Scanning the critical fluctuations: Application to the phenomenology
of the two-dimensional XY model
Ricardo Paredes V.1,2 and Robert Botet3,4
1Instituto Venezolano de Investigaciones Cientìfica, Centro de Fìsica, Laboratorio de Fisica Estadistica Apdo21827
1020A Caracas, Venezuela
2
NanoStructured Materials, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands
3
LPS Orsay, CNRS UMR8502, F-91405 Orsay, France
4
Laboratoire de Physique des Solides Bât. 510, Université Paris-Sud, F-91405 Orsay, France
共Received 19 June 2006; published 19 December 2006兲
We show how applying field conjugated to the order parameter may act as a very precise probe to explore the probability distribution function of the order parameter. Using this “magnetic-field scanning” on large-scale numerical simulations of the critical two-dimensional XY model, we are able to discard the conjectured double-exponential form of the large-magnetization asymptote.
DOI:10.1103/PhysRevE.74.060102 PACS number共s兲: 05.50.⫹q, 75.10.Hk, 64.60.Fr
I. INTRODUCTION
Derivation of the complete equation of state of a many-body system is generally a formidable task. When the system may appear under various phases at the thermodynamic equi-librium, this problem requires knowledge of the exact prob-ability distribution function共PDF兲 of its order parameter. De-spite a number of attempts, just a few instances are available 关1兴. Even the exact PDF for the two-dimensional 共2D兲 Ising model is still unknown.
Within this context, the critical point is very particular, since the universality concept tells us that only limited infor-mation is needed to obtain the complete leading critical be-havior. For instance, general arguments give precisely the tail of the critical PDF, P共m兲, for the large values of the order parameter, m, namely关2兴,
P共m兲 ⬃ e−cm␦+1, 共1兲
with c a positive constant and ␦ the magnetic field critical exponent, or the distribution of the zeros of the Ising parti-tion funcparti-tion in the complex magnetic field 关3兴 共such a partition function is Fourier transform of the PDF兲.
In the present work, we explain how the real magnetic field can be generally used as a very accurate probe to scan quantitatively the zero-field PDF tail, exemplifying the method with the critical 2D XY model. By the way, we will see that the popular double-exponential approximation of the PDF for this system in the low-temperature range cannot be correct at the critical temperature, and we provide alternative approximation which is consistent with the critical behavior. Consequently, our results discard possible fundamental con-nection between the behavior of this magnetic model at the critical point and the field of extreme value statistics.
II. FORMER APPROXIMATION OF THE MAGNETIZATION PDF FOR THE CRITICAL 2D
XY-MODEL
It was argued关4–7兴 that the PDF P共m兲 of the magnetiza-tion m of the 2D XY model, at temperatures smaller but close to the Berezinskii-Kosterlitz-Thouless 共BKT兲 critical
tem-perature, could be approximated by the generalized Gumbel form,
P共m兲 ⬀ exp共bz−eaz兲, 共2兲 where the reduced magnetization z=共m−具m典兲/ is used. From low-temperature spin-wave theory and direct numerical simulations, one obtains关5兴
a⬇ 1.105; b⬇ 1.74; ⬇ 0.69. 共3兲 It was regularly noticed关5兴 that the form 共2兲 cannot be the exact solution of the corresponding statistical problem, and the alternative form共1兲 was proposed 关5兴. Indeed, Eq. 共2兲 is inconsistent with the general behavior 共1兲, since ␦= 15 for the 2D XY model. On the other hand, Eq.共2兲 is appealing, as it suggests connection between the 2D XY model close to the critical temperature and the statistics of extreme variables 关8兴. Therefore, the question of a possible bridge between these two active fields of statistical physics should be exam-ined precisely. The additional question to know whether re-lation 共1兲 could fail for this system is also fundamentally important. We will examine hereafter these two questions.
III. TWO ALTERNATIVE HYPOTHESIS
We consider the 2D XY model关9兴 on a square lattice of size L⫻L with periodic boundary conditions. The N=L2 classical spins are confined in the x-y lattice plane, and they interact according to the Hamiltionian H = −J兺具i,j典Si· Sj, where J⬎0 is the ferromagnetic coupling constant and the sum runs over all nearest-neighbor pairs of spins. Eventual critical features are characterized by the singular behavior of the scalar magnetization per site: m⬅N1
冑共兺i
Si兲2, which is a non-negative real number. We define also the instantaneous magnetization direction as the angle such that 兺iSix
= mN cos and兺iSi y
= mN sin.
There is a continuous line of critical points for any tem-perature below the critical BKT temtem-perature TBKT 关10兴. In this region, 0艋T艋TBKT, the system is critical, and asymptotic共i.e., L→⬁兲 self-similarity results in the so-called first-scaling law关11兴:
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具m典P共m兲 = ⌽T共z1兲, with z1⬅ m
具m典, 共4兲
and ⌽T is a scaling function which depends only on the actual temperature T. Under this form, the hyperscaling rela-tion,具m典/= constant term, is automatically realized. Equa-tion 共4兲 is sequel of the standard finite-size scaling theory 关12兴, but it is highly advantageous that Eq. 共4兲 does not re-quire knowledge of any critical exponent. Figure1 gives a numerical exemplification of the first-scaling law at TBKT, and illustrates the overall shape of the distribution ⌽c共z1兲 共hereafter, the index“c” refers to the BKT critical point, T = TBKT兲.
We separate the free energy F of the 2D XY system at equilibrium共temperature T=1/兲 into the sum of a regular part describing the small values of the magnetization, a sin-gular part关14兴 vanishing as the essential singularity 关15,16兴 when T→TBKT, and a regular part for the large values of the magnetization, namely,
F共m兲 =0共m/具m典兲 +S共m/具m典兲 +⬁共m/具m典兲. 共5兲 Clearly, discussion on the system behavior can be carried out either through the free energy共5兲 or the first-scaling law 共4兲, since ln P共m兲=−F共m兲+constant term.
A. The regular small-m tail
As the singular behavior should vanish at the BKT tran-sition, we study first the regular small-m behavior of P共m兲 at T = TBKT. Numerical results for Pc共m兲 are shown in Fig.2in the form共4兲. They suggest the leading form
ln Pc共m兲 ⬇ b1共m/具m典c兲2. 共6兲
B. The singular small-m tail
We consider now the singular part of the free energy through the combination ln(具m典P共m兲)−ln(具m典cPc共m兲) vs the reduced magnetization z1⬅m/具m典. The data plotted in Fig. 3, suggest a cubic z13behavior:
S共z1兲 ⬇ c共T兲z13, 共7兲 for every T⬍TBKT, and for the values of m smaller than the mean. Moreover, c共TBKT兲=0.
FIG. 1. PDF of the magnetization for the 2D XY model at the critical temperature TBKT, plotted in the first-scaling form共4兲. The
scaling law is confirmed for L = 64 共stars兲 and L=128 共circles兲, while the L = 16 共continuous line兲 shows finite-size deviation. Wolff’s single-cluster algorithm was used 关13兴. Each data set
corresponds to average over 25 000 000 independent realizations.
FIG. 2. Part m艋具m典cof the logarithm of the scaled PDF共4兲 vs z12, for the 2D XY model at T = TBKT. The solid straight line is
the best fit: ln(具m典cPc共m兲)=b1z12+ constant term for the L = 128,
z1⬍0.8, data. Numerically, b1= 12.7. Same symbols as in Fig.1.
FIG. 3. Part m艋具m典 of the logarithm of the scaled PDF, cor-rected by the regular part of the free energy, for L = 16 and four different temperatures: T = 0.3 共circles兲, T=0.6 共squares兲, T=0.8 共diamonds兲, and T=0.885 共stars兲 which is close to the critical tem-perature共TBKT⬇0.893 关19兴兲. The plot is versus z13⬅共m/具m典兲3. The
straight lines are the best fits Eq.共7兲.
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C. The large-m tail at BKT point
Instead of using multicanonical Monte Carlo simulations 关17兴 which need too large system sizes to conclude 关18兴, we consider static in-plane magnetic field, H, as a probe to study the features of the PDF for the large values of the magneti-zation. Indeed, as the intensity of H increases, the most prob-able magnetization, mH쐓, as well as its mean value, 具mH典, explores larger values of the PDF tail. We consider two al-ternative forms for the critical tail, namely,共a兲 the “Gumbel-like” shape 共2兲—noted below: “hypothesis 共G兲”—which writes in the first-scaling form
⌽c共z1兲 ⬃ exp共− 0ea0z1兲 for z1→ ⬁ 共G兲,
with a0= a⫻共具m典c/c兲 关⬇16.4 from Eq. 共3兲 and Table I兴, and 0=e−a0. It is the form suggested in 关4–6兴. 共b兲 The
“Weibull-like” critical shape—noted below: “hypothesis 共W兲”—which is 关20兴
⌽c共z1兲 ⬃ exp共− 1z1␦+1兲 for z1→ ⬁ 共W兲, with1a positive parameter, and␦+ 1 = 16关21兴.
Let be the direction of H with respect to the x axis 关i.e., H=共H cos, H sin兲兴. According to general thermo-dynamics, the magnetization PDF is given by Pc共m,H兲 ⬀exp关−cF+cL2mH cos共−兲兴, with the fieldless free en-ergyF. The most probable magnetization direction is given by Pc共m,H兲/= 0, leading to =, while the most probable magnetization, mH쐓, is the solution of the equation
Pc共m,H兲/m = 0 for a given value of H. Rewritten in terms of the auxiliary variables X⬅H/具m典c␦ and Y⬅H/mH쐓␦, Eqs.共5兲 and 共6兲, with hypothesis 共G兲 or 共W兲, result, respec-tively, in X A+ 2b1
冉
X Y冊
1/␦ =0a0ea0共X/Y兲 1/␦ 共8兲 or =1共␦+ 1兲X Y, 共9兲which are implicit equations for the most probable magneti-zation, mH쐓 共written in the combination Y兲 vs the magnetic field H and the system size N共written in the combination X兲. The constant A is such that A−1=
cL2具m典c␦+1⬇1.07.
For the large magnetic field, mH쐓 is expected to be much larger than具m典c, that is, X / YⰇ1. Consequently, the solution of Eq.共8兲 is
Y = a0␦X/共ln X + C兲␦, 共10兲 where C = −ln共A0a0兲⬇14.2 is a positive constant.
Within the hypothesis共W兲, one has 共X/Y兲1/␦ⰆX/Y, such that Eq.共9兲 shows that Y is asymptotically a constant
Y⬇ A1共␦+ 1兲. 共11兲
So, increase of Y with the intense magnetic field should be interpretated as a failure of共W兲.
IV. INFERENCE FROM THE NUMERICAL SIMULATIONS Both solutions, Eqs.共10兲 and 共11兲, are drawn in Fig.4in comparison with the results of large-scale numerical simula-tions of the 2D XY model with the in-plane magnetic field at the BKT temperature. It is clear that the numerical simula-tions are consistent with the hypothesis 共W兲, while the double-exponential tail 共G兲 should be discarded. This sug-gests the following form of the critical PDF for the 2D XY model:
TABLE I. Temperature, system size, average magnetization per spin, ratio of average magnetization to standard deviation. The best fit for the latter is具m典c/c= 14.81− 21.5/ L at the BKT temperature
TBKT= 0.893. T L 具m典 具m典/ 0.3 16 0.923218 66.958 0.6 16 0.836307 29.249 0.8 16 0.764091 18.260 0.885 16 0.723259 13.907 0.893 16 0.718814 13.467 0.893 32 0.662819 14.119 0.893 64 0.611181 14.486 0.893 96 0.582217 14.583 0.893 128 0.563209 14.644 0.893 256 0.518921 14.687 0.893 512 0.478045 14.829
FIG. 4. Double-logarithmic plot of H /具mH典␦ vs the reduced
magnetic field HLyH with y
H= 2␦/共␦+ 1兲. These two variables are
convenient for the numerical simulations and simply related to the variables X and Y of the text: H /具mH典␦= Y⫻共mH쐓/具mH典兲␦ and
HLyH= X⫻共A
c兲−␦/共␦+1兲, with the constant values 共mH쐓/具mH典兲␦⬇1
and共Ac兲−␦/共␦+1兲⬇0.96. The dashed curve is the solution of the first
Eq.共8兲 关corresponding to the hypothesis 共G兲兴, while the dotted line
is Eq.共11兲, with 1=in agreement with Eq.共13兲. The system size
goes from L = 16 up to L = 512. Each point corresponds to an aver-age over 100 000 independent realizations关22兴.
SCANNING THE CRITICAL FLUCTUATIONS:… PHYSICAL REVIEW E 74, 060102共R兲 共2006兲
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Pc共m兲 ⬀ eb1z1 2
−1z1 16
, z1⬅ m/具m典. 共12兲
Below the BKT critical temperature, additional term +c共T兲z1 3
should appear in the exponential.
In order to understand the origin of the approximation 共2兲, let us change the reduced magnetization according to z1= 1 + z/共具m典/兲. At TBKT, and for the small values of z/共具m典c/c兲 共recall that 具z典=0 and that 具m典c/c⬇14.8 is a rather large number兲, we obtain
Pc共m兲 ⬀ e2b1z/共具m典c/c兲−1关1 + z/共具m典c/c兲兴16.
Writing then 1 + z/共具m典c/c兲⬇ez/共具m典c/c兲, yields Eq. 共2兲,
provided the following relations are verified:
a= 16 具m典c/c
; b= 2b1 具m典c/c
; =1. 共13兲 So, Eq. 共2兲 appears to be a good approximation around the most probable magnetization, but is inconsistent with the general critical relation具mH典⬀H1/␦, unlike Eq. 共12兲. By the way, the conjectured relation 关5兴 b/ a=/ 2 writes simply b1= 4, that we accept here as a conjecture 共numerically b1⬇12.7, see Fig.2兲.
V. CONCLUSION
In the present work, we explained how the use of the field conjugated to the order parameter provides unique informa-tion on the tail of the probability distribuinforma-tion funcinforma-tion of the order parameter. This is of major importance for the critical systems, since the tail shape is directly linked to the value of the magnetic critical exponent. Therefore, this general method provides an alternative way to calculate or measure the critical exponent␦.
We chose to use this method with the 2D XY model at the critical temperature. Indeed, a double-exponential approxi-mation of the magnetization PDF in the zero magnetic field is found to be inconsistent with the critical behavior of the system—though correct near the most probable magnetiza-tion. Our analysis is only done at the BKT temperature. However, when the temperature is reduced the same type of solution 共W兲, with different values of ␦, is expected along the line of critical points. Finally, the Gumbel distribution becomes a good approximation in the low-temperature range 共spin-wave solution region兲 when excitations of vortex pairs are negligible.
ACKNOWLEDGMENTS
The authors thank CNRS and FONACIT 共Contract No. PI2004000007兲 for their support.
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