# Quantum-tunnelling deep neural networks for sociophysical neuromorphic AI

Ivan S. Maksymov<EMAIL_ADDRESS>
###### Abstract

The discovery of the quantum tunnelling effect—the transmission of particles
through a high potential barrier—was one of the most impressive achievements
of quantum mechanics made in the 1920s. Responding to the contemporary
challenges, I introduce a novel deep neural network (DNN) architecture that
processes information using the effect of quantum tunnelling. I demonstrate
the ability of the quantum tunnelling DNN (QT-DNN) to recognise optical
illusions like a human. Hardware implementation of QT-DNN is expected to
result in an inexpensive and energy-efficient neuromorphic chip suitable for
applications in autonomous vehicles. The optical illusions recognition tests
developed in this paper should lay foundations for cognitive benchmarking
tasks for AI systems of the future, benefiting the fields of sociophysics and
behavioural science.

##  I Introduction

###  I.1 Can AI see optical illusions?

How will a conscious robot [1] see us and our world? Will it understand how
humans behave in natural complex environments such as oceans and space [2, 3,
4] and make decisions in virtual reality systems [5, 6, 7] and video games
[8]? Speaking broadly, will a robot differentiate lie from truth in media,
social networks and politics [9, 10, 11]?

Biological vision is enabled by physical, physiological and psychological
processes [12]. In turn, AI employs machine vision [13] and neural network
models [14] to recognise and classify objects. Hence, even though machine
vision shares certain features with biological vision [15], in general AI and
humans see two very different worlds [16, 17, 18].

For example, how would AI see the Necker cube [19, 20] and Rubin’s vase [21,
22] (Fig. 1a, b)? While a human looking at these paradigmatic optical
illusions [19] can easily report a random switching of their perception from
one possible interpretation of the figure to another [20, 23] (Fig. 1c), AI
will struggle to produce a similar output in spite of the advances in
simulation of human vision [24, 25, 26, 27, 28, 29, 30].

Figure 1: (a) The Necker cube: The answer to the question ‘Is the shaded face
of the cube at the front or at the rear?’ will randomly switch between two
stable perceptual states corresponding to the front ($|0\rangle$) and rear
($|1\rangle$) face of the cube. (b) Rubin’s vase: Do you see two people
looking towards each other ($|0\rangle$) or a vase ($|1\rangle$)?’ (c)
According to the traditional psychological theories, the change from one
perceptual state to another is instantaneous (the dotted line). However,
current research works demonstrate that humans might see a superposition of
the states $|0\rangle$ and $|1\rangle$ (the solid curve).

###  I.2 AI, psychology and quantum mechanics

AI cannot see optical illusions since its algorithms do not take into account
peculiar psychological and neurological aspects of human vision [31, 19, 32,
22]. Yet, although this problem has been stressed in the literature [33, 34,
35, 36], the relevance of visual and biological complexity of natural
environments to AI has not been fully appreciated [37, 38, 4].

Experimental studies [39, 40, 41] demonstrated that the perception of the
Necker cube and other ambiguous figures may not abruptly switch between the
states $|0\rangle$ and $|1\rangle$ (the dotted line in Fig. 1c) but
continuously oscillates between them (the solid curve in Fig. 1c). Those
results were interpreted as the ability of humans to see a quantum-like
superposition of $|0\rangle$ and $|1\rangle$ states [42, 43, 20, 44]. Indeed,
in psychological experiments humans are asked to push an electric button every
time their perception of the Necker cube switches between $|0\rangle$ and
$|1\rangle$. While humans might see the cube in a superposition state [31,
41], by pressing the button the observer ‘collapses’ the superposition to one
of the possible ‘classical’ states [23]. This means that the ability of humans
to access and use environmental information is limited, which may result in
ambiguous interpretation. Subsequently, our perceptual system needs to
disambiguate and interpret the available sensory information to construct
stable and reliable percepts [45]. Yet, this suggests that AI can help humans
to enhance their perception and gain access to information that our senses and
brain cannot process naturally.

Further attempts to understand bistable perception using the methods of
quantum mechanics [43, 20, 44] resulted in the suggestion to study the
superposition of the perceptual states using a quantum oscillator model (QOM)
[20]. Compared with Markov models [20], QOM accounts for multiple outcomes
while processing input data with a large set of constraints, thus describing
human mental states [43, 44] more efficiently than classical models.

The predictions of QOM are in good agreement with a model based on the quantum
Zeno effect [42]. Besides, the ability of QOM to simulate the perception of
optical illusions was improved [46, 23] by combining harmonic motion with the
effect of quantum tunnelling (QT) through a potential barrier [47, 48]. Since
both QOM and quantum Zeno effect underpin quantum neural network models [49],
the predictions made by QOM were additionally validated using a deep neural
network (DNN) model [23] combined with a quantum generator of truly random
numbers [50].

###  I.3 What is this article about?

In this work, I suggest and theoretically validate a novel neuromorphic DNN
model (see Sect. II.1 for state-of-the-art) that processes information using
the physical effect of QT. I showcase the ability of the QT-DNN model to
enable AI to recognise practically important optical illusions, the perception
of which should pave the way for the development of machine vision system
capable of recognising more complex visual effects used in fine arts and
cinematography. I also demonstrate that QT-DNN can be implemented as a
hardware neuromorphic chip.

My findings can be used in AI system and brain-computer interfaces aimed to
enhance the performance of astronauts in long-duration spaceflight [3] and
help airline pilots be aware of disorientation, loss of perspective and
misinterpretation of data provided by flight instruments [51]. Interestingly
enough, the algorithms proposed in this work can also help conduct research on
gender since it has been demonstrated that fluidity of gender identity might
be induced by optical illusions [52].

Finally, concluding the paper with a discussion of the relevance of the effect
of QT to the hypothesised brain-mind link [53, 54], I contribute to bridging
the gap between the theories of quantum mind and decision-making [43, 20] and
the practically important field of sociophysics [9, 11].

##  II QT-DNN architecture

###  II.1 Neuromorphic quantum DNN models

DNNs—networks composed of several layers (Fig. 2a) [14]—are pivotal for
machine vision and human-computer interactions [13]. Yet, they are
mathematically complex [55], require considerable computational resources [56]
and consume significant electric power [57].

The efficiency of DNNs can be increased by constructing the network following
the principles of quantum computing [58, 59, 60] and optimising their
algorithms using the fundamental laws of physics [61]. DNNs can also be built
using the principles of neuromorphic computing [62, 63].

Neuromorphic computers mimic the operation of a biological brain [62, 64].
Several neuromorphic DNN models have been demonstrated [65, 66], where,
instead of relying on conventional Boolean arithmetic, information is
processed using nonlinear dynamical properties of a physical system [64, 63].
Such DNN models can be implemented as an inexpensive and low power consuming
chip [65, 67] suitable for applications in AI and robotics [66].

Figure 2: (a) Sketch of QT-DNN structure. $W^{(n)}$ with $n=1\dots 4$ are the
matrices of the weights of the network connections. (b) QT-DNN model employs
the physical effect of QT as the activation function of its nodes. Panel
(b.i): one-dimensional rectangular potential barrier of thickness $a$ and
height $V_{0}$. Panel (b.ii): unlike in classical mechanics, in quantum
mechanics there is a non-zero probability for an electron with energy
$E<V_{0}$ to be transmitted through the barrier.

###  II.2 Quantum tunnelling effect

In quantum mechanics [48], the rectangular potential barrier is a
computational problem that demonstrates the phenomenon of QT (Fig. 2b). The
problem consists in solving a one-dimensional time-independent Schrödinger
equation for an electron that encounters a rectangular potential energy
barrier [47]

$\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x)\right]\psi(x)=E\psi(x)\,,$
(1)

where $\psi(x)$ is a wave function, $m\approx 9.1093837\times 10^{-31}$ kg is
the mass of the electron, $\hbar\approx 1.054571817\times 10^{-34}$ J$\cdot$s
is Plank’s constant and $E$ is the energy of the electron. The profile of the
potential barrier (Fig. 2b) is

$V(x)=\begin{cases}0&\text{for }\,x<0\\\ V_{0}&\text{for }\,0<x\leq a\\\
0&\text{for }\,x>a\,.\end{cases}$ (2)

In classical mechanics, a counterpart of this physical system is a marble ball
(Fig. 2b). While a ball with energy $E<V_{0}$ cannot penetrate the barrier, an
electron, behaving as a matter wave, has a non-zero probability of penetrating
the barrier and continuing its motion on the other side. Similarly, for
$E>V_{0}$, the electron may be reflected from the barrier with a non-zero
probability.

The electron tunnelling behaviour can be quantified by finding the
transmission coefficient from the solution of Eq. (1) for the potential
barrier given by Eq. (2). The solution of the Schrõdinger equation can be
written as a superposition of left and right moving waves [47]

$V(x)=\begin{cases}\psi_{L}(x)=A_{1}e^{\mathsf{i}kx}+A_{2}e^{-\mathsf{i}kx},&x<0\\\
\psi_{C}(x)=B_{1}e^{\mathsf{i}\kappa x}+B_{2}e^{-\mathsf{i}\kappa x},&0<x\leq
a\\\
\psi_{R}(x)=C_{1}e^{\mathsf{i}kx}+C_{2}e^{-\mathsf{i}kx},&x>a\,,\end{cases}$
(3)

where $\mathsf{i}$ is the imaginary unit, $k=\sqrt{2mE/\hbar^{2}}$ and
$\kappa=\sqrt{2m\alpha/\hbar^{2}}$ with $\alpha=E-V_{0}$ (the special cases
$E=0$ and $E=V_{0}$ are treated separately). The coefficients $A,B,C$ are
found from the boundary conditions at $x=0$ and $x=a$, requiring that
$\psi(x)$ and its derivative have to be continuous. Below, omitting the
intermediate derivations [47], I present the expressions for the probability
of the electron transmission through the barrier.

For electron energies smaller than the barrier height ($E<V_{0}$), there is a
non-zero transmission probability [47]

$T|_{E<V_{0}}=\left(1-\beta\sinh^{2}(\kappa_{1}a)\right)^{-1}\,,$ (4)

where $\beta=\dfrac{V_{0}^{2}}{4E\alpha}$ and
$\kappa_{1}=\sqrt{-2m\alpha/\hbar^{2}}$. For $E>V_{0}$

$T|_{E>V_{0}}=\left(1+\beta\sin^{2}(\kappa a)\right)^{-1}\,,$ (5)

Finally, the expression for $E=V_{0}$ is obtained by taking the limit of $T$
as $E$ approaches $V_{0}$, resulting in

$T|_{E=V_{0}}=\left(1+\frac{ma^{2}V_{0}}{2\hbar^{2}}\right)^{-1}\,.$ (6)

###  II.3 Quantum tunnelling as an activation function

Figure 3 shows the probability of transmission of a single electron with
energy $E$ through a potential barrier with the nondimensionalised thickness
$\sqrt{2mV_{0}}a/\hbar$. The result predicted by the classical theory is
denoted by the straight dash-dotted lines. While physical implications of
these results are well-known [47], in the following I will focus only on their
meaning in the context of QT-DNN.

A close inspection of the curve plotted in Figure 3 reveals that the QT
transmission function $T$ can be employed as an activation function of the
nodes of an DNN, provided that the energy of the electron $E$ is interpreted
as the input value of the node. In fact, similarly to the standard activation
functions used in DNNs [14], $T$ fulfils a number of essential mathematical
criteria pertinent to the field of machine learning, including nonlinearity
and differentiability, thus enabling its application with gradient-based
optimisation methods [14, 68]. Although the negative values of $E/V_{0}$ have
no strict meaning in the framework of the physical model, in the QT-DNN model
they can be interpreted as the inputs that produce a zero output. The special
case $E=0$ can be avoided programmatically. It is also noteworthy that the
energy of the electron can be effectively amplified (see Sect. IV.2 for
technical details), which is one of the features of the computational code
that accompanies this paper.

Differentiating Eqs. (4–5) with respect to $E$, I obtain

$\begin{gathered}T^{\prime}|_{E<V_{0}}=-\beta\biggl{(}\frac{\sinh^{2}(\delta_{1})}{E}+\\\
\frac{\sinh^{2}(\delta_{1})-\delta_{1}\cosh(\delta_{1})\sinh(\delta_{1})}{\alpha}\biggr{)}T^{2}|_{E<V_{0}}\,,\\\
T^{\prime}|_{E>V_{0}}=\beta\biggl{(}\frac{\sin^{2}(\delta)}{E}+\frac{\sin^{2}(\delta)-\delta\cos(\delta)\sin(\delta)}{\alpha}\biggr{)}T^{2}|_{E>V_{0}}\,,\\\
T^{\prime}|_{E=V_{0}}=\frac{4V_{0}a^{4}m^{2}+6a^{2}\hbar^{2}m}{3V_{0}^{2}a^{4}m^{2}+12V_{0}a^{2}\hbar^{2}m+12\hbar^{4}}\,,\end{gathered}$
(7)

where $\delta=\kappa a$ and $\delta_{1}=\kappa_{1}a$.

Figure 3: Transmission probability of an electron with energy $E$ through a
rectangular potential barrier with a nondimensionalised thickness
$\sqrt{2mV_{0}}a/\hbar$. The straight dash-dotted lines denote the result
predicted by the classical theory.

###  II.4 QT-DNN algorithm

The QT-DNN network used in this paper (Fig. 2a) consists of an input layer
that has $L=100$ input nodes, three hidden layers each of which has $N=20$
nodes and an output layer that has $M=2$ output nodes that are used to
classify the input dataset. The weights of the connections of the network are
updated using a back-propagation training algorithm [14].

The activation function $\phi_{QT}$ of the nodes of the hidden layers and its
derivative are given by Eqs. (4–7). The output nodes are governed by the
Softmax function [14]

$\phi_{smax}(v_{i})=\frac{\exp(v_{i})}{\sum_{k=1}^{M}\exp(v_{k})}\,,$ (8)

where $v_{i}$ is the weighted sum of input signals to the $i$th output node
and $M$ is the number of the output nodes.

The network is trained and exploited as follows. First, I construct the output
nodes that correspond to the correct answers to the training datasets. Then, I
initialise the weights of the neural network in the range from –1 to 1 using a
random number generator. Entering the input data $x_{j}$ and the corresponding
training data points $d_{i}$, I calculate the error $e_{i}$ between the output
$y_{i}$ and target $d_{i}$ as $e_{i}=d_{i}-y_{i}$. Then, propagating the
output $\delta_{i}=e_{i}$ in the backward direction of the network, I compute
the respective parameters $\delta_{i}^{(n)}$ of the hidden nodes using the
equations $e_{i}^{(n)}=W^{{(n)}^{\top}}\delta_{i}$ and
$\delta_{i}^{(n)}=\phi_{QT}^{\prime}\left(v_{i}^{(n)}\right)e_{i}^{(n)}$,
where the index $n$ denotes the sequential number of the hidden layer, prime
denotes the derivative of the activation function and $W^{\top}$ is the
transpose of the matrix of weights corresponding to each relevant layer of the
network. I continue the back-propagation process until the algorithm reaches
the first hidden layer and then I update the weights using the learning rule
$w_{ij}^{(n)}\leftarrow w_{ij}^{(n)}+\Delta w_{ij}^{(n)}$, where
$w_{ij}^{(n)}$ are the weights between an output node $i$ and input node $j$
of the $n$th layer and $\Delta w_{ij}^{(n)}=\alpha\delta_{i}^{(n)}x_{j}$.

The computational steps outlined above are sequentially applied to all
training data points. Typically, to obtain convergent results it suffices to
complete 1000 epochs, using the learning rate parameter $\alpha=0.01$.

Figure 4: Neuromorphic algorithm involving the training of the network on
distinguishable images of objects and its further exploitation aimed to
recognise optical illusions.

###  II.5 Neuromorphic QT-DNN model of perception

A biological brain is a nonlinear dynamical system that exhibits chaotic
behaviour [69]. This property has motivated the development of neuromorphic AI
that mimics the operation of the brain by exploiting nonlinear dynamical
properties of diverse physical systems [62, 63, 64].

The principal components of the human vision system, including the retina and
visual cortex, also exhibit nonlinear dynamical properties that can be used to
create neuromorphic computers [70]. The visual input to the retina is
modulated by eye blinks and movements, which effectively converts spatial
information in the temporal one [71], also playing an important role in
cognition and visual perception [72, 73, 74]. Yet, the dynamics of eye blink
is also nonlinear and it may exhibit phase changes and chaotic behaviour [75],
which are the processes that underpin the perception of optical illusions
[76].

Based on these facts, it was demonstrated that the introduction of chaotic
changes in the architecture of a neural network enables modelling the dynamics
of information perception [77, 24, 25, 26, 27, 23]. Thus, I randomly generate
the weights $W^{(n)}$, then I train QT-DNN on the $10\times 10$ pixel
unambiguous images of the Necker cube and then I exploit it to predict the
perceptual state of the ambiguous Necker cube. This procedure is repeated in a
loop to plot the dynamics of the perceived states (Figure 4).

Rubin’s vase training images had $20\times 20$ pixels. Although some works
studying the Necker cube claim that similar results would be obtained for
Rubin’s vase [46], recent research demonstrated that Rubin’s vase has an
increased contextual complexity [22]. Hence, I train QT-QNN using the figures
with the shaded faces and vase, respectively, and then exploit it using a
contour version of the drawing (Figure 4). Both shaded and contour versions
have been used in the literature [21] and they represent an intriguing
benchmarking task, especially because the shaded training images are also
ambiguous.

Figure 5: Perceptual switching curves produced by QT-DNN trained on the Necker
cube. The data points with probability $P_{|0\rangle}=0$ or $P_{|1\rangle}=1$
correspond to the fundamental perceptual states of the Necker cube. The
remaining data points are in a superposition of states $|0\rangle$ and
$|1\rangle$ with $P_{|0\rangle}+P_{|1\rangle}=1$.

##  III Results and Discussion

Figure 5 plots the simulated probability of perceiving $|0\rangle$ and
$|1\rangle$ states of the Necker cube as a function of time (arbitrary time
units are used in this work; for a relevant discussion of the cognitive
timescale see Refs. [42, 78]). In each panel of Fig. 5, the nondimensionalised
thickness of the potential barrier that is used as the activation function of
QT-DNN (Fig. 3) is $\sqrt{2mV_{0}}a/\hbar=$ 0.5, 1 and 1.5.

The probability curves were obtained after 50 consecutive computational runs
of the algorithm outlined in Fig. 4. The states of the two output nodes of QT-
DNN were recorded at the end of each run. Every pair of those data points was
computed using a unique set of neural weights $W^{(n)}$. The same procedure
was followed to simulate the perception of Rubin’s vase (Fig. 6). The same
value of barrier thickness and sets of $W^{(n)}$ were used in the respective
panels of Fig. 5 and Fig. 6.

In Fig. 5a, we can observe a time-dependent switching between the fundamental
perceptual states $|0\rangle$ and $|1\rangle$. The switching is not abrupt, as
depicted in Fig. 1c, but involves intermediate states that are a superposition
of $|0\rangle$ and $|1\rangle$. A similar switching pattern is observed in
Fig. 5b obtained for a thicker potential barrier. Although the probability of
perceiving the states $|0\rangle$ and $|1\rangle$ decreases to approximately
0.85, in this case QT-DNN can readily distinguish the two states of the Necker
cube.

Further increase in the thickness of the potential well (Fig. 5c) results in a
smaller difference between the probabilities of perceiving $|0\rangle$ and
$|1\rangle$. The so-configured QT-DNN operates similarly to perception by a
human observer [78], who can quickly decode ambiguous visual information in
just a few loops of recurrent biological neural activity [79], but needs much
more time to decide which perceptual outcome should be reported.

In fact, I established that the same model with 2000 epochs produces a
substantially more distinguishable perception pattern, which effectively means
that the model requires more time to decide which perceptual state to report.
Such a behaviour is beneficial for modelling the perception of illusions by
different age and gender groups [80, 81]. Indeed, since the dynamics of eye
blink (and, therefore, the dynamics of visual information processing [71])
slows down with age [80], QT-DNN with thicker potential barriers appears to
adequately model the perception of senior and visually impaired persons.

As can be seen in Fig. 6, the perception switching of Rubin’s vase is similar
to that of the Necker cube. Yet, QT-DNN better distinguishes the two
perceptual states of Rubin’s vase, including in the case of a thick barrier
(Fig. 6c), without the need of increasing the number of epochs. This result is
consistent with the hypothesis that states that Rubin’s vase combines an
optical illusions with the perception of background [22]. Indeed, observers
may interpret this figure not only as faces-vase but also as a white (black)
vase on black (white) background, which gives them an additional point of
reference that can be used to disambiguate the drawing.

Figure 6: Perceptual switching curves produced by QT-DNN trained on Rubin’s
vase.

Overall, the results presented in Fig. 5 and Fig. 6 are in good agreement with
the predictions made by the alternative quantum cognition models [42, 20, 46].
Below it will also be shown that the operation of QT-DNN is consistent with
the broader models of brain-mind interaction.

##  IV Implementation and Discussion

###  IV.1 Software

Since one of the envisioned applications of the QT-DNN model is its
integration with robotic platforms and autonomous vehicles, I implemented its
algorithm in Python 3 and tested the resulting software on a Raspberry Pi 4
embedded computer (1.8 GHz CPU, 8 GB RAM, Linux Ubuntu Desktop 24.04 LTS) that
is often used in onboard AI systems [82, 83]. Using limited embedded
computational resources, QT-DNN demonstrated the ability to be trained and
exploited in less than 1 second of real time, which is an encouraging result
that can be considered as on-the-fly operation. It is also worth noting that
the Raspberry computer consumes just 3 W of onboard power compared with more
expensive GPU-based onboard computers that are required to run traditional DNN
algorithms and draw 25–30 W [82]. Therefore, QT-DNN algorithm also fulfils the
criterion of energy efficiency.

The advantages of QT-DNN over the traditional neural network models amplify
when software that implements it runs on a high-performance workstation
computer. For example, significantly lower accuracy of neural network models
trained on regular images but tested on negative ones has been reported in the
previous works [84]. The nominal ‘vase’ state of the training images of (Fig.
4) is the negative image of the ‘faces’ state. Subsequently, since the
algorithm of QT-DNN demonstrated good performance in this image recognition
scenario, it is plausible to assume that QT-based activation functions can be
employed in image recognition models trained on negative images. Such models
have important practical applications, including ship detection and
recognition in maritime environments [85] and creative arts [86].

QT-DNN can also enhance the capabilities of models employed in the emergent
field of sociophysics [9]. Indeed, since the effect of QT also underpins the
operation of several physics-inspired models of human opinion formation and
radicalisation in social networks [10], the algorithm of QT-DNN can be adopted
in the studies of fake news [11] and research on gender and racial biases
[10].

###  IV.2 Potential hardware implementations

The effect of QT has been exploited in semiconductor electronic devices [87,
88, 89, 90] and different modalities of spectroscopy [91, 92] and microscopy
[93, 92]. It has also been demonstrated that electron devices exploiting the
effect of QT can serve as a building block of neuromorphic computers [94, 95].
However, in those computers QT has not been exploited directly. Instead,
nonlinear dynamics of the whole QT-based electron devices and circuits formed
by them has been employed as a means of computation.

In practice, the architecture of QT-DNN can be implemented using tunnelling
[87] and resonant tunnelling [89] diodes. Moreover, there exist neuromorphic
computing schemes that exploit negative differential resistance [96, 97], a
prominent feature of tunnelling diodes. Importantly, systems based on
tunnelling diodes and other QT-based devices consume low electric power
compared with the traditional integrated electronic circuits [94].

Alternatively, QT-DNN can be embodied using the instrumentation of scanning
tunnelling microscopy (STM) [93]. For example, amplifiers developed for STM
operate at microwave frequencies, enabling high signal-to-noise ratios and
facilitating differential conductance spectroscopy measurements [98]. Finally,
QT of individual electrons was measured in quantum dots [96]—building blocks
of quantum neuromorphic systems [99].

##  V Conclusions and outlook

In this paper, I pointed out the possibility of employing the effect of
quantum tunnelling as an activation function of artificial neural networks. I
demonstrated that the so-constructed neural networks enable accurate modelling
of human perception of optical illusions. Yet, I suggested that low power
consuming, hardware neuromorphic chips built using quantum tunnelling diodes
or other devices and systems based on the tunnelling effect can be employed in
conscious robots and autonomous vehicles. Finally, I traced a link between
quantum tunnelling and sociophysical models designed to understand human
behaviour in social and political groups.

So there is just one question that needs to be clarified: how can the effect
of quantum tunnelling be related to cognition? This question is important
because the current works on quantum models of cognition and decision-making
mostly exploit the laws and methods of quantum mechanics to create
phenomenological models [10, 100]. While this approach has produced
practically important results [9, 11], addressing this question in more detail
promises to attract the attention of a larger number of experts to the
emergent field of sociophysical AI.

Tunnelling can be observed in all quantum systems, which explains why many
Nobel prizes in physics were awarded for research involving it [101]. This
physical effect is also crucial for chemical and biological evolution [102].
Most likely, it also played an essential role in the formation of the early
universe and life [101, 102]. Subsequently, since quantum tunnelling remains
an important topic of scientific research, there have been theoretical and
experimental attempts to explain the function of biological brain neurons
relying on the principles of quantum mechanics [103, 54] (arguably, some of
those ideas could also contribute to Nobel prize-winning discoveries [104]).

Drawing on the difference between a conscious mind and a biological brain, it
was suggested that the mind and brain can interact by means of quantum
tunnelling [53]. Indeed, based on the cited paper and references therein, it
can be said that a mental state becomes neurally effective when a large number
of biological potential barriers in neurons produce high transmission
probability amplitudes, which, in turn, results in a coherent action.

If proven correct, that hypothesis justifies the design of the QT-DNN model,
showing that the use of tunnelling as an activation function is more than just
an phenomenological mathematical approach but a model based on the fundamental
principles of cognition and brain organisation. The same should be the case of
the other sociophysical and quantum mind theories.

## Data Availability Statement

The computational code that implements QT-DNN can be found at
https://github.com/IvanMaksymov/DeepTunnel (accessed on 26 June 2024)

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# Missing Shapiro steps and the $8\pi$-periodic Josephson effect
in interacting helical electron systems

Christopher J. Pedder  Physics and Materials Science Research Unit, University
of Luxembourg, L-1511 Luxembourg  Tobias Meng  Institut für Theoretische
Physik, Technische Universität Dresden, 01062 Dresden, Germany  Rakesh P.
Tiwari  Department of Physics, University of Basel, Klingelbergstrasse 82,
CH-4056 Basel, Switzerland  Department of Physics, McGill University,
Montréal, Québec, Canada H3A 2T8  Thomas L. Schmidt<EMAIL_ADDRESS>Physics and Materials Science Research Unit, University of Luxembourg, L-1511
Luxembourg

###### Abstract

Two-particle backscattering in time-reversal invariant interacting helical
electron systems can lead to the formation of quasiparticles with charge
$e/2$. We propose a way to detect such states by means of the Josephson effect
in the presence of proximity-induced superconductivity. In this case, the
existence of $e/2$ charges leads to an $8\pi$-periodic component of the
Josephson current which can be identified through measurement of Shapiro steps
in Josephson junctions. In particular, we show that even when there is weak
explicit time-reversal symmetry breaking, which causes the two-particle
backscattering to be a sub-leading effect at low energies, its presence can
still be detected in driven, current-biased Shapiro step measurements. The
disappearance of some of these steps as a function of the drive frequency is
directly related to the existence of non-Abelian zero-energy states. We
suggest that this effect can be measured in current state-of-the-art Rashba
wires.

###### pacs:

71.10.Pm, 74.45.+c, 05.30.Pr

##  I Introduction

One of the most promising experimental avenues for the discovery of Majorana
fermions Nayak et al. (2008); Leijnse and Flensberg (2012); Alicea (2012);
Beenakker (2013); Majorana (1937) as quasiparticles in solid-state systems
Read and Green (2000); Ivanov (2001); Kitaev (2001); Fu and Kane (2008, 2009);
Sato and Fujimoto (2009); Sato et al. (2009); Oreg et al. (2010); Lutchyn et
al. (2010) has been quantum wires with spin-orbit coupling. In such one-
dimensional (1D) wires, the combined effects of Rashba spin-orbit coupling
(RSOC) or helical nuclear order,Braunecker et al. (2009, 2010); Braunecker and
Simon (2015) proximity-coupling to a superconductor, and an externally applied
magnetic field Oreg et al. (2010); Lutchyn et al. (2010) open a band gap in
the single-particle spectrum of the wire. This gap only closes at the ends of
the wire, leading to localized zero-energy states which are believed to behave
as Majorana fermions. Related, but more exotic bound states have been
predicted to descend from strongly correlated phases which display charge
fractionalization, such as the edges of fractional quantum Hall
systems.Fendley (2012); Lindner et al. (2012); Clarke et al. (2013); Alicea
and Fendley (2016) or fractionalized helical states.Meng et al. (2014a); Oreg
et al. (2014)

However, unambiguous confirmation of such Majorana quasiparticles remains
elusive. Several experimental systems have shown signatures that are
consistent with MBS,Deng et al. (2012); Mourik et al. (2012); van Weperen et
al. (2013); Nadj-Perge et al. (2014) but a common objection is that such tests
are not directly sensitive to the specific properties of Majorana states.
Recently, more targeted experimental tests have been used to rule out other
possible causes. Measured current-voltage curves for proximitized edge states
of HgTe, and nanowires with RSOC have shown the disappearance of odd numbered
“Shapiro steps”.Rokhinson et al. (2012); Maier et al. (2015); Wiedenmann et
al. (2016) The disappearance of the odd steps is consistent with theoretical
predictionsDomínguez et al. (2012, 2017) that there are tunnelling states with
charge $e$, rather than the usual $2e$ for Cooper pairs, leading to a
$4\pi$-periodic component of the Josephson current.

Crucial to the localization of bound states in these Majorana systems is the
opening of a gap in the spectrum at the Dirac point at $k=0$, see Fig. 2. This
so-called _helical_ gap can be opened by applying a magnetic field. However,
previous work on the edge states of two-dimensional topological insulators has
shown the possibility of a helical gap even without the application of a
magnetic field, but instead due to two-particle backscattering.Orth et al.
(2015) Proximity coupling such a time-reversal invariant system to an $s$-wave
superconductor then leads to the appearance of fourfold-degenerate zero-energy
bound states, i.e., $\mathbb{Z}_{4}$ parafermions. Zhang and Kane (2014); Orth
et al. (2015); Peng et al. (2016) This system hosts quasiparticles with
fractionalized $e/2$ charge, which will result in an $8\pi$ periodic component
of the Josephson current in a weak-link S-N-S junction. Recently, we showed
that an analogous _spin-umklapp_ scattering can open a helical gap without
magnetic field in strongly interacting wires with RSOC.Pedder et al. (2016)
This prompts the question of whether an $8\pi$-periodic Josephson current can
be detected in Rashba wires as well.

Theoretical works have also turned to unproximitized wires with RSOC,Aseev et
al. (2017); Schmidt (2013); Meng et al. (2014b); Klinovaja and Loss (2014);
Schmidt and Pedder (2016); Kornich et al. (2017) and experimental studies have
investigated transport in these wires too. Without induced superconductivity,
the system only has a partial gap, and so does not contain localized bound
states. Recent data Heedt et al. (2017) shows a dip in the conductance of InAs
wires with an applied magnetic field, consistent with the appearance of a
helical gap. The same authors also find a similar signature without applied
field, possibly arising from a partial spin-umklapp gap. We note that the
theoretically-required interaction strengths have already been reached in
similar quantum wires,Hevroni et al. (2016) and that the opening of the spin-
umklapp gap may also have been seen in other systems (e.g. Ref. [Scheller et
al., 2014]).

In this paper, we investigate a measurement scheme for an $8\pi$-periodic
Josephson current in systems which support two-particle backscattering. This
unusual periodicity of the Josephson current can be seen in Shapiro step
experiments, even when we allow for a weak breaking of time reversal symmetry
(TRS). Breaking TRS lifts the fourfold degeneracy, so we no longer have
tunnelling of $e/2$ quasiparticles in the ground state. However, we will show
that even allowing for dominant tunnelling of Cooper pairs and charge-$e$
quasiparticles, driven Shapiro step experiments can still be tuned to a regime
where the disappearance of Shapiro steps reveals the presence of the
$8\pi$-periodic term in the Josephson current.

The structure of this paper is as follows. In section 2, we introduce the
specific model we choose to study - an interacting Rashba wire proximity
coupled to a superconductor. We then bosonize this model in section 3, and
present a renormalization group (RG) analysis for the flow of the system
parameters. We find there exist regimes in which fourfold degenerate localized
states can be found. In section 4, we describe the implications of the
degenerate ground state for Josephson effect measurements, and suggest a
definitive Shapiro step measurement to identify the $8\pi$ contribution to the
Josephson current arising from these states, even when this contribution is
sub-dominant. In section 5, we describe the situation in the absence of TRS
breaking, and construct the $\mathbb{Z}_{4}$ exchange statistics of the
localized states, described by non-Abelian, parafermionic operators. Finally,
in section 6, we discuss the relevance of our work for experimental
investigations.

##  II Model

We focus on the specific case of a long, quasi-1D nanowire aligned along the
$x$ direction which is harmonically confined in the $y$ and $z$ directions.
The interplay of the intrinsic spin-orbit coupling of a material and the
breaking of inversion symmetry by a particular geometric arrangement, due to
either the presence of a substrate, or to the application of an out of plane
electric field, gives rise to RSOC.Winkler (2003) In the latter case, the
strength of the electric field may be used to tune the magnitude of the RSOC,
denoted as $\alpha_{R}$. The dynamics in the $z$-direction is not affected by
the Rashba coupling and can be safely ignored. Thus, we can model a finite-
width wire by using the following 2D Hamiltonian including RSOC with strength
$\alpha_{R}$ Moroz et al. (2000a, b); Gangadharaiah et al. (2008)

$H=\frac{p_{x}^{2}+p_{y}^{2}}{2m}+\frac{1}{2}m\omega^{2}y^{2}+\alpha_{R}(\sigma_{x}p_{y}-\sigma_{y}p_{x}),$
(1)

where $p_{x}$ and $p_{y}$ are the momentum components in the $x$\- and
$y$-directions, and $\sigma_{x,y}$ are Pauli matrices. The transversal
confinement is modelled as a harmonic potential with frequency $\omega$. The
system has translational invariance along the $x$-direction but is strongly
confined along the $y$-direction, which leads to the appearance of higher
excited bands separated from the lowest band by a spacing determined by the
inverse width of the wire. As described in Ref. [Pedder et al., 2016], changes
of sub-band come with a spin flip, which in tandem with a spin-conserving
interaction can generate _spin-umklapp_ scattering when the chemical potential
is at the Dirac point. This scattering converts e.g. two spin-up particles
into two spin-down particles (see Fig. 1).

Figure 1: (color online) A plot of single-particle spectrum of a Rashba spin-
orbit coupled nanowire showing the two lowest transverse sub-bands, whose spin
character as a function of momentum is shown by the color-coding. Virtual
transitions between sub-bands come with a spin-flip, and electron-electron
interactions allow us to couple states in different sub-bands. The inset shows
the exact process responsible for spin-umklapp scattering, where the virtual
transition between sub-bands (denoted by $g$) changes the spin character of
two incoming particles, and then the four-particle interaction vertex (denoted
by $V$) couples these, resulting in a process in the lowest sub-band where two
spin-$\uparrow$ particles are scattered into two spin-$\downarrow$ particles.

Putting the chemical potential at the Dirac point leads to four low-energy
modes at momenta $k=0,\pm k_{F}$ (where $k_{F}=2m\alpha_{R}$) as shown in Fig.
2. Correspondingly, for small energies, we can split the field operators up
into four modes,

$\displaystyle\psi_{\uparrow}(x)$ $\displaystyle\approx
e^{ik_{F}x}\psi_{R\uparrow}(x)+\psi_{L\uparrow}(x),$ (2)
$\displaystyle\psi_{\downarrow}(x)$
$\displaystyle\approx\psi_{R\downarrow}(x)+e^{-ik_{F}x}\psi_{L\downarrow}(x).$
(3)

Next, we express $\psi_{\alpha s}(x)$ for $\alpha\in\\{L,R\\}$ and
$s\in\\{\uparrow,\downarrow\\}$ in terms of its Fourier components,

$\psi_{\alpha s}(x)=\frac{1}{\sqrt{L}}\sum_{k}e^{ikx}\psi_{\alpha s,k},$ (4)

where $L$ is the length of the wire. In terms of these operators, the
interaction Hamiltonian after projection to the lowest sub-band can be written
as follows. The density-density interaction term readsPedder et al. (2016)

$V_{\rho}=\frac{\tilde{V}(0)}{L}{\sum_{\alpha,s}}\int\,dx\,\rho_{\alpha
s}(x)\rho_{\alpha s}(x),$ (5)

where the fermionic densities are defined as usual as $\rho_{\alpha
s}(x)=\psi^{\dagger}_{\alpha s}(x)\psi_{\alpha s}(x)$ and $\tilde{V}(q)$ is
the Fourier transform of the interaction potential $V(\mathbf{r})$ projected
to the lowest sub-band

Due to momentum conservation, the spin-flip terms only mix terms near $k=0$,

$V_{\rm{sf}}=v_{\rm{sf}}\int\,dx\left[\psi^{\dagger}_{R\uparrow}(\partial_{x}\psi^{\dagger}_{R\uparrow})(\partial_{x}\psi_{L\downarrow})\psi_{L\downarrow}+\rm{h.c.}\right],$
(6)

where $v_{\rm{sf}}$ is a dimensionful parameter characterizing the strength of
the spin-umklapp scattering.

Figure 2: (color online) Panel (a) shows the position of the chemical
potential and the four low-energy linearized modes used to bosonize the
system. Panel (b) shows the reconstruction of the band structure as a result
of the RG-relevant interaction processes between these modes. Umklapp
scattering opens a helical gap of size $g_{\textrm{U}}$ between the two modes
near $k=0$ and proximity-induced $s$-wave superconductivity opens gaps for the
outer modes at $k=\pm 2m\alpha_{R}$ of size $g_{\textrm{SC}}$, resulting in a
fully gapped system.

Most of the spin-exchange terms arising from the Coulomb interaction can be
expressed as density-density interactions, leading merely to changes in the
coefficients of the terms in Eq. (5). We separate out the single non-density-
density term

$V_{\rm{S}}=v_{\rm{S}}\int\,dx\biggl{(}\psi_{L\downarrow}^{\dagger}\psi_{R\uparrow}^{\dagger}\psi_{L\uparrow}\psi_{R\downarrow}+\rm{h.c.}\biggr{)},$
(7)

which corresponds to an interaction between inner and outer bands with
strength $v_{\rm{S}}$. In the limit in which this term dominates, it results
in a spin-density wave state at $q=2k_{F}$, and has been discussed in detail
in Ref. [Gangadharaiah et al., 2008].

Finally, we allow for the possibility of proximity-induced coupling to an
$s$-wave superconductor, which pairs spin-up and spin-down electrons, and so
has an effect at all the Fermi points in Fig. 2a. This pairing contribution to
the Hamiltonian is

$V_{\rm{SC}}=v_{\rm{SC}}\int\,dx\left(\psi_{R\uparrow}^{\dagger}\psi_{L\downarrow}^{\dagger}+\psi_{L\uparrow}^{\dagger}\psi_{R\downarrow}^{\dagger}+\rm{h.c.}\right),$
(8)

where $v_{\rm{SC}}$ is determined by the strength of the proximity-coupling to
the superconductor. We now analyse the competition between these three
possible interaction channels, parametrised by $v_{\rm{sf}}$, $v_{\rm{S}}$ and
$v_{\rm{SC}}$.

##  III Bosonization & renormalization group analysis.

To further analyze the interacting system, we write the Hamiltonian in terms
of bosonic operators $\phi_{\pm}$ and $\theta_{\pm}$ by defining

$\displaystyle\psi_{R\uparrow}$
$\displaystyle=\frac{\eta_{R\uparrow}}{\sqrt{2\pi
a}}e^{-i(\phi_{+}-\theta_{+})},\hskip
14.22636pt\psi_{R\downarrow}=\frac{\eta_{R\downarrow}}{\sqrt{2\pi
a}}e^{-i(\phi_{-}-\theta_{-})},$ $\displaystyle\psi_{L\uparrow}$
$\displaystyle=\frac{\eta_{L\uparrow}}{\sqrt{2\pi
a}}e^{i(\phi_{-}+\theta_{-})},\hskip
14.22636pt\psi_{L\downarrow}=\frac{\eta_{L\downarrow}}{\sqrt{2\pi
a}}e^{i(\phi_{+}+\theta_{+})},$ (9)

where $\eta_{\alpha\sigma}$ are Klein factors and $a$ denotes the short-
distance cutoff, related to the large momentum cutoff $\Lambda$ by
$a\sim\Lambda^{-1}$. Here, $\phi_{+}(x)$ and $\theta_{+}(x)$ are canonically
conjugate bosonic operators for degrees of freedom near $k=\pm k_{F}$ where
$k_{F}=2m\alpha_{R}$, whereas $\phi_{-}(x)$ and $\theta_{-}(x)$ describe modes
near $k=0$. In these variables, the Hamiltonian consists of two Luttinger
Hamiltonians for the $+$ and $-$ species, with approximately equal Luttinger
parameters $K_{\pm}$, and interaction terms reflecting Eqs. (6) and (7).
Moreover, one obtains derivative terms which couple the two species
$\kappa_{\phi}\partial_{x}\phi_{+}\partial_{x}\phi_{-}$ and
$\kappa_{\theta}\partial_{x}\theta_{+}\partial_{x}\theta_{-}$.

Following Ref. [Gangadharaiah et al., 2008], we diagonalize the quadratic
parts of the Hamiltonian by going to the charge-spin basis
$\phi_{\rho,\sigma}=(\phi_{+}\pm\phi_{-})/\sqrt{2}$ and
$\theta_{\rho,\sigma}=(\theta_{+}\pm\theta_{-})/\sqrt{2}$ so that

$H_{0}=\sum_{a=\rho,\sigma}\frac{v_{a}}{2\pi}\int\,dx\left[\frac{(\partial_{x}\phi_{a})^{2}}{K_{a}}+K_{a}(\partial_{x}\theta_{a})^{2}\right],$
(10)

where $v_{\rho,\sigma}$ are the respective sound velocities of the modes. For
repulsive interactions, we have $K_{\rho}<1$ and $K_{\sigma}<1$ Gangadharaiah
et al. (2008). In addition to $H_{0}$, we obtain the two competing interaction
terms,

$\displaystyle V_{\rm{S}}$ $\displaystyle=\frac{g_{\rm{S}}}{(2\pi
a)^{2}}\int\,dx\,\cos[2\sqrt{2}\theta_{\sigma}],$ (11) $\displaystyle
V_{\rm{sf}}$ $\displaystyle=\frac{g_{\rm{U}}}{(2\pi
a)^{2}}\int\,dx\,\cos[2\sqrt{2}(\phi_{\rho}-\phi_{\sigma})].$ (12)

Proximity-induced $s$-wave superconductivity gives a contribution which reads
in bosonized form,

$V_{\rm{SC}}=\frac{g_{\rm{SC}}}{(2\pi
a)^{2}}\int\,dx\,\biggl{(}\cos[\sqrt{2}(\theta_{\rho}+\theta_{\sigma})]+\\{\theta_{\sigma}\rightarrow-\theta_{\sigma}\\}\biggr{)}.$
(13)

For weak interactions, the parameters of the model can be determined precisely
in the bosonization procedure (see Ref. [Pedder et al., 2016]). However, for
strong interactions it is more convenient to regard the parameters
$v_{\rho,\sigma}$ and $K_{\rho,\sigma}$ as well as the three coupling
strengths $g_{\rm{S}}$, $g_{\rm{U}}$ and $g_{\rm{SC}}$ as effective
parameters, which may flow independently under renormalization as we change
the cutoff $a$.

We calculate the flow of the various coupling constants using real-space RG
calculation based on operator product expansions.Cardy (1996) We find the
following first-order RG equations for the coupling constants of the cosine
terms

$\displaystyle\frac{dg_{\rm{S}}}{d\ell}$
$\displaystyle=\left(2-\frac{2}{K_{\sigma}}\right)g_{\rm{S}},$ (14)
$\displaystyle\frac{dg_{\rm{U}}}{d\ell}$
$\displaystyle=2(1-K_{\sigma}-K_{\rho})g_{\rm{U}},$ (15)
$\displaystyle\frac{dg_{\rm{SC}}}{d\ell}$
$\displaystyle=\left(2-\frac{1}{2K_{\sigma}}-\frac{1}{2K_{\rho}}\right)g_{\rm{SC}},$
(16)

implying that the spin-density wave term is always irrelevant for repulsive
interactions ($K_{\sigma}<1$).Gangadharaiah et al. (2008) The spin-umklapp
term, by contrast, can become relevant for strong interactions where
$K_{\rho}+K_{\sigma}<1$. Finally, the superconducting term is relevant for
$K_{\rho}^{-1}+K_{\sigma}^{-1}<4$.

We would like to point out that for $\alpha_{R}=0$, the system becomes
$\rm{SU}(2)$ invariant. In that case, the spin-umklapp term vanishes because
spin is conserved and one finds the well-known Kosterlitz-Thouless RG flow
which brings $K_{\sigma}\to 1$ as $g_{\rm{S}}\to 0$. In contrast, for
$\alpha_{R}\neq 0$, $K_{\sigma}$ is not constrained and strong repulsive
interactions lead to $K_{\sigma}\ll 1$.

We start the RG flow from an initial value $a=a_{0}$ and flow towards $a\sim
L$, the length of the wire. Generically, the RG flow will stop at a finite
value $a_{\infty}<L$ as soon as one of the dimensionless coupling constants
$g_{\rm{SC,U}}$ approaches one. The bare value of $g_{\rm{SC}}(a_{0})$ is
determined by the strength of the proximity coupling to the superconductor,
which can be experimentally optimized. The bare value $g_{\rm{U}}(a_{0})$
depends on the separation between the lowest sub-bands, and so depends on the
transverse confinement (i.e. the physical width) of the wire.

To generate zero-energy bound states, spin-umklapp scattering must gap out the
modes near $k=0$, whereas proximity-induced superconductivity should open a
gap for the modes at $k=\pm k_{F}$, see Fig. 2(b). Superconductivity affects
all modes, so this is only possible if at the end of the RG flow
$|g_{\rm{U}}(a_{\infty})|>|g_{\rm{SC}}(a_{\infty})|>0$. Strong electron-
electron interactions result in $K_{\rho}<1/2$ and $K_{\sigma}<1/2$, which a
priori makes the spin-umklapp term relevant and the superconducting term
irrelevant. However, since the RG flow is cut off at a finite length scale,
one will generally find a non-zero $|g_{\rm{SC}}(a_{\infty})|>0$ at the end of
the RG flow, meaning that a superconducting gap will still open.

##  IV Josephson effect & Shapiro steps.

A zero-bias conductance peak is a possible experimental signature of localized
Majorana fermions associated to a helical gap. However, such peaks could arise
from other mechanisms, e.g. disorder,Pikulin et al. (2012); Liu et al. (2012)
and do not directly indicate the degeneracy of the states involved. In
particular, the umklapp gap, and associated fractionally-charged states we
previously proposed in Ref. [Pedder et al., 2016] would lead to a similar
zero-bias anomaly in a system with proximity-induced superconductivity as that
caused by Majorana fermions, albeit at vanishing magnetic field. To uniquely
discriminate these particular bound states, we instead propose to discover
their presence via the periodicity of the Josephson effect, similar to the
corresponding proposal for Majorana fermions.Fu and Kane (2009); Rokhinson et
al. (2012)

Figure 3: Experimental setup for the measurement of the $8\pi$ periodic
Josephson effect. Two superconductors underneath a Rashba wire are held at a
phase difference $\phi$. The blue portions of the wire are gapped by
superconductivity, whereas the grey section is gapped by umklapp scattering.
Fractionally charged bound states are indicated by red spheres.

To investigate the role of fractionally-charged tunnelling states on the
Josephson effect, we follow the logic of Refs. [Zhang and Kane, 2014; Orth et
al., 2015] and consider an arrangement with two superconducting contacts with
phase difference $\phi$ placed under a Rashba wire partially gapped by spin-
umklapp scattering (see Fig. 3), in an analogous arrangement to the
experimental setup of Ref. [Rokhinson et al., 2012]. In the absence of TRS
breaking, the wire adjacent to the edges of the superconductors will host
zero-energy modes with charge $e/2$, which will dominate the transport at low
energies and for a short junction.Pedder et al. (2016) Tunnelling of a single
quasiparticle through the junction will change the parity of the end states.
In order to satisfy the boundary conditions due to the applied superconducting
phase difference, _four_ $e/2$ quasiparticles must tunnel via the bound
states, leading to an $8\pi$ periodic Josephson effect Zhang and Kane (2014).
The TRS breaking caused by the superconducting phase difference causes a
slight lifting of the fourfold degeneracy, but for realistic parameters this
shift is negligible.Zhang and Kane (2014)

Figure 4: Numerical solutions to Eq. (20) with a d.c. bias current
($\alpha_{1}=0$), showing different periodicity behaviour depending on the
amplitude of the d.c. current. We have made the generic choice of
$\alpha_{m}=1/10$ and $\alpha_{p}=1/100$ for the relative size of the
contributions to the Josephson current from Majorana fermions and
parafermions. Panel (a) shows the superconducting phase changing in $2\pi$
steps for the choice $\alpha_{0}=1.5$, whereas panel (b) is plotted for
$\alpha_{0}=1.1$, and shows residual small steps at $2\pi$ and dominant steps
at $4\pi$ which corresponds to the dominant transport being via tunneling of
Majorana fermions. In panel (c), we choose $\alpha_{0}=1.08$, and recover
jumps of $8\pi$ in the superconducting phase due to tunnelling of
parafermions.

In real material samples, TRS may also be weakly broken by magnetic
impurities, thereby lifting the fourfold degeneracy to a twofold one. In our
finite-length Rashba wire, there will generically be a non-zero overlap of the
modes at each end of the wire and so the degeneracy between the modes will be
split, although this effect is exponentially suppressed in the length of the
wire. In any case, the un-driven Josephson current is no longer $8\pi$
periodic. This raises the question of whether remnants of the $8\pi$
periodicity can be observed in such a non-ideal setting.

A possible answer was proposed for similar problems in Majorana
nanowires,Domínguez et al. (2012) where a $4\pi$ periodicity is reduced by
parity-flipping perturbations to a trivial $2\pi$ periodicity: by driving the
current in the junction at a finite frequency, we allow Landau-Zener
tunnelling between the different low-lying states. Then Shapiro step
measurements Shapiro (1963) can still distinguish higher periodic components
even when those are very weak, as in the case of a $4\pi$ periodicity recently
reported in experiments on Majorana bound states Rokhinson et al. (2012);
Wiedenmann et al. (2016).

In the case of a driven junction, Landau-Zener tunnelling gives us access to
all the low-energy modes, even when they are subject to a small splitting.
Allowing for the possibility of Josephson tunnelling of Cooper pairs, as well
as for the tunnelling of charge $e$ and charge $e/2$ quasiparticles through
the weak link in our system, the total Josephson current flowing is given by

$I(\varphi(t))=i_{c}\sin[\varphi(t)]+i_{m}\sin\left[\frac{\varphi(t)}{2}\right]+i_{p}\sin\left[\frac{\varphi(t)}{4}\right].$
(17)

The amplitude $i_{c}$ accounts for the current due to the tunnelling of Cooper
pairs (this is the critical current above which the junction becomes
“normal”). The parameters $i_{m}$ and $i_{p}$ similarly account for the
tunnelling of fractionalized charge $e$ and charge $e/2$ quasiparticles
through the junction. Note that these contributions to the Josephson current
are periodic under shifts $\varphi(t)\rightarrow\varphi(t)+2\pi$,
$\varphi(t)\rightarrow\varphi(t)+4\pi$ and
$\varphi(t)\rightarrow\varphi(t)+8\pi$ respectively. The Josephson equation
relating the rate of change of the superconducting phase $\varphi$ to the
voltage across the junction reads

$\dot{\varphi}(t)=\frac{2e}{\hbar}V(t).$ (18)

We _current-bias_ our Josephson junction using a constant current with a small
a.c. component $I_{0}+I_{1}\sin(\omega t)$. The current through the junction
consists of two parallel components, a _tunnel current_ given by
$I(\varphi(t))$ and a _resistive current_ due to ohmic quasiparticle transport
in the junction. Equating the sum of these two contributions to the bias
current, the gauge-invariant phase $\varphi(t)$ is described by the dynamical
equation

$I_{0}+I_{1}\sin(\omega
t)=i_{c}\sin[\varphi(t)]+i_{m}\sin\left[\frac{\varphi(t)}{2}\right]+i_{p}\sin\left[\frac{\varphi(t)}{4}\right]+\frac{\hbar\dot{\varphi}(t)}{2eR}.$
(19)

Writing Eq. (19) in terms of the rescaled variables $\tau=2eRi_{c}t/\hbar$,
and $\tilde{\omega}=\hbar\omega/2eRi_{c}$, we find the equation

$\dot{\varphi}(\tau)=\alpha_{0}+\alpha_{1}\sin(\tilde{\omega}\tau)-\sin[\varphi(\tau)]-\alpha_{m}\sin\left[\frac{\varphi(\tau)}{2}\right]-\alpha_{p}\sin\left[\frac{\varphi(\tau)}{4}\right]$
(20)

This equation must be solved numerically.

In order to see that the $8\pi$ periodicity can win out, even when
$\alpha_{c}=1>\alpha_{m}>\alpha_{p}$, we first solve Eq. (20) without the a.c.
drive current, i.e. $\alpha_{1}=0$ (see Fig. 4). For a generic choice of the
d.c. bias current, $\alpha_{0}$, the superconducting phase is $2\pi$ periodic,
reflecting the dominance of the Cooper pair tunnelling (Fig. 4a). However, we
find that as we approach a critical value of $\alpha_{0}$, we first see the
$4\pi$ periodic term due to the charge-$e$ quasiparticles (Fig. 4b) and then
the $8\pi$ contribution from the charge-$e/2$ quasiparticles (Fig. 4c) become
dominant.

Figure 5: A plot of the current-voltage relationship for the system driven
with a small amplitude a.c. current with a large d.c. offset (with
corresponding magnitudes $\alpha_{0}=1.08+\delta I$ and $\alpha_{1}=0.01$).
The excess current $\delta I$ over the critical value at zero drive is plotted
on the vertical axis. The Josephson junction has a dominant $2\pi$ periodic
component, and sub-leading $4\pi$ and $8\pi$ periodic components
($\alpha_{m}=1/10$ and $\alpha_{p}=1/100$). (a) At the fastest drive frequency
(comparable to that of the $2\pi$ periodic component of the Josephson current)
shown in the top curve, we find all integer Shapiro steps. As we drive at a
slower frequency (middle curve), the first Shapiro step shrinks, and then
disappears, leaving only Shapiro steps which are multiples of 2. (b) Driving
at smaller frequencies, we find a double-step structure (top curve), but the
size of the intermediate steps shrink as we reduce $\tilde{\omega}$ (middle
curve). Eventually, at the lowest frequencies, we see a $4n\omega$ step
structure. Note that the extra steps at $V\approx 2\omega/3$ in the top two
lines in panel (a) result from the fact that we are driving non-adiabtically,
but are not relevant to experimental measurements as they do not occur at
integer multiples of the driving frequency.

The winding up of the superconducting phase is not a property which is easy to
directly measure, so to give an experimentally accessible measurement, we must
drive the system to see the degeneracy of the low-lying modes. To do this, we
switch on the small a.c. component to the bias current $\alpha_{1}$
(experimentally achieved by irradiating the junction with microwaves). Tuning
the driving frequency allows us to access three distinct regimes, in which
steps occur in the experimentally-measurable I-V curves for the junction at
distinct multiples of the driving frequency $\tilde{\omega}$. For high
frequencies, we find Shapiro steps are present at _all_ integer multiples of
the driving frequency, indicating that the transport is dominated by
conventional Josephson tunnelling of Cooper pairs through the junction.
Reducing the frequency of the drive, we recover the results of Ref. [Domínguez
et al., 2012], that only the _odd_ -numbered Shapiro steps remain. Finally,
for lower frequencies still, there exists a regime in which only every
$(4n)^{th}$ Shapiro step survives (see Fig. 5).

The appearance of _extra_ steps can be caused e.g. by disorder, or strongly
non-adiabatic driving. However, mechanisms other than the one described by
which Shapiro steps may _disappear_ seem to be unknown. As the disappearing
Shapiro steps are robust even when the $2\pi$ and $4\pi$ contributions to the
Josephson current are dominant over the $8\pi$ component, they therefore
provide a highly selective test of the existence of an interaction-generated
helical gap, even in the presence of weak TR symmetry breaking.

##  V Bound states

Several works Fidkowski and Kitaev (2011); Turner et al. (2011) have suggested
that in a spinless, one-dimensional system, the greatest achievable
topological degeneracy is twofold, leading to the statement that only Majorana
fermions can exist in one-dimensional systems. Our system does not contradict
this theorem because in our case the ground state degeneracy is not entirely
topological. Indeed, it can be viewed as a twofold topological degeneracy
combined with a two-fold degeneracy due to TRS Sela et al. (2011). This second
degeneracy can be lifted by local TRS-breaking perturbations such as a
magnetic field. In that case, only the topological part of the ground state
degeneracy survives, and one recovers the $4\pi$ periodicity of the Josephson
effect seen for Majorana bound states.Fu and Kane (2009)

The Shapiro step measurement suggested in the previous section will
demonstrate the existence of a spin-umklapp generated gap even in disordered
nanowires which may have weak TRS breaking. However, the level of disorder in
current state-of-the-art nanowires and nanowire junctions is so low as to
demonstrate ballistic electron transport.Heedt et al. (2017); Fadaly et al.
(2017); Zhang et al. (2016) In such clean wires, and without explicit TRS
breaking, it may even be possible to see the existence of degenerate,
localized zero-energy bound states. To investigate the form of these bound
states, we use the unfolding transformation described in Refs. [Oreg et al.,
2014; Giamarchi, 2004]. This transformation can be used to map our system with
length $L$ and open boundary conditions to a system of length $2L$ and
periodic boundary conditions. Explicitly, we construct the unfolded _chiral_
fields

$\displaystyle\xi_{R}({\tilde{x}})=\left\\{\begin{array}[]{ll}\varphi_{R+}({\tilde{x}})&{\rm{for}}\
0\leq{\tilde{x}}\leq L\\\
\varphi_{L-}(2L-{\tilde{x}})&{\rm{for}}\,\,L\leq{\tilde{x}}\leq
2L\end{array}\right.$ (23)
$\displaystyle\xi_{L}({\tilde{x}})=\left\\{\begin{array}[]{ll}\varphi_{L+}({\tilde{x}})&{\rm{for}}\
0\leq{\tilde{x}}\leq L\\\ \varphi_{R-}(2L-{\tilde{x}})&{\rm{for}}\
L\leq{\tilde{x}}\leq 2L\end{array}\right.$ (26)

where $\varphi_{\alpha\nu}=\alpha\phi_{\nu}-\theta_{\nu}$ with $\alpha=R,L$
and $\nu=\pm$. In order that the fermionic fields obey the vanishing boundary
conditions at $x=0$ and $x=L$, we find that the bosonic fields must obey

$\displaystyle\varphi_{L+}(0)=\varphi_{R-}(0),$
$\displaystyle\qquad\varphi_{L+}(L)=\varphi_{R-}(L)$
$\displaystyle\varphi_{L-}(0)=\varphi_{R+}(0),$
$\displaystyle\qquad\varphi_{L-}(L)=\varphi_{R+}(L)$ (27)

Note that in this transformation, the degrees of freedom
$(\phi_{+},\theta_{+})$ are mapped on the range ${\tilde{x}}\in[0,L]$, whereas
the $(\phi_{-},\theta_{-})$ fields are mapped on the range
${\tilde{x}}\in[L,2L]$. Since the original chiral fields satisfy
$[\varphi_{\alpha\nu}(x),\varphi_{\alpha^{\prime}\nu^{\prime}}(x^{\prime})]=i\pi\alpha\delta_{\alpha\alpha^{\prime}}\delta_{\nu\nu^{\prime}}{\rm{sgn}}(x-x^{\prime})$,
we find that the unfolded fields obey the correct chiral commutation relations

$[\xi_{\alpha}({\tilde{x}}),\xi_{\alpha^{\prime}}({\tilde{x}}^{\prime})]=i\pi\alpha\delta_{\alpha\alpha^{\prime}}{\rm{sgn}}({\tilde{x}}-{\tilde{x}}^{\prime}),$
(28)

on the whole interval ${\tilde{x}},{\tilde{x}}^{\prime}\in[0,2L]$. In terms of
these unfolded fields, our Hamiltonian for the relevant perturbations arising
from umklapp scattering and superconductivity reads

$\displaystyle V_{\rm{sf}}+V_{\rm{SC}}$ $\displaystyle=$
$\displaystyle\int_{0}^{2L}d{\tilde{x}}\biggl{[}g_{\rm{U}}({\tilde{x}})\cos(2[\xi_{R}({\tilde{x}})-\xi_{L}({\tilde{x}})])$
$\displaystyle\hskip
34.1433pt+g_{\rm{SC}}({\tilde{x}})\cos(\xi_{R}({\tilde{x}})+\xi_{L}({\tilde{x}}))\biggr{]}$
(29)

where the position-dependent couplings $g_{\rm{U}}({\tilde{x}})$ and
$g_{\rm{SC}}({\tilde{x}})$ have support on ${\tilde{x}}\in[0,L]$ and
${\tilde{x}}\in[L,2L]$ respectively. The unfolded system consists of two
adjacent regions. Between ${\tilde{x}}=0$ and ${\tilde{x}}=L$, we have a
region of superconductor where the field
$\theta({\tilde{x}})=-(\xi_{R}+\xi_{L})/2$ is pinned by the term
$\cos[2\theta]$ to the value $\theta^{\star}=(n+1/2)\pi$ for integer $n$. From
${\tilde{x}}=L$ to ${\tilde{x}}=2L$, there is a region of “Mott insulator”
where the field $\phi=(\xi_{R}-\xi_{L})/2$ is pinned by the term $\cos[4\phi]$
to the value $\phi^{\star}=1/2(m+1/2)\pi$ for integer $m$. The spectrum is
completely gapped, except possibly at the boundaries between the two regions,
${\tilde{x}}=0$ and ${\tilde{x}}=L$, where the parafermion states we describe
emerge. The unfolded system is identical to the topological insulator edge
state system studied in Ref. [Orth et al., 2015], which tells us that our
original system contains $\mathbb{Z}_{4}$ parafermion state at its ends.

Let us reproduce the essential parts of the derivation here. We define the
total charge and total spin operators for the system, according to

$\displaystyle\pi S$ $\displaystyle=\theta(2L)-\theta(0),$ (30)
$\displaystyle\pi Q$ $\displaystyle=\phi(2L)-\phi(0).$ (31)

Despite the fact that the fermionic fields must be continuous, the bosonic
fields $\phi$ and $\theta$ may jump by integer multiples of $2\pi$, so that
$S$ and $Q$ can be non-zero in spite of the periodic boundary conditions. The
spin $S$ of the system takes integer values, and is conserved $mod(4)$ so
$s=\\{0,1,2,3\\}$ (measured in units of $\hbar/2$), whereas the charge $Q$
takes half-integer values and is conserved $mod(2)$ so
$q=\\{0,\frac{1}{2},1,\frac{3}{2}\\}$ (measured in units of $e$). Since we may
only add integer amounts of electronic charge to our junction, we must
restrict our value of the charge to be $q\in\\{0,1\\}$. Note that in a system
where there are several junctions between superconducting and Mott insulating
regions, it is perfectly acceptable to have states with half-integer charge,
as long as the total charge of the complete system is restricted again to
$q\in\\{0,1\\}$. The state of our system is then defined by $|s,q\rangle$.
Since every physical electron carries one unit of spin, this means that for
the charge state $q=0$, only the two total spin states $s\in\\{0,2\\}$ are
permissible. Similarly, $q=1$ requires $s\in\\{1,3\\}$. Hence, we have a total
fourfold degeneracy of the ground state.

To see explicitly the parafermionic statistics of the bound states at $x=0$
and $x=L$, we write the pinned values of the fields in terms of integer-
spectrum operators $m,n_{1},n_{2}$ as

$\displaystyle\phi_{1}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\pi
m+\frac{\pi}{2}\right),$ $\displaystyle\theta_{1,2}$ $\displaystyle=$
$\displaystyle\pi n_{1,2}+\frac{\pi}{2}.$ (32)

These operators then have commutation relations

$[m,n_{1}]=-\frac{i}{\pi},$ (33) $[m,n_{2}]=0.$ (34)

The total spin operator is given by

$\hat{S}=e^{i\pi m/2}=e^{-i\pi/4}e^{i\phi_{1}}$ (35)

and the total spin in the system is
$\langle\hat{S}\rangle=(\theta_{1}-\theta_{2})/\pi=n_{2}-n_{1}$. Then the
parafermion states obeying

$\chi_{1}\chi_{2}=e^{-i\pi/2}\chi_{2}\chi_{1}$ (36)

are given by

$\displaystyle\chi_{1}$ $\displaystyle=T_{Q}e^{i\pi m/2},$
$\displaystyle\chi_{2}$ $\displaystyle=e^{i\pi/4}T_{Q}e^{i\pi
m/2}e^{i\pi(n_{2}-n_{1})/2},$ (37)

where $T_{Q}$ is the raising operator for charge
$T_{Q}|s,q\rangle=|s,q+1\rangle$.

##  VI Discussion.

The opening of the spin-umklapp gap is rather generic. In a wire with RSOC and
electron-electron interactions this gap will lead to a reduction of the
normal-state conductance of from $2e^{2}/h$ to $e^{2}/h$ as the chemical
potential approaches the Dirac point, which may already have been seen in InAs
nanowires.Heedt et al. (2017) We would like to point out that the required
strong interactions have already been seen in these wires.Hevroni et al.
(2016) Umklapp scattering has also recently been invoked to explain the
reduction in conductance observed in InAs/GaSb topological insulator edge
states.Li et al. (2015) Introducing a weak superconducting proximity effect in
either these wires or edge states will then lead to the creation of bound
states and allow the observation of the $8\pi$ periodic Josephson current
component.

Our system is susceptible to bulk disorder. Firstly, disorder causes one-
particle backscattering which corresponds to another cosine term in the
bosonized Hamiltonian. For umklapp scattering to win out, we require that the
amplitude of the umklapp term is larger than the disorder potential at the end
of the RG flow. Moreover, disorder can cause local deviations of the chemical
potential away from $\mu=0$. If these fluctuations are small compared to the
umklapp gap, the bound states will persist. Recent measurements on InAs wires
Heedt et al. (2017) and InSb wires and arrays Fadaly et al. (2017); Zhang et
al. (2016) show ballistic transport which is indicative of very low levels of
bulk disorder. Weak magnetic disorder will result in explicit breaking of
time-reversal symmetry, and so the localized $\mathbb{Z}_{4}$ parafermionic
bound states associated to the charge $e/2$ quasiparticles will cease to
exist, but the sub-leading $8\pi$-periodic term will still be visible in
Shapiro step experiments.

To summarize, we propose an experimental scheme allowing us to observe an
$8\pi$ periodic Josephson current occurring via the tunnelling of fractional
$e/2$ charges through the junction. The associated Shapiro step structure in
which only the $4n$ steps survive (where $n\in\mathbb{Z}$) would be a
definitive signature of fourfold degenerate bound states, associated to an
umklapp gap generated by two-particle backscattering in strongly interacting
Rashba wires or edge states of two-dimensional topological insulators. In the
absence of TRS breaking perturbations, the bound state degeneracy is protected
by a combination of time-reversal symmetry and fermion parity symmetry. The
operators describing these bound states in the ground state manifold satisfy a
$\mathbb{Z}_{4}$ parafermionic algebra.

_Acknowledgements._ TLS and CJP are supported by the National Research Fund,
Luxembourg under grant ATTRACT 7556175. TM is funded by Deutsche
Forschungsgemeinschaft through GRK 1621, SFB 1143, and the Emmy-Noether
program ME 4844/1. RPT acknowledges financial support from the Swiss National
Science Foundation.

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