11affiliationtext: London Centre for Nanotechnology, University College
London, London, WC1H 0AH, United Kingdom22affiliationtext:  Faculty of Physics
and Center for Nanointegration Duisburg-Essen (CENIDE), University of
Duisburg-Essen, 47057 Duisburg, Germany33affiliationtext: Institut für Physik,
Johannes Gutenberg-Universität Mainz, Staudingerweg 7, 55128 Mainz,
Germany44affiliationtext: Department of Electronic and Electrical Engineering,
University College London, London, WC1E 7JE, United Kingdom55affiliationtext:
WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1,
Katahira, Sendai 980-8577, Japan

# Perspective on unconventional computing using magnetic skyrmions

Oscar Lee  Robin Msiska  Maarten A. Brems  Mathias Kläui  Hidekazu Kurebayashi
Karin Everschor-Sitte

###### Abstract

Learning and pattern recognition inevitably requires memory of previous
events, a feature that conventional CMOS hardware needs to artificially
simulate. Dynamical systems naturally provide the memory, complexity, and non-
linearity needed for a plethora of novel computational approaches, including
different unconventional computing approaches. In this perspective article, we
focus on the unconventional computing concept of reservoir computing and
provide an overview of key physical reservoir works reported. We focus on the
promising platform of magnetic structures and in particular skyrmions, which
potentially allow for low-power applications. We provide an outlook on the
most important challenges in this field.

##  1 Introduction to reservoir computing and magnetic skyrmions

Modern-day applications of artificial intelligence (AI) have become pervasive
in many aspects of our daily lives, and their importance is only predicted to
increase. Artificial neural networks (ANNs), computational models inspired by
the biological neural network architecture of the human brain, are primarily
responsible for the rapid advancement of AI research [1, 2]. A class of ANN
known as the recurrent neural network (RNN)[3] excels at processing sequential
or time series data. RNNs are distinguished by their "memory", which
incorporates data from previous inputs to process a specific element of an
input sequence.

Reservoir computing (RC) is a general and universal [4] computational
framework descended from RNNs. Its foundations can be traced back to two
independently developed RNN-based models, echo-state networks by Jaeger [5]
and liquid state machines by Maass et al. [6]. The fundamental working
principle of RC is that input data are fed into a nonlinear system called a
reservoir, which then nonlinearly maps these inputs to a higher dimensional
space. This mapping enables spatiotemporal feature selection to be performed
at the readout nodes using relatively simple learning algorithms such as
linear classifiers. In contrast to conventional ANNs which call for the fine-
tuning of a plethora of interconnected node weights across multiple layers,
the internal weights of the reservoir and the input nodes remain fixed and
only the weights of the readout nodes need to be trained. This significantly
reduces the computational cost of learning [7]. Figure 1 schematically depicts
a summary of the concept of reservoir computing. Here, the nonlinear reservoir
is exemplified by a skyrmion fabric system.[8, 9, 10, 11]

Figure 1: Summary of the reservoir computing framework. (a) An arbitrary
temporal input signal excites the (b) reservoir — embodied in this
illustration by a physical system made up of (c) magnetic skyrmions — which
then (d) projects the input data into a linearly separable higher dimensional
space in which hyperplanes can be used to (e) classify different desired
features by only training the output readout. Moreover, the input-excited
reservoir projection also enables other inference tasks such as (f) time-
series prediction.

For RC systems to work effectively, some crucial requirements need to be met:
The reservoir needs to have high complexity, in the sense that the number of
its effective degrees of freedom must be larger than the dimensionality of the
input. Additionally, the reservoir has to have a nonlinear response to a given
stimulus. This ensures the projection of inputs into a linearly separable
high-dimensional feature space, enabling the extraction of nonlinear features
from the input data. Another necessary characteristic is that the reservoir’s
internal state needs to be influenced by recent inputs while remaining
unaffected by inputs from the distant past. This quality is referred to as
fading/short-term memory [12, 13]. The extent of this fading memory has a
profound effect on the information processing capacity of the reservoir [14,
13]. Fading memory is what makes RC well-suited for processing temporal data
with transient dependencies [6] such as stochastic or chaotic time series
prediction [15, 16]. Additionally, a reservoir needs to possess the ability to
respond to a given input uniquely. More concretely, the reservoir should
distinctly map the temporal history of a given input to a specific internal
state. This is termed the echo state property [5, 17, 18].

Although RC does not train the internal weights of the reservoir, it is still
often possible to optimise reservoir performance by tuning hyperparameters.
For systems with many possible parameter choices, task-agnostic metrics help
in identifying excellent hyperparameters[19, 20]. Some reservoir systems
exhibit dynamics that transition between non-chaotic and chaotic regimes upon
adjustment of their intrinsic parameters. Occasionally, such reservoirs can be
optimised by adjusting them to operate at the so-called "edge of chaos" — a
critical phase transition point beyond which the reservoir system’s dynamics
become chaotic [21, 22]. While this approach can be effective in designing
reservoirs with chaotic tendencies, it is not universally applicable, and
there have been exceptions to this hypothesis [23, 24, 25]. For this reason,
it is imperative to understand the dynamical trends of a chosen reservoir
system.

A novel branch of RC called physical RC (PRC) has emerged, in which physical
systems are used as reservoirs [26, 27, 28, 29]. Physical systems often
naturally fulfill the RC criteria of being complex, nonlinear, and possessing
a short-term memory. Please note that for physical reservoirs, system
stability is an additional essential prerequisite which enables reproducible
functioning. Stability allows the system to relax back to a consistent
(meta-)stable state following input signal excitations [20]. Stability also
entails that the reservoir is robust against noisy fluctuations and other
internal transient dynamics that don’t promote the nonlinear transformation of
inputs, but persist even after input signals are removed. A PRC measure of
fading memory needs to include these fleeting dynamics as they have been shown
to support short-term memory [30, 13, 31].

In this Perspective, we describe reservoirs epitomised by skyrmions such as
the skyrmion fabric system shown in Fig. 1. Magnetic skyrmions are localized
magnetic whirls possessing particle-like properties. Skyrmions were originally
proposed by Tony Skyrme in 1961 in a model describing elementary particles
[32]. The magnetic versions studied nowadays were first theoretically
predicted in 1989 [33] and experimentally observed in 2009 [34]. Since then,
they have been shown to occur in many magnetic systems ranging from insulators
to metals at various temperatures even above room temperature [35, 36, 37, 38,
39, 40, 41]. Furthermore, skyrmions have been observed as singular objects
[42, 43, 44, 45], clusters [46, 47], skyrmion lattices[34, 48, 49]and in the
form intermediate skyrmion phases known as skyrmion fabrics (Fig. 1b)) [50].
An example of a single skyrmion is illustrated in Fig. 1c).

Skyrmions have a non-trivial topology which implies that a topological
invariant can be associated with them, quantified by the topological index

$Q=\frac{1}{4\pi}\int\textbf{M}\cdot\left(\frac{\partial\textbf{M}}{\partial
x}\times\frac{\partial\textbf{M}}{\partial y}\right)\mathrm{d}x\mathrm{d}y,$
(1)

where M is the magnetisation unit vector and the integral is taken over a two
dimensional space. $Q=\pm 1$ for skyrmions in particular. Topology has a
profound effect on the physics of skyrmions and influences phenomena such as
transport [51] and thermal [52, 53, 54, 55] motion. Topology massively
increases the skyrmion’s robustness against structural defects and impurities
[56]. Topological properties can be exploited to build stable reservoir
systems [8, 9].

Magnetic spin textures such as skyrmions originate from the interplay of
different competing energetic contributions within a given system [37].
Typically, there is at least one energy term that favours a uniform
ferromagnetic configuration of magnetic moments e.g. exchange or anisotropy,
and there are other terms that promote twisted configurations e.g. DMI and
demagnetization [36]. In magnetic materials with broken inversion symmetry
such as uniaxial non-centrosymmetric ferromagnets [57], cubic helimagnets [58]
and thin film heterostructure systems with structural inversion asymmetry
[36], DMI effects [59] become pronounced.

Skyrmion dynamics can be manipulated by spin torques [60, 61, 51, 62],
magnetic fields [63, 64, 65], electric fields [66, 67, 68], magnons [69, 70,
71], temperature gradients [72, 73] and thermal fluctuations [74, 75, 76].
Through such mechanisms, it is possible to reliably control the creation
(writing) [77], detection (reading) [77], rotation [78, 79] and even
annihilation (deleting) [77] of magnetic skyrmions. These qualities along with
topological stability, small size [80, 81], manoeuvrability around material
defects [82] and ultra-low power operation [60, 51] pave the way for magnetic
skymions to be used in applications such as skyrmion-based racetrack memories
[83, 84], logic devices [85, 86], magnetic tunnel junctions [87, 88], nano-
oscillators [89, 90], and unconventional computing schemes [91, 92, 93, 94]
like neuromorphic [95], probabilistic [96, 53], and reservoir computing [8,
97].

##  2 Overview of key physical reservoir works

Systems from diverse disciplines have demonstrated their capabilities of
constructing a physical reservoir [26, 27, 28, 29], including bioelectronics
[98, 99, 100, 101], electronics [102, 103, 104, 105, 29], magnonics [106, 107,
108], memristors [109, 110, 111, 112], photonics [113, 114, 115, 116, 117],
and spintronics [118, 119, 91, 92, 93, 105, 97, 120], summarised in Table 2.
This brings an advantage from an application standpoint as different
reservoirs could be designed pertinent to a specific application considering
their available inputs and readout mechanisms [121].

Table 1: Examples of experimental physical reservoir systems. Note the
abbreviations are found in the main text.

lcccccc  Year & Discipline Reservoir Input Output Readout Demonstration
2022 [99] Bioelectronics  mBNN Photostimulation Evoked activity Fluorescent
calcium imaging Spoken digit recognition
2021 [98] Bioelectronics  OECT Voltage waveform Current  Voltage Heartbeat
classification
2022 [102] Electronics  FeFET Voltage waveform  Current Current Temporal-XOR &
parity-check
2022 [107] Magnonics  Magnetic phase-space Static magnetic fields Spinwave
spectra Ferromagnetic resonance Signal transformation & forecasting
2022 [106] Magnonics  ASVI Static magnetic fields Spinwave spectra
Ferromagnetic resonance Signal transformation & forecasting
2022 [110] Memristors  DM Voltage pulses  Current RRAM Arrhythmia detection &
gesture recognition
2021 [109] Memristors  Memristive NW networks Voltage pulses Voltage RRAM
Handwritten digit classification
2022 [97] Spintronics  Skyrmion arrangement Electrical currents Voltage TMR
junction Boolean logic operations
2022 [120] Spintronics  Skyrmion dynamics Time-variant magnetic fields Voltage
Hall voltage Waveform & handwritten digit recognition
2017 [118] Spintronics  Spin-torque nano-oscillator Voltage waveform Time-
dependent voltage oscillation Diode-rectified voltage Spoken-digit recognition

In bioelectronic systems, information is processed using biocompatible
materials or organic biological architectures in conjunction with electronic
sensors. While many approaches primarily focus on clinical applications, their
use for neuromorphic computation has begun gaining interest [122, 123, 124]. A
recent study by Sumi et al. [99] have demonstrated RC on a micropatterned
biological neuronal network (mBNN). Here, the input data were transformed to
frequency-dependent photostimulation to create the reservoir using optogenetic
techniques supplied to rat cortical neurons grown on micropatterned
substrates. The readout mechanism incorporated measurements by fluorescent
imaging via calcium probes, where the spontaneous and evoked activities of
mBNN were trained. The system demonstrates spatial pattern and spoken digit
recognition tasks. On the other hand, organic electrochemical transistors
(OECTs) have also sparked promises for RC [100, 121]. Using data-encoded
voltage waveforms as inputs and analogue readings of output time-variant
voltages, some successful examples of signal classification [101] and
heartbeat classification [98] have been shown.

Diverse electronic systems, including analogue circuits, FPGAs, memristors and
ferroelectrics, highlight their potential for RC [104, 105, 29]. Such systems
have flexible scalability and pose a benefit of circuit-level implementation,
compatible with silicon-based CMOS technologies at low power. In particular,
memristive technologies have continued to proliferate for their role in
neuromorphic computation and RC [111, 112]. Diverse architectural designs have
been proposed, and detailed studies have investigated fundamental properties
in building or improving the system efficiency/performance of the reservoirs.
Numerous experiments have explored various design paradigms that take a step
closer to device-level implementations of RC. For example, a recent work by
Milano et al. [109] uses self-organised nanowire (NW) networks with a
memristive architecture as a reservoir. After inputting a sequence of voltage
pulses that encodes the data, it utilises Resistive Random Access Memory
(RRAM) as a readout mechanism to convert the output voltages from the NW
networks to a matrix of currents that could be trained. In this study,
handwritten digit recognition and signal forecasting have been demonstrated.
Similarly, a fully-analogue RC [110] involving non-volatile memristor arrays
used as RRAM in the readout layer with data-translated input voltage pulses to
dynamical memristors (DM) was shown to allow the detection of arrhythmia and
dynamic hand gesture recognition. The study reports that the power consumption
of such a system comprising 24 DMs is 22.2 $\mu$W.

Among silicon-compatible systems, ferroelectric field-effect transistors
(FeFET) have been realised as an alternative approach in designing future
electronic components for in-memory [125] and neuromorphic computation [126,
127, 128]. As a multi-terminal device, while FeFETs share similar nature with
standard field-effect transistors, it uses ferroelectric materials for the
gate insulator. Its nonlinearity stems from the time-dependent polarisation
reversal process on the input gate voltage. Harnessing this property allows
the output currents to exhibit history-dependent and nonlinear dynamics
adequate for RC [102]. On this note, by measuring the drain, source and
substrate currents from a data-mapped voltage waveform input, Toprasertpong et
al. [102] have performed RC on a HfO2-based FeFET to compute logic-based
tasks, including temporal-XOR and parity-checks.

Spintronic systems have also highlighted unique advantages as a physical
platform for RC. Works by Torrejon et al. [118] demonstrated PRC using a
network of spin-torque oscillators (STOs) made from magnetic tunnel junctions
(MTJs) for spoken digit (pattern) recognition, with an accuracy of up to 99.6
% and nonlinear waveform classification tasks. The data-mapped voltages were
input as currents into the STOs, and the output response of rectified time-
dependent voltages was recorded to construct a reservoir. Gartside et al.
[106] utilised nanomagnets by applying magnetic field inputs and measuring the
nonlinear nucleation dynamics of spinwaves spectra in artificial spin-vortex
ice (ASVI). Furtheremore, complex magnetic structures inherently provide all
the important components of a reservoir without creating a system of
interconnected neurons: complexity, nonlinearity, and short-term memory.
Topological magnetic structures, such as skyrmions, which provide additional
stability, are therefore particularly suitable as a reservoir [8].

##  3 Skyrmion-based reservoir and Brownian computing

Skyrmion-based RC, working at the nanosecond timescale with a power
consumption in the microwatt regime, has been theoretically proposed by
Prychyneko et al. [8]. Due to its independence of concrete details of the
reservoir, the input and readout method various approaches and models have
been predicted and analysed [10, 9, 129, 130, 131, 11]. For example,
Prychynenko et al. [8], Bourianoff et al. [9], Pinna et al. [10], Raab et al.
[97] and Msiska et al. [11] have studied the response of a skyrmion fabrics
system (as exemplified in Fig. 1b) to voltage inputs. In these studies,
reservoir computing is based on exploiting the non-linear current-voltage
characteristics of skyrmion systems due to the complex interplay of current-
induced dynamics and pinning effects. A readout is possible, for example, as a
time-traced resistance signal, or a spatially resolved magnetization
measurement, or a combination of both. By adjusting material parameters that
appear as hyperparameters in the reservoir computing model, the skyrmion
reservoir can be customized for tasks that rely more on memory or nonlinearity
[20]. For example, the recently simulated multi-dimensional input skyrmion-
based reservoir solved a standard benchmark spoken digit classification task
with the best performance ever reported for in-materio reservoir computers
[11]. Raab et al. combined the RC principle with the Brownian computing (BC)
concept and demonstrated a skyrmion-based RC experimentally [97]. BC refers to
the broad idea of exploiting intrinsic random dynamics of a physical system to
the benefit of a computing architecture [97, 132, 133, 134, 94]. It is
inspired by noise-exploiting mechanisms in biological processes where e.g.,
Brownian motion drives molecular machines [135, 136] – hence the name.
Skyrmions are a particularly promising system for BC as they have been shown
to undergo thermally activated diffusion [53, 137, 55]. In addition, the
thermal effects compete with different skyrmion interactions at room
temperature [138, 139]. There exists a multitude of ways how exactly thermal
random effects can be exploited for different computing architectures [97, 53,
134, 94, 140]. In particular, RC can be realized by combining the RC concept
with thermally activated skyrmion dynamics [97] which is discussed in the next
subsection along other skyrmion-based RC approaches. The two subsequent sub-
sections then introduce other non-conventional computing approaches based on
thermally activated skyrmion dynamics. These approaches, apart from being
promising future applications on their own, aid in understanding the
versatility of the BC concept in combination with different computing
architectures such as RC.

###  3.1 Reservoir computing using skyrmions

While several of the theoretical skyrmion RC concepts [8, 10, 9] heavily rely
on the presence of local pinning sites [138], Raab et al. first realized
Brownian RC by overcoming pinning effects in confined geometries by using
thermal skyrmion diffusion [97]. Their proof-of-concept reservoir consists of
a single skyrmion in a triangular confinement [141], which provides an
automatic reset mechanism after operation due to the repulsive skyrmion-
boundary interaction [139]. Inputs are encoded by the patterns of voltages at
the corners of the triangle (Fig. 2a&b). The resulting current distribution
acts as a biasing-mechanism for the thermal skyrmion motion and thereby alters
the average spatial distribution of the skyrmion. In a nano-scale device, one
can exploit that the probability for a certain region of the sample to be
occupied by a skyrmion determines the average local tunnel magneto-resistance
(TMR) [142]. For the proof-of-concept device the local occupation probability
was determined in four regions (white circles in Fig. 2b) using Kerr-
microscopy to mimic TMR readout via magnetic tunnel junctions [142]. The study
demonstrates that by training a linear read-out based on the local occupation
probabilities already this minimalistic Brownian RC device suffices to perform
2- and 3-input logic operations including the non-linearly separable XOR.
Moreover, exploiting thermal effects allows for ultra low-current operation
and overcomes pinning effects which would hinder proper operation in a
diffusion-free system.

Another skyrmion-based RC concept was realized by Yokouchi et al. that have
experimentally studied a skyrmion-based RC device capable of complex pattern
recognition [120]. Their reservoir consists of a collection of Hall bars
containing skyrmions, each at a different constant out-of-plane (OOP) field
(Fig. 2c). The input signal is encoded as a time dependent OOP field in
addition to the constant OOP field. Due to the different constant OOP fields,
the skyrmions in each Hall bar potentially react differently to the input
signal. Training is then performed to tune the weights combining the anomalous
Hall voltages of all Hall bars. The reservoir succeeds in high-accuracy
handwritten digit recognition as well as waveform recognition.

The spectral properties of skyrmions can be exploited for constructing a
physical reservoir. Lee et al. recently demonstrated that the GHz dynamics of
skyrmions generated in multiferroic Cu2OSeO3 can provide a scheme of phase-
tunable, task-adaptive PRC [107]. By utilizing rich thermodynamical phases
available in Cu2OSeO3, they show that the single reservoir unit can offer
multiple reservoir properties, hence adaptive to different computational tasks
that require different reservoir properties (i.e., nonlinearity, memory
capacity & complexity). In their work, external magnetic field values and
temperature are controlled to navigate between available phase spaces to
modify the key reservoir properties on demand. Subsequently, translating the
input data (e.g. a sinewave signal) into a sequence of magnetic field values
can encode its information to the spectral states of various spinwave modes to
construct a reservoir. During this process, a particular input protocol, named
“mapped field-cycling”, is incorporated to nucleate metastable magnetic phase
spaces such as low-temperature skyrmions [143, 144, 145]. The study
demonstrates that the computational power for different tasks highly depends
on the choice of the magnetic phase space. For example, as shown in Fig. 2d&e,
while the skyrmion textures excel in future forecasting, their performance
deteriorates substantially for linear-to-nonlinear transformation tasks.
However, the conical modes observe the opposite behaviour, suggesting a
correlation between the intrinsic magnetic phase properties and the task
capability, which is related to the properties of the reservoir. These results
highlight that a single material system can be reconfigured (by adjusting the
magnetic field or temperature) based on the task’s nature without creating
alternative reservoir systems each time and takes a step closer to flexible
on-demand PRC. The task-adaptive nature is important since a typical physical
reservoir is often fixed and inflexible in terms of reservoir properties due
to the constraint of specific response phenomena of a given physical system:
consequently, many physical reservoirs result in severely constrained
computational performance as it lacks the versatility to meet demands
requiring different reservoir properties. For example, a reservoir system
constructed with a high nonlinearity would not be adequate in performing tasks
requiring a high memory capacity, and vice versa.

Figure 2: a)& b) Kerr-microscopy images of the skyrmion-based Brownian RC
device by Raab et al. [97] for different input voltage combinations. The white
circles in b) depict the regions within which the local skyrmion occupation
probability is measured. Adapted from Ref. [97] (CC BY 4.0). c) Schematic
illustration of the skyrmion-based RC by Yokouchi et al. [120] including Kerr-
microscopy images of the Hall bars containing skyrmions at different constant
OOP fields $H_{const}$. Adapted from Ref. [120] (CC BY-NC 4.0). d)& e)
Performance comparison of RC for a d) nonlinear transformation and e) future
forecasting tasks using skyrmion (orange) and conical (blue) magnetic phase
spaces, respectively. Different magnetic modes are accessed by changing the
applied temperature to the magnet. f) Kerr-microscopy image of the skyrmion
reshuffler device by Zázvorka et al. [53] The input signal is constructed as a
time frame in which the skyrmion crosses the blue threshold line. The output
is produced on crossing the orange line. The corresponding input signal is
depicted in blue (top) and the resulting output signal in orange (bottom).
Adapted from Ref. [53] with permission from the authors. Copyright Zázvorka et
al., under exclusive license to Springer Nature Limited 2019. g) Circuit
layout for a Brownian token-based half-adder by Brems et al. [94]. Cjoins
(colored/numbered squares) can only be passed by both singal carriers (bold
dots) together. The colored arrows show the relevant directions of driven
motion to tune the balance of computation speed and energy consumption of the
device. Adapted with permission from Brems et al., Ref. [94]. Copyright 2021
AIP Publishing LLC.

###  3.2 Stochastic computing using skyrmions

Thermal excitations can not only be used in reservoir computing [97], but also
for stochastic computing. In skyrmion-based stochastic computing [96, 53, 146]
numerical values are encoded as the probability of a “1” occurring in random
bit-streams and computation results are obtained by averaging over the results
of bit-wise operations. For instance, multiplication can be realized by bit-
wise application of the AND operation on two input-streams since the
“1”-probability in the resulting bit-stream is equal to the product of the
“1”-proabilities of the two input-streams. Skyrmion-based stochastic computing
can offer very interesting advantages such as high error tolerance and an
accuracy increasing with computation time (length of processed bit-stream).
However, one key challenge in this approach is that strong correlations
between bit streams severely impair computations as the individual bit-wise
computation results will no longer inherit the correct statistical
distribution from the input-streams. In 2019, Zázvorka et al. have
experimentally constructed a skyrmion-based Brownian reshuffler device to
decorrelate bit-streams [53]. As shown in Fig. 2f the device consists of an
upper and a lower channel and a skyrmion passing a certain section of a
channel at a given time sets the bit-stream’s value to “1” or “0”
respectively. Each channel contains a chamber where the effects of the overall
current-induced drift to the right is combined with thermal random diffusion
to decorrelate the pre-chamber (blue) and post-chamber (orange) bit-streams.
It was demonstrated that this reshuffler device leads to very good
decorrelation of the bit-streams, while keeping the value encoded in the bit-
stream constant.

###  3.3 Brownian token-based computing using Skyrmions

In Brownian token-based computing [132, 133, 134, 94, 55], computations are
performed as skyrmions explore a network of computational paths due to their
thermal random motion. Brems et al. have proposed a skyrmion-suitable
crossing-free circuit for a Brownian half-adder (Fig. 2g) along with a
framework to tune the balance of speed and energy consumption of token-based
computers using current-driven skyrmion motion [94]. The key advantage of this
computing method is that energy must not be invested to move the information
carriers but only to synchronize their movement at certain points in the
circuit allowing potentially for low-energy operation. (Matching
colored/labeled squares in Fig. 2g can only be passed across by skyrmions at
the same time.) Experimental advances have been made by Jibiki et al. in
implementing skyrmion-based circuit modules for Brownian token-based computing
[134, 55].

##  4 Perspective of skyrmions for unconventional computing

Skyrmion-based non-conventional computing is an emerging field that aims to
harness the unique properties of magnetic skyrmions. While this field has
already shown several promising results and is expected to play a significant
role in the future of unconventional computing, there are still a multitude of
challenges left to overcome. These include both short-term goals to make
skyrmion-based reservoir computing more competitive, practical, and efficient
as well as long-term challenges that need to be addressed to realize the full
potential of future skyrmion-based computing.

The field of RC currently features a wide range of designs and architectures
with virtually endless potential for further customization and
experimentation, since it encompasses any dynamical phenomenon that can be
harnessed to build reservoirs. While this diversity is a strength of RC, it
also presents one of its biggest challenges as it makes it difficult to
achieve cohesion and standardization among the different systems. This is also
evident in skyrmion RC where a wide range of diverse system designs have
already been put forward, as elaborated in Section 3, despite the field being
in its nascent stages. To date, there are practically no universally applied
measures or standards for evaluating and comparing PRC models. Such methods
would not only facilitate the selection of the most suitable models for a
specific task, but also grant researchers valuable insight into the underlying
principles and dynamics of reservoir systems. This understanding can be
further leveraged to create new systems that incorporate the most advantageous
features of existing models. Additionally, by comparing models with varying
parameters such as reservoir size and connectivity, researchers can determine
the effect of these parameters on system performance and optimize the design
of the reservoir accordingly. Although we are yet to realize a unified
formalism for RC, progress is being made in certain sub-fields of PRC to
introduce reliable measures. In the particular case of skyrmion RC, task-
agnostic local metrics have been proposed [20]. In addition to performance
classifications, it is also important to establish what constitutes fair
comparison among models. There have been instances in PRC where researchers
have selected or modified datasets to achieve favorable benchmark results.
Implementing standardized comparison schemes would help to eliminate such
practices and ensure fair and unbiased evaluations.

Reservoirs possess memory capabilities that enable them to retain information
about past inputs for a certain period of time, due to recurrent connections
within the reservoir [147]. However, nonlinearity in the reservoir’s dynamics
decreases memory capacity [148]. Therefore, the role of nonlinearity, which is
necessary for input mapping, needs to be balanced with the memory capacity for
optimal performance. One way achieve this in RC design is to create a gradient
mixture reservoir, where one section of the reservoir has high
nonlinearity/low memory, while another section has high memory/low
nonlinearity [149]. Additionally, it is possible to achieve output states that
emphasize either memory or nonlinearity by strategically placing readout
contacts [20].

In task-specific RC, it will be important to identify tasks that are
particularly suited to skyrmion-based reservoirs. These tasks will
fundamentally depend on the intrinsic properties of skyrmion systems such as
the time scales of driven dynamics and decorrelation dynamics (fading memory)
as well as the non-linearity of skyrmion interactions. One of the reasons
skyrmions why have emerged as a promising physical reservoir candidate is
their internal and collective dynamics on different time scales as well as
competing interactions on different length scales. This means that the
complexity of the skyrmion reservoirs allows for a range of tasks to be
implemented. Time scales are particularly important to consider since a
computationally expensive conversion algorithm from the relevant time scale of
a specific task to that of the reservoir may prove to be a bottleneck and
potentially impede the high computational speed and energy efficiency
advantages of RC.

Another important factor to take into consideration is the plethora of
excitation methods for skyrmion dynamics like spin-torques [60, 150, 43, 151,
152] and field gradients [64, 65] and read-out methods like magneto-resistance
[142]. These methods must be gauged with regard to their applicability in
skyrmion-based RC. Apart from stimulation and response measurement methods,
the skyrmion pinning effect has played a significant role in recent RC
concepts. Skyrmion pinning is an essential ingredient for some RC approaches
and an obstacle for others. So methods to engineer the strength and
distribution of pinning areas and thus tune it as necessary for a given
reservoir may be major benefits for future RC approaches. This includes both
material engineering as well as methods to manipulate the effective pinning
effect on state-of-the-art samples [153]. Finally, the trade-offs introduced
by thermal effects in skyrmion reservoirs must be further investigated. It has
been demonstrated that thermal dynamics can benefit a device’s energy
efficiency and error tolerance. On the downside, the stochasticity
accompanying thermal dynamics is expected to act to the detriment of the
systems’ short- and long-term memory.

The future of skyrmion RC holds significant promise, with numerous untapped
research avenues to be explored, including skyrmion oscillators, cyclic
reservoirs, and beyond. Figure 3 depicts the predicted importance of
materials, algorithms, and applications, visualised by blue-coloured beams
that vary in size over time. As research progresses, the field will likely
move from a broad range of reservoir materials to only a narrow selection of
the most effective, while algorithms and applications are anticipated to grow
with advancements in research. Additionally, the figure suggests some key
areas for reservoir optimisation and predicts long-term outcomes.

Figure 3: Skyrmion reservoir computing perspectives.

To conclude, integrating skyrmions into existing technological infrastructure
is feasible, as the dynamics of skyrmions can be manipulated by electric
currents, with current-induced spin torques being a major factor. While
significant progress has been made, further research is necessary to identify
more affordable and durable material systems that host stable skyrmions.
Furthermore, it will be crucial to develop readout methods that are both
efficient and cost-effective and can merge seamlessly with I/O components of
electronic systems.

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## Acknowledgments

O.L. and H.K. thank the Leverhulme Trust for financial support via
RPG-2016-391. M.K., K.E-S, R.M. and M.B. are grateful to the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) for funding this
research: Project numbers #403502522 and #403233384-SPP 2137 Skyrmionics and
#320163632 (Emmy Noether) and acknowledge funding from the Emergent AI Centre
funded by the Carl-Zeiss-Stiftung. M.K., K.E.S and M.B. also acknowledge
funding from TopDyn and SFB TRR 173 Spin+X (project A01 #268565370 and project
B12 #268565370. M.K., and M.B. further acknowledge funding from the Horizon
2020 Framework Programme of the European Commission under FET-Open Grant No.
863155 (s-Nebula), Grant No. 856538 (ERC-SyG 3D MAGIC), No. 101070290
(NIMFEIA) and under the Marie Skłodowska-Curie Grant Agreement No. 860060
“Magnetism and the effect of Electric Field” (MagnEFi). M.B. is supported by a
doctoral scholarship of the Studienstiftung des deutschen Volkes and
acknowledges funding from TRR 146 (project #233630050).


# Dynamics of large samples of repulsive Fermi gases at nonzero temperatures

Jarosław Ryszkiewicz, Mirosław Brewczyk, and Tomasz Karpiuk  Wydział Fizyki,
Uniwersytet w Białymstoku, ul. K. Ciołkowskiego 1L, 15-245 Białystok, Poland

###### Abstract

We develop a model of a binary fermionic mixture, consisting of large number
of atoms, applicable at nonzero temperatures, in the normal phase. We use this
approach to study dynamics of degenerate Fermi systems under various
perturbations. For example, we analyze spin-dipole oscillations of a two-
component fermionic mixture, demonstrating that the ferromagnetic phase shows
up at stronger repulsion between components while the temperature raises. We
study as well the radial oscillations of weakly interacting repulsive Fermi
gases. We obtain a good agreement with experimental data when available.
Otherwise, we compare our results with the outcome of the Hartree-Fock orbital
calculations done for the system with small number of fermions.

##  I Introduction

Systems of ultracold fermionic atoms have been already studied for years, both
experimentally and theoretically. Since first experimental achievement of
quantum degeneracy in fermionic potassium gas DeMarco99 , followed by
successful attempts to cooling other elements Truscott01 ; Schreck01 ;
Granade02 ; Hadzibabic03 , the interest in cold fermionic gases has quickly
increased, covering broad range of quantum many-body phenomena including
thermodynamic and transport related effects at unitarity Navon10 ; Ku12 ;
Cao11 , correlations, in particular in optical lattices Rom06 ; Greif13 ;
Kondov15 ; Schreiber15 ; Hart15 ; Cheuk16 ; Salomon19 ; Chiu19  or near
Feshbach resonances Greiner05 ; Sanner12 ; Amico18 , strongly interacting
gases in lower dimensions Vogt12 ; Koschorreck13 ; Luciuk17 ; Murthy19 ;
Luick20 ; Bohlen20 , or dipolar gases Lu12 ; Aikawa14 ; Frisch14 ; Burdick16 .

In particular, dynamics of fermionic gases has been thoroughly investigated.
The measurement of collective mode frequencies and damping rates as a function
of temperature supplied evidences for superfluid behavior of a Fermi gas
Kinast04 , while an observation of a vortex lattice Zwierlein05  provided
direct verification for the superfluidity. By exciting hydrodynamic modes,
such as collective oscillations Kinast04 ; Bartenstein04 ; Altmeyer07 , sound
Joseph07 , or rotational modes Clancy07 , transport properties of a unitary
Fermi gas have been experimentally determined. Careful analysis of
compression, quadrupole, and scissors modes in the unitarity limit in the
range of temperatures above the critical temperature for superfluidity has
been performed in Riedl08  and revealed transition from hydrodynamic to
collisionless behavior with increase of temperature. Recently, oscillations of
repulsive binary fermionic mixture Sommer11 ; Valtolina17 , initially phase-
separated by domain wall, were studied experimentally in connection with the
long-standing problem of Stoner instability Stoner33 .

In this paper we investigate finite temperature dynamics of Fermi-Fermi
mixtures by using the density-functional like description. To derive the
equations of motion we start with introducing the semi-classical distribution
function for fermions. Then we evoke the Kohn and Sham Kohn65  way of treating
nonzero temperatures case within the density-functional methods and replace
the local kinetic energy expression by the one corresponding to the free
energy. Next we switch to the quantum hydrodynamic description Madelung27  of
the system, apply the inverse Madelung transformation Dey98 ; Domps98 ;
Grochowski17 , and follow Dirac prescription Dirac30  to get desired
equations.

The paper is then organized as follows. First, we present the model of a two-
component Fermi gas in the normal phase capable to retrieve dynamics when the
number of atoms is large (Section II). To prove the effectiveness of our model
we compare numerical results to experimental data on dynamics of fermionic
systems in the case of spin-dipole modes Valtolina17  (Section III) and to the
outcome of the Hartree-Fock orbital calculations in the case of radial
oscillations of weakly interacting repulsive Fermi gas (Section IV). We
conclude in Section V.

##  II Equations of motion

A simple description of a one-component gas in terms of a semi-classical
distribution function $f_{\bf{p}}({\bf{r}})$ assumes that
$f_{\bf{p}}({\bf{r}})d{\bf{r}}d{\bf{p}}/(2\pi\hbar)^{3}$ gives the mean number
of particles in the phase-space volume element $d{\bf{r}}d{\bf{p}}$. At
equilibrium, at a given temperature $T$ and a chemical potential $\mu$, one
has for a degenerate Fermi gas

$\displaystyle
f_{\bf{p}}({\bf{r}})=\frac{1}{e^{[\varepsilon_{\bf{p}}({\bf{r}})-\mu]/k_{B}T}+1}$
(1)

with $\varepsilon_{\bf{p}}({\bf{r}})$ being the particle energy at position
${\bf{r}}$. For a single component ideal Fermi gas in a trap this energy
becomes

$\displaystyle\varepsilon_{\bf{p}}({\bf{r}})=\frac{{\bf{p}}^{2}}{2m}+V_{tr}({\bf{r}})\,.$
(2)

The density of particles is obtained by integrating the distribution function
over all momenta $n({\bf{r}})\sim\int f_{\bf{p}}({\bf{r}})d{\bf{p}}$. Other
kinds of energies can be added to Eq. (2), in particular the one related to
the Weizsäcker correction
$E_{W}=\xi(T)\,(\hbar^{2}/2m)\int(\nabla\sqrt{n({\bf{r}})})^{2}\,d{\bf{r}}$,
with a weakly temperature dependent coefficient $\xi(T)$ Perrot79 . Now, when
the other fermionic component comes to the scene, the interaction energy has
to be included as well. Assuming the inter-component interactions depend on
densities only $V_{int}(n_{+},n_{-})$ (hereafter the components are
distinguished by indices ’$+$’ and ’$-$’), Eq. (2) becomes

$\displaystyle\varepsilon_{\bf{p}}({\bf{r}})=\frac{{\bf{p}}^{2}}{2m}+V_{tr}({\bf{r}})+\frac{\delta
E_{W}^{+}}{\delta n_{+}}+\frac{\delta V_{int}}{\delta n_{+}}\,.$ (3)

The density of particles of ’$+$’ component is calculated as

$\displaystyle n_{+}({\bf{r}})$ $\displaystyle=$
$\displaystyle\int\frac{1}{e^{[\varepsilon_{\bf{p}}({\bf{r}})-\mu_{+}]/k_{B}T}+1}\,\frac{d{\bf{p}}}{(2\pi\hbar)^{3}}$
(4) $\displaystyle=$ $\displaystyle\frac{1}{\lambda^{3}}f_{3/2}(z_{+})$

and the energy density related to the local motion as

$\displaystyle\varepsilon_{+}({\bf{r}})$ $\displaystyle=$
$\displaystyle\int\frac{{\bf{p}}^{2}/2m}{e^{[\varepsilon_{\bf{p}}({\bf{r}})-\mu_{+}]/k_{B}T}+1}\,\frac{d{\bf{p}}}{(2\pi\hbar)^{3}}$
(5) $\displaystyle=$
$\displaystyle\frac{3}{2}\frac{k_{B}T}{\lambda^{3}}f_{5/2}(z_{+})\,,$

where $k_{B}$ is the Boltzmann constant,
$\lambda=\sqrt{2\pi\hbar^{2}/mk_{B}T}$ is the thermal wavelength, and
$f_{3/2}(z)$ and $f_{5/2}(z)$ are the standard functions for fermions Huang .
The ’extended fugacity’ equals

$\displaystyle z_{+}({\bf{r}})=e^{(\mu_{+}-V_{tr}({\bf{r}})-\delta
E_{W}^{+}/\delta n_{+}-\delta V_{int}/\delta n_{+})/k_{B}T}\,.$ (6)

The chemical potential $\mu_{+}$ is determined by the normalization condition
$N_{+}=\int n_{+}({\bf{r}})d{\bf{r}}$.

According to the Kohn and Sham proposition Kohn65  on a generalization of the
density-functional formalism to finite temperature case, for further analysis
the energy of the system should be replaced by its free energy, whose density
is

$\displaystyle
f_{+}({\bf{r}})=\frac{k_{B}T}{\lambda^{3}}\left[(\ln{z_{+}})\,f_{3/2}(z_{+})-f_{5/2}(z_{+})\right]\,.$
(7)

Additionally, in a dynamical case the energy functional has to be modified by
adding the energy of a macroscopic flow. Then the part of the functional
related to ’$+$’ component, which is minimized to get the equations underlying
the system’s dynamics, becomes

$\displaystyle F_{+}(n_{+},{\bf{v}}_{+})$ $\displaystyle=$ $\displaystyle\int
f_{+}({\bf{r}})\,d{\bf{r}}+\int n_{+}\frac{1}{2}m{\bf{v}}_{+}^{2}d{\bf{r}}$
(8) $\displaystyle+$ $\displaystyle\int
V_{tr}({\bf{r}})\,n_{+}\,d{\bf{r}}+E_{W}+V_{int}\,.$

Here, the ${\bf{v}}_{+}({\bf{r}})$ is the velocity field of a macroscopic flow
of ’$+$’ fermionic component and the second term on the right-hand side
represents the energy of such motion.

Now we introduce the pseudo-wave function $\psi_{+}({\bf{r}})$
($n_{+}=|\psi_{+}|^{2}$) for ’$+$’ component in such a way that

$\displaystyle\frac{\hbar^{2}}{2m}(\nabla\psi_{+}^{*})\,(\nabla\psi_{+})=\frac{\hbar^{2}}{2m}(\nabla|\psi_{+}|)^{2}+n_{+}\frac{1}{2}\,m\,{\rm
v}_{+}^{2}\,.$ (9)

The functional Eq. (8) is then transformed to

$\displaystyle F_{+}(\psi_{+},\nabla\psi_{+})=\int
f_{+}({\bf{r}})\,d{\bf{r}}+\int\Big{(}-\frac{\hbar^{2}}{2m}\psi_{+}^{*}\nabla^{2}\psi_{+}\Big{)}\,d{\bf{r}}$
$\displaystyle-\frac{\hbar^{2}}{2m}\int(\nabla|\psi_{+}|)^{2}d{\bf{r}}+\int
V_{tr}({\bf{r}})\,n_{+}\,d{\bf{r}}+E_{W}+V_{int}\,.$ (10)

Similar functional applies to the second component. The equations of motion
are

$\displaystyle i\hbar\,\frac{\partial}{\partial t}\psi_{\pm}({\bf
r},t)=\frac{\delta}{\delta\psi_{\pm}^{*}}F_{\pm}[\psi_{\pm},\nabla\psi_{\pm}]\,.$
(11)

Since

$\displaystyle\frac{\delta}{\delta n_{\pm}}\int
f_{\pm}({\bf{r}})\,d{\bf{r}}=k_{B}T\,\ln{z_{\pm}}\,,$ (12)

the equations of motion become

$\displaystyle i\hbar\,\frac{\partial\psi_{\pm}}{\partial t}$ $\displaystyle=$
$\displaystyle\left(-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{\hbar^{2}}{2m}\frac{\nabla^{2}|\psi_{\pm}|}{|\psi_{\pm}|}+k_{B}T\,\ln{z_{\pm}}\right.$
$\displaystyle+$
$\displaystyle\left.\,V_{tr}-\xi(T)\,\frac{\hbar^{2}}{2m}\frac{\nabla^{2}\sqrt{n_{\pm}}}{\sqrt{n_{\pm}}}+\frac{\delta
V_{int}}{\delta n_{\pm}}\right)\psi_{\pm}\,.$

While solving Eqs. (LABEL:equmotion), the extended fugacities
$z_{\pm}({\bf{r}})$ are found from the self-consistency condition
$f_{3/2}(z_{\pm})=\lambda^{3}\,n_{\pm}$, Eq. (4), with
$n_{\pm}=|\psi_{\pm}|^{2}$. The Weizsäcker correction (the one before the last
one) becomes less important when the number of atoms increases.

##  III Spin-dipole oscillations

We first examine our model in the case of experiment on spin-dipole
oscillations of repulsive two-component fermionic mixtures Valtolina17 . This
experiment was aimed to prove the existence of the phase transition from
paramagnetic to ferromagnetic phase in a system of two-component short-range
repulsive Fermi gas, i.e., to resolve the long-standing hypothesis of
itinerant ferromagnetism posed by E. Stoner Stoner33 . It was predicted in
Stoner33  that not localized electrons get into ferromagnetic state when
short-range repulsion between opposite spin electrons becomes large enough to
beat the Fermi pressure.

To minimize the effect of pairing phenomenon Sanner12 ; Amico18 , in the
experiment of Valtolina17  a mixture of 6Li atoms was prepared in a special
state, in which both components were spatially separated. It was realized in
two steps. First, components held in a prolate harmonic trap were spatially
separated by using a magnetic field gradient. Next, when the overlap of two
components was small enough, the optical repulsive barrier separating clouds
was switched on and the magnetic field gradient turned off. Then the optical
barrier was suddenly removed and the spin dynamics, i.e. oscillations of
centers of mass of each component, was studied. Both frequencies and damping
rates were measured, which demonstrated the existence of the critical
repulsion between components. For weak repulsion the effect of softening of
the spin-dipole mode was observed, i.e., the frequency of oscillations was
continuously decreasing with a strength of a repulsion. Simultaneously, both
atomic clouds were passing through each other. For stronger repulsion,
however, qualitatively different behavior was found. Two atomic clouds started
to bounce off each other with frequency higher than the axial trap frequency.

Figure 1: Frequencies of the spin-dipole mode of a repulsive two-component
Fermi gas plotted as a function of $k_{F}a$ for the temperatures
$T/T_{F}=0.12$ (upper frame) and $T/T_{F}=0.25$ (lower frame) – a comparison
with the experiment of Ref. Valtolina17 . Simulations were performed for the
system with number of atoms equal to $N/2=50000$ (as in the experiment), here
the Weizsäcker correction in Eq. (LABEL:equmotion) can be safely neglected,
and $N/2=1000$.

To model the experiment of Ref. Valtolina17  with Eqs. (LABEL:equmotion), we
first obtain the initial state of two-component Fermi gas by solving Eqs.
(LABEL:equmotion) by imaginary time technique Gawryluk18  at the presence of a
cigar-shaped harmonic trap with radial and axial frequencies equal to
$\omega_{\perp}=2\pi\times 265\,$Hz and $\omega_{z}=2\pi\times 21\,$Hz,
respectively, and at the presence of the repulsive optical barrier. Then we
remove the barrier and monitor the dynamics of the system by calculating the
relative distance $d(t)$ between centers of two spin clouds. Analyzing $d(t)$
as a function of time we extract both the frequency and the damping rate of
the mode (see Ref. Ryszkiewicz20  for details).

To get an agreement with experimental data we must include in the model the
many-body correlations due to interactions. This can be achieved by
renormalizing the coupling constant in the two-body contact potential
Stecher07 ; Grochowski17 ; Ryszkiewicz20 . For a uniform system, it is done in
a way to get correct low-density expansion, in parameter $\tilde{k}_{F}a$, of
an energy of a two-component Fermi gas, see Eq. (3) in Ref. Ryszkiewicz20 .
Here, $\tilde{k}_{F}$ is the Fermi wave number and $a$ is the $s$-wave
scattering length. For a trapped gas a local density approximation is used. In
the mean-field approximation the interaction energy density is $gn_{+}n_{-}$,
with $g=4\pi\hbar^{2}a/m$, and the term $\delta V_{int}/\delta
n_{\pm}=gn_{\mp}$ appears in Eqs. (LABEL:equmotion). After renormalization,
$gn_{\pm}$ term is replaced by
$gn_{\pm}+A(4/3\,n_{\mp}^{1/3}\,n_{\pm}+n_{\pm}^{4/3})+B(5/3\,n_{\mp}^{2/3}\,n_{\pm}+n_{\pm}^{5/3})$
with $A=3ga(6\pi^{2})^{1/3}(11-2\ln{2})/35\pi$ and
$B=3ga^{2}(6\pi^{2})^{2/3}\pi/4\times 0.23$ Grochowski17 .

In Fig. 1 we show numerical results for frequencies of the spin-dipole mode
for two temperatures studied experimentally in Valtolina17  (see Fig. 2). In
the upper frame additional experimental point (most right) is included, see
Fig. 3 c in Ref. Valtolina17 . Simulations were performed for the system both
with the number of atoms as in the experiment ($50$ thousand of atoms in each
component) and much smaller ($N/2=1000$). Fig. 1 proves overall agreement
between numerics and experimental data. Our calculations reveal softening of
the spin-dipole mode followed by the transition from the paramagnetic to
ferromagnetic phase. The hydrodynamic model, we developed, gives correct value
of $k_{F}a$ (here, $k_{F}=(24N)^{1/6}/(\hbar/m\,\overline{\omega})^{1/2}$,
where $\overline{\omega}$ is the geometric mean of trap frequencies in all
directions) at which this transition occurs. It is already well understood
that at zero temperature the softening phenomenon depends solely on the
combination $k_{F}a$ Recati11 . Our simulations support this observation also
for nonzero temperatures, see Fig. 1. According to Stoner’s model the
transition to ferromagnetic phase depends on $k_{F}a$ only as well, which
again is exhibited by our simulations.

Above the critical value of $k_{F}a$ both components stop to penetrate each
other and oscillate with frequencies smaller than twice the axial trap
frequency, in agreement with experiment. Numerical results seem to be
consistent for two considered numbers of atoms. Our observation is, however,
that the size of the intermediate regime (the one between the paramagnetic and
ferromagnetic regimes) differs depending on the number of atoms in the sample.
For larger systems the large-$(k_{F}a)$ value of the oscillation frequency is
reached faster, i.e., for smaller $k_{F}a$. This can be understood as follows.
For small samples ($N=1000$ in our case) even after crossing the critical
value of $k_{F}a$ we can still observe the gas transmission through the
intercomponent interface on the perimeter. Hence, in the intermediate regime
the flow is partially still miscible. The full transition into the immiscible
regime (i.e., when the oscillation frequency takes its large-$(k_{F}a)$ value)
is then shifted to stronger interactions. This transition occurs faster (in
terms of $k_{F}a$) for larger systems. It happens because, first, the damping
rates for spin-dipole oscillations are high for the values of repulsion
strengths $k_{F}a$ close to the critical one (see Fig. 3 in Ref. Ryszkiewicz20
), and, second, these rates are bigger for systems with larger number of
atoms. Since damping rates decrease with temperature, the intermediate region
broadens with temperature.

Figure 2: Upper frame: Frequencies of the spin-dipole mode of a repulsive two-
component Fermi gas plotted as a function of $k_{F}a$ for different
temperatures, up to $T/T_{F}=0.53$. Lower frame: Phase diagram showing the
critical value of the repulsive interaction strength at a given temperature.
The red crosses are the experimental data taken from Ref. Valtolina17  (Fig. 3
d), while blue bullets come from numerics. Solid line, which is a power-law
fit to the numerical points, separates the paramagnetic (white area) from the
ferromagnetic (dark area) phase. Blue squares and red diamonds are the
predictions of static Stoner’s model assuming interactions between atoms are
not and are renormalized, respectively.

Fig. 2 summarizes hydrodynamic calculations performed for smaller atomic
samples (Fig. 1 demonstrates that the transition to ferromagnetic phase occurs
actually at the same value of $k_{F}a$, independently of the number of atoms).
In the upper frame we show the frequencies of the spin-dipole mode for
additional two temperatures, $T/T_{F}=0.4$ and $T/T_{F}=0.53$, for the system
consisting of $1000$ atoms in each component. These temperatures were studied
in Valtolina17 , although in a different way – not by following the spin-
dipole oscillations but by analyzing the stability of initially created spin
domains. The lower frame is the phase diagram, it gathers the results
corresponding to transition between the paramagnetic and ferromagnetic phases.
This diagram shows the critical value of $k_{F}a$ as a function of
temperature. The experimental data are marked as red crosses, taken from Ref.
Valtolina17  (Fig. 3 d), while numerical results are put as blue bullets. The
solid line is a power-law fit to the numerical points and separates the
paramagnetic and ferromagnetic phases. At a given value of $k_{F}a$ the
ferromagnetic phase is entered while the system’s temperature is decreased, in
qualitative agreement with Stoner’s model of itinerant ferromagnetism – the
ferromagnetic phase rises when interactions are able to overcome the fermionic
quantum pressure, which gets lower with decreasing temperature.

To check on a quantitative level our numerical (dynamic) results versus
predictions of original (static) Stoner’s model we compare the kinetic energy
of the gas to its interaction energy at equilibrium Zwerger09 . In the
simplest case, i.e., in the mean-field approximation the interaction energy is
$g\int n_{+}n_{-}\,d{\bf r}$ and the above mentioned comparison reads

$\displaystyle\frac{3}{2}\frac{k_{B}T}{\lambda^{3}}\int f_{5/2}(z_{+})\,d{\bf
r}=\frac{4\pi\hbar^{2}}{m\,k_{F}}\left(\int n_{+}n_{-}\,d{\bf
r}\right)\,(k_{F}a)_{cr}\,.$ (14)

The critical value of repulsive interactions $(k_{F}a)_{cr}$ is found assuming
equal component densities $n_{+}=n_{-}=f_{3/2}(z_{+})/\lambda^{3}$ normalized
to $N_{+}=N_{-}=1000$ with $z_{+}=\exp{[(\mu_{+}-V_{tr})/k_{B}T]}$. The
critical values $(k_{F}a)_{cr}$ as a function of $T/T_{F}$ are plotted in Fig.
2, lower frame, as blue squares. At zero temperature $(k_{F}a)_{cr}\approx
1.7$ in agreement with our earlier calculations Grochowski17 , see Fig. 1d.
When interactions between atoms get renormalized, the condition (14) changes
into the third degree polynomial equation for the critical interactions
$(k_{F}a)_{cr}$. The solutions as a function of $T/T_{F}$ are shown in Fig. 2,
lower frame, as red diamonds. Now, at zero temperature $(k_{F}a)_{cr}\approx
0.9$, in agreement with experimental data of Ref. Valtolina17  at the lowest
temperature. Overall behavior of $(k_{F}a)_{cr}$ resembles the one determined
experimentally and obtained in numerical simulations (both representing the
dynamical Stoner effect), especially for lower temperatures.

A note regarding consistency of presented results with those reported already
in Refs. Grochowski17 ; Ryszkiewicz20  is now in order. First, the frequency
of the spin-dipole mode in a ferromagnetic phase strongly depends on the
geometry of the trapping potential, while a density functional method is used.
In elongated trap, as in the experiment of Ref. Valtolina17 , it is about
$1.7$ (see Fig. 2, upper frame). At zero temperature and in a spherically
symmetric trap this frequency approaches the value of twice the trap frequency
Grochowski17 , with recognizable admixture of other frequency
$(\omega_{SD}/\omega_{z}=\sqrt{2})$. On the other hand, within the Hartree-
Fock approach the spin-dipole mode frequency in a ferromagnetic phase remains
twice the axial frequency, independently of temperature Ryszkiewicz20 . This
probably happens because our treatment of Hartree-Fock dynamics at nonzero
temperatures does not allow for atoms to change between single-particle
orbitals during the evolution. In our case only one-particle orbitals change
in time during dynamics, not the populations – populations are chosen by using
a Monte Carlo sampling technique, before the barrier separating components is
removed (for an approach in which populations are treated on the same way as
orbitals, although at equilibrium only, see Lipparini ).

##  IV Oscillations of weakly interacting repulsive fermionic mixtures

In this section we carry out simulations of dynamics of two-component weakly
interacting Fermi gas, initially confined in a spherically symmetric trap,
after a weak disturbance of the trapping potential. Within a weak-driving
regime the system’s response can be treated analytically in the range of high
temperatures and in the limit of an ideal gas. Here, we are verifying our
description of large Fermi systems by studying the monopole oscillations of a
Fermi gas for weak repulsive interactions. As in Section III, we use
renormalized interaction to describe two-component weakly interacting Fermi
gas.

Both gases are perturbed in-phase, i.e., two atomic clouds are first
simultaneously being squeezed as an effect of increasing trap frequencies and
next the trapping potential is slightly attenuated (by decreasing trap
frequencies) to allow the gas to expand. Such a cycle is repeated a few times
after which the system starts to oscillate in a trap. In this way the
spherically symmetric oscillations are excited. We find frequencies of such
excitations by calculating the width of an atomic cloud $\int d^{3}{\bf
r}\,r^{2}n_{\pm}({\bf r},t)$ and analyzing its time dependence. We show
frequencies of in-phase monopole modes as a function of temperature in Fig. 3,
limiting ourselves only to the paramagnetic range of parameters Karpiuk20 ;
Trappe16 ; Trappe21 . At zero temperature (open circles data) our results
perfectly match those obtained within time-dependent Hartree-Fock method (see
Ref. Karpiuk20 ). In the limit of no interaction between components, the
spherically symmetric mode oscillates with frequency $2\omega_{0}$, where
$\omega_{0}$ is the trap frequency. While moving towards the paramagnetic-
ferromagnetic phase crossing this frequency increases to about
$2.2\omega_{0}$. For higher temperatures all mode frequencies are shifted down
and in the limit of $T\gtrsim T_{F}$ can be studied analytically.

Figure 3: Upper frame: Frequencies of in-phase monopole modes as a function of
temperature. The critical interaction for which the phase separation occurs
depends on temperature. Only frequencies of modes excited in the paramagnetic
phase are shown. Both components consist of $1000$ atoms. Lower frame: The
monopole mode frequency as a function of temperature in the limit of no
interaction between components, clearly approaching $\sqrt{2}\,\omega_{0}$ for
high temperatures.

To analyze small in-phase oscillations of two-component interacting Fermi gas
at high temperature limit we utilize the Madelung representation Madelung27
of Eqs. (LABEL:equmotion). For the ’$+$’ component this representation is
expressed as a set of equations for the density and the velocity fields:

$\displaystyle\frac{\partial n_{+}}{\partial t}+\nabla\cdot(n_{+}{\bf
v}_{+})=0$ $\displaystyle m\frac{\partial{\bf v}_{+}}{\partial
t}+\nabla\Big{(}k_{B}T\,\ln{z_{+}}+V_{tr}+\frac{\delta V_{int}}{\delta
n_{+}}+\frac{1}{2}m{\bf v}_{+}^{2}\Big{)}=0\,.$ (15)

At high temperature limit one has $n_{+}\lambda^{3}=f_{3/2}(z_{+})\approx
z_{+}$. Small oscillations are investigated by assuming small deviations of
the state from the equilibrium and by looking for periodic solutions for
deviations. We write the density as $n_{+}=n_{eq}^{+}+\delta n_{+}$, where
$\delta n_{+}$ is the departure from the equilibrium density and assume that
both the velocity and $\delta n_{+}$ are small quantities. Since
$n_{eq}^{+}=\exp{[(\mu_{+}-V_{tr}-(\delta V_{int}/\delta
n_{+})_{eq})/k_{B}T]}/\lambda^{3}$ and $\delta n_{+}=\delta n_{-}$ (in-phase
oscillations), the Eqs. (15) are transformed to

$\displaystyle\frac{\partial}{\partial t}\delta n_{+}$ $\displaystyle=$
$\displaystyle-\nabla\cdot(n_{eq}^{+}\,{\bf v}_{+})$ $\displaystyle
m\frac{\partial{\bf v}_{+}}{\partial t}$ $\displaystyle=$
$\displaystyle-\nabla\left[k_{B}T\,\frac{\delta
n_{+}}{n_{eq}^{+}}+G(n_{eq}^{+})\,\delta n_{+}\right]\,,$ (16)

where

$\displaystyle G(n_{eq}^{+})=\left[\left(\frac{\partial}{\partial
n_{+}}+\frac{\partial}{\partial n_{-}}\right)\frac{\delta V_{int}}{\delta
n_{+}}\right]_{n_{+}=n_{-}=n_{eq}^{+}}\,,$ (17)

and can be combined into a single equation for the density deviation

$\displaystyle m\frac{\partial^{2}}{\partial t^{2}}\delta
n_{+}=\nabla\cdot\left[n_{eq}^{+}\,\nabla\left(k_{B}T\,\frac{\delta
n_{+}}{n_{eq}^{+}}+G(n_{eq}^{+})\,\delta n_{+}\right)\right]\,.$ (18)

Now, the limit of small interactions can be analyzed. For weak inter-species
interactions the second term in the right-hand side of Eq. (18) is neglected
and $n_{eq}^{+}=\exp{[(\mu_{+}-V_{tr})/k_{B}T]}/\lambda^{3}$. Eq. (18) can be
rewritten as

$\displaystyle m\frac{\partial^{2}}{\partial t^{2}}\left(\frac{\delta
n_{+}}{n_{eq}^{+}}\right)=k_{B}T\,\nabla^{2}\left(\frac{\delta
n_{+}}{n_{eq}^{+}}\right)-(\nabla V_{tr})\nabla\left(\frac{\delta
n_{+}}{n_{eq}^{+}}\right)\,.$ (19)

The trapping potential is spherically symmetric,
$V_{tr}=m\,\omega_{0}^{2}\,r^{2}/2$, and we search for periodic solutions
$\delta n_{+}/n_{eq}^{+}\sim e^{-i\omega t}$ of Eq. (19) which are spherically
symmetric as well. The solutions can be found by using the power series
method. The lowest energy mode has a frequency $\omega=\sqrt{2}\,\omega_{0}$,
marked by a horizontal solid line in Fig. 3, lower frame. The mode itself is
$\delta n_{+}\sim(1-m\omega^{2}r^{2}/(6k_{B}T))\,n_{eq}^{+}$.

##  V Conclusions

In summary, we have studied dynamics of mixtures of repulsive Fermi gases
consisting of large number of atoms at nonzero temperatures. We find a
quantitative agreement with experimental results of Valtolina17  on spin-
dipole oscillations. The calculations show the dependence of the critical
repulsion $k_{F}a$ on the temperature. The transition to the ferromagnetic
phase requires larger value of $k_{F}a$ with increasing temperature, in
agreement with Stoner’s picture of itinerant ferromagnetism. We also model
breathing modes of weakly interacting repulsive fermionic mixtures, getting
agreement with low temperature results of Karpiuk20  and showing decrease of
oscillation frequencies with increase of temperature.

###### Acknowledgements.

The authors acknowledge support from the (Polish) National Science Center
Grant No. 2018/29/B/ST2/01308. Part of the results were obtained using
computers at the Computer Center of University of Bialystok.

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