop threaded by a
magnetic flux, $\Phi$, and the superconducting phase difference across the
junction is fixed to the value $\phi=2\pi\Phi/\Phi_{0}$, where $\Phi_{0}$ is
the magnetic flux quantum. In this case the current-phase relation [Eq. 11]
can be experimentally measured by tuning the magnetic flux.

In the absence of a magnetic flux the junction is phase unbiased and the phase
difference self-adjusts in such a way that the free energy of the system is
minimized. The ground-state phase ($\phi_{GS}$) is the superconducting phase
difference that minimizes the free energy of the system, i.e.,

$F(\phi_{GS})=\min_{\phi}F(\phi)$ (16)

and the ground-state spectrum is the energy spectrum of the JJ evaluated at
the ground-state phase, i.e., $E_{n}(\phi_{GS})$. The mathematical conditions
for the free energy to have a minimum at the ground-state phase are,

$\left.\frac{dF}{d\phi}\right|_{\phi=\phi_{GS}}=0\;\;{\rm
and}\;\;\left.\frac{d^{2}F}{d\phi^{2}}\right|_{\phi=\phi_{GS}}>0,$ (17)

which, according to Eq. (11) can be rewritten as,

$I(\phi_{GS})=0\;\;{\rm
and}\;\;\left.\frac{dI}{d\phi}\right|_{\phi=\phi_{GS}}>0.$ (18)

The relations above allow for extracting the ground-state phase from the CPR.

Under forward bias voltage, the critical current of a phase-unbiased JJ is
obtained by maximizing the current amplitude with respect to the phase
difference, i.e.,

$I_{c}=\max_{\phi}I(\phi).$ (19)

However, under reverse bias voltage, the critical current is negative and is
determined by minimizing the supercurrent. In centrosymmetric JJs the
amplitudes of the forward and reverse critical currents are equal. However,
the interrelation between the SOC and the in-plane magnetic field can break
the inversion symmetry and lead to the so-called superconducting diode effect
[85, 86, 87, 88, 89, 90, 91], where the amplitudes of the forward and reverse
critical currents become different. In this work we limit our analysis to the
case of the forward critical current.

###  II.4 Numerical Approach

We use a finite-difference discretization of Eq. (1) to build a tight-binding
version of the BdG Hamitonian, which is then numerically diagonalized to find
the eigenstates and energy spectrum (see Appendix A for more details). The
Pfaffians of the tight-binding Hamiltonian with imposed translational
invariance along the junction direction are numerically calculated to compute
the topological charge, while the energy spectrum is used to calculate the
free energy of the system, CPR, ground state phase, and critical currents.

The numerical simulations of the tight-binding version of the BdG Hamiltonian
are performed by using the Kwant package [92]. To illustrate and compare the
different effects of magnetic and crystalline anisotropies, two type of
junctions are considered: i) Al/HgTe JJs, where only Rashba SOC plays a role
and ii) Al/InSb JJs, where both Rashba and Dresselhaus SOC become relevant
(see Appendix for more details).

##  III Current-phase relation in phase-biased Josephson junctions

###  III.1 Effects of magnetoanisotropy

To investigate the effects of magnetic field orientation on the CPR of a
phase-biased JJ, we consider Al/HgTe JJs (see system parameters in Appendix
A), where Rashba SOC is large and Dresselhaus SOC is negligibly small. In such
systems the spin-orbit angle $\theta_{so}=0$ and the CPR is independent of the
junction crystallographic orientation. For the chosen system parameters, the
estimated zero-field superconducting coherence length of the Al/HgTe JJ is
$\xi=81$nm, which is smaller but comparable to the width of the normal region,
and about 1/4th the size of each lead.

Figure 2: [Color online] (a) Plot of $I(\phi)$ and $Q$ as a function of $\phi$
and $B$ for a HgTe Josephson junction (see Figure 1). Here,
$\theta_{\textrm{SO}}=\theta_{c}=0$ and $\varphi_{B}=\pi/2$. The TB
simulations are made for a translationally invariant system in the
$y$-direction (Appendix A), with a SC lead width of $252$nm each and a
junction width of $96$nm. The vertical dashed lines correspond to $B=0.4$T and
$B=0.8$T (see Figure 3). The shaded (unshaded) areas have $Q=1$ ($Q=-1$). The
TB lattice parameter is $a=6$nm, yielding a hopping parameter of $t=27.9$meV.
(b) and (c) The lowest 200 energy levels corresponding to a fully 2D TB
simulation of the system in (a), with a length $L=4002$nm, at (b) $B=0.4$T and
(c) $B=0.8$T. The red lines correspond lowest energy levels.

The supercurrent (normalized to its maximum value) of a Al/HgTe JJ is shown in
Fig. 2(a) as a function of an in-plane magnetic field perpendicular to the
current (i.e., $\varphi_{B}=\pi/2$) and the superconducting phase difference.
The shaded (unshaded) areas correspond to topological charge $Q=1$ (trivial
state) and $Q=-1$ (TS state), respectively. Both the supercurrent amplitude
and direction can be tuned by changing the phase difference and/or the
magnetic field strength. Note that for magnetic fields larger than the
critical field (1.19 T at 0.7 K), superconductivity is destroyed and the
supercurrent vanishes.

Figure 3: [Color online] [(a) and (b)] Plot of $I(\phi)$ and $Q$ as a function
of $\phi$ and $\varphi_{B}$, with $\theta_{\textrm{SO}}=\theta_{c}=0$, for a
HgTe junction with (a) $B=0.4$T and (b) $B=0.8$T. The shaded (unshaded) areas
have $Q=1$ ($Q=-1$). The TB simulation parameters and the junction geometry
are the same as in Fig. 2. The vertical dashed lines are at
$\varphi_{B}=0.2\pi$ and $\varphi_{B}=0.5\pi$. (c), (d) The current-phase
relation along the dashed lines in (a) and (b), respectively. In (d), note the
difference between the maxima of the forward and reverse supercurrents,
indicated by the horizontal dashed lines for $\varphi_{B}=0.5\pi$ case.

At zero magnetic field the CPR is anti-symmetric under reflection with respect
to $\phi=\pi$, i.e., $I(\phi)=-I(2\pi-\phi)$. In the presence of Rashba SOC
and a finite in-plane magnetic field this symmetry is preserved only when the
field is parallel to the current ($\varphi_{B}=0$). The symmetry breaking for
$B\neq 0$ and $\varphi_{B}=\pi/2$ is clearly seen in Fig. 2(a).

The two white traces in Fig. 2(a) correspond to parameters for which the
supercurrent vanishes. However, the conditions in Eq. (18) are satisfied only
along the lower trace. As shown in the figure, the path of the lower trace is
in good agreement with the magnetic field dependence of the ground-state phase
(black dashed line). This illustrates how the magnetic field dependence of the
CPR in a phase-biased junction can be used to extract the ground-state phase
of the phase-unbiased junction. The method works well when only a single trace
in the $B$ dependence of the CPR satisfies Eq. (18). However, the situation
may not be that clear when there are multiple traces obeying Eq. (18). In such
a case, different traces correspond to different free energy minima, and the
information in the $B$ dependence of the CPR is not enough to decide which one
represents the absolute minimum.

The phase dependence of the energy spectrum is shown in Figs. 2(b) and 2(c)
for $B=0.4$ T and $B=0.8$ T [indicated with vertical dashed lines in (a)],
respectively. The red lines illustrate the formation of MBSs in the TS state.
For $B=0.4$ T, zero-energy MBSs appear in a reduced interval of $\phi$-values,
while at $B=0.8$ T zero-energy MBSs exist for any value of the phase
difference. This is an agreement with the topological region depicted in Figs.
2(a). Although the junction is long enough, and the MBSs are typically well
separated from each other [note that MBSs have zero energy in most of the
topological region, as shown in Figs. 2 (b) and (c)], they start to hybridize
and depart from zero energy when the topological gap becomes too small. This
is particularly noticeable in Fig. 2(c) around $\phi\sim 1.5\pi$, and
emphasizes the fact that, even if the system is in the topological state, one
may still need to optimize the topological gap to realize stable MBSs.

The Rashba SOC is rotationally invariant, however its combination with the in-
plane Zeeman interaction leads to magnetoanisotropic effects. The magnetic
anisotropy of the CPR is shown if Figs. 3(a) and 3(b), where the normalized
supercurrent is shown as a function of the phase difference, $\phi$, and the
magnetic field orientation, $\varphi_{B}$ for the two values of magnetic field
amplitudes corresponding to the vertical dashed lines in Fig. 2(a). In both
cases, the topological region exhibits a strong dependence on $\varphi_{B}$.
When the in-plane magnetic field is parallel to the supercurrent direction,
the system is in the trivial state for any phase difference. As the magnetic
field is rotated towards the junction direction, the range of phase
differences leading to TS increases. The results demonstrate the convenience
of orienting the magnetic field in the direction, $\varphi_{B}=\pi/2$, i.e.,
perpendicular to the supercurrent flow. [48, 65] The CPR at different magnetic
field orientations indicated by vertical dashed lines in Figs. 3(a) and 3(b)
are shown in Figs. 3(c) and 3(d), respectively. In both cases, the CPR shows a
slight deviation from a sinusoidal function, evidencing the low transparency
of the junction. Furthermore, an anomalous phase [93, 94, 90, 95, 51, 87]
emerges due to the combined action of the Rashba SOC and the in-plane magnetic
field, producing a $\varphi_{B}$-dependent shift of the CPR. Although hardly
notable at the scale of the figure, numerical evaluation reveals that for
$\varphi_{B}\neq 0$, the forward (maximum) and reverse (minimum) supercurrents
are slightly different. This manifestation of the superconducting spin diode
effect[89] is perhaps more apparent in the CPR shown in Fig. 3(d) for $B=0.8$
T and $\varphi_{B}=\pi/2$, where the horizontal lines indicate the different
amplitudes of the forward and reverse supercurrents.

###  III.2 Effects of crystalline anisotropy

The linear Rashba SOC exhibits rotational invariance. However, the coexistence
of Rashba and Dresselhaus SOCs reduce the symmetry of the spin-orbit field to
a twofold $C_{2v}$ symmetry [96, 97]. Such a symmetry reduction leads to
various magnetoanisotropic phenomena in both the normal [98, 99, 100, 101] and
superconducting [102, 103, 104, 105, 106, 107] states as well as crystalline
anisotropic phenomena in which the system properties depend on the specific
crystallographic orientation and/or transport direction [108, 65, 109]. The
effects of crystalline anisotropy on the topological gap of planar JJs was
investigated in Ref. 65. In this Section we explore the SOC-induced
crystalline anisotropy of the CPR in a phase-biased, planar JJ. For the
numerical simulations of the crystalline anisotropy, we consider Al/InSb JJs,
which for the considered system parameters (see Appendix A) have an estimated
zero-field superconducting coherence length, $\xi=164$nm, which is larger than
$W_{N}$, but shorter than $W_{S}$.

The CPR as a function of the in-plane magnetic field direction is shown in
Fig. 4 for different supercurrent directions ($\theta_{c}$) and spin-orbit
angles ($\theta_{so}$). The supercurrent direction is fixed by the junction
orientation with respect to the $[100]$ crystallographic axis. The dashed
lines indicate the ground-state phase computed by minimizing the system free
energy and are in good agreement with the CPR contours obeying Eq. (18). When
only Rashba SOC is present, $\theta_{so}=0$ and the CPR is independent of
$\theta_{c}$. However, the presence of Dresselhaus SOC leads to appreciable
changes in the CPR. For $\theta_{so}=\pi/8$, the topological (unshaded) region
changes its size and exhibits a shift in its position with respect to the
magnetic field direction when the supercurrent direction changes, as shown in
Figs. 4(a)–4(c). A similar trend is observed when the strength of the Rashba
and Dresselahus SOC are equal, $\alpha=\beta$ [i.e., $\theta_{so}=\pi/4$ in
Figs. 4(d) and 4(e)] and when only Dresselhaus SOC is present [i.e.,
$\theta_{so}=\pi/2$ in Figs. 4(g)–4(i)].

Figure 4: [Color online] [(a)–(i)] Plot of $I(\phi)$ and $Q$ as a function of
$\phi$ and $\varphi_{B}$ for an InSb junction, for various values of
$\theta_{\textrm{SO}}$ and $\theta_{c}$, for $B=0.6$T [see Figs. 2(b) and
3(c)] . The top row has $\theta_{\textrm{SO}}=\pi/8$, the middle row has
$\theta_{\textrm{SO}}=\pi/4$ and the bottom row has
$\theta_{\textrm{SO}}=\pi/2$. The shaded (unshaded) areas have $Q=1$ ($Q=-1$).
The TB simulation parameters and the junction geometry are given in Appendix.

Previous investigations [37, 39, 42] have shown the importance of properly
tuning the chemical potential, magnetic field strength, and superconducting
phase difference for driving the JJ into the TS state. However, the strong
dependence of the TS state and the ground-state phase on the spin-orbit angle
($\theta_{so}$), the magnetic field orientation ($\varphi_{B}$), and the
supercurrent direction ($\theta_{c}$), shown in Fig. 4, reveals that an
experimental setup with an adequate combination of $\theta_{so}$,
$\varphi_{B}$, and $\theta_{c}$ values is also crucial for inducing TS in
planar JJs.

##  IV Topological superconductivity in phase-unbiased Josephson junctions

###  IV.1 Effects of magnetoanisotropy

Figure 5: [Color online] (a) Plot of $\Delta_{\textrm{top}}/\Delta_{0}$ and
$Q$ as a function of $\phi$ and $B$ for a HgTe Josephson junction (see Fig.
1), with $\theta_{\textrm{SO}}=\theta_{c}=0$ and $\varphi_{B}=\pi/2$. The
shaded (unshaded) areas have $Q=1$ ($Q=-1$). The TB simulation parameters and
the junction geometry are the same as in Fig. 2. The red lines are the ground
state phase $\phi_{GS}$ of the system and the blue lines are
$I_{c}/I_{c}^{\textrm{max}}$, $I_{c}^{\textrm{max}}$ being the maximum
critical current in the junction at $T=0.7$K. The white ring indicates the
field and phase values at which the topological transition occurs. (b) The
lowest 200 energy levels calculated along the red ($\phi_{GS}$) curve in (a).
Here, a fully 2D closed system with $L=4002$nm is used instead of the
translationally invariant system used in (a). For (a) and (b) both, the
vertical dashed line marks the $B=0.79$T value at which
$\Delta_{\textrm{top}}/\Delta_{0}$ is largest along the $\phi_{GS}$ curve
within the nontrivial region. (c) The density plot of $\rho=|\Psi|^{2}$ for
the Majorana mode (b) (red curve) at $B=0.79$T.

In the absence of a magnetic flux, the superconducting phase difference self-
tunes to a value (the ground-state phase, $\phi_{GS}$) that leads to the
minimization of the system free energy. Since the free energy of the junction
varies with the applied magnetic field, both the ground-state phase and the
critical current also change as the strength of the in-plane magnetic field
increases.

The topological gap [see Eq. (7)] of a HgTe JJ (where only Rashba SOC is
relevant) with an in-plane magnetic field perpendicular to the supercurrent
direction is shown in Fig. 5(a) as a function of the phase difference and
magnetic field amplitude. The shaded and unshaded regions correspond to
trivial ($Q=1$) and topological ($Q=-1$) phases. The red and blue lines
correspond to the ground-state phase and normalized critical current,
respectively. The ground-state phase exhibits a jump from 0 to $\pi$ as the
magnetic field increases, as shown in Fig. 5(a) and earlier in Fig. 2(a). The
junction transitions into the topological state at the center of the white
ring, where $\phi_{GS}$ crosses the topological region. The ground-state phase
jump is accompanied by a local minimum in the critical current (blue line).
However, due to the smoothness of the ground-state jump, the critical current
minimum may occur at a magnetic field higher than the transition field
(corresponding to the center of the white ring). Therefore, in the best
situation, the critical current minimum alone can only be an indirect
indication of the topological phase transition. In a more general scenario,
there are situations in which the topological transition occurs without the
current having a local minimum [110, 39].

Figure 5(a) reveals that the topological gap is sizable only on a reduced part
of the topological region. The topological gap is crucial for ensuring the
practical protection of the MBSs. Therefore, only the portion of the parameter
space leading to the TS state with a sizable topological gap is useful from a
practical point of view. The departure of $\phi_{GS}$ from the value 0 as the
magnetic field increases, yields a topological transition at a magnetic field
slightly smaller than the one required at $\phi=0$. Furthermore, the self-
tuned jump of the ground-state phase to values close to $\pi$ allows for
achieving a sizable topological gap in the TS state.

The ground-state spectrum, i.e., the energy spectrum for which the junction
has the minimum free energy, is shown in Fig. 5(b) as a function of the
magnetic field strength. At zero magnetic field the system is in the trivial
state with a gap of about $\Delta_{0}/2$. As the magnetic field increases, the
ground-state phase starts to depart from zero, yielding a decrease in the
energy gap until it closes and reopens at the topological transition. Once the
system enters the TS state, zero energy MBSs (red line) emerge inside the
topological gap. The probability density of the MBSs (normalized to its
maximum value) is shown in Fig. 5(c) for a magnetic field value indicated by
the vertical dashed line in (b). The MBSs are well localized at the ends of
the junction.

Figure 6: [Color online] [(a) and (b)] Plot of
$\Delta_{\textrm{top}}/\Delta_{0}$ and $Q$ as a function of $\phi$ and
$\varphi_{B}$ for a HgTe junction, for (a) $B=0.4$T and (b) $B=1.0$T. The
shaded (unshaded) areas have $Q=1$ ($Q=-1$). The TB simulation parameters and
the junction geometry are the same as in Fig. 2. The red lines are the ground
state phase $\phi_{GS}$ of the system and the blue lines are
$I_{c}/I_{c}^{\textrm{max}}$, $I_{c}^{\textrm{max}}$ being the maximum
critical current in the junction for the respective $B$ value and at $T=0.7$K.
(c) Plot of $\Delta_{\textrm{top}}$ along $\phi=\phi_{GS}$ curve (red line) in
(b) as a function of $\varphi_{B}$. The shaded (unshaded) areas have $Q=1$
($Q=-1$) at the given $\varphi_{B}$. The vertical dashed line corresponds to
$\varphi_{B}^{\textrm{opt}}$ given by Eq. (8).
$\theta_{\textrm{SO}}=\theta_{c}=0$ for all plots.

The magnetic anisotropy of the topological gap, the ground-state phase (red
line), and the critical current is shown in Fig. 6(a) for $B=0.4$ T. The
trajectory of the ground-state phase indicates that for such a field value the
system is unable to self-tune into the TS state for any magnetic field
orientation. The critical current (blue line) exhibits a local minimum but it
is not associated to a topological transition. However, if the field amplitude
is increased to 1 T, the self-tuning of the ground-state phase can drive the
system into the TS state when the magnetic field is rotated away from the
direction of the supercurrent, as shown in Fig. 6(b). In this case, the jump
in the ground-state phase not only allows for the topological transition, but
also for a finite topological gap when $\varphi_{B}=\pi/2$. The jumps in the
ground-state phase are accompanied by critical current minima. Interestingly,
the critical current exhibits a maximum at the magnetic orientation leading to
the TS state with the largest topological gap.

The ground-state topological gap (i.e., the topological gap at the ground-
state phase) as a function of the magnetic field orientation for a phase-
unbiased Al/HgTe JJ is represented by the green line in Fig. 6(c). As the in-
plane magnetic field is rotated from a direction parallel ($\varphi_{B}=0$) to
perpendicular ($\varphi_{B}=\pi/2$) to the supercurrent direction, the
topological gap self-tunes from a negligible small value to a maximum of about
$0.08\Delta_{0}$. The figure illustrates the importance of properly orienting
the magnetic field when driving the system into a robust TS state, and
evidences that even if the ground-state phase self-tuning can drive the system
in the TS state for a wide range of magnetic field orientations (white area),
the topological gap is sizable and stable only within a small window around
$\varphi_{B}=\pi/2$. This is consistent with recent experimental results,
where the TS state deteriorates as the magnetic field deviates from the
direction perpendicular to the supercurrent. [42]

###  IV.2 Effects of crystalline anisotropy

Figure 7: [Color online] (a)-(i) Plot of $\Delta_{\textrm{top}}/\Delta_{0}$
and $Q$ as a function of $\phi$ and $\varphi_{B}$ for an InSb junction, for
various values of $\theta_{\textrm{SO}}$ and $\theta_{c}$. The top row has
$\theta_{\textrm{SO}}=\pi/8$, the middle row has $\theta_{\textrm{SO}}=\pi/4$
and the bottom row has $\theta_{\textrm{SO}}=\pi/2$. For all of the figures,
$B=0.6$T. The shaded (unshaded) areas have $Q=1$ ($Q=-1$). The TB simulation
parameters and the junction geometry are specified in the Appendix. The red
lines are the ground state phase $\phi_{GS}$ of the system and the blue lines
are $I_{c}/I_{c}^{\textrm{max}}$, $I_{c}^{\textrm{max}}$ being the maximum
critical current in the junction at $T=0.7$K. The vertical dashed lines
correspond to $\varphi_{B}^{\textrm{opt}}$ given by Eq. (8), for the
respective $\theta_{\textrm{SO}}$ and $\theta_{c}$.

Figure 8: [Color online] (a)-(i) Plot of $\Delta_{\textrm{top}}/\Delta_{0}$
and $Q$ as a function of $\phi$ and $B$ for an InSb junction, for various
values of $\theta_{\textrm{SO}}$ and $\theta_{c}$. The shaded (unshaded) areas
have $Q=1$ ($Q=-1$). The TB simulation parameters and the junction geometry
are provided in the Appendix. The red lines are the ground state phase
$\phi_{GS}$ of the system and the blue lines are $I_{c}/I_{c}^{\textrm{max}}$,
$I_{c}^{\textrm{max}}$ being the maximum critical current in the junction at
$T=0.7$K. For each plot, $\varphi_{B}=\varphi_{B}^{\textrm{opt}}$ for the
respective $\theta_{\textrm{SO}}$ and $\theta_{c}$, corresponding to (a)
$\varphi_{B}=3\pi/8$, (b) $\varphi_{B}=\pi/2$, (c) $\varphi_{B}=\pi/2$, (d)
$\varphi_{B}=\pi/8$, (e) $\varphi_{B}=0$, (f) $\varphi_{B}=\pi/2$, (g)
$\varphi_{B}=3\pi/4$, (h) $\varphi_{B}=\pi/2$ and (i) $\varphi_{B}=\pi/2$.

To explore the effects of crystalline anisotropy on phase-unbiased JJs, we
consider an Al/InSb junction (see Appendix A for system parameters), where
Rashba and Dresselhaus SOCs coexist. The topological gap as a function of the
magnetic field orientation is shown in Fig. 7 for $B=0.6$ T, different
crystallographic orientations ($\theta_{c}$) of the junction, and various
values of the spin-orbit angle ($\theta_{so}$). The top row corresponds to
$\theta_{so}=\pi/8$, i.e., to a situation in which Rashba SOC is about 2.4
times stronger than Dresselhaus SOC. The middle row displays the case
$\theta_{so}=\pi/4$, in which Rashba and Dressellhaus SOCs have equal
strength, and the bottom row with $\theta_{so}=\pi/2$ corresponds to a
junction in which only Dresselhaus SOC is present. The case of junctions in
which only Rashba SOC is present has been omitted because such junctions do
not exhibit crystalline anisotropy. Shaded and unshaded regions represent
trivial ($Q=1$) and topological ($Q=-1$) states, respectively. The blue lines
represent the normalized positive branch of the critical current, while the
red lines indicate the ground-state phase behavior. The topological regions,
critical current, and ground-state phase exhibit a strong dependence on both
the magnetic field and junction orientations. Furthermore, the critical
current dependence on the magnetic field direction can be used to determine
whether both Rashba and Dresselhaus or only one SOC interaction is present in
the system. Indeed, as long as the junction is not oriented along the spin-
orbit field symmetry axes [i.e., $\theta_{c}\neq(2n+1)\pi/4$] and only Rashba,
only Dresselhaus, or both SOCs are present, the absolute maxima of the
critical current occur for magnetic field orientations
$\varphi_{B}=(2n+1)\pi/2$ [see Fig. 6(b)], $\varphi_{B}=n\pi$ [see Fig. 7(g)],
or $\varphi_{B}\neq n\pi/2$ (with $n$ being an integer number) [see Figs. 7(a)
and 7(d)], respectively.

The spin-orbit field in zinc-blende semiconductor quantum wells grown along
the $[001]$ crystallographic direction exhibits a $C_{2v}$ symmetry, with
symmetry axes along the $[110]$ and $[\bar{1}10]$ directions. [97] Although
the specific crystallographic direction of the junction may lower the symmetry
to $C_{2}$, the $C_{2v}$ symmetry is still preserved as long as the junction
direction coincides with one of the spin-orbit field symmetry axes (i.e., when
$\theta_{c}=(2n+1)\pi/4$, with $n$ being an integer number). This is the
situation in the middle and right columns in Figs. 7, where the topological
gap, ground-state phase, and critical current exhibit a $C_{2v}$ symmetry with
respect to the magnetic field orientation with a symmetry axis
$\varphi_{B}=\pi/2$. Note that for $\theta_{c}=\pi/4$ and $\theta_{c}=3\pi/4$,
$\varphi_{B}=\pi/2$ corresponds to magnetic fields along the $[\bar{1}10]$ and
$[\bar{1}\bar{1}0]$ directions, respectively (i.e., to magnetic field
directions along symmetry axes of the spin-orbit field).

Unlike the critical current dependence on the magnetic field strength, which
may exhibit minima (accompanied by ground-state phase jumps) when the system
transits from the trivial to the topological superconducting state [37, 42]
(see also Fig. 8), the minima of the critical current dependence on the
magnetic field direction (see Fig. 7) is not an indicator of topological phase
transitions.

The results shown in Fig. 7 reveal that the realization of the topological
superconducting state with a sizable topological gap requires an adequate
orientation of the magnetic field, according to the junction crystallographic
direction. Indeed, even if the system is in the topological state, the
topological gap protecting the MBSs can be very small when the optimal
magnetic field orientation (vertical, dashed lines) calculated from Eq. (8) do
not cross the topological region, as shown in Figs. 7(a), 7(d), and 7(e).
However, a sizable topological gap is achieved when the system is in the TS
state and the magnetic field orientation fulfills Eq.(8), as illustrated in
Figs. 7(b), 7(c), and 7(f)–7(i).

The topological gap as a function of the magnetic field strength is shown in
Fig. 8 for for the corresponding optimal magnetic field orientations
($\varphi_{B}$), different crystallographic orientations ($\theta_{c}$) of the
junction, and various values of the spin-orbit angle ($\theta_{so}$). For
$\theta_{so}=\pi/8$ and $\theta_{so}=\pi/2$ (upper and lower rows,
respectively), as the magnetic field strength is increased, the system
transits into the TS state when the ground-state phase (red line) jumps and
crosses into the topological region (unshaded zones). The proper orientation
of the magnetic field allows for the existence of a sizable topological gap
when the system enters in the TS state. The jump of the ground-state phase is
accompanied by a minimum in the critical current. Such a behavior has
previously been used as a signature of TS phase transitions. [37, 42] Note,
however, that in junctions with narrow S regions critical current minima may
not necessarily signal topological transitions [38, 110] and the observation
of more reliable signatures in other physical quantities, such as the spin
susceptibility might be required [110]. When the Rashba and Dresselhaus SOCs
have equal strengths (i.e., $\theta_{so}=\pi/4$) and $\theta_{c}=\pi/8$ or
$\theta_{c}=\pi/4$, the system remains in the trivial state for all magnetic
field strengths and no ground-state jump nor local critical current minimum
occur. However, the system can still reach the TS state when
$\theta_{c}=3\pi/4$, even without a ground-state phase jump (nor associated
critical current minimum). However, the absence of the phase jump does not
allow the system to reach the TS state with an optimal topological gap. The
results suggest that coexisting Rashba and Dresselhaus SOCs with equal
strength may not be favorable for realizing stable TS in phase-unbiased JJs.

##  V Summary

We show that a rich interplay of magnetic field direction, crystalline
orientation, and the relative strengths between Rashba and Dresselhaus SOCs
(parametrized by a spin-orbit angle), all play an important role in defining
the optimal set of parameters that lead to a large topological gap in a
topologically nontrivial state. We provide examples of the effects of
magnetoanisotropy on the current-phase relation of a proximitized Al/HgTe
planar JJ, where only Rashba SOC is sizable, and show that both the
topological phase diagram and the CPR strongly depend on the direction of the
applied magnetic field. We demonstrate that for a phase-unbiased planar JJ in
which the phase is allowed to self-tune to its ground-state value (i.e., the
phase value that minimizes the system free energy), changing the magnetic
field direction can cause $\pi$-jumps in the ground-state phase. We also
consider the case of an Al/InSb junction, where both Rashba and Dresselhaus
SOCs are relevant and not only the magnetic field direction, but the junction
crystallographic orientation and spin-orbit angle also affect the topological
superconducting state. We show that to realize stable MBSs, the system
parameters must be tuned such that the junction is in the topological
superconducting state and a sizable topological gap is achieved by properly
orienting the magnetic field, in dependence of the junction crystallographic
orientation and the spin-orbit angle parametraizing the relative strengths
between Rashba and Dresselhaus SOCs. In phase-unbiased JJs the ground-state
phase self-tunes and can exhibit $\pi$-jumps as the magnetic field is rotated.
When the magnetic field orientation is optimal and the field strength is
varied, the ground-state selftuning $\pi$-jumps enables the system transition
to the topological superconducting state with a sizable topological gap.

_Acknowledgments._ This work was supported by DARPA Grant No. DP18AP900007.

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##  Appendix A Tight-binding simulations

In this Appendix, we briefly describe the tight-binding (TB) simulation
methods used in the main text to obtain Figures 2-8. We start by discretizing
the Hamiltonian in Eqs. (1) and (II.1) on a square lattice in the usual
manner: [111]

$\displaystyle H_{\textrm{TB}}$
$\displaystyle=\hat{H}^{\textrm{onsite}}+\left(\hat{V}^{\textrm{up}}+\hat{V}^{\textrm{right}}+\textrm{h.c.}\right)$
$\displaystyle\hat{H}^{\textrm{onsite}}$ $\displaystyle=\sum_{j\geq
0,i}h^{\textrm{onsite}}(x_{i})\,\ket{x_{i},y_{j}}\bra{x_{i},y_{j}}$
$\displaystyle\hat{V}^{\textrm{up}}$ $\displaystyle=\sum_{j\geq
0,i}v^{\textrm{up}}(x_{i})\,\ket{x_{i},y_{j}+a}\bra{x_{i},y_{j}}$
$\displaystyle\hat{V}^{\textrm{right}}$ $\displaystyle=\sum_{j\geq
0,i}v^{\textrm{right}}(x_{i})\,\ket{x_{i}+a,y_{j}}\bra{x_{i},y_{j}},$ (20)

where $a$ is the lattice constant and $(x_{i},y_{j})=(i\,a,j\,a)$ are the $x-$
and $y-$ coordinates of the lattice points and $i,j$ are integers representing
the $i^{\mathrm{th}}$ ($j^{\mathrm{th}}$) lattice point along the $x$\- ($y$-)
axis. The onsite ($h^{\textrm{onsite}}$) and hopping
($v^{\textrm{up}},v^{\textrm{right}}$) terms do not have $y_{j}$-dependence
because we consider a system with translational invariance in the
$y$-direction, except in the cases where we consider the energy levels or the
wavefunction amplitude of a 2D closed system [see Figs. 2(b) and 2(c), and
Figs. 5(b) and 5(c)]. These terms are given by

$\displaystyle h^{\textrm{onsite}}(x_{i})$
$\displaystyle=\left(4t+\epsilon(\lambda,\theta_{c})-\mu\right)\,\tau_{z}\sigma_{0}+\mathbf{B}\cdot\bm{\sigma}$
$\displaystyle\qquad+\Delta(x_{i})\tau_{+}+\Delta^{*}(x_{i})\tau_{-},$
$\displaystyle v^{\textrm{right}}(x_{i})$
$\displaystyle=-t\,\tau_{z}\sigma_{0}+\frac{i}{2a}\alpha\tau_{z}\sigma_{y}$
$\displaystyle\qquad-\frac{i}{2a}\,\beta\,\left(\cos{2\theta_{c}}\,\tau_{z}\sigma_{x}-\sin{2\theta_{c}}\,\tau_{z}\sigma_{y}\right),$
$\displaystyle v^{\textrm{up}}(x_{i})$
$\displaystyle=-t\,\tau_{z}\sigma_{0}-\frac{i}{2a}\alpha\tau_{z}\sigma_{x}$
$\displaystyle\qquad+\frac{i}{2a}\,\beta\,\left(\cos{2\theta_{c}}\,\tau_{z}\sigma_{y}+\sin{2\theta_{c}}\,\tau_{z}\sigma_{x}\right).$
(21)

Here, $t=\hbar^{2}/2m^{*}a^{2}$ is the hopping parameter, $m^{*}$ is the
effective mass and the definitions of $\tau$, $\sigma$, $\Delta$, $\lambda$,
$\alpha$, $\beta$, and $\theta_{c}$ are given in the main text.
$\epsilon(\lambda,\theta_{c})=(2m^{*}\lambda^{2}/\hbar^{2})\,\left(1+(\sin{2\theta_{c}})^{2}\right)$
is the minimum single particle energy, which is at least an order smaller than
other relevant energies in the systems we consider. An illustration of the
lattice discretization used for the numerical calculations is shown in Fig. 9.

Figure 9: [Color online] The tight-binding simulation lattice with the lattice
constant $a$. The square sites correspond to the Josephson junction and the
round sites correspond to the superconducting leads. The grayed out copies of
the system represent the translational invariance in the $y$-direction of the
system. The width of the SC leads are $42a=252$nm each and the width of the
Josephson junction is $16a=96$nm.

In order to calculate the free energy in Eq. (10) as well as the topological
gap, we need the energy spectrum of the system. The translational invariance
in the $y$-direction implies the momentum $\hbar k_{y}$ is a good quantum
number. We utilize the Kwant package [92] to solve the relevant eigenvalue
problem for a given $k_{y}$:

$\displaystyle\left(\hat{V}^{\textrm{down}}_{y=0}\,e^{i\,k_{y}a}+\hat{H}_{0}^{y=0}+\hat{V}^{\textrm{up}}_{y=0}\,e^{-i\,k_{y}a}\right)\,\psi$
$\displaystyle=E(k_{y})\,\psi$ (22)

where

$\displaystyle\hat{H}_{0}^{y=0}$
$\displaystyle=\sum_{i}\left[h^{\textrm{onsite}}(x_{i})\ket{x_{i},y=0}\bra{x_{i},y=0}\right.$
$\displaystyle\qquad\left.+\left(v^{\textrm{right}}(x_{i})\ket{x_{i}+a,y=0}\bra{x_{i},y=0}+\mathrm{h.c.}\right)\right],$
$\displaystyle\hat{V}^{\textrm{up}}_{y=0}$
$\displaystyle=\sum_{i}v^{\textrm{up}}(x_{i})\,\ket{x_{i},y=a}\bra{x_{i},y=0}\textrm{
and}$ $\displaystyle\hat{V}^{\textrm{down}}_{y=0}$
$\displaystyle=\left(\hat{V}^{\textrm{up}}_{y=0}\right)^{\dagger}.$ (23)

Finally, to obtain the topological charge $Q$, we make use of the formula [78]

$\displaystyle Q$
$\displaystyle=\mathrm{sgn}\left[\frac{\textrm{Pf}\left(H(k_{y}=\pi/a)\right)}{\textrm{Pf}\left(H(k_{y}=0)\right)}\right],$
(24)

where $\textrm{Pf}(.)$ is the Pfaffian and $H(k_{y}=\pi/a)$, $H(k_{y}=0)$ are
obtained using the Kwant package. [92, 112]

The system parameters used in the numerical simulations are summarized in
Table 1.

Table 1: Material and simulation properties of the planar JJs, used throughout this work. Here, $m_{0}$ is the rest mass of the electron. Parameters are taken from Ref. 48, 71. Name | HgTe | InSb
---|---|---
Effective mass ($m^{*}$ ) | 0.038$m_{0}$ | 0.013$m_{0}$
Landé factor ($g^{*}$) | -10 | -20
Induced SC gap ($\Delta_{0}$) | 0.25 meV | 0.21 meV
SOC strength ($\lambda$) | 16 meV nm | 15 meV nm
Critical field at $0$K | 1.45 T | 1.45 T
Temperature ($T$) | 0.7 K | 0.7 K
Chemical potential in S ($\mu_{S}$) | 1 meV | 1 meV
Chemical potential in N ($\mu_{N}$) | 1 meV | 1 meV
Junction width ($W_{N}$) | 96 nm | 96 nm
Left SC lead width ($W_{S}$) | 252 nm | 252 nm
Right SC lead width ($W_{S}$) | 252 nm | 252 nm
Junction length ($L$) | 4000 nm | 4000 nm
TB lattice constant ($a$) | 6 nm | 6nm
TB hopping parameter($t$) | 27.9 meV | 81.5 meV