Non-reciprocal interactions spatially propagate uctuations in a 2D Ising model Daniel S. Seara

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Non-reciprocal interactions spatially propagate

ﬂuctuations in a 2D Ising model

Daniel S. Seara

James Franck Institute, University of Chicago, Chicago IL, USA

Akash Piya

Thomas Jeﬀerson High School for Science and Technology, Alexandria VA, USA

A. Pasha Tabatabai

E-mail: To whom correspondence should be addressed:

tabatabai@seattleu.edu

Department of Physics, Seattle University, Seattle WA, USA

Abstract. Motivated by the anisotropic interactions between ﬁsh, we implement

spatially anisotropic and therefore non-reciprocal interactions in the 2D Ising model.

First, we show that the model with non-reciprocal interactions alters the system

critical temperature away from that of the traditional 2D Ising model. Further,

local perturbations to the magnetization in this out-of-equilibrium system manifest

themselves as traveling waves of spin states along the lattice, also seen in a mean-

ﬁeld model of our system. The speed and directionality of these traveling waves are

controllable by the orientation and magnitude of the non-reciprocal interaction kernel

as well as the proximity of the system to the critical temperature.

arXiv:2210.11229v3 [cond-mat.soft] 5 Apr 2023

Non-reciprocal interactions spatially propagate ﬂuctuations in a 2D Ising model 2

1. Introduction

When two objects come into contact with each other, the macroscopic forces that

each generate on the other are described by Newton’s Third Law and are equal in

magnitude. While the reciprocity of this type of interaction is common, there are many

instances where interactions between objects are non-reciprocal and lead to interesting

behaviors. For example, metamaterials that exhibit broken symmetries in the bonds

between constituents yield asymmetric responses to mechanical [1,2,3] and optical

waves [4,5] as well as ﬂuid/solid behavior [6,7]. Non-reciprocal interactions are also

used as design principles for sensor optimization [8].

Hallmark examples of non-reciprocal interactions occur within the collective

behavior of animal groups such as locusts, birds, and ﬁsh. In these systems,

visual information diﬀerences between animals lead to this non-reciprocal interaction.

While incorporating non-reciprocal interactions are not required to capture ﬂocking

behavior [9], there is a recent push towards understanding the eﬀects of non-reciprocal

interactions on phase behavior [10,11,12] and the non-trivial motion of ﬂocking

objects [13].

In particular, we are interested in how ﬂuctuations in the local polarization within a

ﬂock are propagated through space as a consequence of these non-reciprocal interactions.

To this end, we simplify the problem and study non-reciprocal interactions within a 2D

Ising model that are motivated by the anisotropic ﬁeld of view within animals such as

ﬁsh, consistent with recent eﬀorts to understand the eﬀects of non-reciprocal interactions

within the continuum Vicsek model [10,12]. This model is not equivalent to the

active Ising model [14,15] since objects are not free to move on the lattice and lack

self-propulsion. As a consequence, our model decouples the non-equilibrium eﬀects of

introducing a non-reciprocal interaction and active energy consumption. We note that

most non-reciprocal interactions within studies of ﬂocking focus on so-called ‘vision-

cones’ where object orientation inﬂuences interactions. Our interaction is a simpliﬁed

version of a vision-cone which does not change orientation.

Our results are presented in three parts. First, the model is introduced and

the inﬂuence of the non-reciprocal interaction on the phase behavior of the system

is described in detail. Comparisons are made to the equilibrium 2D Ising model. Then,

we characterize the spatial propagation of ﬂuctuations that originate from the presence

of this non-reciprocal interaction. We introduce a mean-ﬁeld model which predicts the

propagation of spin ﬂuctuations under these parity-breaking interactions. We show

that this propagation is robust to changes in the algorithm used to generate the system

dynamics. Finally, we ﬁnd that the propagation velocity is maximized near the critical

temperature.

Non-reciprocal interactions spatially propagate ﬂuctuations in a 2D Ising model 3

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Figure 1. Size and Oﬀset Eﬀects on System Magnetization. (a) Magnetization

per spin M/N as a function of temperature Tfor lattices with L= 8 (red), L= 16

(black), L= 32 (blue), and L= 64 (magenta). (a-inset) Representation of a standard

Ising Model interaction kernel. Neighbors of the blue cell are deﬁned as the four red

lattice sites. (b) Magnetization per spin as a function of temperature for oﬀset ∆ = 2

with symbol colors equivalent to (a). (b-inset) Representation of an non-reciprocal

interaction kernel with an oﬀset ∆ = 2 lattice sites. Arrows point in the direction of

increasing system size. (c) Magnetization per spin M/N as a function of temperature

Tfor lattices of size L= 64 at diﬀerent values of oﬀset ∆ = 0 (red), ∆ = 2 (black),

∆ = 4 (blue), ∆ = 8 (magenta), and ∆ = 10 (cyan). (d) Magnetic susceptibility per

spin χ/N for data in (c).

2. Methods

We consider a 2D Ising model on a square lattice of dimension L×Lwith N=L2

spins where the system at a time tis in a conﬁguration denoted by the vector s(t)

with elements si=±1, where i∈Z2indexes the lattice site [16]. In order to generate

spin dynamics on the lattice, we use the energy of an individual spin si, calculated as

[17,18,19]

Ei(s) = −siX

Jijsj(1)

where Jij is an element of an interaction matrix which couples spins iand j.

Note that this is related, but distinct, from the traditional Ising model Hamiltonian,

given by E= (PiEi)/2 [20]. Importantly, while Ei(s) is not an energy functional, we

nevertheless have ∆E= ∆Eiwhen Jij =Jji (see Appendix A for details). When this

symmetry condition on the interaction matrix is satisﬁed, we call the system reciprocal.

Otherwise, if Jij 6=Jji, we call the system non-reciprocal [21].

Non-reciprocal interactions spatially propagate ﬂuctuations in a 2D Ising model 4

To begin, we study an equilibrium reciprocal Ising model where spins iand j

interact with strength Jij =Jif they are nearest neighbors, otherwise Jij = 0 (Fig. 1a-

inset). We can write the interaction matrix as

Jij =J(δi−ˆex,j +δi+ˆex,j +δi−ˆey,j +δi+ˆey,j ).(2)

In the above, ˆeαdenotes the unit vector in the αdirection, with α∈ {x, y}. Unless

otherwise stated, we set J= 1.

Lattices are placed in contact with a heat bath at a temperature T, with

units of the Boltzmann constant kB(Supplemental Movie 1). Assuming periodic

boundary conditions, the lattices are evolved towards their thermodynamic equilibrium

conﬁgurations using a Metropolis Monte Carlo method (Appendix A). Brieﬂy, individual

spins in the lattice are chosen at random and the energy cost/gain of ﬂipping the spin as

given by the variation of Eq. 1determines the probability of ﬂipping the spin as shown

P(s→Fis) = min 1, eβ∆Ei(3)

where Fiis an operator that takes si7→ −siand ∆Eiis given by Eq. A.3 in Appendix

A[22].

We deﬁne a ‘sweep’ as a proxy for time; in a single sweep of a lattice with Nlattice

sites, Nsites are randomly selected sequentially and have the possibility of ﬂipping their

spin. Systems are brought to a steady-state conﬁguration and ensemble measurements

of the system magnetization are made of independent conﬁgurations (Appendix B).

For a given lattice, we calculate the system magnetization from the sum of all spins

si,M=Pisi, which exhibits the expected temperature and system-size dependence

previously described (Fig. 1a) [22]. Since the s7→ −ssymmetry is not broken by the

existence of an external ﬁeld, we quote the absolute value of the magnetization.

Next, we amend the traditional Ising model interaction matrix and introduce a

non-reciprocal interaction (Jij 6=Jji) in Eq. 1. Inspired by the non-local information

processing of active systems, such as ﬁsh within a school or starlings within a

ﬂock [23,24,25,10], we deﬁne an oﬀset vector ∆∈Z2that represents a spatial

translation of the interaction kernel (Fig. 1b-inset),

∆= ∆xˆex+ ∆yˆey,(4)

where ∆α∈Zare integers. The interaction matrix’s elements are now given by

Jij =J(δi+∆−ˆex,j +δi+∆+ˆex,j +δi+∆−ˆey,j +δi+∆+ˆey,j )(5)

We choose to keep ∆constant and uniform throughout space, speciﬁcally ∆≡∆ˆex, and

∆ is not dependent on the sign of the lattice spin. These unidirectional interactions lead

to qualitatively similar temperature and size-dependence of the system magnetization

(Fig. 1b). Note that the traditional Ising model is represented by ∆ = 0, and we keep a

constant number of interactions between spins (4) to keep the connectivity of the lattice

constant for all values of ∆.

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Non-reciprocalinteractionsspatiallypropagateuctuationsina2DIsingmodelDanielS.SearaJamesFranckInstitute,UniversityofChicago,ChicagoIL,USAAkashPiyaThomasJeersonHighSchoolforScienceandTechnology,AlexandriaVA,USAA.PashaTabatabaiE-mail:Towhomcorrespondenceshouldbeaddressed:tabatabai@seattleu.eduDepartme...

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Non-reciprocal interactions spatially propagate uctuations in a 2D Ising model Daniel S. Seara

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