Memristive Ising Circuits Vincent J. Dowling and Yuriy V. Pershin Department of Physics and Astronomy University of South Carolina Columbia SC 29208 USA

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Memristive Ising Circuits

Vincent J. Dowling and Yuriy V. Pershin∗

Department of Physics and Astronomy, University of South Carolina, Columbia, SC 29208 USA

The Ising model is of prime importance in the ﬁeld of statistical mechanics. Here we show that

Ising-type interactions can be realized in periodically-driven circuits of stochastic binary resistors

with memory. A key feature of our realization is the simultaneous co-existence of ferromagnetic

and antiferromagnetic interactions between two neighboring spins – an extraordinary property not

available in nature. We demonstrate that the statistics of circuit states may perfectly match the ones

found in the Ising model with ferromagnetic or antiferromagnetic interactions, and, importantly, the

corresponding Ising model parameters can be extracted from the probabilities of circuit states. Using

this ﬁnding, the Ising Hamiltonian is re-constructed in several model cases, and it is shown that

diﬀerent types of interaction can be realized in circuits of stochastic memristors.

I. INTRODUCTION

The utilization of electronic circuits as an analog to

other physical systems is becoming more and more preva-

lent. It has been recently shown that certain circuits

comprised of only capacitors and inductors [1,2] as well

as circuits combining passive resistive [3,4] or active [5]

components with capacitors and inductors can be used

to realize the same states that are found in topological

phases in condensed matter [6–9], forming a connection

between two, otherwise, distinct systems. For instance,

in the topoelectric Su-Schrieﬀer–Heeger (SSH) circuit [1]

the boundary resonances in the impedance are reminis-

cent of edge states in the SSH model. Here, we introduce

a circuit of stochastic memristors exhibiting the same

statistics of states as in the Ising model.

While the concept of constructing an electric analog

to the Ising model is not novel [10–18] and gaining in-

creasing attention in the context of building Ising ma-

chines [11–18], our approach is. The basic idea is as fol-

lows. We use a resistor and stochastic memristor con-

nected in-series as a memristive spin (Fig. 1(a)), and

couple memristive spins by resistors to induce their in-

teractions (see Fig. 1(b) for the circuit considered in

this Letter). It is assumed that the stochastic mem-

ristor can be found in one of two states, RON and

ROF F (such that RON < ROF F ), and the switching

between these states occurs probabilistically and is de-

scribed by voltage-dependent switching rates (the details

of the model are given below). The circuit is subjected to

alternating polarity pulses that drive the memristive dy-

namics. The states of memristors are read each period of

the pulse sequence (say, at the end of the negative pulse)

and the probabilities of these states are determined. We

note that the circuit in Fig. 1(b) but with deterministic

memristors was introduced in Ref. [19], and a mean-ﬁeld

model of memristive interactions in a similar (but not the

same) deterministic circuit was developed in [20].

Using numerical simulations, we have found that our

circuit is capable of exhibiting an analogous type of or-

∗pershin@physics.sc.edu

dering in memristor conﬁgurations as of those found in

magnetic materials. Meaning, there can exist a strong

bias for a speciﬁc circuit to exist in an antiferromagnetic

(AFM) memristor conﬁguration (−RON −ROF F −RON −

ROF F −) or an ferromagnetic (FM) memristor conﬁgura-

tion (−RON −RON −RON −RON −or −ROF F −ROF F −

ROF F −ROF F −). In fact, a very important aspect of our

circuit is the simultaneous co-existence of AFM and FM

interactions between two neighboring spins. The goal of

this work is to demonstrate the possibility of the stan-

dard magnetic orderings (AFM and FM) in the memris-

tive Ising circuits.

This paper is organized as follows. In Sec. II we intro-

duce the Ising Hamiltonian, stochastic model of mem-

(a)

RM,1=

ROFF (RON)

=( )

(c)

(b)

RM,1 RM, 2 RM, 3 RM, N

V(t)

RR R

FIG. 1. (a) Memristive spin sub-circuit: the high- and

low-resistance states of stochastic memristor correspond to

spin-down (0) and spin-up (1) states, respectively. (b) One-

dimensional memristive Ising circuit with a periodic boundary

condition. Here, r-s denote the resistance of coupling resis-

tors. (c) The scheme of interactions in the Ising Hamiltonian.

arXiv:2210.04257v1 [cond-mat.mes-hall] 9 Oct 2022

risotrs, and make the connection between the statistical

properties of the circuit and ones of the Ising Hamilto-

nian. In the same section, we brieﬂy discuss the nu-

merical approach used in our work. The results of our

simulations are presented in Sec. II with the emphasise

on the possibility of reaching FM and AFM interactions

in the circuit. The paper ends with a conclusion.

II. METHODS

Mathematically, we utilize an eﬀective Ising-type

Hamiltonian to describe the probabilities observed in

the circuit simulations. For the circuit in Fig. 1(b), the

Hamiltonian has the form

H=−JX

σiσi+1 −J2X

σiσi+2 −hX

σi,(1)

where Jis the interaction coeﬃcient for adjacent spins,

J2is the next-to-adjacent interaction, his the magnetic

ﬁeld, and periodic coupling is assumed. Schematically,

these interactions are presented in Fig. 1(c). We consider

the electronic circuit as a physical system described by

the Boltzmann distribution

pi=1

Ze−Ei

kT .(2)

Here, Z=P

e−Ej

kT is the statistical sum, and Ej-s are the

“energies” of circuit states. We argue that for the circuit

in Fig. 1and similar circuits these “energies” correspond

to the Ising Hamiltonian (1).

To explain the co-existence of AFM and FM interac-

tions, consider a set of identical memristors in ROF F sub-

jected to a positive voltage pulse driving the OFF-to-ON

transition. Each memristor will have an equivalent prob-

ability of being the ﬁrst to switch states. When one of

these memristors swaps states, it reduces the probability

to switch of its neighbors (reducing the voltage across

them). In this scenario, memristors with neighbors both

in the ROF F state will have the highest chance of switch-

ing. This leads to the tendency of antiferromagnetic or-

dering in the memristors under a positive voltage pulse.

However, under a negative voltage pulse the RON state

memristors with neighboring ROF F state memristors will

be favored to change states. Meaning, the conﬁguration

will tend towards ferromagnetic ordering under a nega-

tive voltage pulse. The overall ordering of a memristive

circuit driven by an AC source will then be dependent

on the choice of model parameters for the memristors.

Based on the parameters, one type of ordering may be

dominant.

Next, we introduce the model of stochastic memris-

tors. According to experiments with certain electrochem-

ical metallization (ECM) cells [21,22] and valence change

memory (VCM) cells [23], the probability of switching be-

tween resistance states of these devices can be described

1000 1500 2000 2500 3000

1 0

1 2

1 4

1 6

S t a t e

T i m e ( a r b . u n i t s )

FIG. 2. Dynamics of the states in a circuit with N= 4

memristive spins. The circuit has 24= 16 states that are

labeled from 0 to 15. The 0000 state (all memristors are in

ROF F ) is labeled by 0, 0001 by 1, and so on. This plot was

obtained using the following set of parameters: R=r=

ROF F = 1 kΩ, RON = 100 Ω, τ01 = 3 ·105s, τ10 = 160 s,

V01 = 0.05 V, V10 = 0.5 V, Vpeak = 1 V, and T= 2 s.

by switching rates of the form

γ0→1(V) = τ01e−V /V01 −1, V > 0

0,otherwise ,(3)

γ1→0(V) = τ10e−|V|/V10 −1, V < 0

0,otherwise ,(4)

where Vis the voltage across the device, and τ01(10) and

V01(10) are device-speciﬁc parameters. Here, 0 and 1 cor-

respond to the high (ROF F ) and low (RON ) resistance

states, respectively. Under a constant voltage, the prob-

ability to switch follows the distribution [21,22]

P(t) = ∆t

τ(V)e−t/τ(V),(5)

where τ(V) is the inverse of the switching rate given by

Eq. (3) or (4) (depending on the sign of V). Previously,

we have developed a master equation approach for the

circuit of stochastic memristors [24] and designed its im-

plementation in SPICE [25].

Most of the results presented here are obtained through

numerical simulations of the circuit in Fig. 1(b) contain-

ing Nmemristive spins. The set of parameters deﬁning

the circuit and the simulations such as the model con-

stants, voltage period, duration, resistances, etc. are ﬁrst

set. The memristors are then initialized to their starting

states (typically all OFF). The voltages across each mem-

ristor are calculated for the current time-step through

Kirchhoﬀ’s laws. The switching time is then generated

for each memristor randomly with Eq. (5) distribution.

The fastest switching time is extracted and compared to

the remaining time in the current voltage pulse. If there

is suﬃcient time remaining in the pulse, that memristor

switches states and the the time remaining in the period

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MemristiveIsingCircuitsVincentJ.DowlingandYuriyV.Pershin*DepartmentofPhysicsandAstronomy,UniversityofSouthCarolina,Columbia,SC29208USATheIsingmodelisofprimeimportanceintheeldofstatisticalmechanics.HereweshowthatIsing-typeinteractionscanberealizedinperiodically-drivencircuitsofstochasticbinaryresist...

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Memristive Ising Circuits Vincent J. Dowling and Yuriy V. Pershin Department of Physics and Astronomy University of South Carolina Columbia SC 29208 USA

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