Master M2 Sciences de la Matiere – ENS de Lyon – 2015-2016

Phase Transitions and Critical Phenomena

Numerical study of the Kosterlitz-Thouless transition inthe 2D XY Model

Boattini Emanuele

January 10, 2016

Abstract

In this paper I present a numerical study of the two-dimensional XY model and its Kosterlitz-Thouless transition.Two different Monte Carlo algorithms were used to simulate the system: the single-spin-flipMetropolis algorithm and the Wolff algorithm. The former performs very well far from thecritical point, while the latter allows to strongly reduce correlation times and increase statisti-cal accuracy close to the critical temperature.In this study, I show: the different behavior of the correlation function between the low-temperature and the high-temperature phase; how the transition is characterized by the bindingand unbinding of vortices, and how vortex excitations proliferate for T ≥ TKT ; the divergenceof the magnetic susceptibility in the thermodynamic limit for all temperatures T ≤ TKT , and Iestimate the exponent η(T ) for different temperatures below TKT using different techniques.Finally, I estimate the critical temperature TKT using different methods. My best estimate ofTKT in the thermodynamic limit is TKT = 0.894±0.002, which is in perfect agreement withthe reference value in [4].

1 IntroductionThe two-dimensional XY model is a lattice model described by the Hamiltonian

H = −J∑〈i,j〉

~si · ~sj

= −J∑〈i,j〉

cos(θi − θj),(1)

where∑〈i,j〉 indicates the sum over nearest neighbors, J > 0 is the coupling constant1, and

the spins are two-dimensional vectors, ~si = (cos(θi), sin(θi)), arranged on a square latticeof linear size L (N = L2 spins). This model can describe planar XY-anisotropic magnets,as well as critical properties of superfluid and superconducting films, the two-dimensionalCoulomb gas, and so on.

For such a model in two dimensions, with a continuous symmetry and short range inter-actions, there is no transition from ordered to disordered phase according to the Mermim-Wagner theorem. Fluctuations kill any form of order at any finite temperature. However,this model shows a particular phase transition, called Kosterlitz-Thouless transition, which ischaracterized by a low-temperature phase with quasi-long range order, and a completely dis-ordered high-temperature phase. At a critical temperature, TKT , the transition between thesetwo phases occurs. My reference value for the critical temperature is the estimate made byOlsson in [4], TKT = 0.89213(10).

In the following sections, I will discuss the main features of this transition and presentnumerical results obtained from Monte Carlo simulations.

2 Computational MethodsThe simplest Monte Carlo (MC) algorithm to study this model is the single-spin-flip Metropo-lis algorithm [1]. A trial move in this algorithm consists in choosing a spin at random on thelattice, proposing a new direction for it by randomly generating its new angle between 0 and2π, and accepting the trial move with the following acceptance probability

acc(o→ n) = min(e−β∆E , 1), (2)

where ”o” represent the old state, ”n” the new proposed state, and ∆E = En − Eo is theenergy difference between the two states. A Monte Carlo step consists in N (number ofspins) MC trial moves.

Another MC algorithm that can be used for this model is the Wolff algorithm [1], whichis a cluster algorithm. The basic idea is to choose at random both a seed spin, from which thecluster starts being built, and a random direction, denoted by a unit vector n. A neighbor ofthe seed spin whose component in this direction has the same sign as that of the seed spin canbe added to the cluster with a certain probability Padd. If the components have different sign,then the spin is not added to the cluster. When the complete cluster is built, it is ”flipped” byreflecting all the spins in the plane perpendicular to n. If one chooses Padd as follows

Padd(~si, ~sj) = 1− exp [−2β(n · ~si)(n · ~sj)] , (3)

1in the simulations J = 1.

1

then all cluster moves are accepted with probability 1 and detailed balance is satisfied. In thiscase one MC step consists in one cluster move.

The single-spin-flip Metropolis algorithm performs very well in simulating the system farfrom the critical temperature TKT . However, as we approach the critical temperature, thecombination of large critical fluctuation and long correlation times makes the errors on themeasured quantities grow enormously. Nothing can be done about the critical fluctuations,since they are an intrinsic property of the system near its critical point, but by using the Wolffalgorithm we can strongly reduce the increase in the correlation times close to the phasetransition, as shown in Fig.1 for a system of linear size L = 16. Therefore, I used the Wolffalgorithm to simulate the system at temperatures close to the critical temperature, and theMetropolis algorithm for all the other temperatures considered.

0

10

20

30

40

50

60

70

0.6 0.7 0.8 0.9 1 1.1 1.2 1.3

τ

T

WolffMetropolis

Figure 1: Comparison between correlation times for the Metropolis and the Wolff algorithms as functions of temper-ature for a system of linear size L = 16.

For all the observables measured directly in the simulation (energy, magnetization...) Icomputed the errors using autocorrelation analysis, while for other quantities, such as thespecific heat and the magnetic susceptibility, I used the bootstrap method [1].

In the simulations, for convenience, one works in dimensionless units. Hence, all theresults presented in this paper are in these units.

3 Preliminary resultsBefore discussing the phase transition of the two-dimensional XY Model, I present somepreliminary results and discuss some effects due to the finite size of the systems considered inthe simulations.

Fig.2 and 3 show the average energy per spin

〈e〉 =1

N〈E〉 (4)

and the specific heat per spin

c =kBβ

2

N

(⟨E2⟩− 〈E〉2

)(5)

as functions of the temperature for different sizes L of the system. As one can see, the valuesof 〈e〉 are weakly affected by the size, and the results look very similar for different values of

2

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

e

T

L = 8

L = 12

L = 16

L = 20

L = 32

L = 64

L = 100

Figure 2: Average energy per spin as a functionof the temperature for different sizes L. Error barsare often smaller than symbols.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

c

T

L = 8

L = 12

L = 16

L = 20

L = 32

L = 64

L = 100

Figure 3: Specific heat per spin as a function ofthe temperature for different sizes L. Error bars areoften smaller than symbols.

L. The specific heat has a peak, but it does not diverge in the thermodynamic limit becauseno second order phase transition occurs in this model. The main effect of the size on thespecific heat is the change of the position of the peak: the temperature at which c is maximumdecreases as the size of the system increases (and it converges to a certain temperature in thethermodynamic limit).

According to the Mermin-Wagner theorem, the magnetization per spin

〈m〉 =1

N

⟨√M2x +M2

y

⟩(6)

vanishes at all finite temperatures. However, due to finite size effects, the system appears to bemagnetized at finite temperatures, as shown in Fig.4. Notice that the magnetization decreasesas the size L increases, and in the limit of L → ∞ it will correctly be zero for any T > 0.The magnetic susceptibility per spin is computed as

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

m

T

L = 8

L = 12

L = 16

L = 20

L = 32

L = 64

L = 100

Figure 4: Average magnetization per spin as a function of the temperature for different sizes L. Error bars are oftensmaller than symbols.

χ = βN(⟨m2⟩− 〈m〉2

)(7)

3

0

20

40

60

80

100

120

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χ =

βN

(<m

2>

- <

m>

2)

T

L = 8L = 12L = 16L = 20L = 32L = 64L = 100

Figure 5: Magnetic susceptibility per spin, com-puted as χ = βN

(⟨m2

⟩− 〈m〉2

), as a function

of temperature for different sizes L. Error bars areoften smaller than symbols.

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χ =

βN

<m

2>

T

L = 8

L = 12

L = 16

L = 20

L = 32

L = 64

L = 100

Figure 6: Magnetic susceptibility per spin, com-puted as χ = βN

⟨m2

⟩, as a function of tem-

perature for different sizes L. Error bars are oftensmaller than symbols.

and since 〈m〉 = 0, this expression reduces simply to

χ = βN⟨m2⟩

(8)

However, using expression 7 or 8 will give different results because, even for large sizes, themagnetization computed in the simulation does not vanish. This difference is shown in Fig.5and 6. From now on, I will refer to the magnetic susceptibility as the one computed withEq.8. Moreover, according to Kosterlitz-Thouless theory, the susceptibility diverges as thetransition is approached from above, and remains infinite for all lower temperatures. On theother hand, this divergence is never observed in the simulations because of the finite size ofthe systems considered. In the next section i will discuss this problem in more details.

4 Kosterlitz-Thouless transitionIn this section I present the main aspects of the Kosterlitz-Thouless transition in the two-dimensional XY model.

4.1 Correlation functionAs announced in the introduction, this model presents a low-temperature and a high-temperaturephase.

In the low-temperature phase, T < TKT , all spins are almost aligned but the long-rangeorder is destroyed by spin fluctuations, and the correlation function decays with a power law

〈~si · ~sj〉 ∼ r−η(T ) (9)

This phase is said to possess quasi-long range order.In the high-temperature phase, T > TKT , even neighbor spins are very weakly correlated

and the correlation function decays exponentially

〈~si · ~sj〉 ∼ e−rξ (10)

In Fig.7 the correlation functions obtained for a system of linear size L = 32 for differenttemperatures are shown. One can notice that the decay of the correlation function is faster as

4

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 5 10 15 20

Corr

ela

tion function

r

T = 0.1

T = 0.2

T = 0.3

T = 0.4

T = 0.5

T = 0.6

T = 0.7

T = 0.8

T = 0.9

T = 1.0

T = 1.1

T = 1.2

T = 1.3

T = 1.4

T = 1.5

T = 1.6

Figure 7: 〈~si · ~sj〉 as a function of the distancebetween spins for several temperatures. Lines areonly a guide for the eyes.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 2 4 6 8 10 12 14 16

Corr

ela

tion function

r

T = 0.4T = 1.6fit f(r)fit g(r)

Figure 8: 〈~si · ~sj〉 for T = 0.4 and T = 1.6and corresponding fits, where f(r) ∼ r−η(T ) andg(r) ∼ e−

rξ .

the temperature increases. In order to stress the different behavior of this decay for T < TKTand T > TKT , Fig.8 shows the correlation function for one temperature below and one aboveTKT . Notice that the former is well fitted by a function of the type of Eq.9, while the latter iswell fitted by a function of the type of Eq.10.

4.2 Role of vorticesThe magnetization is, as noticed, not a good order parameter to characterize the phase transi-tion in the two-dimensional XY model. Kosterlitz and Thouless showed indeed that the phasetransition is driven by the binding and unbinding of vortices.

Vortices are energetically stable configurations which affect the spin configuration of thesystem in a highly non-local way (they decrease the correlation of the system). A vortex (orantivortex) describes what happens with the spin angles as we go around an closed path. Fig.9shows a typical configuration with a vortex and an antivortex of vorticity |n| = 1 (or unitcharge). In order to compute the vorticity in the simulation, one goes around an elementaryplaquette and measures the difference of the angles2 at the end and begining of each bond inthe anti-clockwise direction. Finally, one takes the sum of these differences. For instance, fora vortex with n = 1 this sum is equal to 2π, while for an antivortex with n = −1 it is equalto −2π (for a general vortex of vorticity n, the sum is equal to 2nπ).

As said, vortices are energetically stable, meaning that, in order to get rid of them, one hasto overcome an energy barrier. Hence, a configuration with vortices is a metastable configu-ration at low temperatures. Indeed, if one starts from a configuration at T = ∞ (a randomconfiguration) and quench to T = 0 (accepting only moves that decrease the energy), severalvortex-antivortex pairs will be visible in the system, as shown in Fig.10.

In a simplified picture where we have just one vortex of vorticity (or charge) n in a systemof radius R, the energy cost of the vortex is [3]

βEnvortex ≈ βεn0 + πKn2ln(R

a

), (11)

where K = βJ , a is the radius of the core of the vortex, and εn0 is the energy cost associatedto the core region. The dominant part of the energy comes from the region outside the core,

2making sure that these differences are always between −π and +π.

5

30

35

40

45

50

20 25 30 35 40 45 50

Figure 9: Typical configuration with a vortex(blue) and an antivortex (green) of unit charge.

0

10

20

30

40

50

0 10 20 30 40 50

Figure 10: Typical configuration with severalvortices after quenching to T = 0.

and diverges with the size of the system, R. The entropy associated with the vortex is

Svortex = kB ln(R

a

)2

. (12)

Therefore, in the case of n = 1, the free energy difference between a configuration with andone without a vortex is

β∆F = (πK − 2)ln(R

a

)+ βε0. (13)

Equation 13, even though it represents an approximation, clearly shows that for temperatureshigher than a particular temperature (T > TKT ) ∆F < 0, while for lower temperatures (T <TKT ) ∆F > 0. This means that for temperatures above the critical temperature there willbe proliferation of vortex excitations, while for temperatures below there will be no vortices.This is clearly shown Fig.11, where the computed number of vortices with n = 1 (normalizedby the number of spins) is represented as a function of temperature for different system sizes.

-0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

N+

/N

T

L = 8

L = 12

L = 16

L = 20

L = 32

L = 64

L = 100

Figure 11: Number of vortices (n = 1) per site as a function of temperature for different system sizes.

Actually, in order to minimize the energy, vortices and antivortices always come in pairs.For any vortex there will be a clearly associated close by antivortex (see for instance Fig.10).

6

Indeed, computing the number of antivortices with n = −1, one gets exactly the same resultsshown in Fig.11 for the vortices. In the low-temperature phase, vortex-antivortex pairs arebounded toghether, while in the high-temperature phase they are unbounded. Hence, thehigher the temperature, the easier to find ”isolated” vortices.

It can be shown [3] that the energy of a system with two vortices of opposite vorticity isgiven by

Hvortices = V (~r1 − ~r2) = −2πKn1n2ln(|~r1 − ~r2|

a

)+ constant, (14)

where the constant takes into account the core energies of the two vortices. If n1 = 1 andn2 = −1, Eq.14 reduces simply to

V (~r1 − ~r2) = 2πKln(|~r1 − ~r2|

a

)+ constant, (15)

which is nothing else but the electrostatic interaction of two opposite unit charges in twodimensions. Hence, a vortex and an antivortex of unit vorticity attract each other via thepotential in Eq.15, like if they were quasi-particle of opposite charge in a two-dimensionalworld3. In order to verify qualitatively Eq.15, I generated several configurations with a vortex(n = 1) and an antivortex (n = −1) by equilibrating a 50 × 50 system at T = 1, and thenquenching to T = 0. A plot of the energy as a function of the distance between the vortex andthe antivortex is shown in Fig.12. Notice, that the energy globally increases logarithmicallyas the distance increases, even though data are not in perfect agreement with the fitted curve.This is due to the fact that, for a system of finite size, the energy also depends on the relativeorientation between the vector ~r = ~r1 − ~r2 and the axes of the lattice.

-1.992

-1.991

-1.99

-1.989

-1.988

-1.987

-1.986

-1.985

-1.984

0 5 10 15 20 25 30

E

r

datalogarithmic fit

Figure 12: Computed energy as a function of the distance between vortex and antivortex and corresponding logarith-mic fit.

In the next subsection, I will mainly focus on the methods to estimate the critical temper-ature TKT .

4.3 Magnetic susceptibility, critical temperature and exponent η(T )As said in the previous section, in the thermodynamic limit the magnetic susceptibility di-verges for T = TKT and remains infinite for all lower temperatures. Given the finite size of

3from here the parallelism between the two-dimensional XY model and the two-dimensional Coulomb gas.

7

the systems considered, this divergence is never observed in the simulations. However, onecan see it by analyzing the dependence of the results on the linear size, L, as discussed below.

The susceptibility at the critical temperature may be expressed in terms of the correlationlength [2], ξ, as

χ ∼ ξ2−η(T=TKT ), (16)

where η(T = TKT ) = 1/4. In the thermodynamic limit, when T approaches TKT fromabove, T → T+

KT , the correlation length diverges [2]

ξ ∼ exp(

b

t1/2

), (17)

where t = (T −TKT )/TKT and b is a constant. For a finite size system this means that at thecritical temperature ξ ∼ L. Therefore, we can rewrite Eq. 16 in terms of L as

χ(TKT ) ∼ L7/4 (18)

Hence, the quantity χL/L7/4 at the critical temperature should be independent of L. Thismeans that if we plot χL(T )/L7/4 for different values of L, the graphs should intersect at agiven value of the temperature (as shown in Fig.13), which represents an estimate of TKT .Actually, the point of intersection still has a weak dependence on the particular sizes consid-

0

1

2

3

4

5

6

7

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

χL/L

7/4

T

L = 8

L = 12

L = 16

L = 20

L = 32

L = 64

L = 100

L = 200

Figure 13: χL(T )/L7/4 for different sizes. Error bars are often smaller than symbols.

ered. However, for large enough sizes this dependence can roughly be neglected. For instance,considering the point of intersection between χ100 and χ200, I get TKT = 0.90±0.014, whichis in agreement with the estimate in [4].

The divergence of the magnetic susceptibility for T ≤ TKT in the thermodynamic limit(L → ∞) can be seen by plotting the logarithm of χL(T ) versus the logarithm of L. Fig.14clearly shows that χL increases as a certain power of L for T ≤ TKT , while for T > TKT itreaches a plateau. This means that, in the limit of L→∞, χ diverges for T ≤ TKT , while itis finite for T > TKT .

In particular, one can fit the results for T ≤ TKT with a straight line f(T ) = αT + β,where the slope α is nothing else but 2− η(T ), obtaining therefore an estimate for η(T ). Theresults for η(T ) are shown in Fig.15.

4I chose the error simply as 1/2 of the resolution on the temperature.

8

2

3

4

5

6

7

8

9

10

11

12

13

2 3 4 5 6 7

log(χ

L)

log(L)

T = 0.1T = 0.2T = 0.3T = 0.4T = 0.5T = 0.6T = 0.7T = 0.8T = 0.9T = 1.0T = 1.1T = 1.2

Figure 14: log(χL(T )/L7/4) Vs log(L) for dif-ferent temperatures.

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

η

T

Figure 15: η as a function of T with correspond-ing error bars.

In order to estimate the critical temperature, one could also compute the so called spinstiffness, or helicity modulus, which measures the energy change caused by a twist on theboundary conditions (in one direction). It can be calculated as

ρs = −1

2〈E〉 − 1

T

⟨ ∑<i,j>

sin(θi − θj) ~rij · ~x

2⟩, (19)

where ~rij is the vector from spin i to spin j and ~x is a unit vector in a fixed direction. Fortemperatures above TKT the twist has little influence due to the exponentially decreasingcorrelation. From the relation (with kB = 1, like in the simulations)

ρs(TKT ) =2

πTKT (20)

we see that TKT (L) is given by the intersection of the spin stiffness with the line f(T ) = 2πT .

The results are shown in Fig.16 and Tab.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Spin

Stiffness

T

L = 8L = 12L = 16L = 20L = 32L = 64L = 100L = 200f(T)=2T/πâ��

Figure 16: Spin stiffness as a function of temperature for different sizes. The line f(T ) = 2πT is also shown.

Taking ξ ∼ L in Eq.17, it can be shown that

TKT (L) ∼ TKT +b2

(lnL)2. (21)

9

Table 1: Estimates of TKT (L) and corresponding errors.

L TKT error

8 0.962 0.01012 0.945 0.01016 0.937 0.01020 0.931 0.01032 0.923 0.01064 0.916 0.010100 0.912 0.010200 0.906 0.010

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24

TK

T(L

)

ln(L)-2

TKT(L = ∞)= 0.898 +/- 0.001

datafit

Figure 17: TKT (L) Vs ln(L)−2 and correspond-ing linear fit considering all data.

0.895

0.9

0.905

0.91

0.915

0.92

0.925

0.93

0.935

0.03 0.04 0.05 0.06 0.07 0.08 0.09

TK

T(L

)

ln(L)-2

TKT(L = ∞)= 0.894 +/- 0.002

datafit

Figure 18: TKT (L) Vs ln(L)−2 and correspond-ing linear fit considering only L ≥ 32.

Therefore, plotting TKT (L) versus ln(L)−2 data should be distributed in a straight line. Fit-ting the results, TKT (L→∞) can be estimated as the intersection of the line with the y-axis.Considering all data, I got TKT = 0.898±0.001, which is very close but not compatible withthe reference value in [4]. However, fitting only data withL ≥ 32, I got TKT = 0.894±0.002,which is in perfect agreement with the reference value in [4]. The results are shown in Fig.17,18.

Finally, from the spin stiffness we can also get qualitative estimates of the exponent η(T )through the relation [5]

η(T ) =T

2πρs(T ). (22)

Qualitative for two reasons: first, these estimates will be size dependent; second, no errorshave been computed for the spin stiffness and, therefore, no errors can be computed for η. Theresults obtained from the spin stiffness with L = 200 are shown in Fig.19. The estimates arein good agreement with the ones showed in Fig.15. Another possible way to estimate η(T )is by fitting the correlation function for temperatures below TKT . However, one has to becareful and take into account that in the simulation we are dealing with systems with periodicboundary conditions. Hence, also the correlation function is periodic and it is necessary toadd a correction to a simply power law decay. One possible solution is using the following

10

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

η

T

Figure 19: Qualitative estimates of η(T ) ob-tained for a system of linear size L = 200 throughEq.22. Lines are only a guide for the eyes.

0

0.05

0.1

0.15

0.2

0.25

0 0.2 0.4 0.6 0.8 1

η

T

Figure 20: Estimates of η(T ) obtained by fittingthe correlation function for T ≤ TKT with Eq.23.Lines are only a guide for the eyes.

function

f(r) = A

[L

πsin(πLr)]−η

. (23)

Fig.20 shows the estimates of η(T ) obtained by fitting the correlation function shown previ-ously with the function in Eq.23. The results are in very good agreement with the previousones.

5 ConclusionI performed a numerical study of the Kosterlitz-Thouless transition in the two-dimensionalXY model using Monte Carlo simulations. First, I showed that the correlation function inthe low-temperature phase decreases with the distance between spins with a power law, whileit decreases exponentially in the high-temperature phase. Fitting the results for T ≤ TKT ,I could also estimate the exponent η(T ). Then, I showed how the transition is character-ized by the binding and the unbinding of vortices, and that vortex excitations proliferate fortemperatures above TKT . Then, I showed how the magnetic susceptibility diverges in thethermodynamic limit for T ≤ TKT , by plotting the logarithm of χL(T ) versus the logarithmof L. This also allowed me to estimate the exponent η(T ) for temperatures below TKT .

Finally, I estimated the critical temperature using two different methods. The first es-timate, TKT = 0.90 ± 0.01, was derived as the point of intersection between the curvesχL/L

7/4 with L = 100 and L = 200. This is in perfect agreement with the reference valuein [4], even though it is still weakly affected by finite size effects. In order to eliminate theseeffects, I estimated the critical temperature for different sizes L of the system as the pointof intercection between the spin stiffness and the line f(T ) = 2

πT . I finally fitted the re-sults for L ≥ 32 to get an estimate of TKT in the thermodynamic limit (L → ∞), obtainingTKT = 0.894± 0.002. This final estimate is in perfect agreement with the reference value in[4] and more accurate than the previous one.

11

References[1] M. E. J. Newman, G. T. Barkema, Monte Carlo Methods in Statistical Physics. Oxford

University Press, 1999.

[2] M. Le Bellac, F. Mortessagne and G. G. Batrouni, Equilibrium and Non-EquilibriumStatistical Thermodynamics. Cambridge University Press, 2004.

[3] Mehran Kardar, Statistical Physics of Fields. Cambridge University Press, 2007.

[4] Peter Olsson, Monte Carlo analysis of the two-dimensional XY model. II Comparisonwith the Kosterlitz renormalization-group equations, Phys. Rev. Lett. 17, 1133 - 1136(1995).

[5] P. M. Chaikin and T. C. Lubensky, Principles of Condensed Matter Physics. CambridgeUniversity Press, 2000.

12