The BKT Transition and its Dynamics in a Spin Fluid Thomas Bissingeraand Matthias Fuchsb Fachbereich Physik Universit at Konstanz 78457 Konstanz Germany

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The BKT Transition and its Dynamics in a Spin Fluid

Thomas Bissingera) and Matthias Fuchsb)

Fachbereich Physik, Universit¨at Konstanz, 78457 Konstanz, Germany

(Dated: 6 October 2022)

We study the eﬀect of particle mobility on phase transitions in a spin ﬂuid in two dimensions. The presence

of a phase transition of the BKT universality class is shown in an oﬀ-lattice model of particles with purely

repulsive interaction employing computer simulations. A critical spin wave region 0 < T < TBKT is found

with a non-universal exponent η(T) that follows the shape suggested by BKT theory, including a critical

value consistent with ηBKT = 1/4. One can observe a transition from power-law decay to exponential decay

in the static correlation functions at the transition temperature TBKT, which is supported by ﬁnite-size scaling

analysis. A critical temperature TBKT = 0.17(1) is suggested. Investigations into the dynamic aspects of the

phase transition are carried out. The short-time behavior of the incoherent spin autocorrelation function

agrees with the Nelson-Fisher prediction, whereas the long-time behavior diﬀers from the ﬁnite-size scaling

known for the static XY model. Analysis of coherent spin wave dynamics shows that the spin wave peak is a

propagating mode that can be reasonably well ﬁtted by hydrodynamic theory. The mobility of the particles

strongly enhances damping of the spin waves, but the model lies still within the dynamic universality class

of the standard XY model.

I. INTRODUCTION

The Berezinskii-Kosterlitz-Thouless (BKT)

transition1–3is ubiquitous in two-dimensional (2D)

systems with a continuous symmetry. The symmetry

cannot be broken in a conventional sense due to the

Mermin-Wagner theorem,4i.e. the mean order pa-

rameter must vanish at all ﬁnite temperatures. Phase

transitions driven by topological excitations (like vortices

for magnetic systems5or lattice defects for crystals6)

circumvent this constraint. The excitations are bound at

low temperatures, renormalizing the coupling constants

of the system, and become unbound at a transition

temperature TBKT.7A characteristic feature of this type

of transition is the spatial decay of order parameter ﬂuc-

tuations, which changes from a power-law behavior at

all low temperatures to an exponential one above TBKT.

In the language of critical phenomena, this is described

by a continuous line of critical points in 0 < T ≤TBKT.

When crossing TBKT, the stiﬀness constant Kof the

eﬀective Gaussian Hamiltonian displays a jump value

from KBKT = 2/π to zero.7,8

A paragon example of this behavior is the classi-

cal XY model of planar rotators on a 2D lattice, also

known as the O(2) model. In fact, the model goes by

a lot of names and some authors distinguish the rotator

model from the XY model as a quantum model for an

easy-plane Heisenberg ferromagnet with the typical spin

algebra.9,10 We will use the term XY model synonymous

with the rotator model,11 the key feature is the sym-

metry of the relevant order parameter. Kosterlitz theo-

retically predicted the BKT scenario for XY systems.2,3

a)Electronic mail: thomas.bissinger@uni-konstanz.de

b)Electronic mail: matthias.fuchs@uni-konstanz.de

Further theoretical investigations and simulations con-

ﬁrmed this prediction.12–14 The Mermin-Wagner theorem

only holds in the thermodynamic limit of an inﬁnitely

extended system, as only there inﬁnitely long-ranged

Nambu-Goldstone modes suppress global order. Concur-

rently, the BKT transition in the XY model shows strong

ﬁnite-size scaling, which was analyzed by an RG treat-

ment and compared to simulation and experiment by

Bramwell, Holdsworth et al.15–17 They predicted ﬁnite-

size scaling of the magnetization with a universal ﬁnite-

size exponent β≈0.23, which is in agreement with ex-

perimental observation.18

These early results on the static properties of the clas-

sical XY model were soon supplemented by discussions

of the dynamics. Villain5put forward an approach to

studying the hydrodynamics of spin waves on which Nel-

son and Fisher9based their “ﬁxed-length“ hydrodynamic

theory of dynamical critical scaling for easy-plane Heisen-

berg magnets. They predict divergent spin-wave peaks,

a projection operator approach due to Menezes et al.19

arrives at a similar conclusion. The Nelson-Fisher predic-

tions are also partially supported by works by Lepri and

Ruﬀo20, who conﬁrmed their short-time validity but also

discovered further universal ﬁnite-size features in a rota-

tor model. Mertens et al.21 also investigated the spin dy-

namics of easy-plane Heisenberg magnets and predicted

spin waves with a central peak and a propagating peak

at low temperature that features a ﬁnite damping in the

q→0 limit. In an extensive numerical study, Evertz

and Landau10 analyzed the neutron scattering function

of the XY model, conﬁrming the short-time prediction of

the Nelson-Fisher approach while discovering structure

beyond a single spin wave peak, including a central peak

and ﬁne-structure which they hypothesized to be due to

scattering of multiple spin waves.

Mobile XY (MXY) models, also called spin ﬂuids, pro-

arXiv:2210.01838v1 [cond-mat.stat-mech] 4 Oct 2022

vide oﬀ-lattice counterparts to the standard XY model

discussed above.22,23 They have been employed as phe-

nomenological classical models to describe liquid-vapor

interfaces24 or the eﬀect of phonon-magnon couplings on

the dynamics of bcc iron25. With the current interest

in critical phenomena of active matter26,27, such systems

are also interesting as inactive counterparts to models for

living systems like the famed Vicsek model28. Recently,

spin-ﬂuids were studied by Casiulis et al.29–31 They found

a ferromagnetism-induced phase separation (FIPS) re-

placing the BKT scenario. Beside the lattice-free spin

ﬂuids, lattice-gas generalizations of the XY model have

also been studied, and BKT transitions were observed in

lattice simulations.11

In this paper, we argue that in an equilibrium 2D MXY

model with purely repulsive interaction, no phase separa-

tion occurs at the transition temperature and the system

undergoes a BKT-type transition at a critical tempera-

ture TBKT. To that end, we employ microcanonical MD

simulations up to N= (256)2particles. We compare

data of a mobile model to that of a similar model with

disordered yet ﬁxed particle positions, which we call a

disordered XY (DXY) model. We will investigate the

static properties of these models as well as their dynam-

ical features, which are aﬀected by the mobility of the

spins.

In Section II, we present the mobile XY model, dis-

cussing some key properties, including the interaction po-

tential, as well as introducing the thermodynamic quanti-

ties we use for the analysis. In Section III, we give a short

overview over the simulation parameters and some details

of the implementation, then the results of the simulation

are presented in Section IV. We will consider the prop-

erties of the total magnetization and its susceptibility in

Section IV A 1, the eﬀect of structural order in Section

IV A 2, then study the eﬀect of ﬁnite size on static spin

correlation functions in real and reciprocal space in Sec-

tions IV A 3 and IV A 4, respectively. Considering the dy-

namic properties of the system, we compare data for the

incoherent spin autocorrelation function to the Nelson-

Fisher9and Lepri-Ruﬀo20 results in Section IV B 1. Fi-

nally, we study the coherent spin autocorrelation function

in Section IV B 2 and their compatibility with a damped

oscillator ﬁt to the spin wave peak. Concluding remarks

are given in Section V.

II. DEFINITIONS

This section summarizes the main deﬁnitions used

throughout the paper. We introduce the model and then

discuss various quantities that are relevant in the study of

the critical phenomena. We conclude with a short discus-

sion regarding the distinction between longitudinal and

transversal ﬂuctuations.

A. The MXY and the DXY Model

Our main focus is on the MXY model, or spin ﬂuid22,29,

with Hamiltonian

H=X

ω2

2I−X

j6=k

J(rjk) cos(θjk)

2m+X

j6=k

U(rjk).

(1)

This describes Nisotropic particles with positions and

momenta {rj,pj}(bottom line) in 2D supplemented by

a spin interaction depending on the interparticle dis-

tance. Spins are two-dimensional unit vectors si=

(cos(θi),sin(θi))|with angle −π < θi≤π. We use

the shorthand rjk =rk−rjand similarly for θjk, and

rjk =|rjk|. After additionally deﬁning Ujk =U(rjk)

and Jjk =J(rjk), we can write the resulting equations

of motion as

dtθj=ωj

I,d

dtrj=pj

dtωj=X

k6=j

Jjk sin(θjk)

dtpj=−X

k6=j

∇rj[Ujk −Jjk cos(θjk)] .

(2)

We choose soft interactions for the two interaction po-

tentials J(r) and U(r), giving them the form

J(r) = J0(σ−r)2Θ(1 −r/σ),

U(r) = U0(σ−r)2Θ(1 −r/σ),(3)

with energy scales J0and U0, an interaction range σand

the Heaviside function Θ. Throughout this paper, dimen-

sionless quantities are used, which we obtain by choosing

scales in the system such that we can set m= 1, J0= 1

and σ= 1 in (1), (2) and (3). The remaining parameters

of the system are then the particle number N, the (di-

mensionless) density ρ=Nσ2/L2, the ratio U0/J0and

the coeﬃcient I/(mσ2). For our simulations, we will use

various Nat ﬁxed ρ= 2.99, and set U0/J0= 4 as well

as I/(mσ2) = 1. Temperatures will be given in units of

energy as well, that is kB= 1.

Note an important diﬀerence between the interaction

potentials (3) and those of the spin ﬂuids and Hamil-

tonian polar particles recently studied by Casiulis et

al.Casiulis et al.29–32 While having the same J(r), they

chose U(r) = U0(σ−r)4Θ(σ−r). In our case, the overall

interaction potential, U(r)−J(r) cos(θ), is repulsive ev-

erywhere for all choices of rand θbecause of the shared

square power (σ−r)2in the spatial dependence in (3).

In contrast, Casiulis et al. have an attractive region at r

close to σfor spins with cos(θ)>0. This diﬀerence has

important consequences for the model: While Casiulis

et al. ﬁnd the system dominated by a melting transition

that suppresses the BKT transition, we ﬁnd BKT physics

in the absence of melting.

We are particularly interested in the eﬀect of mobility

on the dynamic aspects of the phase transition. To iso-

late the eﬀect of spatial disorder of the particles, we will

sometimes work with what we call a disordered XY model

(DXY), obtained by equilibrating the MXY model and

then freezing the particles’ positions and only allowing

for the spin dynamics of the Hamiltonian (1).

B. Basic Diagnostic Tools

To analyze the static properties, the standard order

parameter is the magnetization per particle,

m=1

sj(4)

and we will be interested in its modulus m=|m|. Asso-

ciated to mis a susceptibility per spin33

χm=βN Dm2− hmi2E=βNσ2

m(5)

with the variance σ2

m. Equally interesting is the maxi-

mum value of the variance

σ2

max = max

Tσ2

m(6)

Both χmand σ2

max exhibit useful ﬁnite-size properties

that are deemed characteristic of the BKT transition.33

The spin stiﬀness Kis related to the helicity modulus

Υ via K=βΥ. Υ describes the elasticity of the spin

alignment to twist distortions. Explicit formulae for Υ

can be derived from the response of the system’s free

energy to a twist ﬁeld,34 cf. Appendix A. For the MXY

model, the standard relation35

Υ = 1

2AhHx+Hyi − βI2

x+I2

y.(7)

with A=LxLy=L2can be generalized according to the

deﬁnitions36

Hx=1

i6=j

J(rij ) cos(θij )x2

ij ,

Ix=1

i6=j

J(rij ) sin(θij )xij ,

(8)

with xij =rij ·exthe x-component of the distance vector

rij . Analogous relations hold for Hyand Iy.36

We will further consider the magnetization and spin

momentum densities

m(r) = 1

√NX

sjδ(r−rj),

w(r) = 1

√NX

ωjδ(r−rj)

(9)

and their collective ﬂuctuations

δmq=1

√NX

(sj−m)e−iq·rj,

δwq=1

√NX

ωje−iq·rj.

(10)

Note that m(r) has a diﬀerent normalization than m.

C. Longitudinal and Transversal Magnetization

Fluctuations

In the deﬁnition (10), we encounter a peculiarity of

the ﬁnite size behavior of XY type systems. While in

an inﬁnitely expanded system, hmi= 0 by the Mermin-

Wagner theorem (which holds true for mobile particles

as well36), ﬁnite systems exhibit a well-deﬁned value of

hmi 6= 0 at low temperatures. This is readily seen in

Figure 1for the MXY model, and similar results have

been reported for the standard XY model.33,37 At low

temperatures, the modulus of the magnetization ﬂuctu-

ates around an average value, while its orientation dif-

fuses over the circle with time. Increasing the system

size leads to smaller ﬂuctuations around the mean value

and a reduced diﬀusion over the circle. At high temper-

atures, disorder sets in and the magnetization values are

no longer bound to a circle.

This means that, eﬀectively, one has to treat ﬁnite XY

systems as if they underwent symmetry breaking to an

ordered state. We incorporate this by subtracting the

spontaneous magnetization min the deﬁnition of δmqin

(10). Note that subtracting hmiis not sensible, as hmi=

0 due to the degeneracy of the “broken symmetry“ state.

One could instead subtract hmi0for some non-ergodic

average h···i0that averages only over times where the

orientation of mdoes not change considerably, but we

found that this does not yield diﬀerent results than the

ones obtained from (10).

The inﬁnite-system critical behavior can be extracted

from the ﬂuctuations transversal to the spontaneous or-

der of the system. We will discuss this in more detail

later, for now consider this a motivation to deﬁne

δmk,q=δmq·ˆ

m, δm⊥,q=δmq·ˆ

m⊥,(11)

where the hat marks a unit vector m=mˆ

mand the

orthogonal vector m⊥= (−my, mx) is mrotated by 90◦.

Note that the siare not vectors with respect to spatial

transformations, so there is no use in splitting into com-

ponents parallel and perpendicular to q. This is a con-

sequence of the spin orientation degree of freedom being

independent of rotations of the particle position in (1).

-1 -0.5 0 0.5 1

-1

-0.5

0.5

hmi= 0:80

T= 0:14

N= (16)2

-1 -0.5 0 0.5 1

hmi= 0:74

T= 0:17

-1 -0.5 0 0.5 1

-1

-0.5

0.5

hmi= 0:63

T= 0:20

-1

-0.5

0.5

hmi= 0:73

N= (64)2

hmi= 0:63 -1

-0.5

0.5

hmi= 0:36

-1 -0.5 0 0.5 1

-1

-0.5

0.5

hmi= 0:64

N= (256)2

-1 -0.5 0 0.5 1

hmi= 0:54

-1 -0.5 0 0.5 1

-1

-0.5

0.5

hmi= 0:07

FIG. 1. The ﬁnite-size magnetization below, at and above the transition in an MXY model. Evolution for a simulation time

over tmax = 3 ·103. Temperature increases from left to right, system size increases from top to bottom.

D. Correlation Functions

With these deﬁnitions, let us consider the relevant cor-

relation functions in our system. For spatial order, we are

interested in the pair distribution function

ρg(r) = 1

i6=jhδ(r−rij )i,(12)

ρbeing the particle density. We will also consider the

spin-spin correlation function

ρg(r)Cm(r) = 1

N*X

i6=j

si·sjδ(r−rij )+,(13)

which measures the correlations of a spin’s orientation

to that of another spin a distance rapart. The factor

ρg(r) is scaled out, it accounts for the packing of par-

ticles. It is known from Kosterlitz’s BKT theory of the

standard XY model2that the behavior of Cm(r) changes

from exponential decay to power-law decay at the tran-

sition temperature, and we will see how that transfers to

the MXY model in Section IV A 3.

We can deﬁne q-dependent susceptibilities

χm(q) = D|δmq|2E=χmk(q) + χm⊥(q),

χm⊥(q) = D|δm⊥,q|2E,

χmk(q) = Dδmk,q

2E.

(14)

The susceptibility χm(q) is related to the Fourier spec-

trum of Cm(r) via36

ρg(r)Cm(r)−m2

=Zddq

(2π)deiq·rχm(q)−1−m2.(15)

The susceptibility χw(q) follows from equipartition,

hωiωji=T δij , it is χw(q) = h|wq|i =T.

A characteristic quantity of the dynamic BKT univer-

sality class is the incoherent spin autocorrelation function

Cinc

m(t) = hsi(0) ·si(t)i,(16)

which compares the orientation of a spin to its orientation

after a time thas passed. Note that in the MXY model,

the spin iwill change position with t.

Finally, to study the collective dynamics of the system,

we shall study the spin-spin time correlation function

Cm(q, t) = hδmq·δmq(t)i(17)

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TheBKTTransitionanditsDynamicsinaSpinFluidThomasBissingera)andMatthiasFuchsb)FachbereichPhysik,UniversitatKonstanz,78457Konstanz,Germany(Dated:6October2022)Westudytheeectofparticlemobilityonphasetransitionsinaspinuidintwodimensions.ThepresenceofaphasetransitionoftheBKTuniversalityclassisshowninano...

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The BKT Transition and its Dynamics in a Spin Fluid Thomas Bissingeraand Matthias Fuchsb Fachbereich Physik Universit at Konstanz 78457 Konstanz Germany

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