# Anisotropic MagnetoMemristance

$\ {}^{1}$F. Caravelli, $\ {}^{2}$ E. Iacocca , $\ {}^{3}$G.-W. Chern,$\
{}^{1}$C. Nisoli,$\ {}^{4}$C. I. L. de Araujo

$\ {}^{1}$ Theoretical Division (T4), Los Alamos National Laboratory, Los
Alamos, New Mexico 87545, USA
$\ {}^{2}$Center for Magnetism and Magnetic Materials, University of Colorado,
Colorado Springs, CO 80918, USA
$\ {}^{3}$Department of Physics, University of Virginia, Charlottesville, VA
22904, USA
$\ {}^{4}$Departamento de Fìsica, Universidade Federal de Viçosa, Viçosa,
36570-900, Minas Gerais, Brazil

###### Abstract

The development of efficient beyond-von Neumann, bio-inspired, or
unconventional computing hinges on the realization of novel devices that can
be integrated into traditional circuitry. For this reason, memristive devices
are an interesting option. Magnetic materials are promising in this regard
because of their integration with CMOS and relatively simple device
production. However, memristance in magnetic materials has been limited to
quasi-static cases that prohibit their operation at GHz frequencies. Here, we
analytically and numerically show that a single ferromagnetic layer can
possess GHz memristance, due to a combination of two factors: a current-
induced transfer of angular momentum (Zhang-Li torque) and the anisotropic
magnetoresistance (AMR). We term the resulting effect the anisotropic
magnetomemristance (AMM). We show that AMM is of infinite order, because it
depends on an infinite number of internal states associated with the
magnetization vector field. We connect the AMM to the topology of the
magnetization state, within a simple model of a 1-dimensional ring-shaped
magnetic layer. We then confirm the analytical results with micromagnetic
simulations on a permalloy ring that includes nonlocal dipole fields. We show
that for topological magnetic states the memristance effect could be
measurable at the nanoVolt scale. Our results open a new path towards the
realization of single-layer magnetic memristive devices operating at GHz
frequencies and harnessing topological protection for unconventional
computing. From this perspective, the device is ideal for neuromorphic Random
Access Memory (nRAM).

## Introduction

A memristive device is 1-port electrical component that satisfies Ohm’s law,
$V=R\ I$, and moreover has a dynamical resistance of the form
$\frac{dR}{dt}=f(R,I)$ [1, 2, 3]. An ideal memristor, in which the resistance
depends only on the charge $q=\int I(t)dt$ has not been realized yet. Instead,
any generic electrical device that exhibits a pinched hysteresis loop in the
current-voltage diagram, whether the memory is stored for long or short term,
is currently regarded as a memristive device [4].

From an immediate practical perspective, memristive devices are interesting
because they can be used as a memory. More broadly, they are studied for their
collective behavior and applications in neuromorphic engineering [5]. For
instance, the Strukov-Williams memristor, initially identified while studying
titanium dioxide [6], can be roughly approximated, far from the resistive
boundaries, by the functional form 111Here we use an explicit equation for the
resistance, but this is equivalent to a first-order memristor model with
$R(w)=(1-w)R_{\mathrm{off}}+wR_{\mathrm{on}}$ and $\frac{dw}{dt}=\alpha
w-\beta J$. Note that if $\alpha=0$, this is an ideal memristor.

$\frac{dR}{dt}=\beta J-\sigma R+\eta,\quad R_{\mathrm{on}}\leq R(t)\leq
R_{\mathrm{off}},$ (1)

where $J$ is typically the total current density flowing into the device,
while $\beta$, $\sigma$, and $\eta$ are constants which need to be determined
experimentally. The equation above describes a resistive device whose
resistance $R$ is constrained between two values
$R_{\mathrm{off}}>R>R_{\mathrm{on}}$. In general, close to the boundaries
$R_{\mathrm{off}}$ and $R_{\mathrm{on}}$, eqn. (1) loses its validity.
Nonetheless, in a memristive device the history of its resistance is not
uniquely determined by the applied voltage, but depends also on the device
internal state.222Another way of expressing the Strukov-Williams memristor is
via the introduction of a dynamical internal parameter, which is however not
physically measurable.

The use of memory effects in resistive materials has been suggested for a
variety of computing applications ranging from logical gates [7, 8, 9, 10] to
unconventional computing [11, 12, 13]; the long term view is that resistive
type of memories can perform a variety of tasks ranging from Boolean
computation to machine learning [5, 2, 3]. Within the context of purely
electronic memristive devices, their application has focused on hybrid analog-
CMOS devices such as crossbar arrays, in which the memristive effect is
exploited as memory devices (switches). These are often used in neuromorphic
applications via a field-programmable gate array (FPGA) controller, ranging
from neural networks to optimization [5]. Such architectures however do not
harness the full analog properties of memristive devices, being used as an
independent memory unit, with the advantage of being controllable with a
single bus. There is a general interest in harnessing the analog power of
memristor devices because they exhibit emergent behavior in complex
interconnected circuits [5]. For instance, network of memristors [12, 13] can
be used for reservoir computing [14] and emergent phenomena such as symmetry
breaking and tunneling [15, 16].

This work introduces a magnetic memristor which, due to the absence of a
tunnel-junction, leads to a smooth response to current, thus making it
suitable for analog use. Interestingly, as we discuss in this paper, any soft
magnetic material that exhibits anisosotropic magnetoresistance, if the the
spin transfer torque is non-negligible, should exhibit a memristance. The
effect is due to the spin-transfer torque [17], and can access the intrinsic
magnetization precession frequencies in the GHz regime [18, 19]. Having a GHz
memristive device implies that these can operate at frequencies compatible
with current CPU speeds, and have the potential to be used symbiotically in
novel neuromorphic-von Neumann computer architectures.

Different types of memristive devices use magnetic materials and rely on
magnetoresistance effects. An initial approach was based on the giant
magnetoresistance (GMR) in magnetic tunnel junctions [20], relying on current-
induced domain wall motion. Moreover, such a resistance was shown to depend on
differently accessed minor loops, offering a robust multi-level resistive
device. More recently, it has been observed that current-induced domain-wall
formation in connected Kagomé artificial spin ices [21, 22, 23] may give rise
to memristive effects.

The use of magnetic materials as a memristive functional element has led to
the development of the rapidly growing field of bio-inspired and
unconventional computing. Applications and device design include neuromorphic
computing with MTJ-oscillators [24, 25]; reservoir computing via domain walls
in arrays of magnetic rings inducing “fading memory” [26], superparamagnetic
ensembles [27], and magnetic topological objects [28, 29]; and inverse-design
or magnonic neural networks [30, 31].

These applications, however, do not rely on a memristive effect as defined in
Eq. (1). On this regard, recent evidence for a memristive effect has been
presented for exchange-biased bilayers with mHz field sweeps relying on
“viscous” dynamics [32] and for a rather exotic spin-glass heterostructure
driven by spin-transfer torque in MHz regime [33]. While conforming more
closely with Eq. (1), these devices do not achieve GHz memristive effects,
thus preventing their integration with current computer architectures.

A naïve expectation is that the intrinsic precessional GHz frequencies of
trivial magnetic materials provides a natural environment for memristive
effects. Such frequencies can be practically accessed by compensating the
magnetic damping with spin-transfer torques [34], e.g., using spin-torque
nano-oscillators [24]. It is therefore desirable to fully describe memristive
effects originating in a single magnetic material due to current. This is
further advantageous from the point of view of scalability and operation speed
of fully analog 1-port devices [5].

Here, we analytically demonstrate that a single, trivial ferromagnet exhibits
memristive effects at GHz frequencies when an in-plane current both exerts
torque via the Zhang-Li mechanism [35] and induces anisotropic
magnetoresistance, a combined effect we term anisotropic magnetomemristance
(AMM). By considering a simple toy model of a 1D ferromagnetic ring, we
analytically identify that the memristive effect is enabled by homochiral
textures that exhibit a nontrivial topology, akin to spin superfluids [36, 37,
38]. This implies that the memristive effect depends on the magnetization
state and is, in principle, _reconfigurable_. We extend our analysis to
micromagnetic simulations where we consider a ferromagnetic ring with finite
thickness. In such a ring, a magnetization state composed of homochiral pairs
of domain walls is readily accessible by simple field relaxation, conserving
the magnetization’s topology, and affording the system with AMM. This allows
us to conclude that memristive effects originate from the structural dynamics
of the domain walls rather than their current-induced translation. We expect
our results to foster research on analog and reconfigurable memristive devices
based on magnetic materials.

## Results

Let us now present the main results in the paper. We first develop a
theoretical framework to analyze certain magnetic states and formally
establish their relationship to resistive and memristive effects. The ideal
magnetization states considered in this section allows us to identify the
requisites to enable memeristance. We will then use this general framework to
analyze magnetic states obtained from micromagnetic simulations that take into
account nonlocal dipole fields.

### Analytical formulation

We begin by considering the first-order correction to the electric field
arising in a soft ferromagnet due to AMR [21, 39, 22]. This correction appears
as a combination of current and the magnetization according to [39, 22],

$\vec{\mathbf{E}}=\rho_{0}\vec{\mathbf{J}}+\hat{\mathbf{m}}(\rho_{||}-\rho_{\perp})(\hat{\mathbf{m}}\cdot\vec{\mathbf{J}}),$
(2)

where $\rho_{||}$ and $\rho_{\perp}$ are the resistivity parallel and
perpendicular to the normalized magnetization vector $\mathbf{m}$,
respectively, and $\vec{\mathbf{J}}$ is the current density. We assume
$\rho_{\|}>\rho_{\perp}$. The first term in Eq. (2) is simply Ohm’s law in the
continuous medium, while the second term is the first order contribution to
the electric field due the AMR effect [39]. In Ref. [23], it was shown that it
can be recasted as a resistor network with voltage generators in series. Then
it is exactly solvable and mapped into an effective memristive effect. While a
network analysis is certainly desirable, we focus here on the simple case of a
single magnetic element to rigorously derive the memristive effect due to
current-induced magnetization dynamics and the magnetization texture.

The electrical voltage along a certain path $\gamma$ is given by
$V(t)=\int_{\gamma}\mathbf{\vec{E}(\mathbf{\vec{J}}(t))}\cdot\hat{\mathbf{T}}(\gamma)dl$,
where $\hat{\mathbf{T}}$ is the tangent to the path $\gamma$. We consider a
ring geometry, whose coordinate system and current physical properties are
shown in Fig. 1. Let us thus consider a closed, quasi-one-dimensional path
such as a ring of radius $r$. In this case, $\gamma$ coincides with the ring
itself, so that $\hat{\mathbf{T}}(\theta)=[-\sin\theta,\cos\theta,0]$ which is
a vector oriented clockwise and tangent to the ring at all points, and where
$\theta=0$ corresponds to the positive $\hat{x}$ axis. Assuming that the
current circulates tangent to the path,
$\mathbf{J}(t,\theta)=j(t)\hat{\mathbf{T}}(\theta)$, then the effective
resistivity is given by $\tilde{\rho}=V(t)/(2\pi rj(t))$. Analogously and for
later convenience, we also define $\hat{\mathbf{R}}(\theta)$, the in plane
vector such that $\hat{\mathbf{R}}(\theta)\cdot\hat{\mathbf{T}}(\theta)=0$,
i.e. parallel to the radius of the ring and directed from the center to the
ring point at $\theta$ (Fig. 1). In this formulation, we can identify a
particular point on the ring via the angular value $\theta$.

Using Eq. (2), we obtain the effective resistivity

$\displaystyle\tilde{\rho}=\frac{V(t)}{2\pi
rj(t)}=\rho_{0}+\frac{\Delta\rho}{2\pi}\int_{0}^{2\pi}(\hat{\mathbf{T}}(\theta)\cdot\hat{\mathbf{m}}(\theta))^{2}\
d\theta.$ (3)

where $\Delta\rho=\rho_{||}-\rho_{\perp}$ and $\hat{\mathbf{m}}(\theta)$ is
the normalized magnetization vector expressed at the point $\theta$ on the
ring. The second term in Equation (3) depends on the magnetic state and
originates from the AMR. From it, we see that the resistivity is constrained
by two limiting values, $\rho_{\mathrm{off}}=\rho_{0}+\Delta\rho$ and
$\rho_{\mathrm{on}}=\rho_{0}$, the typical resistivity of the material. These
two values are typical for the AMR effect. The value $\rho_{\mathrm{off}}$ is
obtained if $\hat{\mathbf{T}}(\theta)\|\hat{\mathbf{m}}(\theta)$, while
$\rho_{\mathrm{on}}$ if
$\hat{\mathbf{T}}(\theta)\perp\hat{\mathbf{m}}(\theta)$.
$(\rho_{\mathrm{off}}-\rho_{\mathrm{on}})/\rho_{\mathrm{on}}=\frac{\Delta\rho}{\rho_{0}}$.
Values for $\Delta\rho/\rho_{0}$ are documented for a variety of materials
[39] and it is around 3% for Permalloy.

We are interested in how the effective resistivity of the material changes
dynamically. We thus must consider the time evolution of the magnetization
vector into Eq. (3). For a ferromagnet subject to an in-plane current, the
magnetization dynamics is described by the Landau-Lifshitz-Gilbert (LLG) [40]
equation extended with the Zhang-Li torque [35], given by

$\displaystyle\frac{\partial\hat{\mathbf{m}}}{\partial
t}=-\gamma_{g}\mu_{0}\hat{\mathbf{m}}\times\vec{\mathbf{H}}_{\text{eff}}+\alpha\hat{\mathbf{m}}\times\frac{\partial\hat{\mathbf{m}}}{\partial
t}-\xi\nu\left(\hat{\mathbf{m}}\times(\mathbf{\vec{J}}(t)\cdot\nabla)\hat{\mathbf{m}}\right)-\nu\hat{\mathbf{m}}\times\left(\hat{\mathbf{m}}\times(\mathbf{\vec{J}}(t)\cdot\nabla)\hat{\mathbf{m}}\right),$
(4)

where $\nu=\frac{p\mu_{0}}{eM_{S}(1+\xi^{2})}$, and where we consider the
gyromagnetic ratio $\gamma_{g}$, saturation magnetization $M_{\text{S}}$, the
vacuum permeability $\mu_{0}$, Gilbert damping coefficient $\alpha$,
polarization ratio of the electric current $p$, electric charge $e$, and the
degree of adiabaticity $\xi$ [35]. The effective field
$\vec{\mathbf{H}}_{\text{eff}}$ may include an external magnetic field,
magnetocrystalline anisotropy, exchange interaction, and nonlocal dipole
fields interaction.

Figure 1: The geometry under study. We consider a nanoscale magnetic device
which has the structure of a thick ring, with the dimensions of $10$nm in
thickness and $55$nm radius. The current configuration,
$\vec{\boldsymbol{J}}(\theta,t)$ runs along the ring, either clockwise or
anticlockwise depending on the sign of $j(t)$. The ring coordinate system is
given by $\hat{\boldsymbol{T}}(\theta)$, $\hat{\boldsymbol{R}}(\theta)$ and
$\hat{\boldsymbol{z}}$. In the toy model, the thickness of the ring is assumed
to be negligible, while in numerical simulations finite size effects are
considered.

As we show in the Supplementary Material, the combination of both the AMR
effect of Eqs. (2) and (4) leads to a time-evolution of the resistivity of the
form

$\displaystyle\frac{d\tilde{\rho}}{dt}$ $\displaystyle=$
$\displaystyle\beta(\hat{\mathbf{m}})j(t)-\eta(\hat{\mathbf{m}},\vec{\mathbf{H}}_{\mathrm{eff}}),$
(5a) $\displaystyle\beta(\hat{\mathbf{m}})$ $\displaystyle=$
$\displaystyle\beta_{a}(\hat{\mathbf{m}})+\beta_{na}(\hat{\mathbf{m}})$
$\displaystyle\beta_{na}$ $\displaystyle=$
$\displaystyle-\frac{\xi\nu\Delta\rho}{2\pi r}\int_{0}^{2\pi}d\theta\
\frac{\partial\hat{\mathbf{m}}}{\partial\theta}\cdot\left(\hat{\mathbf{T}}\times\hat{\mathbf{m}}\right)t_{m},$
(5b) $\displaystyle\beta_{a}$ $\displaystyle=$
$\displaystyle\frac{\nu\Delta\rho}{2\pi
r}\int_{0}^{2\pi}t_{m}(\theta)r_{m}(\theta)d\theta$ (5c)
$\displaystyle\eta(\hat{\mathbf{m}})$ $\displaystyle=$
$\displaystyle\frac{\Delta\rho}{2\pi}\int_{0}^{2\pi}d\theta\
(\gamma_{g}\mu_{0}\vec{\mathbf{H}}_{\mathrm{eff}}-\alpha\frac{\partial\hat{\mathbf{m}}}{\partial
t})\cdot\left(\hat{\mathbf{T}}\times\hat{\mathbf{m}}\right)t_{m},$ (5d)
$\displaystyle t_{m}(\theta)$ $\displaystyle=$
$\displaystyle\hat{\mathbf{T}}(\theta)\cdot\hat{\mathbf{m}}(\theta),\ \ \ \
r_{m}(\theta)=\hat{\mathbf{R}}(\theta)\cdot\hat{\mathbf{m}}(\theta),$ (5e)

where $\beta(\hat{\mathbf{m}})$ and $\eta(\hat{\mathbf{m}})$ are functions of
the magnetization state in the ring. They indicate how the device reacts to
currents and in their absence, respectively. The details of the derivation are
presented in the Supplementary Material.

Crucially, Eq. (5a) takes the form of the Strukov-like memristor, described in
Eq. (1) in which $\sigma=0$, while $\beta$ and $\eta$ are not constants but
depend on the magnetization state and are thus tunable. In particular, the
$\beta$ constant of Eq. (5) takes the form of a memristive device whose
properties depend on the geometry of the ring, such as the radius $r$, the
parameters of the magnetic material, and the magnetization state
$\hat{\mathbf{m}}(\theta)$. From the point of view of the theory of
memristors, Eqs. (5a) represent an infinite-order memristor, i.e., the device
depends on a vector field rather then a single parameter (as for instance the
doping depth in oxide-based devices).

Equations (5) are the central result of this paper as they rigorously link
current-driven magnetization dynamics to memristive effects in magnetic
materials subject to AMR effect and Zhang-Li torque. We then call this
combined effect the anisotropic magnetomemristance (AMM).

A few comments on the equations above are in order. First, the quantities
$\beta_{na}$ and $\beta_{a}$ are the magnetization-dependent non-adiabatic and
adiabatic coefficients which, if non-zero, induce a memristive effect. Another
interesting feature is that the smaller the ring radius, $r$, the larger the
$\beta_{a}$ and $\beta_{na}$ coefficients are; on the contrary, $\eta$ does
not depend on the ring radius. This observation implies a crossover between
two regimes: a memristive behavior for small radii and a current-independent
resistivity change due to the $\eta$. In addition, the parameter $\eta$
contains the balance between the conservative and dissipative terms of the LLG
equation. This implies that memristive behavior is primarily due to the
magnetization texture present in the material.

In addition to the technical comments above, let us briefly provide here a
physical interpretation of the equations. The $\beta$ term is the term that
leads to a memristive effect, as the resistance changes as a function of the
current history, and can lead to a pinched hysteresis loop in the current-
voltage diagram. Since this term has two components, originating from the
adiabatic and non-adiabatic terms in LLG, both contribute to memristance.
Because of the vectorial dependence of both these terms, the direction of the
magnetization with respect to the plane spanned by the ring is of key
importance. In order for a memristance to be present, the magnetization will
have to not lay on the plane, and in particular revolve around the ring,
originating from the $\hat{\mathbf{T(\theta)}}\times\hat{\mathbf{m}}$
dependence. In addition to this, the magnetization will also have to be
dynamical in a non-trivial manner. For instance, domain walls that can
annihilate in the ring can lead only to a transient memristance, but not a
stable one. While these conditions appear to be restrictive, we find that
physically achievable magnetic states that cannot be easily removed via spin
transfer torque lead to a more stable memristance effect. This is one of the
key features of the ring geometry, which naturally can induce states that are
“topological”. We will see below that magnetic states that we find to lead to
a noticeable memristive states are imperfect, e.g. arising from the interplay
of the magnetodynamics of the LLG equation with non-trivial magnetic states.

Naively, one would expect an ideal memristor would be obtained if $\eta=0$ for
any magnetization state and time. However, we must recall that $\beta$ is not
a constant, and thus $\eta=0$ does not imply that the internal parameter can
be directly be associated to the charge. Moreover, $\eta=0$ is in general
unlikely by pure dissipative action on the conservative term since the
effective field is nonlocal. However, we will discuss a particular magnetic
state in which this is possible. The geometrical terms in the integrand
provide other possibilities to nullify $\eta$. If $t_{m}=0$, which implies a
magnetization state perpendicular to the ring’s circumference, i.e., pointing
radially or fully out-of-plane. The latter state could be easily achieved by a
saturating magnetic field applied normal to the plane. However, this condition
will also nullify $\beta$ and the memristive effect would be suppressed. If
$\hat{\mathbf{T}}(\theta)\times\hat{\mathbf{m}}=0$ then the only allowed
magnetization state is the ground state, i.e., a magnetization parallel to the
ring’s circumference. This case also imply that $r_{m}(\theta)=0$ and again
$\beta=0$ for all cases. This analysis demonstrates that a magnetic ring
cannot be an ideal memristor.

### Magnetization states, resistivity and memristance

We now analyze how certain magnetization patterns could affect the resistance
and memristance in this toy model. First, let us note that the resistance
depends on $t_{m}(\theta)$, as $\rho=\rho_{0}+\frac{\Delta\rho}{2\pi}\int
t_{m}(\theta)^{2}d\theta$ and that there are magnetic states that would lead
to no resistance and no memristance. We emphasize that the memristive effect
is purely due to the interplay between AMR and spin-transfer torque.

Here we focus on the non-adiabatic term, which is larger in real materials.
However, an analysis of which states contribute to memristance for the
adiabatic term of eqn. (5c) is provided in the Supplementary Material. In the
case of the non-adiabatic term in eqn. (5c), the correction depends on
$r_{m}(\theta)$ and $t_{m}(\theta)$, which means that the the magnetization
must not be perpendicular to both the radius and tangent to the ring at all
times.

Thus, a memristive behavior is nullified if the magnetization is everywhere
parallel to the current or, equivalently, the ring’s circumference. A similar
fate befalls onto magnetization states where
$\partial_{\theta}\hat{\mathbf{m}}=0$, i.e., when the the device is statically
magnetized in a particular direction. This means that memristive effects can
only happen as a function of current if the magnetization texture along the
ring is non-trivial.

The typical magnetodynamics scale set by the gyromagnetic ratio and the
effective magnetic field is contained in the term $\eta(\theta)$ which
produces a change in the device’s resistivity in non-equilibrium independent
of the current density. In a steady state, the damping will exactly cancel out
any dynamic contribution and $\eta(\theta)=0$ so that the device will have a
finite resistivity. In addition, the effective magnetic field is given by
$\vec{\mathbf{H}}_{\text{eff}}=\vec{\mathbf{H}}_{0}+\lambda\partial^{2}_{\theta\theta}\hat{\mathbf{m}}-H_{k}(\hat{\mathbf{m}}\cdot\mathbf{\hat{T}}(\theta))\mathbf{\hat{T}}(\theta)$.
Consequently, the main factor determining the memristance is the static
magnetization state in the ring.

In addition to trivial magnetization states, it is theoretically possible to
have textures in our toy model. Non-trivial magnetization states in rings have
been theoretically suggested as means to store energy in the form of angular
momentum [38]. Such a state exhibits a smooth, continuous rotation of the
magnetization about its perpendicular-to-plane axis and is formally analogous
to mass superfluidity or superconductivity [36]. Indeed, a ring provides a
suitable geometry to realize periodic boundary conditions that, in principle,
would stabilize a spin superflow with a quantized number of periods $n$ in the
absence of dipolar fields [37].

We now analyze the predicted resistivity and memristance of a variety of
possible magnetization states.

Fully magnetized states. Since both $\eta$ and $\beta$ depend on
$t_{m}(\theta)$ and $r_{m}(\theta)$ which are the projection on the tangent
and radial vectors of the ring, any out-of-plane or uniformly in-plane
magnetization will lead to a pure resistive state. This can be easily
achieved, e.g, by an external field saturating the magnetization.

In real materials, if the sample is prepared in a state in which $\mathbf{m}$
has most of its components in the plane, then the effect of an external field
${\mathbf{H}}_{0}$ produces only a small correction to the resistivity. We
found (see Supplementary Material) that the correction is of the same order of
the out-of-plane component of $\mathbf{m}$, implying that the sample is robust
to in-plane external field perturbations.

Topological states. As we have seen, certain natural states on the ring do not
support a memristive effect. Let us now attempt at introducing a set of states
with full rotations of the magnetization. We introduce a generic magnetization
texture of the form
$\hat{\mathbf{m}}(\theta)=\left(\sin(\phi)\cos(n\theta),\sin(\phi)\sin(n\theta),\cos(\phi)\right)$,
where $\phi(\theta)$ can be an arbitrary function that represents the out-of-
plane component of the texture and $n$ quantifies the topology of the state.
When $\phi(\theta)=\pi/2$, the magnetization is fully in-plane and the
parameter $n$ defines the number of rotations around the ring. For $n=1$, we
recover the ground or vortex state discussed above. For $n\geq 2$, such an
equation describes a smooth, coherent rotation of the magnetization known as a
spin superflow stemming from its formal analogy to mass superfluidity [36]. In
these states, the homochiral phase rotations along $\theta$ lead to a net pure
spin current which has lead to proposed applications in storage [38] and long-
distance spin transport [41, 42, 37, 43, 44, 45]. These states are interesting
because they should be in principle dynamical and non-easily removable via the
spin transfer torque. Unfortunately, from Eq. (5), a spin superflow leads to
$\beta(\theta)=0$. Another state of interest is obtained when
$\phi=\omega\theta$, which we call a helical state. Helical states preclude
any AMR change, i.e., $\tilde{R}=R_{0}$ from Eq. (3). Therefore, neither pure
spin superflows nor perfect “helical states” produce memristive effects
(details of this calculations are shown in the Supplementary Material).

This analysis implies that memristive effects can be expected for states that
are both topological and possess a “helical” character. While a smooth state
of this form is unlikely to exist in nature, a topologically equivalent state
is simply a pair of homochiral domain walls [46, 47] as we discuss below.

Magnetic states dominated by shape anisotropy. Here we would like to discuss
how the interplay between long-range interactions and topology can lead to
non-trivial memristive relevant states. In realistic magnetic materials, often
the long-range dipolar interaction is the dominating force, leading to the so-
called shape anisotropy. Combined with the exchange interaction which favors
parallel spins, there are two degenerate ground states on a magnetic ring with
magnetization $\hat{\mathbf{m}}(\theta)=\pm\hat{\mathbf{T}}(\theta)$. The
magnetization is curled around the ring either clockwise or counterclockwise.
Such minimum energy states are also known as the vortex state [48, 49, 50].
Interestingly, the vortex state is an example of magnetic state which
possesses AMR, but no memristance. In fact, while this magnetic state
contributes to the resistivity, it is not hard to see that $\beta_{a}$ is zero
as $r_{m}(\theta)$ is zero everywhere; $\beta_{na}$ is alzo zero as
$\hat{\mathbf{T}}(\theta)\times\hat{\mathbf{T}}(\theta)=0$. Thus, the ground
state has no memristance, as it is confirmed below via micromagnetics of
extensive rings.

The magnetic texture that interpolates the two different vortex states with
opposite circulation is called the domain wall (DW). Since DWs have to be
created and destroyed in pairs on a ring, they are also topological defects on
top of the minimum energy state. In fact, each DW can be viewed as a
mesoscopic object that carries a net magnetic charges: positive charge for a
tail-to-tail DW, and negative charge for head-to-head DW. Metastable states on
the ring can thus be classified according to the number $n_{\rm
DW}=2,4,\cdots$ of DWs. Importantly, DWs play a crucial part in the memristive
effect of a magnetic ring by allowing for a non-zero non-adiabatic
$\beta_{na}$ term. Since the magnetization $\hat{\mathbf{m}}(\theta)$ changes
from $+\hat{\mathbf{T}(\theta)}$ to $-\hat{\mathbf{T}}(\theta)$ in the
vicinity of a DW, thus allowing for a finite gradient
$\partial\hat{\mathbf{m}}/\partial\theta$. Moreover, magnetization is forced
to deviate from the tangent direction $\pm\hat{\mathbf{T}}(\theta)$, giving
rise to a nonzero $\hat{\mathbf{T}}\times\hat{\mathbf{m}}$ as well. As
indicated in Eq. (5c), these are the two ingredients for a nonzero
$\beta_{na}$ term. As it will be demonstrated in the next Section, the two
domain-wall states (2DW) indeed exhibits a significant memristance.
Experimentally, these states have been observed in magnetic nanorings [48, 49,
50]. Numerically, such a pair of domain walls can be readily stabilized by
including nonlocal dipole fields.

### Simulations for magnetic rings

The analytical treatment described above provides a useful guide into the
phenomenology, but itis limited to an ideal one-dimensional case. In reality,
a finite cross-section of a ring would produce a non-local dipole field that
in turn modifies both the stabilized magnetization states and the dynamics. To
account for the non-local dipole contribution, we perform micromagnetic
simulations with use of the open-source package MuMax3 [52].

We simulate a Py ring with a mean radius of $55$ nm, a ring width of $10$ nm,
and thickness $10$ nm. We use standard material parameters $M_{s}=790$ kA/m,
$A=10$ pJ/m, and $\alpha=0.01$. The Zhang-Li torque polarization parameter is
set to $0.56$ and we also include a small non-adiabatic term $\xi=0.1$, and
for permalloy we have $\rho_{0}=123~{}\Omega$ nm.

We consider four initial magnetic configurations. In all cases, we follow the
relaxation protocol described in the Methods to quench spurious dynamics. The
ground state is the magnetization along the ring, stabilized by shape
anisotropy, as shown in Fig. 2 a obtained numerically. In-plane and out-of-
plane magnetized states are achieved applying an external magnetic field of
magnitude 1 T along the $x$ and $z$ directions, respectively. Finally, we
consider homochiral domain walls stabilized by initializing the magnetization
with a finite number of topological defect, and minimizing the energy. The
case of two domain walls is shown in Fig. 2 b. The final magnetic state is a
localized kink of the form
$\hat{\boldsymbol{m}}(\theta)=a(\theta)\hat{\boldsymbol{T}}(\theta)+b(\theta)\hat{\boldsymbol{K}}(\theta)$
where we assume $\hat{\boldsymbol{K}}\cdot\hat{\boldsymbol{T}}=0$, and thus
$|\hat{\boldsymbol{m}}|^{2}=a(\theta)^{2}+b(\theta)^{2}=1$.

Figure 2: Stabilized a ground state and b 2DW state utilizing the tiered
energy minimization approach described in the Methods. The $m_{z}$ component
is displayed in colorscale on the order of $10^{-7}$. The predominant in-plane
component is shown by black arrows.

We have shown in the Supplementary Material that such a magnetic state can
have both AMR induced resistance and memristance. This type of states have
been obtained seen both numerically and experimentally in the past [46, 48,
47, 49]. These states are topologically distinct from the ground state [51].

We analyze the simulation results by computing the change in voltage and the
change in resistivity for each state upon the application of a time-varying
current density of magnitude $j$ and frequency $f$. In the following, we
discuss variations in voltage, deviating from the resistive state. These can
be studied via $\Delta V=(R-\langle R\rangle)j$, with $\langle R\rangle$ being
the time average resistance. We see that if $R$ is constant, then $\Delta
V=0$, which is associated to a pure resistance.

Figure 3: Resistivity of the ground state (black curve), the in-plane
magnetization state (blue) and the out-of-plane magnetic state (light grey).
Left: We plot the I-$\Delta V$ diagram of the resistive device as a function
of the magnetization. We can see that for currents up to a 100 $\mu$A the
resistive states do not possess a noticeable hysteresis in the nanovolts. Yet,
the AMR leads to a nonlinear resistance state. Right: We plot the change in
resistivity as a function of the current density, which is in units of
$10^{12}$ A m-2. From this plot we can see that the ground state and the in-
plane do in fact have a memristance effect, which is however very small. For
the out-of-plane state instead, the memristance is zero, as predicted by our
model.

For the ground state, in-plane, and out-of-plane magnetization states, the
change in voltage versus the current density in a cycle of frequency $f=1$ GHz
is shown in Fig. 3. For these simulations a rather large current density of
$10^{12}$ A/m2 was used to discern effects above numerical noise. The $I-V$
curves are shown in Fig. 3(a), where we see that a memristance effect is not
noticeable at a $nV$ scale. This is consistent with the predictions of the toy
model.

However, because the thickness of the ring gives a small memristance effect
that can be seen when looking at $\frac{d\rho}{dt}$. First,
$\frac{d\rho}{dt}=0$ corresponds to pure resistance. That is the case of out-
of-plane magnetization. We should expect this property to be true also in real
materials with a finite thickness. For the cases of the in-plane and the
ground state we note that a small change in resistivity is indeed present, and
depends on the current. While a memristance in the ground state and in-plane
magnetized state appears to be at odds with the symmetry arguments presented
above, we note that the strong current density used in this simulation induces
a large enough magnetization dynamics.

The curves we observe for $\frac{d\rho}{dt}$ are Lissajous curves.
Intuitively, this is due to the fact that while the magnetic ground state
follows the geometry of the ring, the non-adiabatic coupling leads to a
rotation of the magnetization vector. As discussed in the Supplement, we
associate this memristance to an adiabatic effect only accessible at large
current densities.We wish to stress that the fact that we observe a non-zero
area associated with these curves is indicative of a memristive effect which
depends on an internal state. The pinched hysteresis loop in the
$\frac{d\rho}{dt}$ vs $j$ diagram implies a zero decay constant $\eta$.
Indeed, if the change in resistivity was proportional to the current density
$j$, we should expect to see a line in these plots, with a slope given by a
constant $\beta$ parameter. The fact that we see a Lissajous curve means that
the parameter $\beta$ is not constant in time. In particular, the periodicity
of $\beta$ is exactly the one of the driving $j$.

While for the cases of magnetic in-plane, out-of-plane and ground states we
observe a negligible memristive state, we argue that a non-trivial and
measurable effect could be seen for topological magnetic states. In
particular, we focus on the domain wall states. The results for two such
states with two domain walls (2DW) and 4 domain walls (4DW), and utilizing a
current density of $10^{11}$ A/m2 are shown in Fig. 4. In particular, in Fig.
4 (a) we plot the current-$\Delta$ Voltage diagram, where this time a non-
negligible hysteresis can be observed at the nano-volts scale. For the 2DW
case, a hysteresis is clearly visible. In Fig. 4 (b) we plot
$\frac{d\tilde{\rho}}{dt}$ as a function of the current. We can see that we
have a memristance effect with a time varying $\beta$ constant in both the
cases 2DW and 4DW. However, while the curve for the 2DW is pinched, we have a
non-pinched hysteresis for the 4DW state, which implies a non-zero (and
positive) decay $\eta$. This suggests that we can in principle engineer the
behavior of the memristor device depending on the ring internal magnetic
state.

A way to estimate and compare how much dependence on the magnetic state we
have in the four magnetic states above, is by analysing the area spanned by
the Lissajous curves in $\frac{d\rho}{dt}$ vs. $j$. In Fig. 5 we plot the
estimated area for curves of the ground state and the 2DW state of Fig. 3 and
4 respectively, which are those with the largest hysteresis in the two sets,
as a function of the frequency of the input current density. In both cases,
the Lissajous’ area for the ground state grows with frequency, a further
indication that the memristance in this case is produced by current-induced
dynamics. We do not observe a peak up until $10$ GHz. In contrast, the 2DW
state peaks at around 600 MHz and exhibits a finite magnitude at 1 GHz, which
is larger than the magnetic ground state. It is worth stressing that while the
current densities used for the 2DW state simulations are one order of
magnitude smaller than those used for the ground state, the Lissajous area at
600 MHz is approximately two orders of magnitude larger. This demonstrates
that the memristance of the 2DW state is primarily due to the magnetization’s
texture and in good agreement with a memristive device. The key reason why the
memristive effect is amplified is that the magnetic kink cannot be easily
removed at small currents, which is due to a form of topological protection.
In fact, as the current flows, the kink moves due to the Zhang-Li torque, but
cannot be easily destroyed.

Based on the 2DW state, we now explore the current-density dependence on the
Lissajous area. The results are presented in Fig. 6. We observe an exponential
increase in the area for low current densities that appears to start
saturating at $10^{12}$ A m-2. The sudden jump at large current densities is
indicative of a sufficient distortion of the domain walls to untwist the
magnetization, i.e., the topological protection is lost and the system relaxes
into the ground state. The results of this section provide the intuition that
at sufficiently low currents, the magnetic states with a finite rotations
about the ring’s easy axis are those for which we observe a larger memristive
effect. We argue that this observation is associated with the distinct
topology of such states when compared to the ground state in the ring. As
such, this suggests that in real materials topological protection of the
magnetic states is an important ingredient for the observation of a dynamical
resistance rooted in the AMR effect. The origin of this protection is indeed
due to the fact that we are considering a ring where these kinks are allowed
and not easily annihilated via singularities or phase slips.

A few minor comments are in order. In addition to the analysis above, we have
also tested the dependence on the memristance on the ring radius and the
thickness. Our numerical studies (see Supp. Mat.) confirms that the thicker
the ring, the larger the memristive effect, measured in terms of Lissajous
area. Moreover, we have also confirmed that for smaller radii the effect is
amplified, in line with the toy model result. However, It is worth mentioning
that the Lissajous areas were only slightly affected for larger ring radii.
While more in depth studies are required, this is encouraging for experiments
relying on nanolithography.

Figure 4: Resistivity of the 2DW and 4DW states. Left: Pinched hysteresis
obtained from plotting the device current versus the change in voltage $\Delta
V$, in nanovolts. As we can see, the 2DW state has a larger change in voltage,
which is of the order of $500$ nV. Right: We plot the change in resistivity as
a function of the current density, which is in units of $10^{12}$ A m-12. As
we can see, the domain-wall states have a larger memristive effect that the
ground state and the in-plane states.  Figure 5: Area spanned by the curves of
the ground state and the 2DW state in Figs. 3 and 4. The area provide an
estimate of how large is the dependence of $\beta$ on the magnetic state, as a
function of the frequency of the voltage input. The peak in Lissajous area for
the 2DW state is at $f=600$ GHz. The error bars denote standard deviation of
the area based on numerical fluctuations and transient effects in the
Lissajous’ curves. Figure 6: Fragility of the 2DW state as a function of the
current density. As we can see, the area of the Lissajous curve depends
strongly on the current density, and increases as a function of it. At low
densities, the two domain walls are mostly static and the area is small,
indicating a negligible memristive effect. At higher densities (yellow area),
the domain-walls experience a noticeable distortion accompanied by a small yet
finite periodic translation of the domain walls. At much higher current
densities (gray area), translation dominates and phase slips can occur,
leading to the destruction of the domain-wall pair such that the state of the
system stabilizes into its ground state. These cases are evidenced by the low
Lissajous areas. The stochastic nature of this regime is due to the cell’s
large aspect ratio. The error bars denote standard deviation of the area based
on numerical fluctuations and transient effects in the Lissajous’ curves.

## Discussion

We have analyzed the memristive effects which occur in realistic magnetic
materials due to the interplay between the anisotropic magneto-resistive
effect, an electric current via the Zhang-Li coupling, and the magnetodynamics
of the material. Our theoretical and numerical results suggest the existence
of a memristive effect, of which we have characterized the behavior. Our study
focused on a ring geometry, and we have obtained a good agreement between
theory and numerical simulations. Specifically, we have introduced a simple
one-dimensional model of a magnetic ring with an applied voltage and a
resulting current, in which the memristive effect due to the Zhang-Li spin-
torque interaction between the current and the local magnetization has been
derived analytically. This model characterizes explicitly the type of
memrisive behavior the device has, and how the non-adiabatic and adiabatic
couplings affect the memristive dynamics, e.g. how the resistance changes with
the magnetic state.

Our results are based on Landau-Lifshitz-Gilbert evolution for the
magnetization applied to the Anisotropic-Magnetoresistance effect in
continuous magnetic media. We focused on the study of a common magnetic
material such as permalloy, which is readily available. The model explicitly
shows that a memristive effect should be present in soft-magnetic material. We
predict that for permalloy, it should be measurable at a nano-Volt scale and
in the range of $600-1000$ MHz.

One of the most striking features of this device is that the behavior of the
memristance can be engineered by inducing particular magnetic states. While
the effect can be considered small by the standard of typical oxide-devices,
we argue that a larger effect should be present in ring-based lattices, as the
change in voltage should accumulate in a coherent fashion. From an
experimental perspective, our results are valid for AC currents, implying that
for such effect to be observable any DC voltage should be filtered. However,
this should be possible using a setup with capacitors or lock-in amplifiers,
also given the fact that the effect is in the nV regime. The main experimental
challenge might be to stabilize a full rotation of the magnetization in a
device that necessarily will depart from a perfect ring, in order to
manipulate the magnetization and to inject a microwave current.

From the perspective of the memristive device, engineering the hysteresis is
an exciting development. Typically, the hysteresis curves of memristive
devices strongly depend on the type of underlying physical properties of the
device [4]. While even in numerical experiments we can see that our device is
not ideal, we have observed that depending on the magnetic states we can have
a non-current dependent memristance decay which is zero.

Crucially, the underlying feature of the amplified memristive effect is a
topological protection, enacted by the similar chirality of the domain walls
that are only removable by the formation of singularities. The proposed
memristor is therefore topological in nature. The possibility of a posteriori
re-programming the memristor by changing the magnetic properties of the device
makes this type of device ideal for a fast neuromorphic Random Access Memory
(nRAM).

This proposal is a departure from previous works on the subject. In fact,
other memristive devices based on magnetic states used viscosity to obtain a
memristive state, for instance via magnetic pinning [26], domain-wall motion
[23], inclusion of MTJ [24], which induce a slower dynamics. Instead, here we
have shown that at the natural timescale (GHz) typical of magnetic dynamics
there can be a memristive effect insofar as the stabilized magnetic state is
topological.

The present study opens new analyses in this direction. We have analyzed only
a handful of magnetic states of a ring. In ring heterostructures many more
possible and non-trivial interactions between the magnetic states are
possible, and include the formation and propagation of spin waves. Moreover,
we have considered the case of permalloy, but different and harder magnetic
materials may behave differently and provide a larger memristive effect.

Overall, while this paper does not provide an exhaustive answer of which
magnetic states maximize such complex interplay of phenomena, it provides the
foundation for the study of this effect. As we have discussed, the memristance
effect arises from the combination of dynamical magnetic and non-trivial
magnetic states which cannot be easily annihilated due to their topological
nature. It would be interesting to explore if other topological states exhibit
memristance, for example, spin superfluids or dissipative exchange flows in
nanowires [41, 42, 37, 43, 44, 45] or even skyrmion lattices in extended films
[53, 54].

## Methods

### Sketch of the theoretical derivations

We provide a sketch of the proof of the main formulae provided in the paper,
in particular eqns. (5a)-(5d). The full derivations are provided in the
Supplementary Material. The two key equations are the LLG equation:

$\displaystyle\frac{\partial\hat{\mathbf{m}}}{\partial t}$ $\displaystyle=$
$\displaystyle-\gamma\mu_{0}\hat{\mathbf{m}}\times\vec{\mathbf{H}}_{\text{eff}}+\alpha\hat{\mathbf{m}}\times\frac{\partial\hat{\mathbf{m}}}{\partial
t}$ (6) $\displaystyle-$
$\displaystyle\xi\nu\left(\hat{\mathbf{m}}\times(\vec{\mathbf{J}}(t)\cdot\nabla)\hat{\mathbf{m}}\right),$
$\displaystyle-$
$\displaystyle\nu\hat{\mathbf{m}}\times\left(\hat{\mathbf{m}}\times(\vec{\mathbf{J}}(t)\cdot\nabla)\hat{\mathbf{m}}\right),$

and the AMR correction to the electric field

$\vec{\mathbf{E}}=\rho_{0}\vec{\mathbf{J}}+\hat{\mathbf{m}}(\rho_{||}-\rho_{\perp})(\hat{\mathbf{m}}\cdot\vec{\mathbf{J}}).$
(7)

The voltage drop along a certain path is given by

$\displaystyle V(t)$ $\displaystyle=$
$\displaystyle\int_{\gamma}\vec{\mathbf{E}}(\vec{\mathbf{J}}(t))\cdot\hat{\mathbf{T}}(\gamma)d\gamma.$
(8)

Substituting eqn. (7) in eqn. (9) one obtains the voltage in terms of the
magnetization.

$\displaystyle V(t)$ $\displaystyle=$
$\displaystyle\int_{\gamma}\big{(}\rho_{0}\vec{\mathbf{J}}+\hat{\mathbf{m}}(\rho_{||}-\rho_{\perp})(\hat{\mathbf{m}}\cdot\vec{\mathbf{J}})\big{)}\cdot\hat{\mathbf{T}}(\gamma).$
(9)

If the current is along the ring then as discussed in the main text, we have

$\displaystyle\vec{\mathbf{J}}(t)=j(t)\hat{\mathbf{T}}(\theta).$ (10)

Thus, we can define the resistance along a ring, we can define the path
$\gamma$ as $d\gamma=rd\theta$, obtaining

$\displaystyle R(t)=\frac{V(t)}{j(t)}$ $\displaystyle=$ $\displaystyle 2\pi
r\rho_{0}+r\Delta\rho\int
d\theta\big{(}\hat{\mathbf{m}}\cdot\hat{\mathbf{T}}(\theta)\big{)}^{2}.$ (11)

which is the formula we provided in the main text. Eqns. (5a)-(5d) can now be
derived taking a time derivative, and replacing
$\frac{\partial\hat{\mathbf{m}}}{\partial t}$ with the LLG equation.

### Numerical Simulations

Micromagnetic simulations using MuMax3 [52] were performed on a GPU NVIDIA
Geforce 940MX. The system is discretized in cells of approximately 1 nm
$\times$1 nm $\times$10 nm. While the aspect ratio is large, it has been shown
in previous works that the numerical effects are negligible for near-uniform
states across the thickness [55]. We note that possible instabilities across
the thickness would be hidden, but we do not expect such instabilities to
exist in uniform or topological states insofar as the magnetization is largely
in plane or relatively large in extent [37]. For large current densities, we
do observe the compression of the domain wall to an extent where the
magnetization is out-of-plane and a phase-slips occur. Such a phase-slip could
occur at smaller currents for cubic cells, i.e., via a texture distortion
across the thickness. The simulation is set to use a Runge-Kutta 45 stepper
and the maximum step time is set to 100th of a period. This is essential to
avoid noise for simulations excited by currents of different frequencies.

It is fundamental to stabilize a robust initial condition. The general
protocol followed features a “tiered” energy minimization approach to quench
spurious dynamics down to numerical accuracy: we first use MuMax’s built-in
energy minimization function relax, then we run for $20$ ns with $\alpha=1$,
$30$ ns with $\alpha=0.1$ and $50$ ns with $\alpha=0.01$. The current density
is implemented as being parallel to $\mathbf{T}(\theta)$. Such current density
is generated in Matlab with and then imported to MuMax3 as a mask whose
magnitude is time-dependent.

Dynamic simulations are run for 20 to 100 periods, depending on the frequency.
In general, larger frequencies require more time to stabilize as sudden
introduction of current launches spin waves from the texture that need to
attenuate to discern a memristive effect. The resulting time-dependent
magnetization is then processed in MATLAB to obtain the resistivity according
to Eq. (3). The change in resistivity $d\tilde{\rho}/dt$ is calculated as the
numerical derivative of Eq. (3) while the voltage is simply extracted from Eq.
(3) and we consider the average radius of the ring.

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## Acknowledgements.

The work of FC and CN was carried out under the auspices of the US Department
of Energy through the Los Alamos National Laboratory, operated by Triad
National SecurityLLC (Contract No. 892333218NCA000001).

## Funding

CN was funded by DOE-LDRD grant 2017014ER. FC was also financed via DOE-LDRD
grants PRD20170660 and PRD20190195. CILA was funded by CNPq 432029/2018-1,
FAPEMIG, Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) -
Finance Code 001 and Newton Fund NAF-R2-192040.

## Author contributions

C.I.L.A and E.I performed the numerical simulations and F.C developed and
worked on the theoretical model. All authors contributed to the discussion on
analytical and numerical results, as well as writing the paper. F.C. and E. I.
contributed equally to this work.

## Supplementary Material

### Spin-Transfer torque memristor: Ring model

Let us consider the following simplified model. We consider a ring of length
$L=2\pi r$, where $r$ is the radius of the ring. We consider a magnetization
vector dependent on the angle on the ring,
$\vec{\boldsymbol{M}}_{t}(\theta)=M_{x}(t,\theta)\boldsymbol{\hat{x}}+M_{y}(t,\theta)\boldsymbol{\hat{y}}+M_{z}(t,\theta)\boldsymbol{\hat{z}}$.
The resistivity of the alloy per unit of length is $\rho$, such that
$R=2\pi\rho r$. We also consider a voltage generator $V_{0}$ on the ring,
which introduces a current $j$ in the ring, such that $V_{0}=Rj$. Vectorially,
we have that the vector of the radius is given by

$\displaystyle\boldsymbol{r}=r[\cos\theta,\sin\theta].$ (12)

Since $j$ flows on the ring, we must have
$\boldsymbol{r}\cdot\vec{\boldsymbol{J}}=0$, which implies
$\vec{\boldsymbol{J}}(t,\theta)=j(t)[-\sin\theta,\cos\theta,0]=j\hat{\boldsymbol{T}}(\theta)$,
asssuming that the current is flowing clockwise if $j>0$, and where
$\hat{\boldsymbol{T}}(\theta)$ is the tangent to the wire, with the property
$\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{T}}(\theta)=1$. The
equation for the magnetization is given by the LLG equation [40] plus the
Zhang-Li coupling with the current [35]:

$\displaystyle\frac{\partial\vec{\boldsymbol{M}}_{t}}{\partial t}$
$\displaystyle=$
$\displaystyle-\gamma\vec{\boldsymbol{M}}_{t}\times\vec{\boldsymbol{H}}_{\text{eff}}+\frac{\alpha}{M_{s}}\vec{\boldsymbol{M}}_{t}\times\frac{\partial\vec{\boldsymbol{M}}_{t}}{\partial
t}$ (13) $\displaystyle-$
$\displaystyle\frac{\nu}{M_{S}}\Big{(}\xi\vec{\boldsymbol{M}}_{t}\times+\frac{1}{M_{S}}\vec{\boldsymbol{M}}_{t}\times\vec{\boldsymbol{M}}_{t}\times\Big{)}\Big{(}\big{(}\vec{\boldsymbol{J}}(\boldsymbol{\theta})\cdot\triangledown\big{)}\vec{\boldsymbol{M}}_{t}\Big{)},$

which we need to reduce on an effective one dimensional model by converting
the $\nabla$ operator on the ring. We have
$\vec{\boldsymbol{J}}(\boldsymbol{\theta})\cdot\triangledown=j_{x}\partial_{x}+j_{y}\partial_{y}$,
and we can write
$\partial_{\cdot}\vec{\boldsymbol{M}}_{t}(\theta)=(\partial_{\cdot}\theta)\partial_{\theta}\vec{\boldsymbol{M}}_{t}(\theta)$.

We have $\theta=\arctan\frac{y}{x}$, from which we obtain
$\partial_{x}\theta=-\frac{y}{x^{2}+y^{2}}=-\frac{1}{r}\sin\theta$ and
$\partial_{y}=\frac{x}{x^{2}+y^{2}}=\frac{1}{r}\cos\theta$. It follows that
$(j_{x}\partial_{x}+j_{y}\partial_{y})\vec{\boldsymbol{M}}_{t}(\theta)=\frac{j}{r}(\cos\theta^{2}+\sin\theta^{2})\partial_{\theta}\vec{\boldsymbol{M}}_{t}(\theta)=\frac{j}{r}\partial_{\theta}\vec{\boldsymbol{M}}_{t}(\theta)$.
We thus obtain that, on the ring, we have the reduced equation

$\displaystyle\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial
t}=\vec{\boldsymbol{M}}_{t}(\theta)\times\Big{(}$ $\displaystyle-$
$\displaystyle\gamma\vec{\boldsymbol{H}}_{\text{eff}}+\frac{\alpha}{M_{s}}\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial
t}-\frac{\xi\nu
j}{M_{\text{S}}r}\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}$
(14) $\displaystyle-$ $\displaystyle\frac{\nu
j}{M_{S}^{2}}\vec{\boldsymbol{M}}_{t}(\theta)\times\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}\Big{)},$

where we neglect all finite-size effects, and in the last line we used the
notation employed by MuMax.

The AMR effect is given by the vector field in the wire

$\displaystyle\vec{\boldsymbol{E}}=\rho_{0}\vec{\boldsymbol{J}}+\Delta\rho(\vec{\boldsymbol{J}}\cdot\hat{\boldsymbol{m}}_{t}(\theta))\hat{\boldsymbol{m}}_{t}(\theta).$
(15)

The effective voltage is given by

$\displaystyle
V=r\int_{0}^{2\pi}\vec{\boldsymbol{E}}\cdot\hat{\boldsymbol{T}}(\theta)\
d\theta.$ (16)

If we replace now the AMR correction, we find

$\displaystyle V$ $\displaystyle=$ $\displaystyle
r\int_{0}^{2\pi}\big{(}\rho_{0}\vec{\boldsymbol{J}}+\Delta\rho(\vec{\boldsymbol{J}}\cdot\hat{\boldsymbol{m}}_{t}(\theta))\hat{\boldsymbol{m}}_{t}(\theta)\big{)}\cdot\hat{\boldsymbol{T}}(\theta)\
d\theta$ (17) $\displaystyle=$ $\displaystyle
r\int_{0}^{2\pi}\big{(}\rho_{0}j\hat{\boldsymbol{T}}(\theta)+j\Delta\rho(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))\hat{\boldsymbol{m}}_{t}(\theta)\big{)}\cdot\hat{\boldsymbol{T}}(\theta)\
d\theta$ $\displaystyle=$ $\displaystyle
r\int_{0}^{2\pi}\big{(}\rho_{0}j+j\Delta\rho(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))^{2}\big{)}\
d\theta$ $\displaystyle=$ $\displaystyle
V_{0}+jr(\Delta\rho)\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))^{2}\
d\theta$

where we used
$\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{T}}(\theta)=1$. We can
write explicitly

$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(t,\theta)=M_{y}(t,\theta)\cos\theta-
M_{x}(t,\theta)\sin(\theta),$ (18)

where $M_{y}^{2}+M_{x}^{2}=M_{S}^{2}$. Let us now note that

$\displaystyle V=j\Big{(}\rho_{0}2\pi
r+r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))^{2}\
d\theta\Big{)}=j\tilde{R},$ (19)

where $\tilde{R}$ is an effective resistance. If
$\vec{\boldsymbol{M}}_{t}(\theta)$ does not depend on $j$, then $\tilde{R}$ is
only a constant with respect to the history of the voltage, and thus is simply
a resistance. If on the other hand in eqn. (14) we have $p\neq 0$, clearly
there is a history dependence in $M$ on $j$, and the system has a memory.

### Derivation of memristance equations.

Let us now look at the effective resistance, which is

$\displaystyle\tilde{R}=\frac{V(t)}{j(t)}=R_{0}+\frac{r\Delta\rho}{M_{S}^{2}}\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(\theta))^{2}\
d\theta$ (20)

If we take a derivative with respect to time, we obtain, and define
$\Delta\vec{\boldsymbol{M}}=-\gamma\vec{\boldsymbol{H}}_{\text{eff}}+\frac{\alpha}{M_{s}}\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial
t}-\frac{\xi\nu}{M_{\text{S}}r}\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}-\frac{\nu}{M_{S}^{2}r}\vec{\boldsymbol{M}}_{t}(\theta)\times\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}$

$\displaystyle\partial_{t}\tilde{R}$ $\displaystyle=$
$\displaystyle\frac{r\Delta\rho}{M_{S}^{2}}\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(\theta))(\hat{\boldsymbol{T}}(\theta)\cdot\partial_{t}\vec{\boldsymbol{M}}_{t}(\theta))\
d\theta$ (21) $\displaystyle=$
$\displaystyle\frac{r\Delta\rho}{M_{S}^{2}}\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(\theta))(\hat{\boldsymbol{T}}(\theta)\cdot\Big{(}\vec{\boldsymbol{M}}_{t}(\theta)\times\Delta\vec{\boldsymbol{M}}\Big{)}\Big{)}\
d\theta$ $\displaystyle=$
$\displaystyle\frac{r\Delta\rho}{M_{S}^{2}}\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(\theta))(\Delta\vec{\boldsymbol{M}}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\vec{\boldsymbol{M}}_{t}(\theta)\Big{)}\Big{)}\
d\theta.$

From which we see that if the magnetization is either parallel or
perpendicular to $\hat{\boldsymbol{T}}$ the resistance does not change.
Similarly, if the vector $\Delta\vec{\boldsymbol{M}}$ is perpendicular to
either $\hat{\boldsymbol{T}}$ or $\vec{\boldsymbol{M}}$ along the ring, the
resistance does not change. The memristive effect is possible in the regime in
which
$\|\vec{\boldsymbol{H}}_{eff}\|\ll\frac{\nu}{M_{\text{S}}r\gamma}\|\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}\|$
and the system’s dissipation is small. Thus, the effective memristor equation
is given by

$\displaystyle\frac{\partial\tilde{R}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{\nu
j(t)\Delta\rho}{M_{\text{S}}^{3}}\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(\theta))\Big{(}\big{(}\xi\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}+\frac{1}{M_{S}}\vec{\boldsymbol{M}}_{t}(\theta)\times\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial\theta}\big{)}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\vec{\boldsymbol{M}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$ (22) $\displaystyle-$
$\displaystyle\frac{r\Delta\rho}{M_{S}^{2}}\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\vec{\boldsymbol{M}}_{t}(\theta))((\gamma\vec{\boldsymbol{H}}_{\text{eff}}-\frac{\alpha}{M_{s}}\frac{\partial\vec{\boldsymbol{M}}_{t}(\theta)}{\partial
t})\cdot\big{(}\hat{\boldsymbol{T}}(\theta)\times\vec{\boldsymbol{M}}_{t}(\theta)\big{)}\Big{)}\
d\theta$ $\displaystyle\equiv$
$\displaystyle\beta(\vec{\boldsymbol{M}}_{t})j(t)-\eta(\vec{\boldsymbol{M}}_{t},\vec{\boldsymbol{H}}_{eff})$

where we used the identity
$\boldsymbol{a}\cdot(\boldsymbol{b}\times\boldsymbol{c})=\boldsymbol{c}\cdot(\boldsymbol{a}\times\boldsymbol{b})=\boldsymbol{b}\cdot(\boldsymbol{c}\times\boldsymbol{a})$.
We see that we have an dynamical memristor equation in which the internal
parameter is the magnetization vector along the ring. It is interesting to
note that the geometry of the ring, which is its radius $r$, enters only in
the parameter $\alpha$ and cancels in the parameter $\beta$. We can now
rescale the value of
$\vec{\boldsymbol{M}}_{t}(\theta)=M_{S}\hat{\boldsymbol{m}}_{t}$ (or
equivalently setting $M_{S}=1$). In terms of the normalized magnetic field
$\hat{\boldsymbol{m}}(\theta)$ , we have

$\displaystyle\eta(\hat{\boldsymbol{m}}_{t},\vec{\boldsymbol{H}}_{eff})$
$\displaystyle=$ $\displaystyle
r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)(\theta))((\gamma\vec{\boldsymbol{H}}_{\text{eff}}-\alpha\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)(\theta)}{\partial
t})\cdot\big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)(\theta)\big{)}\Big{)}\
d\theta$ (23) $\displaystyle\beta(\hat{\boldsymbol{m}}_{t}(\theta))$
$\displaystyle=$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)(\theta))\Big{(}\big{(}\xi\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}+\hat{\boldsymbol{m}}_{t}(\theta)\times\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\big{)}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$

Let us now use the fact that

$\displaystyle\big{(}\hat{\boldsymbol{m}}_{t}(\theta)\times\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\big{)}\cdot\big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\big{)}$
$\displaystyle=$
$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\Big{(}\hat{\boldsymbol{m}}_{t}(\theta)\times\big{(}\hat{\boldsymbol{m}}_{t}(\theta)\times\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\big{)}\Big{)}$
(24)

We now use the identity
$\boldsymbol{a}\times(\boldsymbol{b}\times\boldsymbol{c})=(\boldsymbol{a}\cdot\boldsymbol{c})\boldsymbol{b}-(\boldsymbol{a}\cdot\boldsymbol{b})\boldsymbol{c}$,
to obtain

$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\Big{(}\hat{\boldsymbol{m}}_{t}(\theta)\times\big{(}\hat{\boldsymbol{m}}_{t}(\theta)\times\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\big{)}\Big{)}=(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t})(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{m}}_{t}(\theta))-\|\hat{\boldsymbol{m}}_{t}(\theta)\|^{2}(\hat{\boldsymbol{T}}(\theta)\cdot\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}).$
(25)

If we use that $||\hat{\boldsymbol{m}}_{t}(\theta)||^{2}=1$, we define
$t_{m}(\theta)=\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)$
and from which we see that we can rewrite the parameter $\beta$ as the
expression

$\displaystyle\frac{\beta(\hat{\boldsymbol{m}}_{t})}{\nu\Delta\rho}$
$\displaystyle=$ $\displaystyle-\xi\
\int_{0}^{2\pi}t_{m}(\theta)\Big{(}\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$ $\displaystyle-$
$\displaystyle\int_{0}^{2\pi}t_{m}(\theta)^{2}(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{m}}_{t}(\theta))d\theta.$
$\displaystyle+$
$\displaystyle\int_{0}^{2\pi}t_{m}(\theta)(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{T}}(\theta))d\theta.$
(27)

All terms above depends explicitly on $t_{m}(\theta)$, and thus if
$\hat{\vec{\boldsymbol{M}}}_{t}(\theta)\cdot\hat{\boldsymbol{T}}(\theta)=0$
everywhere (e.g. the magnetization is out of the ring’s plane only) there can
be no resistance change.

We now observe that a term proportional to $\xi$, the degree of non-
adiabaticity, the memristance change depends on
$\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}$,
while the second term which is independent from $\xi$, on
$(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{m}}_{t}(\theta))$
and
$\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{T}}(\theta)$.
For systems which are fully magnetized, however, the second term is always
zero. This can be seen by noticing that

$\displaystyle
0=\partial_{\theta}1=\partial_{\theta}\sum_{i}\hat{m}_{i}(\theta)^{2}=2\big{(}\partial_{\theta}\hat{m}_{i}(\theta)\big{)}\hat{m}_{i}(\theta)=2\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{m}}_{t}(\theta).$
(28)

We now use another identity. We have

$\displaystyle
t_{m}(\theta)(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{T}}(\theta))=\frac{1}{2}\partial_{\theta}t_{m}(\theta)^{2}-t_{m}(\theta)\big{(}\frac{\partial\hat{\boldsymbol{T}}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{m}}_{t}(\theta)\big{)}$
(29)

Clearly, since
$\frac{1}{2}\int_{0}^{2\pi}\partial_{\theta}t_{m}^{2}(\theta)d\theta=\frac{t_{m}(2\pi)^{2}-t_{m}(0)^{2}}{2}=0$
and, moreover, we have

$\displaystyle\partial_{\theta}\hat{\boldsymbol{T}}(\theta)=-\hat{\boldsymbol{R}}(\theta)$
(30)

where $\hat{\boldsymbol{R}}(\theta)=\big{(}\cos(\theta),\sin(\theta),0)$ is
the vector oriented from the center of the ring towards the point at angle
$\theta$ on the ring, we have

$\displaystyle\int_{0}^{2\pi}t_{m}(\theta)(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{T}}(\theta))d\theta$
$\displaystyle=$
$\displaystyle-\int_{0}^{2\pi}t_{m}(\theta)\big{(}\frac{\partial\hat{\boldsymbol{T}}(\theta)}{\partial\theta}\cdot\hat{\boldsymbol{m}}_{t}(\theta)\big{)}d\theta$
(31) $\displaystyle=$
$\displaystyle\int_{0}^{2\pi}t_{m}(\theta)\big{(}\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)\big{)}d\theta$

Let us call now
$r_{m}(\theta)=\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)$.

It follows that if the magnetization density is constant across the material,
and changes in direction, then the second term can be neglected. It follows
that in the approximation in which
$\|\hat{\boldsymbol{m}}_{t}(\theta)\|^{2}1$, then

$\displaystyle\frac{\beta(\hat{\boldsymbol{m}}_{t})}{\nu\Delta\rho}$
$\displaystyle=$ $\displaystyle-\xi\
\int_{0}^{2\pi}t_{m}(\theta)\Big{(}\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$ (32) $\displaystyle+$
$\displaystyle\int_{0}^{2\pi}t_{m}(\theta)r_{m}(\theta)d\theta\equiv\frac{\beta_{a}(\hat{\boldsymbol{m}}_{t})}{\nu\Delta\rho}+\frac{\beta_{na}(\hat{\boldsymbol{m}}_{t})}{\nu\Delta\rho}$
(33)

where we implicitly defined the adiabatic and non-adiabatic quantities
$\beta_{a}$ and $\beta_{na}$.

### Perturbation to external field

The effective field with (micromagnetic) exchange interaction can be written
as
$\vec{\boldsymbol{H}}_{\text{eff}}=\vec{\boldsymbol{H}}_{0}+\lambda\partial^{2}_{\theta\theta}{\vec{\boldsymbol{M}}}_{t}(\theta)-H_{sh}(\hat{\boldsymbol{m}}_{t}(\theta)\cdot\hat{\boldsymbol{T}}(\theta))\hat{\boldsymbol{T}}(\theta)$,
where $\lambda$ and $H_{sh}$ are effective parameters [37]. Let us now replace
$\vec{\boldsymbol{H}}_{eff}$ in
$\eta(\hat{\boldsymbol{m}}_{t},\vec{\boldsymbol{H}}_{eff})$.

We have, since $a\cdot(a\times b)=0$, that

$\displaystyle\eta(\hat{\boldsymbol{m}}_{t})=r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))((\gamma(\vec{\boldsymbol{H}}_{0}+\lambda
M_{S}\partial^{2}_{\theta\theta}\hat{\boldsymbol{m}}_{t}(\theta))-\alpha\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial
t})\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$ (34)

Let us now assume that $M_{t}(\theta)$ and $H_{0}$ are in plane. Then,
$\vec{\boldsymbol{H}}_{0}\cdot(\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t})=0$,
since $\hat{\boldsymbol{T}}\times\hat{\boldsymbol{m}}_{t}$ is out of plane. As
a result, for highly biasing external fields, the coefficient $\eta$ is
independent from the strength of $\vec{\boldsymbol{H}}_{0}$. We thus have

$\displaystyle\eta(\hat{\boldsymbol{m}}_{t})=r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))((\gamma(\vec{\boldsymbol{H}}_{0}+\lambda
M_{s}\partial^{2}_{\theta\theta}\hat{\boldsymbol{m}}_{t}(\theta))-\alpha\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial
t})\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$ (35)

If we instead assume that, up to a normalizing constant,
$\hat{\boldsymbol{m}}_{t}\approx\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)$,
where we assume that $\boldsymbol{P}$ is in plane and
$\vec{\boldsymbol{Q}}_{t}$ is a generic perturbation. We have

$\displaystyle\frac{\partial\tilde{R}}{\partial t}$ $\displaystyle=$
$\displaystyle-\xi\nu\Delta\rho
j(t)\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)))(\frac{\partial}{\partial\theta}(\vec{\boldsymbol{H}}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}\Big{)}\
d\theta$ $\displaystyle+$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}\big{(}\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\big{)}\big{(}\hat{\boldsymbol{R}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\big{)}$
$\displaystyle-$ $\displaystyle
r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))((\gamma\vec{\boldsymbol{H}}_{\text{eff}}-\alpha\frac{\partial}{\partial
t}(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}\Big{)}\
d\theta$ $\displaystyle=$ $\displaystyle-\xi\nu\Delta\rho
j(t)\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\big{(}\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)\big{)})(\frac{\partial}{\partial\theta}(\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}\Big{)}\
d\theta$ $\displaystyle+$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}\big{(}\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}\big{(}\hat{\boldsymbol{R}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}$
$\displaystyle-$ $\displaystyle
r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot(\vec{\boldsymbol{H}}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)))((\gamma\vec{\boldsymbol{H}}_{\text{eff}}-\alpha\frac{\partial}{\partial
t}(\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}\Big{)}\
d\theta$

where we used the fact that
$\vec{\boldsymbol{Q}}_{t}(\theta)\cdot\hat{\boldsymbol{R}}(\theta)=\vec{\boldsymbol{Q}}_{t}(\theta)\cdot\hat{\boldsymbol{T}}(\theta)=0$
if the perturbation is out-of-plane. We now keep terms that are only up to the
order $O(\epsilon^{2})$.

$\displaystyle\frac{\partial\tilde{R}}{\partial t}$ $\displaystyle=$
$\displaystyle-\epsilon\xi\nu
j(t)\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\boldsymbol{P}_{0})(\frac{\partial}{\partial\theta}(\vec{\boldsymbol{Q}}_{t}(\theta))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\boldsymbol{P}_{0}\Big{)}\Big{)}\
d\theta$ $\displaystyle+$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}\big{(}\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}\big{(}\hat{\boldsymbol{R}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}$
$\displaystyle-$ $\displaystyle
r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)))((\gamma\vec{\boldsymbol{H}}_{\text{eff}}-\alpha\frac{\partial}{\partial
t}(\epsilon\vec{\boldsymbol{Q}}_{t}(\theta)))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}\Big{)}\
d\theta$

We now replace
$\vec{\boldsymbol{H}}_{\text{eff}}=\vec{\boldsymbol{H}}_{0}+\lambda\partial^{2}_{\theta\theta}\vec{\boldsymbol{M}}_{t}=\vec{\boldsymbol{H}}_{0}+\lambda
M_{S}\partial^{2}_{\theta\theta}\hat{\boldsymbol{m}}_{t}=\vec{\boldsymbol{H}}_{0}+\epsilon\lambda
M_{s}\partial^{2}_{\theta\theta}\boldsymbol{Q_{t}}(\theta)$, where we assume
that $\boldsymbol{H_{0}}$ is in plane. Then, if we define
$\boldsymbol{G}=\gamma\lambda
M_{s}\partial^{2}_{\theta\theta}\boldsymbol{Q_{t}}(\theta)-\alpha\frac{\partial}{\partial
t}\vec{\boldsymbol{Q}}_{t}(\theta)$, we have

$\displaystyle\frac{\partial\tilde{R}}{\partial t}$ $\displaystyle=$
$\displaystyle-\xi\epsilon\nu
j(t)\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\boldsymbol{P}_{0})(\frac{\partial}{\partial\theta}(\vec{\boldsymbol{Q}}_{t}(\theta))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\boldsymbol{P}_{0}\Big{)}\Big{)}\
d\theta$ $\displaystyle+$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}\big{(}\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}\big{(}\hat{\boldsymbol{R}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}$
$\displaystyle-$ $\displaystyle
r\Delta\rho\int_{0}^{2\pi}\Big{(}\big{(}\hat{\boldsymbol{T}}(\theta)\cdot(\vec{\boldsymbol{H}}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\big{)}(\gamma\boldsymbol{H_{0}}+\epsilon\boldsymbol{G})\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}\Big{)}\
d\theta$

Let us focus on

$(\gamma\vec{\boldsymbol{H}}_{0}+\gamma\epsilon\lambda
M_{s}\partial^{2}_{\theta\theta}\boldsymbol{Q_{t}}(\theta)-\epsilon\alpha\frac{\partial}{\partial
t}\vec{\boldsymbol{Q}}_{t}(\theta))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}$
(39)

We can expand this term as,

$\displaystyle(\gamma\vec{\boldsymbol{H}}_{0}+\epsilon\boldsymbol{G})\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}_{t}(\theta))\Big{)}$
$\displaystyle=$ ~~γ→H0⋅(^T(θ)×P0)~~ $\displaystyle+$
$\displaystyle\epsilon\gamma\vec{\boldsymbol{H}}_{0}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\vec{\boldsymbol{Q}}_{t}(\theta)\Big{)}$
$\displaystyle+$
$\displaystyle\epsilon\boldsymbol{G}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\boldsymbol{P}_{0}\Big{)}$
$\displaystyle+$
$\displaystyle\epsilon^{2}\cancel{\boldsymbol{G}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\vec{\boldsymbol{Q}}_{t}(\theta)\Big{)}}$

where the first term cancels because it is identically zero since both
$\vec{\boldsymbol{H}}_{0}$ and $\boldsymbol{P}_{0}$ are assumed to be in
plane, while the last term is of order $\epsilon^{2}$. Since we assume that
$\vec{\boldsymbol{H}}_{0}$ is on plane, it follows that these terms are non-
zero only if $\vec{\boldsymbol{Q}}_{t}(\theta)$ has a component out-of-plane.
We then have

$\displaystyle\frac{\partial\tilde{R}}{\partial t}$ $\displaystyle=$
$\displaystyle-\epsilon\nu\xi
j(t)\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\boldsymbol{P}_{0})(\frac{\partial}{\partial\theta}(\vec{\boldsymbol{Q}}^{\perp}_{t}(\theta))\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\boldsymbol{P}_{0}\Big{)}\Big{)}\
d\theta$ $\displaystyle+$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}\big{(}\hat{\boldsymbol{T}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}\big{(}\hat{\boldsymbol{R}}(\theta)\cdot(\boldsymbol{P}_{0}+\epsilon\vec{\boldsymbol{Q}}^{\|}_{t}(\theta))\big{)}$
$\displaystyle-$ $\displaystyle\epsilon
r\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\boldsymbol{P}_{0})\Big{(}\boldsymbol{G}\cdot\big{(}\hat{\boldsymbol{T}}(\theta)\times\boldsymbol{P}_{0}\big{)}+\gamma\vec{\boldsymbol{H}}_{0}\cdot\big{(}\hat{\boldsymbol{T}}(\theta)\times\vec{\boldsymbol{Q}}^{\perp}_{t}(\theta)\big{)}\Big{)}\
d\theta$

where $\boldsymbol{G}=\gamma\lambda
M_{s}\partial^{2}_{\theta\theta}\boldsymbol{Q_{t}}(\theta)-\alpha\frac{\partial}{\partial
t}\vec{\boldsymbol{Q}}_{t}(\theta)$. We thus expect that, if
$\boldsymbol{H_{0}}$ is strong enough to bias the magnetization
$\hat{\boldsymbol{m}}_{t}$ along a certain direction in the same plane of
$\boldsymbol{P}_{0}$, the change in the behavior of the resistance should not
be consistent. For the normalization to be consistent, we need to choose
$\epsilon\rightarrow\frac{\epsilon}{||\boldsymbol{P}_{0}||}$.

As a result, we find that for $\beta_{a}$ and $\eta$, if both
$\vec{\boldsymbol{H}}_{0}$ and $\boldsymbol{P}_{0}$ are in the same plane, the
effect of $\vec{\boldsymbol{H}}_{0}$ is a smaller correction, and not of order
$1$. For $\beta_{na}$ instead, only perturbations in plane are relevant. We
thus predict that magnetomemristors are going to be robust to in plane
perturbations.

### Constant magnetic states and resistance

Let us note that the extra resistance is zero only if eqn. (18) is equal to
zero, e.g. if

$\displaystyle\frac{M_{y}(t,\theta)}{M_{x}(t,\theta)}=\tan(\theta).$ (41)

Assume then that $M_{x}(\theta)=M_{r}(t,\theta)\cos(\theta)$ and
$M_{y}(\theta)=M_{r}(t,\theta)\sin(\theta)$, and assume that
$M^{2}=M_{x}^{2}+M_{y}^{2}+M_{z}^{2}$ is conserved. A first-order
resistanceless state is thus given by

$\displaystyle M^{2}=M_{r}^{2}(t,\theta)+M_{z}^{2}(t,\theta),$ (42)

from which it follows that $M_{z}=M\cos(f_{t}(\theta))$ and
$M_{r}(\theta)=M\sin(f_{t}(\theta))$, for an arbitrary function of
$f_{t}(\theta)$. This implies that there is an infinite family of magnetic
configurations which has effectively zero magnetically induced resistance. As
an example, consider $M_{z}=0$. In this case we must have
$M_{x}=M\sin(\theta)$ and $M_{y}=M\cos(\theta)$. These are states pointing
towards or away from the center of the ring at all $\theta$, e.g. radial
magnetic states along the ring.

Constant magnetic states If $\hat{\boldsymbol{m}}_{t}(\theta)$ is constant
instead, we have

$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)=M_{y}\cos\theta-
M_{x}\sin(\theta).$ (43)

from which we obtain

$\displaystyle\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))^{2}\
d\theta=\pi\left(M_{x}^{2}+M_{y}^{2}\right)=\pi(1-M_{z}^{2})$ (44)

assuming that the norm is conserved. We thus see here that if the magnetic
field is perpendicular to the ring plane, then the resistance does not have
any AMR correction.

### Topological memristance protection

We now study some magnetic states within the context of the ring toy model, in
which perfect magnetic states do not lead to memristance.

#### Non-adiabatic correction

Let us first analyze the non-adiabatic correction. This is given by

$\displaystyle\beta_{na}(\hat{\boldsymbol{m}}_{t})=\int_{0}^{2\pi}t_{m}(\theta)r_{m}(\theta)$
(45)

It is easy to analyze the states for which the non-adiabatic term vanishes.
These are those for which either $t_{m}(\theta)=0$ or for which
$r_{m}(\theta)=0$.

These are given by magnetization states for which
$\hat{\boldsymbol{m}}_{t}\perp\hat{\boldsymbol{T}}(\theta)$ or
$\hat{\boldsymbol{m}}_{t}\perp\hat{\boldsymbol{R}}(\theta)$, which are
respectively given by, setting $\hat{\boldsymbol{z}}=(0,0,1)$ and introducing
the constants $a$, $b$ such that $a^{2}+b^{2}=1$,

$\displaystyle\hat{\boldsymbol{m}}_{t}^{\perp\hat{\boldsymbol{T}}}\propto
a\hat{\boldsymbol{R}}(\theta)+b\hat{\boldsymbol{z}},and$ (46)
$\displaystyle\hat{\boldsymbol{m}}_{t}^{\perp\hat{\boldsymbol{R}}}\propto
a\hat{\boldsymbol{T}}(\theta)+b\hat{\boldsymbol{z}}.$ (47)

Since $t_{m}(\theta)$ is also contained in $\eta$ and in the adiabatic
correction, it follows that states of the form
$\hat{\boldsymbol{m}}_{t}^{\perp\hat{\boldsymbol{T}}}$ are stable
memristanceless states. We stress that this is true only for the case of a
ring of zero thickness, and in the absence of dipole interactions. Indeed, as
we show in the main text, the interplay between magnetic kinks and dipoles are
the origin of the memristance in the realistic finite size simulations.

Non-adiabatic maximal and minimal states. There are however some states which
present a maximum value of $\beta$. In particular, $\beta_{na}$ would be
dominating in several experiments, given that typically $\xi\ll 1$. For this
purpose, we need to maximize
$\hat{\boldsymbol{m}}^{*}=\text{max}_{\hat{\boldsymbol{m}}}\beta_{na}(\hat{\boldsymbol{m}})$.
It is not hard to see that the maximum value that both
$t_{m}(\theta),r_{m}(\theta)$ can have is $1/2$ if
$\hat{\boldsymbol{m}}_{t}(\theta)=\pm\frac{1}{\sqrt{2}}[\hat{\boldsymbol{R}}(\theta)+\hat{\boldsymbol{T}}(\theta)]$
(non-adiabatic maximum state), while the minimum value is $-\frac{1}{2}$ if
$\hat{\boldsymbol{m}}_{t}(\theta)=\pm\frac{1}{\sqrt{2}}[\hat{\boldsymbol{R}}(\theta)-\hat{\boldsymbol{T}}(\theta)]$
(non-adiabatic minimum state). Since
$\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{T}}(\theta)=0$, we have in
both cases $\|\hat{\boldsymbol{m}}_{t}\|^{2}=1$. These states, shown in Fig.
7, correspond to a maximal rate of change of the resistivity due to the non-
adiabatic interaction, for a non-zero current. We thus obtain a reference
value for $\xi$: if $\xi\gg\frac{1}{2}$, then the non-adiabatic term
dominates, while if $\xi\ll\frac{1}{2}$ the adiabatic term dominates. Such
static states would be difficult to stabilize in a trivial ferromagnet.

Figure 7: The two magnetic states from which a maximum absolute change in
resistivity is attained in the non-adiabatic term of $\beta$.

#### Adiabatic correction

Let us consider the constant $\beta$ for certain particular helical states of
the magnetization along the ring.

$\displaystyle\beta_{a}(\hat{\boldsymbol{m}}_{t})$ $\displaystyle=$
$\displaystyle\nu\Delta\rho\int_{0}^{2\pi}(\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta))(\frac{\partial\hat{\boldsymbol{m}}_{t}(\theta)}{\partial\theta}\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}\Big{)}\
d\theta$

We recall that

$\displaystyle\mathbb{T}(\theta)=[-\sin\theta,\cos\theta,0].$ (49)

We note that

$\displaystyle\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)=\Big{(}T_{y}M_{z}-T_{z}M_{y},T_{z}M_{x}-T_{x}M_{z},T_{x}M_{y}-T_{y}M_{x}\Big{)}.$
(50)

#### x-y walls

First, we consider a magnetic configuration which corresponds to $N$ planar
Neél walls. The magnetic state is given by

$\displaystyle\hat{\boldsymbol{m}}_{t}(\theta)=\Big{(}\cos(\omega\theta),\sin(\omega\theta),0\Big{)}$
(51)

where we impose that
$\hat{\boldsymbol{m}}_{t}(0)=\hat{\boldsymbol{m}}_{t}(2\pi)$. Thus, we can
write $\omega=N$ up to translations in the $\theta$ direction. We have

$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)$
$\displaystyle=$
$\displaystyle-\sin(\theta)\cos(\omega\theta)+\sin(\omega\theta)\cos(\theta)$
(52) $\displaystyle\partial_{\theta}\hat{\boldsymbol{m}}_{t}(\theta)$
$\displaystyle=$
$\displaystyle\omega\Big{(}-\sin(\omega\theta),\cos(\omega\theta),0\Big{)}$
(53)
$\displaystyle\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)$
$\displaystyle=$
$\displaystyle-\Big{(}0,0,\cos(\theta)\cos(\omega\theta)+\sin(\theta)\sin(\omega\theta)\Big{)}$
(54)

It follows that
$\partial_{\theta}\hat{\mathbb{m}}_{t}(\theta)\cdot(\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta))=0$,
and thus $\beta_{a}(\hat{\mathbb{m}}_{t})=0$ identically for any $\omega$.

#### y-z walls

Let us consider now a Neél wall in the y-z direction.

The magnetic state is given by

$\displaystyle\hat{\boldsymbol{m}}_{t}(\theta)=\Big{(}0,\cos(\omega\theta),\sin(\omega\theta)\Big{)}$
(55)

where we impose again that
$\hat{\boldsymbol{m}}_{t}(0)=\hat{\boldsymbol{m}}_{t}(2\pi)$. We have

$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}_{t}(\theta)$
$\displaystyle=$ $\displaystyle\cos(\omega\theta)\cos(\theta)$ (56)
$\displaystyle\partial_{\theta}\hat{\boldsymbol{m}}_{t}(\theta)$
$\displaystyle=$
$\displaystyle\omega\Big{(}0,-\sin(\omega\theta),\cos(\omega\theta)\Big{)}$
(57)
$\displaystyle\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)$
$\displaystyle=$
$\displaystyle\Big{(}\cos(\omega\theta)\cos(\theta),0,-\sin(\theta)\cos(\omega\theta)\Big{)}.$
(58)

It follows that
$\partial_{\theta}\hat{\mathbb{m}}_{t}(\theta)\cdot(\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta))=0$,
and thus $\beta_{na}(\hat{\mathbb{m}}_{t})=0$ identically for any $\omega$.
Thus,

$\displaystyle\partial_{\theta}\hat{\boldsymbol{m}}_{t}(\theta)\cdot\Big{(}\hat{\boldsymbol{T}}(\theta)\times\hat{\boldsymbol{m}}_{t}(\theta)\Big{)}=-\sin(\theta)\cos^{2}(\omega\theta).$
(59)

We have

$\displaystyle\beta_{a}(\hat{\boldsymbol{m}}_{t})$ $\displaystyle=$
$\displaystyle-\nu\Delta\rho\int_{0}^{2\pi}\cos^{3}(\omega\theta)\sin(\theta)\cos(\theta)d\theta=0,\
\forall\omega$

#### Helical states.

Next, we consider a helical magnetic state along the ring, given by

$\displaystyle\hat{\boldsymbol{m}}_{t}(\theta)=(M_{x},M_{y},M_{z})$
$\displaystyle=$ $\displaystyle
m_{0}\Big{(}\sin(\phi_{1}(\theta))\cos(\phi_{2}(\theta)),\sin(\phi_{1}(\theta))\sin(\phi_{2}(\theta)),\cos(\phi_{1}(\theta))\Big{)}$

where we require that the functions $\phi_{1}=\omega_{1}\theta$ and
$\phi_{2}=\omega_{2}\theta$ to be linear, with
$\hat{\boldsymbol{m}}_{t}(0)=\hat{\boldsymbol{m}}_{t}(2\pi)$, up to some
translation in the $\theta$ direction. It follows that $\omega_{1}2\pi=n2\pi$,
and $\omega_{2}2\pi=k2\pi$ for integers $n$ and $k$. It is not hard to see
that $\hat{\boldsymbol{m}}_{t}(\theta)\cdot\hat{\boldsymbol{T}}(\theta)=0$
everywhere. This implies that the resistivity reduces to $\rho_{0}$, the
material natural resistivity. This is a special case of the memristanceless
case. Since $\beta$’s and $\eta^{\prime}s$ also depend on $t_{m}(\theta)$,
these states also lead to $\partial_{t}\tilde{\rho}=0$. Examples of these
states are shown in Fig. 8.

Figure 8: Ring magnetic states in which the resistivity $\rho_{0}$ is
protected from the magnetomemristive effect.

#### Perfect spin superflows

For spin superflows we have

$\hat{{\boldsymbol{m}}}(\theta,\phi)=\sin(\phi)\cos(n\theta)\hat{x}+\sin(\phi)\sin(n\theta)\hat{y}+\cos(\phi)\hat{z}.$
(61)

We have

$\displaystyle t_{m}(\theta)$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{T}}(\theta)\cdot\hat{\boldsymbol{m}}=-\sin(\phi)\cos(n\theta)\sin(\theta)+\sin(\phi)\cos(\theta)\sin(n\theta)$
(62) $\displaystyle r_{m}(\theta)$ $\displaystyle=$
$\displaystyle\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{m}}=\cos(\theta)\cos(n\theta)\sin(\phi)+\sin(\theta)\sin(\phi)\sin(n\theta).$
(63)

These states, do contribute to the resistivity. We have in fact that

$\displaystyle\rho=\rho_{0}+\frac{\Delta\rho}{2\pi}\int_{0}^{2\pi}d\theta\
t_{m}^{2}(\theta)=\rho_{0}+\frac{\Delta\rho}{2\pi}\left(\pi-\frac{\sin(4\pi
n)}{4(n-1)}\right)\sin^{2}(\phi)$ (64)

Clearly, for $n=1$ the latter term in equation above is zero, as $t_{m}=0$ in
that case. For any other integer $n$ however, we have

$\displaystyle\rho=\rho_{0}+\frac{\Delta\rho}{2\pi}\int_{0}^{2\pi}d\theta\
t_{m}^{2}(\theta)=\rho_{0}+\frac{\Delta\rho}{2}\sin^{2}(\phi)$ (65)

which we see depends explicitly on the angle $\phi$. We now test whether
$\beta$’s are non zero, in which case we have a memristive effect.

From which we can perform the integral analytically for $\beta_{a}$ for the
resistivity change, obtaining

$\displaystyle\beta_{a}(\hat{\boldsymbol{m}}_{t})$ $\displaystyle=$
$\displaystyle-\nu\Delta\rho\frac{1}{2\pi}\int_{0}^{2\pi}\cos^{3}(\omega\theta)\sin(\theta)\cos(\theta)d\theta=0,\
\forall\omega$ (66) $\displaystyle=$
$\displaystyle\nu\Delta\rho\frac{\sin^{2}(2\pi n)\sin^{2}(\phi)}{4\pi(n-1)}$

from which we see that if $n$ is integer, we have
$\beta_{a}(\hat{\boldsymbol{m}}_{t})=0$. This implies that while we have an
AMR effect contributing to a static resistance, there is no memristive effect.

#### Localized kink

We now discuss a localized magnetic kink, which is a model for the 2DW state.

This state is sum of two magnetization vectors,

$\displaystyle\hat{\boldsymbol{m}}(\theta)=a(\theta)\hat{\boldsymbol{T}}(\theta)+b(\theta)\hat{\boldsymbol{K}}(\theta)$
(67)

where we assume $\boldsymbol{K}\cdot\hat{\boldsymbol{T}}=0$, and thus
$|\hat{\boldsymbol{m}}|^{2}=a(\theta)^{2}+b(\theta)^{2}=1$.

We assume, in particular, that $a(\theta)$ and $b(\theta)$ are non-zero over
almost non-overlapping regions, meaning that the magnetic state transitions
over a localized region from the ground state (which is
$\hat{\boldsymbol{T}}$) to $\boldsymbol{K}$, which is why we call this state a
localized kink. A state formally of this type is obtained from the relaxation
of domain walls in the extended system in the main text.

It is not hard to see that the resistive state is connected to
$t_{m}(\theta)=a(\theta)$, and

$\displaystyle\tilde{\rho}=\rho_{0}+\frac{\Delta\rho}{2\pi}\int_{0}^{2\pi}d\theta\
a(\theta)^{2}.$ (68)

Then, we assume that
$r_{m}(\theta)=b(\theta)\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{K}}(\theta)$
is non-zero. Then

$\displaystyle\beta_{a}=-\frac{\nu\Delta\rho}{2\pi}\int_{0}^{2\pi}a(\theta)b(\theta)\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{K}}(\theta)\
d\theta.$ (69)

From the equation above, we see that if the transition from $a(\theta)=1$ to
$b(\theta)=1$ is smooth, as long as
$\hat{\boldsymbol{R}}(\theta)\cdot\hat{\boldsymbol{K}}(\theta)\neq 0$, then
there can be a memristive behavior. This is in essence why the 2DW state can
have both AM resistance and memristance, unlike a perfect spin superfluid.

## Further details on numerical simulations

### Thickness dependence in micromagnetic simulations

It is understood that the high aspect ratio of the micromagnetic cells can
significantly affect the results when the magnetic textures are potentially
distorted. For low current densities, when the domain walls are approximately
rigid, the qualitative behavior remains unchanged. For example, we reproduce
the AMM in a ring of thickness $5$ nm (as opposed to $10$ nm in the main
text), a current density of $10^{11}$ A m-2, and a frequency $f=1$ GHz. The
resulting Lissajous curves are shown in Fig. 9, where we immediately observe
the AMM effect. The quantitative difference in the curves may be associated in
part with the cell’s aspect ratio but we must stress that the reduced magnetic
volume will also decrease the magnitude of the non-local dipole and, in turn,
this will lead to a reduced out-of-plane magnetization components in the
domain wall (helical distortion) that ultimately gives rise to memristance.

Figure 9: Lissajous curves for rings with $10$ nm and $5$ nm thickness. The
current in the $5$ nm is smaller because of the reduced cross-area by a factor
$2$. Despite this difference, the maximum voltage has a similar magnitude.

### Dependence on the ring radius

The ring radius can play an important role in the memristance, as seen by the
inverse proportionality of $\beta$ in Eqs. (5). While exploring the effect of
varying $r$ can be considered to be a project on its own, initial simulations
suggest that indeed the inverse proportionality trend holds. We maintained the
ring width to $10$ nm throughout the simulations and modified the average ring
radius $r$. For $r=45$ nm the largest change in $d\tilde{\rho}/dt$ was $22.2$
k$\Omega$ s-1 while for $r=300$ nm we obtained a change of $0.58$ k$\Omega$
s-1. However, the Lissajous area in the I-V characteristics appeared to vary
only slightly on the order of $1.6$ $\mu$W. This is encouraging for
experiments where the effect could be measured for much larger rings that can
be realistically patterned with traditional lithography techniques.