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Topological Defects in the XY -Model

Condensed Matter Physics - Term Paper Report

Pavithran S Iyer, 3rd yr BSc Physics, Chennai Mathematical Institute

Guide: Dr. Radha D Banhatti

Email: [email protected]

Typeset Using LATEX

Jan 15th, 2011

Contents

1 Introduction 31.1 Spin systems and the XY model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 General features of the XY model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Elastic free energy of the XY model . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Correlation functions and Phase Transitions . . . . . . . . . . . . . . . . . . . . . . . 6

1.4.1 High temperature regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4.2 Low temperature regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Topological defects in the XY-model 102.1 Properties of winding numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Energy costs associated with defects . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Defects and the order parameter space . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Conclusions 15

A Functional Derivative [9] 16

1

mailto:[email protected]

Contents Contents 2

This term paper was done in the partial fulfillment of Condensed Matter Physics course re-quirement. I declare that I have worked through this project along with the help of Dr. RadhaBanhatti, who I would like to thank for having guided me through this with patience, along withher numerous suggestions and questions which helped in bettering this report.

Date: Jan 15th, 2011

Signature:

Table of Contents ↑ Topological defects in XY -model: 2 of 17 Jan 15th, 2011

1 Introduction 3

1 Introduction

1.1 Spin systems and the XY model

Spin systems on a lattice have two independent degrees of freedom, one being the dimensionality ofthe Spin space (or the spin-degrees of freedom) and the other, the dimensionality of the underlyinglattice. The former is usually denoted by n and the latter by d. In general, spin systems on ad-dimensional lattice are called n-vector models. For various values of n, we have[1]:

n = 0: The Self-Avoiding Walks (SAW)n = 1: The Ising modeln = 2: The XY modeln = 3: The Heisenberg modeln = 4: Toy model for the Higgs sector of the Standard Model

Here onwards, special attention is given to the n = 2, d = 2 model or the 2-D XY model. Thereasons for this will become apparent in the coming sections.

1.2 General features of the XY model

XY Model is a regular arrangement of a set of continuous valued (or classical) spins (spins whichcan be oriented along any direction on a 2D plane) on a d-dimensional lattice. Spin of each particleat position ~x can be described as a vector in the 2D-plane: ~S (~x) = (s cos θ (~x) , s sin θ (~x)). Thespin can point along any direction, corresponding to some value of θ between 0 and 2π. Notice thatθ (~x) for all ~x would define the entire model and we expect the HamiltonianI of this model to be afunction of θ only.

H = −J∑〈ij〉

~Si · ~Sj ⇒ −J∑〈ij〉

(cos θi cos θj + sin θi sin θj)⇒ −J∑〈ij〉

cos (θi − θj) (1)

H is invariant under global continuous rotations of all spins, and hence has a U(1) gauge symmetry.The ordered state of this model would correspond to the case where all the spins are aligned parallel.Unlike the Ising model, the magnetization: S = 〈~S〉 can take any value between 1 and -1 since theHamiltonian ∼ cos θ where 0 ≤ θ ≤ 2π, corresponding to the angular orientation of the spins.Different states of the system can have the same energy, with all of them related to each other bya global rotation of all spins and hence with different values of magnetization. Clearly, the numberof ground states for this system is infinite.

IOne can also think of this model as a special case of the Heisenberg Model: H =∑〈ij〉

Si, where the Sz interactions

are suppressed.


1 Introduction 1.2 General features of the XY model 4

Figure 1: Different ground state configurations of the XY -model. Each differs by a spatially uniformrotation of the spins. Since the Hamiltonian is invariant under this rotation, all these have the sameenergy.

When all the spins point in random directions (disordered state), the magnetization (〈~S〉) of thesystem is zero and the ordered case has a finite magnetization. Hence 〈~S〉 is the order parameterfor the phase transition from ordered to disordered state of spins. Since 〈~S〉 = (cos θ, sin θ), theorder parameter has only one degree of freedom: θ. Taking values of θ from 0 to 2π, we find that

〈~S〉 traces a circle with radius∣∣∣〈~S〉∣∣∣ with different values of the order parameter lying on this circle.

A unit circle, or more formally S1, is the order parameter space of the XY -Model. The mapping ofspin configurations in the order parameter space is particularly useful in examining vortices in theXY -Model.

However, configurations where the order parameter is spatially non-uniformII, the energy is notsame and finite energy is required to go from one configuration to another.

Figure 2: Figure showing spatially non-uniform spin alignments. Each of these have an energygreater than the ground state energy, corresponding to the spatially uniform spin alignments. Figuretaken from [2].

IIIn such configurations, the magnetization per spin, for a spin, is defined by the direction of orientation of thatparticular spin. We can show this explicitly for a 1D lattice, as a similar procedure is adopted for lattices in largerdimensions. If we consider neighboring spins to be aligned parallel (no topological defects), there are only ferromagneticinteractions i.e J < 0 and is fixed for all interactions. To compute the magnetization per spin, we take the expression

for the Hamiltonian in the absence of external fields: H = −J∑〈ij〉

cos (θi − θj) and turn on an external field, whose

value at the ith lattice point is hi. Hence the hamiltonian takes the value: H = −J∑〈ij〉

cos (θi − θj) − hi∑i

Si and

Z[h1, h2, . . . hN ] = TrS1,S2,...,SN e−βH . Now it can be seen that 〈~Si〉 =

∂ lnZ[h1, h2, . . . hN ]

∂hi

∣∣∣∣h1=h2=···=hN=0

. In other

words, the magnetization per spin, for the spin at position i, in a lattice where the spatial distribution of spins isnon-uniform, denotes the change in the energy of the system in response to turning on a small field at the latticeposition i.


1 Introduction 1.3 Elastic free energy of the XY model 5

1.3 Elastic free energy of the XY model

Assuming the direction of spins to be slowly varying across the lattice results in spatially non-uniform spin arrangements (fig. 2). This is equivalent to the low temperature limit of the XY-Hamiltonian. From Landau theory[4], we see that contribution to the free energy of this systemcomes both from the local order parameter as well as the change in the order parameter, overthe entire model, with the approximation that this change is slow. The local contribution to the

free energy is given byIII: f (s, h,m) ≡ f (θ (~x) , h, T ) = g0 (h, T ) +1

2g1 (h, T ) s2 +

1

4g2 (h, T ) s4.

Contribution from the variation in order parameter (across the system) depends on ~∇θ (~x) and is

treated like a kinetic energy termIV [4]:

∫C(~∇θ (x)

)2ddx, where C is a parameter with dimensions:

[C] = [Energy] [L]2−d. Since this is valid only for slow variations in s, and hence higher order terms

like∣∣∣~∇θ (x)

∣∣∣4, ∣∣∣~∇θ (x)∣∣∣6 are neglected[2].

F = Flocal + Fdisloc ⇒ F =

∫f (θ (~x)) ddx+

1

2

∫C[~∇θ (~x)

]2︸︷︷︸

[L]−2

ddx︸︷︷︸[L]d

(3)

C is denoted by ρS and is known as the Stiffness constant. It is a measure of change in the groundstate energy due to the spatially varying order parameter. The lowest free energy configurationscorrespond to those where all spins are aligned parallel. For these configurations, only the local(spatially uniform) value of the order parameter gives a non-zero contribution and the free energyis minimum for them. Alignment of spins in these configurations breaks the U(1) symmetry of theXY Model, thereby indicating a phase transition from a state with zero to a state with finite or non-zero magnetization. In order to obtain the free energy, we need to minimize the Landau Potentialw.r.t the order parameter, which in this case corresponds to setting the functional derivativeVVI

δFdislocδθ (~x)

to 0. Using the integral form of Fdisloc (eq. 5), the function derivative of the integral can be

computed using standard techniques (for an explicit calculation, see appendix A) and is obtained

IIIComparing the liquid-gas transition with the magnetic phase transition, we find the magnetization correspondsto volume and the external field to pressure. Hence m → v and h → p. Therefore the locally invariant free energyis expanded in terms of h and T instead of p and T . Furthermore, when no magnetic field is considered, i.e; U(1)-symmetry is not broken, we have h = 0. Hence f (s, h, T ) depends only s and T .

IVAn alternate way to look at this would be: considering only small variations in order parameter θ, taking a lowtemperature approximation of the Hamiltonian in (eq. 1), to leading order, the cos (θi − θj) can be replaced by:

1 − (θi − θj)2

2. In the continuum limit, sum over (θi − θj)2 is replaced by the (∇θ)2 operator and the hamiltonian

becomes:

H = −J

2N −

(~∇θ)2

2

⇒ E0 +J

2

∫d2x

(~∇θ)2

(2)

where E0 = −2JN and we have used that the number of nearest neighbor pairs in the 2-D model is 2N , as opposedto N in the 1-D model.

V~x is a vector, specifying a position on a N -dimensional lattice. Hence it will have N components.VIWe would generally consider

∂F

∂〈~S〉= 0, where 〈~S〉 is the order parameter. But in this case 〈~S〉 =

(cos θ (~x) , sin θ (~x)) is itself a function of θ (~x). Therefore a derivative w.r.t 〈~S〉 turns into a functional derivativeof F w.r.t θ(~x).


1 Introduction 1.4 Correlation functions and Phase Transitions 6

as:

δFdislocδθ(~x)

= −ρs~∇2θ (~x)⇒ Minimum free energy condition: − ρs~∇2θ (~x) = 0 (4)

Till now we haven’t anywhere assumed a two-dimensional (d = 2) lattice. In the following fewstatements, the reasons for this assumption would be stated. The above equation can be solved toobtain the free energy of the lowest energy configuration. Since it is a second order equation, at leasttwo independent initial conditions are required to solve for θ. The lattice is d dimensional, but theboundary conditions which we will assume is on one of the dimensions. Choosing this dimension asei-axis, we have: on the ei = 0 plane, θ = 0 and on the ei = L plane, θ = θ0. This states that one ofthe solutions for θ (~x) could be independent of all components of ~x, except for the one perpendicularto the ei = 0 and ei = L planes (which is any component parallel to the eiaxis.). In other words, weconsider the case where θ(~x) ≡ θ(ei). Hence θ ≡ θ (ei) and as we are still in the cartesian coordinate

system, ~∇2θ =d2θ

de2i. Solving the equation along with these boundary conditions, we get:

~∇2θ (~x) =d2θ(z)

de2i= 0 ⇒ θ(ei) = Aei + C

Boundary conditions : θ(0) = 0⇒ C = 0 and θ(L) = θ0 ⇒ θ0 = AL⇒ ei =θ0L

∴ θ (ei) = eiθ0L, which on substituting gives: Fdisloc =

1

2

∫C

(θ0L

)2

ddx

On performing the integral with the above solution for θ (~x), we get:

Fdisloc =Cθ20

2L−2

∫ddx ⇒ Fdisloc =

Cθ202Ld−2 (5)

Hence we see the free energy, in general is some function of the stiffness constant and the spatialdimensions of the lattice. Notice that we have a new variable L, which we had used to solve for θ (~x)now appears in the free energy of the system. If F must not dependVII on any macroscopic size ofthe system, it must be independent of L, i,e; d = 2. Therefore, for a two dimensional XY model,the elastic free energy is independent of the sizeVIII of the lattice and for this purpose, we wouldconsider, from now onwards, a two dimensional lattice (d = 2) for the XY model. The minimum

value for the elastic free energy in this two dimensional XY model is given by Fel =Cθ20

2.

1.4 Correlation functions and Phase Transitions

To see the ordering in the high and low temperature phases, spin-spin correlation function hasto be computed. We would now compute the spin-spin correlation function in both, the high aswell as the low temperature regimes. Asymptotic behavior of the correlation function (exponentialdecay at high temperatures and power law decay at low temperatures) suggest the presence of afinite temperature phase transition. It was observed by Stanley and Kaplan (1971) that at highbut finite temperatures the magnetic susceptibility diverges[6], thereby indicating the presence of a

VIIWe require the elastic free energy Fdicloc not to depend on the size of the lattice and depend only on the elasticproperties of the lattice. The total free energy Flocal + Fdisloc, in (eq. 5) however, is extensive and its dependence onthe size of the lattice comes from Flocal.VIIIThe size of the lattice is proportional to the number of lattice points in each dimension.



phase transition without the presence of spontaneous symmetry breaking[6]. This later led Wignerto explore the possibility of a phase transition without symmetry breaking, which are known asTopological Phase transitions. We now look in detail at the presence of a topological phase transitionfor the case of a two dimensional n-vector model. The lattice is assumed to be two-dimensional, sothat the elastic free energy is independent of the lattice size (by size, we mean the number of latticepoints), which is seen in the previous section. Though only two spin degrees of freedom need notbe assumed, as we go along working with the XY model in 2D, its relevance will become evident.

1.4.1 High temperature regime

In order to find statistical properties of the system, it is necessary to obtain an expression forthe partition function first. Using the form of the Hamiltonian in (eq. 1), we compute the ex-

ponent in the partition function: H = −J∑〈ij〉

cos (θr − θ0) ⇒ βH = − J

kbT

∑〈ij〉

cos (θr − θ0) ⇒

−K∑〈ij〉

cos (θr − θ0). Since K is inversely related to temperature, the high temperature approxima-

tion corresponds to small K. The partition function Z as a function of K, can be Taylor expandedin powers of K. We would now consider only upto leading order in K.

Z =

∫ 2π

0

dθ0 . . . dθn(2π)n

∏〈ij〉

[1 +K cos (θi − θj) +O

(K2)]

⇒∫ 2π

0

dθ0 . . . dθn2π

[1 +K cos (θ0 − θ1)] [1 +K cos (θ1 − θ2)] . . . [1 +K cos (θn−1 − θn)] (6)

To find the spin-spin correlation function, we can use the identity:⟨~S(~0) · ~S(~r)

⟩= 〈(cos(θ0), sin(θ0)) · (cos(θr), sin(θr))〉 = 〈cos(θ0) cos(θr) + sin θ0 sin θr〉 ⇒ 〈cos (θ0 − θr)〉

This averaging is done by multiplying cos (θr − θ0) to the integrand in the expression for the partitionfunction, and integrating this over all θi for i = 1 . . . n.

〈cos (θr − θ0)〉 =

∫ 2π

0

dθ0 . . . dθn2π

cos (θr − θ0) [1 +K cos (θ0 − θ1)] [1 +K cos (θ1 − θ2)] . . . [1 +K cos (θn−1 − θn)]

(7)

Each term in (eq. 6), cos(θi − θj), can be represented[7] as a bond between the lattice sites i andj. The integrals over cosines follow some properties:∫ 2π

0

dθi2π

cos (θi − θj) = 0 and

∫ 2π

0

∫ 2π

0

dθi2π

dθj2π

cos(θi − θj) cos(θi − θj) =1

2∫ 2π

0

dθj2π

cos (θi − θj) cos (θj − θk) =cos (θi − θk)

2

with which, we see that the integrand must reduce to a product of squares of cosines, for the integralto produce a non-vanishing value. Hence, in the bond picture, we must have a closed bonds fromlattice point 0 to ~r, in order for the term 〈cos(θ~r − θ0)〉 to be non vanishing. Hence to determine



〈cos (θr − θ0)〉, we need to integrate over all terms which correspond to bonds from site 0 to site ~r.

〈cos (θr − θ0)〉 =

∫ 2π

0

∫ 2π

0. . .

∫ 2π

0

dθ0 . . . dθr(2π)r

K cos (θ0 − θ1) ·K cos (θ1 − θ2) . . .K cos (θr−1 − θr) cos (θr − θ0)

= K |r|(

1

2

)|r|−1 ∫ 2π

0

∫ 2π

0

dθ0dθr(2π)2

cos (θ0 − θr) · cos (θ0 − θr)

∴ 〈 ~Sr · ~S0〉 ∼(K

2

)|r|∼ e−

|r|ξ

where ξ = ln

(2

k

). Hence we see that the correlation function diverges exponentially[7], which it

should, since we are looking at the high temperature regime, corresponding to the disordered phase.

1.4.2 Low temperature regime

Let us now see the more interesting case corresponding to the low temperature regime. In this case,the variation in the spin angle can be assumed to be continuous and hence (eq. 5):

−βH =ρS2

∫d2x

(~∇θ (~x)

)2(8)

and the partition function: Z = exp

[ρS2

∫d2x

(~∇θ (~x)

)2]. The spin-spin correlation function can

be computed using the identity in (eq. 7). Since the underlying Hamiltonian is quadratic, thedistribution to be integrated with, to find the average of cos(θ0 − θr), is Gaussian. Using standard

Gaussian rules of integration[7]:⟨eA⟩

= e〈A〉+12〈A2〉:

〈cos (θ0 − θr)〉 = <⟨ei(θ0−θr)

⟩⇒ <

[e〈i(θ0−θr)〉−

12〈(θ0−θr)2〉

](9)

We assume a random distribution of phase and hence the average phase is zero: 〈i (θ0 − θr)〉 = 0.

The quantity that needs to be computed is⟨

(θ0 − θr)2⟩

. Notice that:⟨

(θ0 − θr)2⟩

=⟨θ20 + θ2r − 2θ0θr

⟩⇒

2⟨θ20 − θ0θr

⟩, which leaves the computation of the functions 〈θ0θr〉 and

⟨θ20⟩.

It suffices to compute an expression for 〈θ~rθ0〉 since ~r = 0 would give the case: 〈θ20〉. The twopoint correlation functions, in general is given by the green’s function, G, defined by[7]: G(~x, ~x′) =〈θ(~x)θ(~x′)〉, which further simplifies in our case to:

G(~r, 0) = 〈θ~rθ0〉

where G satisfies the property[7]: −~∇2G(~r, 0) =1

ρsδd(~r). Introducing a function Cd(~r, 0) with the

property ~∇2Cd(~r, 0) = δd(~r), we get[7]:

〈θ~rθ0〉 = G(~r, 0) = −Cd(~r)ρs

where: ~∇2Cd = 0 (10)

Using the gauss divergence theorem:∫VdV∇2Cd =

∮S

~dS · ~∇Cd(~r)



and (eq. 10), we find that

∮S

~dS · ~∇Cd(~r) = 1. Hence ~∇Cd(~r) is like a d− 1 dimensional dirac delta

function:

dCddx

=1

xd−1Sd

where Sd denotes the contribution from the d-dimensional solid angle[7]: Sd =2π

d2(

d2 − 1

)!

which

would cancel the solid angle measure of the surface integration of Cd. Notice that Cd depends only

on x, the radial coordinate of ~r. We have: 〈θ0θr〉 =Cdρs

. On further integration,

Cd =1

(2− d)xd−2Sd

∴ 〈θ0θr〉 =−1

(2− d)xd−2Sdρs(11)

From the above expression for 〈θ0θr〉, we see that for d ≥ 2, as x → 0: 〈θ0θr〉 → ∞. Hence atsmall length scales, there is an ultraviolet divergence. The divergence has to be removed and can bedone either by adding an explicit term to the expression of free energy or by introducing a cut-offlength scale a such that x > a always.[7] The smallest possible length on the spin lattice is thelattice parameter a, which is also defines the the cut-off length scale. The fluctuation correspondingto x = 0,

⟨θ20⟩

would be replaced by the that corresponding to x = a,⟨θ2a⟩, where a is the lattice

parameter. Using (eq. 11), we have:⟨θ2a⟩

=−1

(2− d)ad−2Sdρs. The correlation function now

becomes, replacing the radial component x by |~r|:

⟨(θ0 − θr)2

⟩=

2(|~r|2−d − a2−d

)ρs(2− d)Sd

For d > 2 we see that there is an exponential decay of the spin correlations. We now see the d = 2case which gives a rather interesting result. Putting d = 2 we find:

Sd =⟨

(θ0 − θr)2⟩

=2π

(1− 1)!= 2π

⇒⟨

(θ0 − θr)2⟩

=1

πρs

[|~r|2−d − a2−d

2− d

]d=2

Replacing the quantity in the square brackets by its series expansionIX around d = 2, we obtain:[|~r|2−d − a2−d

2− d

]d=2

→ [ln |~r| − ln a]

IXRewritingxn

nas

en ln x

n, we have:

en ln x

n=

1

n

[1 + (n lnx) +

(n lnx)2

2!+

(n lnx)3

3!+ . . .

]

Hence, we find:xn

n=

1

n+ lnx+

1

2n ln2 x+

1

6n2 ln3 x+

1

24n3 ln4 x+

1

120ln5 x+ . . .


2 Topological defects in the XY-model 10

〈(θ0 − θr)2〉 =1

πρsln

(|~r|a

)Substituting the above expression in the spin-spin correlation function (eq. 9), we obtain:

⟨~S(~0) · ~S(~r)

⟩=

(a

|~r|

) 12πρs

(12)

The above equation indicates that correlation function, for a two dimensional n-vector model, decaysas a power law. A power law decay usuallyX denotes the existence of a phase transition that isassociated with a critical point[6]. This is true for any spin system arranged on a two dimensionallattice, in the low temperature, Gaussian approximation. To show that this feature is due to theGaussian nature of the underlying distribution, we need to show that adding higher order terms

(like

∫ddx

(~∇θ)4

) to the Hamiltonian, still produces the same results as above. This is done using

perturbative renormalization group methods[6]. On finding the fixed points of this transformation(which correspond to points of phase transitions), it is observed that the zero temperature fixedpoint for d = 2, is stable for n = 2 and unstable for all n > 2. Hence we talk only about then = 2 case which is the XY -model. The low temperature phase of the XY -model therefore is seento behave like an ordered state, but we know that the correlation function decays as a power law.Hence this phase is described as a quasi long-range order state[6].

2 Topological defects in the XY-model

In the previous section, we observed that there is change in the asymptotic behavior of the spin-spin correlation function, which brings about a phase transition in the model, destroying the quasilong-range ordering into a completely disordered state. We have not yet seen the mechanism of thisdisordering. Notice that the disordering mechanism cannot involve any continuous deformation ofthe spin orientations, since these deformations are taken care of by the low temperature gradientexpansion of the Hamiltonian (eq. 8). It was Berezinskii and later Kosterlitz and Thouless whoXI

suggested that the mechanism of disordering involved discontinuous deformation in spin orientations.Such a defect is called a Topological Defect, and for the case of the XY-model, they are calledVortices.

XSignatures of power law decay showing phase transitions with critical points have been observed in the case ofliquid-gas transitions (where, the spin degree of freedom is n = 1), fluid-superfluid transitions (with n = 2) and theparamagnetic-ferromagnetic transitions (with n = 3) cases.

XIJohn Michael Kosterlitz and David James Thouless: received the the 2000 Lars Onsager Prize “for the introductionof the theory of topological phase transitions ...”.


2 Topological defects in the XY-model 11

Figure 3: Vortices in the XY -model. These configurations cannot be transformed to the completelyordered state by any continuous deformations of the spin-orientations. Figure taken from: http:

//www.ibiblio.org/e-notes/Perc/xy.htm

Since the spin-orientations are defined in the interval [0, 2π), the changeXII in this angle alonga closed loop on the lattice would be zero, if the spins are continuously varying (~∇θ (~x) is contin-uous), i,e; if the closed path does not encircle any vortex. If the loop encloses a vortex, ~∇θ (~x)is discontinuous along the loop and hence the change in θ would be 2πn, where n is an integer.By convention, the contour integral is taken in the anti-clockwise direction. If the spin orientationincreases in the clockwise direction, corresponding value of n would be negative and the defect iscalled an anti-vortex. ∮

C

~∇θ (~x) · ~dl =

{0 if C encloses no vortex

2πn if C encloses a vortex(13)

A vortex is characterized by the value of n, which is obtained by a contour integral around thevortex. This number is called the Winding Number of the defect. The defect in (fig. 3) has defectswith winding numbers +1,+1 and −1.

(a) Two n = 1 vortices (b) A vortex (n = 1) anti-vortex (n =−1) pair.

(c) A vortex with n = 2.

Figure 4: Schematic: Vortices and their winding numbers. Figures taken from: http://www.

ibiblio.org/e-notes/Perc/xy.htm

XIIChange in the spin orientation along any closed loop would be 2πn, where n is an integer, since we return to thesame point as started.


http://www.ibiblio.org/e-notes/Perc/xy.htm




2 Topological defects in the XY-model2.1 Properties of winding numbers 12

2.1 Properties of winding numbers

The winding number characterizes the vortex and hence the defect in the model. It can be notedimmediately that the winding number is a discontinuous function: it only varies in integer steps.We now discuss few of the important properties of winding numbers:

1. Suppose there is a defect and C a curve (of arbitrary size) enclosing it. If the curve is shrunkcontinuously, the winding number n too must vary continuously. n being discontinuous, staysconstant for any size of the loop, as long as the defect is enclosed by it. Hence the windingnumber of the defect cannot change by any continuous deformation of the loop enclosed. Asa consequence, in order to correct a topological defect (vortex, in this case), local continuoustransformations of spins is not enough, in which case the defect might be displaced accordingly,but a continuous deformation of spin orientations, at lattice sites arbitrarily far from theoriginal position of the defect is required.

2. If a loop enclosed more than one defect, the total winding number of the defects enclosed isjust the sum of the individual winding numbers. If there are equal number of vortices as anti-vortices, total winding number = 0. During a Kosterlitz Thouless phase transitions, vorticesand anti-vortices are bound (the model having no defects) below a critical temperature whilethey unbind above this temperature, thereby leading to spontaneous formation of vortices.

2.2 Energy costs associated with defects

The free energy associated with the creation of a vortex, whose contributions come from the vortexcore F core itself as well as from the elastic free energy due to continuous distortion of spins awayfrom the vortex F el. It is assumed that θ(~x) is continuous for all ~x, unless there are no topologicaldefects, in which case, θ(~x) is discontinuous at some point ~x′, where the core of the defect is said tobe at ~x′. However, θ(~x) must still be continuous everywhere else (at all ~x 6= ~x′). Hence the regionaround the core would contain spins whose new orientation (after the formation of the defect) differsfrom its previous direction, by a continuous change. The free energy contribution from the core,Fcore, is given by[2]:

F core ∼ fcond

where fcond is the condensation energy of the XY model.Computation of F el is done using certain elastic properties of the XY model (eq. 5). From (eq.

13), we have: 2πn =

∮C~∇θ(~x) · ~dl ⇒

∮C~∇θ(~x) · (drr + rdφφ) = 2nπ. The form of ~∇θ(~x) can be

taken as:n

rφ . Using the low temperature gradient expansion of the energy (eq. 8), in the case of

the 2D XY model gives, denoting the elastic free energy stored in the lattice containing a vortexwith winding number n as F eln :

F eln =ρS2

∫d2~x

(~∇θ(~x)

)2⇒ −ρSn

2

2

∫ L

a

∫ 2π

0rdrdφ

1

r2⇒ ρSn

2π

∫ L

a

dr

r

∴ F eln = πρSn2 ln

(L

a

)(14)

where ρS is the stiffness constant as in (eq. 5). Both upper and lower cut off length scales areused in this integration. a stands for the lattice spacing and L for the system size, L >> a. Noticethat the energy of a vortex is quadratic in its winding number and diverges logarithmically with


2 Topological defects in the XY-model2.3 Defects and the order parameter space 13

system size. Hence free energy of macroscopically large systems with even a single vortex would belarge[8][2]. It is known that[6] a Heisenberg or any model with n > 2 (number of spin degrees offreedom greater than 2), free energy associated with a defect would be finite, even for macroscopicsystems, thereby causing spontaneous appearance and disappearance of vortices (as a vortex canbe formed due to fluctuations). On the other hand, in the XY model, the formation (and alsothe disappearance) of vortices need an energy which scales with the size of the lattice. Thereforefor macroscopic systems, the elastic free energy of the vortex would be large compared to that ofthermal fluctuations. The high energy cost for formation of defects in the XY model prevents the(spontaneous) formation or removal of vortices, due to thermal fluctuations. As a result, statesof the XY model with different number of vortices are distinguished from each other and defectsremove the degeneracy of the energy levels, to some extent.

2.3 Defects and the order parameter space

The order parameter for the magnetic transitions in XY model is magnetization, which takes valuesbetween 0 to 1. Magnetization per spin can be expressed in a polar form, assuming all the spinsto have unit moduli. We then have each spin state expressed as (cos θ (~x) , sin θ (~x)), characterizedby an angle θ (~x). All vectors of the above mentioned form can be plotted on a unit circle. Thecircle is hence called the order parameter space for the magnetic transitions of XY model. Thespin orientations along any closed loop on the XY model can be mapped to points in the orderparameter space, which is the unit circle.

Figure 5: Mapping spins onto the unit circle. Γ is a closed loop in the coordinate space and spinson the loop, oriented at an angle α are mapped on the unit circle (or the order parameter space)to the point corresponding to the polar angle α. Figure taken from [2].

This mapping provides a quick tool for computing the winding number of a defect in the model.The sign for a defect is that the net change in the spin orientation, along a closed loop (enclosingthe defect) is 2nπ, for some integer n. This means that on plotting the orientation of all spinsalong the closed loop, onto the unit circle, the plot would enclose the unit circle n times in theanti-clockwise direction. Note that if the spin orientations along the loop change in the clockwisedirection, we would be retracing the plot on the unit circle. Hence this method gives the net changein spin orientation, which is the winding number. Examples are given below:


2 Topological defects in the XY-model2.3 Defects and the order parameter space 14

(a) All spins pointing in the same di-rection corresponds to a point in theorder parameter space. Point also im-plies that the net change in spin ori-entation along with closed loop is 0.Figure taken from [2].

(b) Spin-orientations whose net direc-tion does not change. The spin ori-entation increases anti-clockwise until180o and then clockwise by 180o. No-tice the retracing of the plot (on theorder parameter space) correspondingto the clockwise movement of the spin-orientation. Figure taken from [2].

(c) Spin-orientations whose net direction changes by 8π, and its plot on the unit circle.

(d) Net direction changes by 2π along the closed loop. The spin-orientation increasesanti-clockwise by 90o until B, clockwise by 900 until C, thereby bringing a net changeof 0. It further increases anticlockwise (through D, E, F, G and H) by 450o andfinally increases clockwise by 90o until back to A, thereby bringing a net increaseof 2π along the closed loop.

Figure 6: Schematic: Loops containing changes in spin orientations and their corresponding plot inthe order parameter space.


3 Conclusions 15

3 Conclusions

Taking the most general case of the n-vector model, we have derived the elastic free energy of thespin system using a Landau-Ginsburg free energy. Following this, we saw the advantage of takingthe underlying lattice to be two dimensional, as the elastic free energy then became independent ofthe size of the lattice. We then looked at the behavior the XY model in low and high temperatureregimes, where its asymptotic behavior in both the regimes turned out to be different, therebyindicating the existence of a phase transition between the high (disordered) and low (quasi-longrange order) temperature phases. The peculiar nature of the disordering mechanism causing thisphase transition, is that these disorders cannot be removed by any continuous transformations ofthe spin orientation. This led to accounting for the possibility of a new type defect in the XYmodel, known as the topological defect. It can be shown, though in this report explicit calculationsweren’t shown, using perturbative renormalization group calculations that topological defects arenot stable for spin models (n-vector models) with n ≥ 3. Following this, topological defects andmethods used to characterize them, in the 2D XY model, were studied. The elastic free energyassociated with the formation of a defect was also discussed. Finally, a few examples of defects werelooked at, illustrating the method used to characterize them.


A Functional Derivative [9] 16

A Functional Derivative [9]

Let F [~φ] be a functional, where φ ≡ φ(~x). The change in F [~φ] can be expressed as sum, over all ~x,of all local changes in F [~φ(~x)]:

δF [~φ] =

∫d~x′

δF [~φ]

δ~φ(~x′)δ~φ(~x′) (15)

The functional derivativeδF [~φ]

δ~φ(~x)is a measure of the change in F [φ] when φ(x′) is varied by a small

amount, at x′ = x. Let this small amount of change in φ(x′) at the point x′ = x be ε. Hence wehave δφ(x′) = ε, at x′ = x. In other words:

δφ(x′) = εδ(x′ − x) (16)

Using the first principles definition of a derivative:

δF [~φ] = F [~φ(~x) + δ~φ(~x)]− F [φ]⇒ F [~φ(~x) + εδ(~x′ − ~x)]− F [φ]

Putting (eq. 16) in (eq. 15), we get: δF [φ] =

∫dx′

δF [φ]

δφ(x′)εδ(x′ − x), which on comparing with the

previous equation gives:

δF [φ]

δφ(x)= lim

ε→0

F [φ+ ε(x′ − x)]− F [φ]

ε(17)

We now have: F [φ] =1

2

∫d~x′(~∇x′θ

(~x′))2

and we wish to computeδF [θ]

δθ(~x). Using the general

prescription in (eq. 17), we have:

δF [θ]

δθ(~x)=

1

2limε→0

1

ε

[∫d~x′(~∇(θ(~x′) + εδ(~x′ − ~x)

))2−∫d~x′(∇θ(~x′)

)2]=δF [θ]

δθ(~x)= lim

ε→0

1

ε

[∫d~x′(~∇θ(~x′) + ε~∇~x′δ(~x′ − ~x)

)2−∫d~x′(∇θ(~x′)

)2]Expanding the square and ignoring terms quadratic (or in higher powers of) ε, we get:

δF [θ]

δθ(~x)=

1

2limε→0

1

ε

[∫d~x′((

~∇θ(~x′))2

+ 2ε~∇θ(~x′)~∇~x′δ(~x′ − ~x)

)−∫d~x′(∇θ(~x′)

)2]⇒ 1

2limε→0

2

ε

[∫d~x′ε~∇θ(~x′)~∇~x′δ(~x′ − ~x)

]The delta function gives a non-vanishing value only for ~x′ = ~x. Putting this, we have:

δF [θ]

δθ(~x)=

∫d~x′[~∇θ(~x)~∇~x′δ(~x′ − ~x)

]Keeping ~∇~x′δ(~x′ − ~x)d~x′ = dv and ~∇θ(~x) = u, we get an integral of the form

∫udv, which

can be expanded, by integration by parts, as uv −∫vdu. From the definition, we see that uv =


References References 17

~∇θ(~x′)∫~∇~x′δ(~x′−~x)d~x′, which gives: uv = ~∇θ(~x′)δ(~x′−~x). Hence we are left with the computation

of

∫vdu, which on substitution gives:

∫δ(~x′ − ~x)~∇ · ~∇θ(~x′)d~x′ ⇒

∫δ(~x′ − ~x)~∇2θ(~x′)d~x′. Since

the integration is over the entire lattice, we take the limits of the integration to be values of ~x′

corresponding to lattice boundaries and assume that θ(~x) vanishes at the boundaries. We assumethat at these lattice boundaries, the value of θ is equal to a fixed constant at the boundaries[3], andhence, the difference between the its values on the two boundaries of the integration gives 0.

δF [θ]

δθ(~x)=

[~∇θ(~x′)δ(~x′ − ~x)−

∫δ(~x′ − ~x)~∇2θ(~x′)d~x′

]lattice boundaries

∴δF [θ]

δθ(~x)= −~∇2θ(~x)

References

[1] Wikipedia Resource: n-vector model, Wikipedia: The Free Encyclopedia. Wikimedia Foun-dation Inc. Updated 25 June 2009, 18:10 UTC. Encyclopedia on-line. Available from http:

//en.wikipedia.org/wiki/N-vector_model . Internet. Retrieved 29 January 2011, 3

[2] BOOK: Principles of condensed matter physics (Oxford graduate texts), Paul M. Chaikin andT. C. Lubensky, Cambridge University Press, 2000 4, 5, 12, 13, 14

[3] The topological theory of defects in ordered media, Mermin, N. D, Rev. Mod. Phys., Vol. 51,Article no. 3, July 1979, DOI: 10.1103/RevModPhys.51.591, http://link.aps.org/doi/10.1103/RevModPhys.51.591 17

[4] BOOK: Principles of condensed matter physics (Oxford graduate texts), L.E. Reichl, Wiley-VCH,2009 5

[5] BOOK: Information, physics, and computation (Oxford graduate texts), Marc Mzard and AndreaMontanari, Oxford University Press, 2009 - 569 pages, http://www.stanford.edu/~montanar/BOOK/book.html

[6] Topological Phase Transitions, Ben Simons, Lectures on Phase Transitions and Collective Phe-nomena, http://www.tcm.phy.cam.ac.uk/~bds10/phase/topol.pdf 6, 7, 10, 13

[7] Ginzburg-Landau Phenomenology, Ben Simons, Lectures on Phase Transitions and CollectivePhenomena, http://www.tcm.phy.cam.ac.uk/~bds10/phase/gl.pdf 7, 8, 9

[8] The Kosterlitz-Thouless Transition, Henrik Jeldtoft Jensen, Department of Mathematics, Impe-rial College, http://www.mit.edu/~levitov/8.334/notes/XYnotes1.pdf 13

[9] Functional Derivatives Manuel Berrondo, Lectures notes for the course Quantum Field Theory -II (C247ESC), Brigham Young University, http://www.physics.byu.edu/faculty/berrondo/wt752/functional%20derivative.pdf


http://en.wikipedia.org/wiki/N-vector_model

http://en.wikipedia.org/wiki/N-vector_model

http://link.aps.org/doi/10.1103/RevModPhys.51.591

http://link.aps.org/doi/10.1103/RevModPhys.51.591

http://www.stanford.edu/~montanar/BOOK/book.html

http://www.stanford.edu/~montanar/BOOK/book.html

http://www.tcm.phy.cam.ac.uk/~bds10/phase/topol.pdf

http://www.tcm.phy.cam.ac.uk/~bds10/phase/gl.pdf

http://www.mit.edu/~levitov/8.334/notes/XYnotes1.pdf

http://www.physics.byu.edu/faculty/berrondo/wt752/functional%20derivative.pdf

http://www.physics.byu.edu/faculty/berrondo/wt752/functional%20derivative.pdf

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