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Inhomogeneity of vortex charge in the 2D classical XY model
20 April 1998
PHYSICS
Physics Letters A 241 (1998)
LETTERS
A
127-130
Inhomogeneity of vortex charge in the 2D classical XY model S.B. Ota, Smita Ota Institute
of Physics, Sachivalaya Marg, Bhubaneswar 751005, fndiu
Received 5
November 1997;
accepted for publication 13
January 1998
Communicated by J. Flouquet
Abstract
~~icrocanonical Monte Carlo simulations have been carried out on the 2D classical XY model which shows both KosteditzThouless (KT) as well as first-order transitions. Simulations on a 30 x 30 spin system show that the vortex charge distribution is inhomogeneous. The maximum inhomogeneity occurs at a temperature which coincides with the position of the specific heat peak in the KT case and in the coexistence region in the first-order case. The inhomogeneity is found to be larger in the KT case as compared to the first-order one. The vortices are found to be dist~buted in a more space-fii~ing manner in the latter case as compared to the former one. @ 1998 Elsevier Science B.V. PACS: 75.IO.Hk;02.7O.Lq:05.20.-y; 0.5.70.Fh
An important statistical spin model is the 2D classical XY system [ I 1, with Hamiltonian given by
where the symbols have their usual meaning. For p* = 1 the Hamiltonian reduces to the usual classical 2D XY system, which admits the Kosterlitz-Thouless (KT) tr~sitio~. By increasing the value of p’, the potential well can be made narrower and for p* > 10 the transition becomes first order in nature [ 21. The system does not show any long-range order [ 31 and there are two types of excitations present in the system; namely, the vortices and the spin waves 14-71. The vortices are topological defects. When p* = 1, the low-temperature phase has only bound vortexantivortex pairs, and the KT transition is associated with the unbinding of the vortex-antivortex pairs. Being of a topological nature, no specific heat anomaly is observed at the KT transition temperature
Tc. But the temperature dependence of the specific heat shows a peak at a temperature, which is about 15% higher than T,. The vortex unbinding transition temperature (7;,) increases by reducing the potentiaI well width. On the other hand, local disorder will set in at tem~ratures near the potential well height TI, N V(v) - V(0) = 2f. Therefore, for a sufficiently narrow potential well TI, < T,,, we expect the continuous transition to yield to a first-order vacancy condensation ~ansition. The classical 2D XY system has been extensively studied in the context of superfluids [ 81 and superconducting films [ 91. Various theoretical techniques have been used to study the spin waves, the vortices and the nature of the transition in the 2D classical XY system [ 10,l1 ] . Several Monte Carlo (MC) simulations have been carried out to study the KT transition and the first-order transition in this system [ 12-23 I. Here we report the inhomogeneity of the vortex distribution of the classical 2D XY system, which has so far not been found.
037%9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO37S-9601(98)00034-6
128
S.B. Otu. S. Otu/Pl~~.sks
We performed microcanonical MC simulations on a square lattice having 30 x 30 spins [ 24,251. We used periodic boundary conditions and have calculated the system temperature (T), vorticity, the magnetisation curve and their respective standard deviations at each given total energy (E) . We used 1 x 1O5 MCSS for equilibration and I x IO5 MCSS for averaging. The accuracy of the mean value of the physical quantities was estimated by performing block averages consisting of 5 x 10” MCSS each and then finding the standard deviation of the block averages. The standard deviation of the calculated vortex density is typically less than 0.5%. The KT transition temperature occurs at 0.9 and the maximum of the temperature dependence of the specitic heat occurs at I .09. We analysed the Monte Carlo configurations evolved during the simulation to study the inhomogeneity of the vortices across the transition. To this end, we estimated the average number (V,(r) ) of vortices-antivortices at a distance r from any given vortex. This is the vortex-vortex correlation function times r2. From this we obtained the number of positive (negative) vortices (V(,+ (V,_ ) ) surrounding a
Letter.7 A 241 (1998) 127-130
Fig. 1. The temperature dependence of the net vortex charge V,, in a radius of v% (+) and 2&&r (X ) for p’ = I. The data represent averages over I x lo5 MCSS. (A ) and (0) represent averages over 2 x 10’ MCSS.
+
-._
%
‘i
positive vortex within a distance of &LZ and 2&a ( CI is the lattice spacing). For p’ = 1, it is seen that V&, when determined over a distance of &a, decreases with temperature initially up to T = 1.1 and then increases with further increase in temperature. A decrease of V,_ with temperature is not observed, when determined over a larger distance of 2&a, whereas V,+ shows a steady increase with increase in temperature. On the other hand, for p’ = 50, both V,+ and v,_ are seen to increase with increase in energy even when determined over a smaller distance of &a. ’ We next determined the net vortex charge (V,,) within a distance of &a and 2&a from a vortex as functions of T (or E). Fig. 1 shows the temperature dependence of V,, for p2 = 1, and comments on some features of this graph are in order. Firstly, V,, goes through a maximum. Secondly, the magnitude of &f is reduced as the distance increases. Finally, the maximum occurs at T = 1.1, which corresponds to the specific heat peak. A similar graph is shown in Fig. 2, for p2 = 50. The behaviour is qualitatively similar to the
’ The energy is used instead of the temperature to clearly present the points in the coexistence region of the first-order transition.
Fig. 2. The energy dependence of net vortex charge V’, in a radius of x/% (+) and 2v’% ( x ) for p’ = SO. The data represent averages over I x IO5 MCSS. (A) and (0) represent averages over 2 x lo5 MCSS.
case of p2 = 1, except for the following differences. The magnitude of V,, is comparatively smaller. The maximum of V,, occurs in the coexistence region of the first-order transition. The vortices are therefore. not distributed homogeneously in the lattice. Examination of the configurations revealed the presence of vortex clusters in the critical region, which has also been pointed out by Tobochnik and Chester [ 151. In order to characterise the clusters, we determined
S.3. &a, S. Ota/Pbysics Lrttem A 24i (19981 127-130
Fig. 3. The temperature dependence of the fractal dimensions (df, ) ( x ) and ((if,,) (+) of the vortex distribution for p2 = I. The data represent averages over I x 10s MCSS. (A) and (0) represent averages over 2 x I@ MCSS. The arrow indicates the maximum difference in dfr and dfP.
the fractal dimension V(v) cx #:,
Fig. 4. The energy dependence of the fractal dimensions (d t, ) ( x ) and (dfP ) (+) of the vortex distribution for p2 = SO. The data represent averages over 1 x 10s MCSS. (A) and (0) represent averages over 2 x IO5 MCSS. The arrow indicates the maximum difference
in df, and dfP.
df using the following definition, (2)
where V(r) is the total number of vortices in a radius r. We choose the origin on a vortex and include only a vortex or both vortex and antivortex, to determine diii or dfr, respectively. In( V,) versus ln( rf showed a linear behaviour for ln( r) up to 1.5 and thereby df was determined. Fig. 3 shows the energy dependence of df for p’ = 50. It is seen that df falls far below 2. The maximum difference in dfp and df, occurs at a system energy corresponding to the maximum position of Vqf. At higher system energy df,, becomes equal to dp. A similar behaviour in the temperature dependence of df was observed for p2 = 1, which is shown in Fig. 4. In this case, the maximum difference in dfp and dfr occurs at a temperature corresponding to the maximum position of V,,. This observation implies a difference in the distribution of a vortex with respect to an antivortex. This difference is found to be less for p2 = 50. Moreover, we observed that dft as a function of the number of vortices (u) is not the same for p’ = 1 and 50. This is displayed in Fig. 5. It is seen that for p2 = 50, the vortices are distributed in a more space-filling manner compared to that for p’ = I. We have not yet found an explanation of the observed inhomogeneity of the vortices. However, cer-
Fig. 5. The fractal dimension (df,) as a function of the total number of vortices for p* = 1 (X ) and p2 = 50 (+). The data represent averages over I x IO” MCSS. (A) and (0) represent averages over 2 x IO’ MCSS.
tain features can be understood as follows, In the low temperature (energy) insulating phase the vortices are bound tightly which results in a small value of t&. In the high temperature (energy) Debye-Htickel regime, V,, is also small due to the presence of a large number of free charges in the liquid phase. We speculate that clusters of vortices in the critical region are responsible for the peak in V,,. The difference between the
130
S.B. &a. S. &a/Physics
50 and 1 cases can be attributed to the change in interaction from ln( r/a) to r/a as p2 changes from 1 to50 [12]. In conclusion, we have studied the vortices in the classical 2D XY model undergoing the KT and firstorder transitions. We have reported a subtle aspect of the vortices; that is, the inhomogeneity of the vortex distribution. p2 =
SBO acknowledges discussions with Dr. A. Batimgartner. The authors thank Dr. V.C. Sahni for helpful suggestions on this manuscript.
References [I] M.N. Barber. Phys. Rep. 59 (1980) 376. 121 E. Domany, M. Schick, R.H. Swendsen. Phys. Rev. Lett. 52 (1984) 1535. [ 3 I N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 ( 1966) 1133. 141 V.L. Berezinskii. Sov. Phys. JETP 32 ( 1970) 493. [ 5 ] J.M. Kosterlitz, D.J. Thouless, J. Phys. C: Solid State Phys. 5 (1972) L124; J. Phys. C: Solid State Phys. 16 (1973) 1181; J.M. Kosterlitz, J. Phys. C: Solid State Phys. 7 (1974) 1046. 161 F.J. Wegner. Z. Phys. 206 (1967) 46.5.
Letters A 241 (1998) 127-130 171 V.L. Berezinskii. Sov. Phys. JETP 34 (1971) 610. Lectures on Phase Transition and I81 N. Goldenfeld. Renormahsation Group ( Addison-Wesley. Reading, MA. 1992). 191 P.M. Chaikin. T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University, New York, 1995). 1101 J. Zittartz, Z. Phys. B 23 (1976) 55; 23 (1976) 63. IllI J.V. Jose. L.P. Kadanoff. S. Kirkpatrick, D.J. Nelson. Phys. Rev. B 16 (1977) 1217. 1121 F. Mila, Phys. Rev. B 47 (1993) 442. I131 C. Kawabata. K. Binder, Solid State Commun. 22 ( 1977) 705. 1141 S. Miyashita. H. Nishimori. A. Kuroda. M. Suzuki, Prog. Theor. Phys. 60 (1978) 1669. 1151 J. Tobochnik. G.V. Chester, Phys. Rev. B 20 (1979) 3761. 1161 W.J. Shugard et al., Phys. Rev. B 21 (1980) 5209. L171 J.E. van Himbergen. S. Chakravarty, Phys. Rev. B 23 ( 1984) 359. 1181 H. Betsuyaku, Physica A 106 ( 198 I ) 3 I 1. 119J J.F. Fernandes, M.F. Ferreira, J. Stankiewicz, Phys. Rev. B 34 (1986) 292. 1201 R. Gupta. J. Delapp, G.G. Batrouni, G.C. Fox. C.F. Baillie, 1. Apostolakis, Phys. Rev. Lett. 61 (1988) 1996. 1211 R. Gupta, CF. Baillie, Phys. Rev. B 45 (1992) 2883. [221 U. Wolff, Nucl. Phys. B 322 (1989) 759. [231 J. Kogut, J. Polonyi, Nucl. Phys. B 265 IFS151 (1986) 313. [241 M. Creutz, Phys. Rev. Lett. 50 (1983) 1411. I251 S. Ota. S.B. Ota, M. FPhne, J. Phys. Condens. Matter 4 (1992) 5411.
PHYSICS
Physics Letters A 241 (1998)
LETTERS
A
127-130
Inhomogeneity of vortex charge in the 2D classical XY model S.B. Ota, Smita Ota Institute
of Physics, Sachivalaya Marg, Bhubaneswar 751005, fndiu
Received 5
November 1997;
accepted for publication 13
January 1998
Communicated by J. Flouquet
Abstract
~~icrocanonical Monte Carlo simulations have been carried out on the 2D classical XY model which shows both KosteditzThouless (KT) as well as first-order transitions. Simulations on a 30 x 30 spin system show that the vortex charge distribution is inhomogeneous. The maximum inhomogeneity occurs at a temperature which coincides with the position of the specific heat peak in the KT case and in the coexistence region in the first-order case. The inhomogeneity is found to be larger in the KT case as compared to the first-order one. The vortices are found to be dist~buted in a more space-fii~ing manner in the latter case as compared to the former one. @ 1998 Elsevier Science B.V. PACS: 75.IO.Hk;02.7O.Lq:05.20.-y; 0.5.70.Fh
An important statistical spin model is the 2D classical XY system [ I 1, with Hamiltonian given by
where the symbols have their usual meaning. For p* = 1 the Hamiltonian reduces to the usual classical 2D XY system, which admits the Kosterlitz-Thouless (KT) tr~sitio~. By increasing the value of p’, the potential well can be made narrower and for p* > 10 the transition becomes first order in nature [ 21. The system does not show any long-range order [ 31 and there are two types of excitations present in the system; namely, the vortices and the spin waves 14-71. The vortices are topological defects. When p* = 1, the low-temperature phase has only bound vortexantivortex pairs, and the KT transition is associated with the unbinding of the vortex-antivortex pairs. Being of a topological nature, no specific heat anomaly is observed at the KT transition temperature
Tc. But the temperature dependence of the specific heat shows a peak at a temperature, which is about 15% higher than T,. The vortex unbinding transition temperature (7;,) increases by reducing the potentiaI well width. On the other hand, local disorder will set in at tem~ratures near the potential well height TI, N V(v) - V(0) = 2f. Therefore, for a sufficiently narrow potential well TI, < T,,, we expect the continuous transition to yield to a first-order vacancy condensation ~ansition. The classical 2D XY system has been extensively studied in the context of superfluids [ 81 and superconducting films [ 91. Various theoretical techniques have been used to study the spin waves, the vortices and the nature of the transition in the 2D classical XY system [ 10,l1 ] . Several Monte Carlo (MC) simulations have been carried out to study the KT transition and the first-order transition in this system [ 12-23 I. Here we report the inhomogeneity of the vortex distribution of the classical 2D XY system, which has so far not been found.
037%9601/98/$19.00 @ 1998 Elsevier Science B.V. All rights reserved. PII SO37S-9601(98)00034-6
128
S.B. Otu. S. Otu/Pl~~.sks
We performed microcanonical MC simulations on a square lattice having 30 x 30 spins [ 24,251. We used periodic boundary conditions and have calculated the system temperature (T), vorticity, the magnetisation curve and their respective standard deviations at each given total energy (E) . We used 1 x 1O5 MCSS for equilibration and I x IO5 MCSS for averaging. The accuracy of the mean value of the physical quantities was estimated by performing block averages consisting of 5 x 10” MCSS each and then finding the standard deviation of the block averages. The standard deviation of the calculated vortex density is typically less than 0.5%. The KT transition temperature occurs at 0.9 and the maximum of the temperature dependence of the specitic heat occurs at I .09. We analysed the Monte Carlo configurations evolved during the simulation to study the inhomogeneity of the vortices across the transition. To this end, we estimated the average number (V,(r) ) of vortices-antivortices at a distance r from any given vortex. This is the vortex-vortex correlation function times r2. From this we obtained the number of positive (negative) vortices (V(,+ (V,_ ) ) surrounding a
Letter.7 A 241 (1998) 127-130
Fig. 1. The temperature dependence of the net vortex charge V,, in a radius of v% (+) and 2&&r (X ) for p’ = I. The data represent averages over I x lo5 MCSS. (A ) and (0) represent averages over 2 x 10’ MCSS.
+
-._
%
‘i
positive vortex within a distance of &LZ and 2&a ( CI is the lattice spacing). For p’ = 1, it is seen that V&, when determined over a distance of &a, decreases with temperature initially up to T = 1.1 and then increases with further increase in temperature. A decrease of V,_ with temperature is not observed, when determined over a larger distance of 2&a, whereas V,+ shows a steady increase with increase in temperature. On the other hand, for p’ = 50, both V,+ and v,_ are seen to increase with increase in energy even when determined over a smaller distance of &a. ’ We next determined the net vortex charge (V,,) within a distance of &a and 2&a from a vortex as functions of T (or E). Fig. 1 shows the temperature dependence of V,, for p2 = 1, and comments on some features of this graph are in order. Firstly, V,, goes through a maximum. Secondly, the magnitude of &f is reduced as the distance increases. Finally, the maximum occurs at T = 1.1, which corresponds to the specific heat peak. A similar graph is shown in Fig. 2, for p2 = 50. The behaviour is qualitatively similar to the
’ The energy is used instead of the temperature to clearly present the points in the coexistence region of the first-order transition.
Fig. 2. The energy dependence of net vortex charge V’, in a radius of x/% (+) and 2v’% ( x ) for p’ = SO. The data represent averages over I x IO5 MCSS. (A) and (0) represent averages over 2 x lo5 MCSS.
case of p2 = 1, except for the following differences. The magnitude of V,, is comparatively smaller. The maximum of V,, occurs in the coexistence region of the first-order transition. The vortices are therefore. not distributed homogeneously in the lattice. Examination of the configurations revealed the presence of vortex clusters in the critical region, which has also been pointed out by Tobochnik and Chester [ 151. In order to characterise the clusters, we determined
S.3. &a, S. Ota/Pbysics Lrttem A 24i (19981 127-130
Fig. 3. The temperature dependence of the fractal dimensions (df, ) ( x ) and ((if,,) (+) of the vortex distribution for p2 = I. The data represent averages over I x 10s MCSS. (A) and (0) represent averages over 2 x I@ MCSS. The arrow indicates the maximum difference in dfr and dfP.
the fractal dimension V(v) cx #:,
Fig. 4. The energy dependence of the fractal dimensions (d t, ) ( x ) and (dfP ) (+) of the vortex distribution for p2 = SO. The data represent averages over 1 x 10s MCSS. (A) and (0) represent averages over 2 x IO5 MCSS. The arrow indicates the maximum difference
in df, and dfP.
df using the following definition, (2)
where V(r) is the total number of vortices in a radius r. We choose the origin on a vortex and include only a vortex or both vortex and antivortex, to determine diii or dfr, respectively. In( V,) versus ln( rf showed a linear behaviour for ln( r) up to 1.5 and thereby df was determined. Fig. 3 shows the energy dependence of df for p’ = 50. It is seen that df falls far below 2. The maximum difference in dfp and df, occurs at a system energy corresponding to the maximum position of Vqf. At higher system energy df,, becomes equal to dp. A similar behaviour in the temperature dependence of df was observed for p2 = 1, which is shown in Fig. 4. In this case, the maximum difference in dfp and dfr occurs at a temperature corresponding to the maximum position of V,,. This observation implies a difference in the distribution of a vortex with respect to an antivortex. This difference is found to be less for p2 = 50. Moreover, we observed that dft as a function of the number of vortices (u) is not the same for p’ = 1 and 50. This is displayed in Fig. 5. It is seen that for p2 = 50, the vortices are distributed in a more space-filling manner compared to that for p’ = I. We have not yet found an explanation of the observed inhomogeneity of the vortices. However, cer-
Fig. 5. The fractal dimension (df,) as a function of the total number of vortices for p* = 1 (X ) and p2 = 50 (+). The data represent averages over I x IO” MCSS. (A) and (0) represent averages over 2 x IO’ MCSS.
tain features can be understood as follows, In the low temperature (energy) insulating phase the vortices are bound tightly which results in a small value of t&. In the high temperature (energy) Debye-Htickel regime, V,, is also small due to the presence of a large number of free charges in the liquid phase. We speculate that clusters of vortices in the critical region are responsible for the peak in V,,. The difference between the
130
S.B. &a. S. &a/Physics
50 and 1 cases can be attributed to the change in interaction from ln( r/a) to r/a as p2 changes from 1 to50 [12]. In conclusion, we have studied the vortices in the classical 2D XY model undergoing the KT and firstorder transitions. We have reported a subtle aspect of the vortices; that is, the inhomogeneity of the vortex distribution. p2 =
SBO acknowledges discussions with Dr. A. Batimgartner. The authors thank Dr. V.C. Sahni for helpful suggestions on this manuscript.
References [I] M.N. Barber. Phys. Rep. 59 (1980) 376. 121 E. Domany, M. Schick, R.H. Swendsen. Phys. Rev. Lett. 52 (1984) 1535. [ 3 I N.D. Mermin, H. Wagner, Phys. Rev. Lett. 17 ( 1966) 1133. 141 V.L. Berezinskii. Sov. Phys. JETP 32 ( 1970) 493. [ 5 ] J.M. Kosterlitz, D.J. Thouless, J. Phys. C: Solid State Phys. 5 (1972) L124; J. Phys. C: Solid State Phys. 16 (1973) 1181; J.M. Kosterlitz, J. Phys. C: Solid State Phys. 7 (1974) 1046. 161 F.J. Wegner. Z. Phys. 206 (1967) 46.5.
Letters A 241 (1998) 127-130 171 V.L. Berezinskii. Sov. Phys. JETP 34 (1971) 610. Lectures on Phase Transition and I81 N. Goldenfeld. Renormahsation Group ( Addison-Wesley. Reading, MA. 1992). 191 P.M. Chaikin. T.C. Lubensky, Principles of Condensed Matter Physics (Cambridge University, New York, 1995). 1101 J. Zittartz, Z. Phys. B 23 (1976) 55; 23 (1976) 63. IllI J.V. Jose. L.P. Kadanoff. S. Kirkpatrick, D.J. Nelson. Phys. Rev. B 16 (1977) 1217. 1121 F. Mila, Phys. Rev. B 47 (1993) 442. I131 C. Kawabata. K. Binder, Solid State Commun. 22 ( 1977) 705. 1141 S. Miyashita. H. Nishimori. A. Kuroda. M. Suzuki, Prog. Theor. Phys. 60 (1978) 1669. 1151 J. Tobochnik. G.V. Chester, Phys. Rev. B 20 (1979) 3761. 1161 W.J. Shugard et al., Phys. Rev. B 21 (1980) 5209. L171 J.E. van Himbergen. S. Chakravarty, Phys. Rev. B 23 ( 1984) 359. 1181 H. Betsuyaku, Physica A 106 ( 198 I ) 3 I 1. 119J J.F. Fernandes, M.F. Ferreira, J. Stankiewicz, Phys. Rev. B 34 (1986) 292. 1201 R. Gupta. J. Delapp, G.G. Batrouni, G.C. Fox. C.F. Baillie, 1. Apostolakis, Phys. Rev. Lett. 61 (1988) 1996. 1211 R. Gupta, CF. Baillie, Phys. Rev. B 45 (1992) 2883. [221 U. Wolff, Nucl. Phys. B 322 (1989) 759. [231 J. Kogut, J. Polonyi, Nucl. Phys. B 265 IFS151 (1986) 313. [241 M. Creutz, Phys. Rev. Lett. 50 (1983) 1411. I251 S. Ota. S.B. Ota, M. FPhne, J. Phys. Condens. Matter 4 (1992) 5411.