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Quantum Monte Carlo study of a two-dimensional Heisenberg antiferromagnet with nonmagnetic impurities
Journal of Magnetism
and Magnetic
Materials
104-107
(1992) 863-864
North-Holland
Quantum Monte C ar 1o st u d y o f a t wo-dimensional antiferromagnet with nonmagnetic impurities J. Behre ‘, S. Miyashita b and H.-J. Mikeska a a lnstitut fir Theoretische Physik, Unicersitiit Hatmover, Hannol:er, ‘Physics
Department,
CLAS,
Kyoto University,
Heisenberg
Germany
Kyoto 606, Japan
Enhancements of spin correlation functions due to nonmagnetic impurities are investigated by both diagonalization method and a quantum Monte Carlo method. Spatial distributions of spin correlation functions near impurities are obtained where large enhancements are observed. On the other hand, correlations between large separated spins seem to be reduced by impurities.
The nature of the order of the Heisenberg antiferromagnet on the square lattice has been studied extensively. It is generally believed that this model has long-range staggered order in the ground state. But the value of the order parameter is much reduced by quantum fluctuations due to the noncommutativity between the order parameter and the Hamiltonian. We are then interested in the influence of nonmagnetic impurities in this system because we may expect some peculiar phenomena due to the combination of effect of impurity and quantum effect. We have found that correlation functions are not necessarily reduced but sometimes enhanced by impurities [l]. Similar phenomena have been recently reported in many systems [2]. In order to study the effect more carefully, we investigate the microscopic distribution of spin correlation functions. For local effects of impurity, we studied small systems on a 4 X 4 lattice by exact diagonalization. In fig. 1 distribution of the nearest-neighbour (nn) spin correlation functions are shown. We find enhancements near impurities very clearly. The enhancement of nn correlations can be understood generally [3] from the variational principle which state that I(G,l~,,,,,lG,)l < IGil~~,,IGi)I where IGJ is the ground state wave function for the pure system (without impurities) and I Gi) the ground state wave function of Z,mp. Because the Hamiltonian is the summation of nn correlations, the average value of the nn correlation should be enhanced. As we see in table 1, however, some of further correlations are also found to be enhanced. For larger lattices, we have performed quantum Monte Carlo simulation based on Suzuki Trotter decomposition. In fig. 2, nn and nnn correlations are shown, where the correlations indicated by * are enhanced. Data are obtained with an extrapolation of data for the Trotter numbers n = 8, 12, 18, 24 at T = 0.2 J. Of course the data have error bars but the 0312.8853/92/$05,00
0 1992 - Elsevier
Science
Publishers
I
I
1.073 I
1.127
----a-
I I * 1.019 -*- 1.127 -+- 1.019 -*- 0.962 I 1.127
*I + 0.962
I ----*-
I.019
I * 0.962 * I ----*I * 1.073 *I ----* * I * I
I 0.962 *
I 1.073 *
I -* I
I l *- 1.073 I *
0
1.073 * 1.127 I I 1.019 -*- 1.127 -*- 1.019 -*- 0.962 I I * 1.127 0.962 * I I *- 1.019 -*- 1.073
0
I
I I
I
Fig. 1. Nearest-neighbour
spin correlation functions, normal-
ized to the pure-case
value f - 0.4678).
Table 1 Further-neighbour enhanced correlation functions for the L = 4 lattice, normalized to the pure-case value (r = (s,s,)/ (SLS,)purc) Site i Site j (1,1)-(4,4) (1 ,l)-(4,2)
(4,3)-(1,2) (4,3)-(1,4) (l,l)-(1,3) (4,3)-(1,4) (l,l)-(4,3) (l,l)-(3,2)
(l&(2,3) (l,l)-(3,4) (4,3)-(3,l) (4,3)-(2,2) (4,3)-(2,4)
B.V. All rights
reserved
Pure case 1.023 1.023 1.023 1.023 1.023 1.023 1.163 1.037 1.037 1.037 1.037 1.037 1.037
-
0.285 0.285 0.285 0.285 0.285 0.285 0.270 0.270 0.270 0.270 0.270 0.270 0.270
J. Behre et al. / Monte Carlo study of Heisenberg antiferromagnet
864 I l I * I * I I* *I I * I * _ - ~___~___l___~___~___~_--~---~-l l
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Table 2 The values of ( N:>/N2
l___~___~___~_-_~__-~***~***~--
I I I
t
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I I
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l
I
netic impurities. The value has now turned out to be comparable to the value of the pure case. Namely, the previous data at low temperatures were frozen in some metastable states, maybe due to the nonergodic nature of the Monte Carlo steps. We will report on the methological point elsewhere. Here we conclude that correlations for long separated spins are reduced because (N,‘)/N*, which is the summation of all correlations, is not enhanced although the short-range correlations are enhanced. In table 2, values of (Nz>/N’ are listed. We may expect that quantum fluctuations could reduce the classical percolation limit of dilution, which will be reported elsewhere.
Fig. 2. Positions of nn and nnn correlation functions which are enhanced by more than 2% are indicated by *.
References
qualitative nature does not change within it. We see that their nature is similar to the 4 x 4 lattice case. In order to see the nature of long separate correlations, we investigate the value of ( NZz)/N2 where N is the number of spins. In the previous report [l], large enhancement of this value was shown. We have then performed Monte Carlo simulation with a more elaborate algorithm which concerns the ergodicity problem for the winding number in the systems with nonmag-
[l] J. Behre, S. Miyashita and H.-J. Mikeska, J. Phys. A 23 (1990) L1175. [2] D.D. Betts and J. Oitmaa, private communication. N. Nagaosa, Y. Hatsugai and M. Imada, J. Phys. Sot. Jpn. 58 (1989) 978. R.E. Camley, W. von der Linden and V. Zevin, Phys. Rev. B 40 (1989) 119. N. Bulut, D. Hone, D.J. Scalapino and E.Y. Loh, Phys. Rev. Lett. 62 (1989) 2192. K.J.B. Lee and P. Schlottmann, Phys. Rev. B 42 (1990) 4426. [3] D.D. Betts and S. Miyashita, Can. J. Phys. 68 (1990) 1410.
and Magnetic
Materials
104-107
(1992) 863-864
North-Holland
Quantum Monte C ar 1o st u d y o f a t wo-dimensional antiferromagnet with nonmagnetic impurities J. Behre ‘, S. Miyashita b and H.-J. Mikeska a a lnstitut fir Theoretische Physik, Unicersitiit Hatmover, Hannol:er, ‘Physics
Department,
CLAS,
Kyoto University,
Heisenberg
Germany
Kyoto 606, Japan
Enhancements of spin correlation functions due to nonmagnetic impurities are investigated by both diagonalization method and a quantum Monte Carlo method. Spatial distributions of spin correlation functions near impurities are obtained where large enhancements are observed. On the other hand, correlations between large separated spins seem to be reduced by impurities.
The nature of the order of the Heisenberg antiferromagnet on the square lattice has been studied extensively. It is generally believed that this model has long-range staggered order in the ground state. But the value of the order parameter is much reduced by quantum fluctuations due to the noncommutativity between the order parameter and the Hamiltonian. We are then interested in the influence of nonmagnetic impurities in this system because we may expect some peculiar phenomena due to the combination of effect of impurity and quantum effect. We have found that correlation functions are not necessarily reduced but sometimes enhanced by impurities [l]. Similar phenomena have been recently reported in many systems [2]. In order to study the effect more carefully, we investigate the microscopic distribution of spin correlation functions. For local effects of impurity, we studied small systems on a 4 X 4 lattice by exact diagonalization. In fig. 1 distribution of the nearest-neighbour (nn) spin correlation functions are shown. We find enhancements near impurities very clearly. The enhancement of nn correlations can be understood generally [3] from the variational principle which state that I(G,l~,,,,,lG,)l < IGil~~,,IGi)I where IGJ is the ground state wave function for the pure system (without impurities) and I Gi) the ground state wave function of Z,mp. Because the Hamiltonian is the summation of nn correlations, the average value of the nn correlation should be enhanced. As we see in table 1, however, some of further correlations are also found to be enhanced. For larger lattices, we have performed quantum Monte Carlo simulation based on Suzuki Trotter decomposition. In fig. 2, nn and nnn correlations are shown, where the correlations indicated by * are enhanced. Data are obtained with an extrapolation of data for the Trotter numbers n = 8, 12, 18, 24 at T = 0.2 J. Of course the data have error bars but the 0312.8853/92/$05,00
0 1992 - Elsevier
Science
Publishers
I
I
1.073 I
1.127
----a-
I I * 1.019 -*- 1.127 -+- 1.019 -*- 0.962 I 1.127
*I + 0.962
I ----*-
I.019
I * 0.962 * I ----*I * 1.073 *I ----* * I * I
I 0.962 *
I 1.073 *
I -* I
I l *- 1.073 I *
0
1.073 * 1.127 I I 1.019 -*- 1.127 -*- 1.019 -*- 0.962 I I * 1.127 0.962 * I I *- 1.019 -*- 1.073
0
I
I I
I
Fig. 1. Nearest-neighbour
spin correlation functions, normal-
ized to the pure-case
value f - 0.4678).
Table 1 Further-neighbour enhanced correlation functions for the L = 4 lattice, normalized to the pure-case value (r = (s,s,)/ (SLS,)purc) Site i Site j (1,1)-(4,4) (1 ,l)-(4,2)
(4,3)-(1,2) (4,3)-(1,4) (l,l)-(1,3) (4,3)-(1,4) (l,l)-(4,3) (l,l)-(3,2)
(l&(2,3) (l,l)-(3,4) (4,3)-(3,l) (4,3)-(2,2) (4,3)-(2,4)
B.V. All rights
reserved
Pure case 1.023 1.023 1.023 1.023 1.023 1.023 1.163 1.037 1.037 1.037 1.037 1.037 1.037
-
0.285 0.285 0.285 0.285 0.285 0.285 0.270 0.270 0.270 0.270 0.270 0.270 0.270
J. Behre et al. / Monte Carlo study of Heisenberg antiferromagnet
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Table 2 The values of ( N:>/N2
l___~___~___~_-_~__-~***~***~--
I I I
t
.I *
I I
I I
I I
I I
l
I
netic impurities. The value has now turned out to be comparable to the value of the pure case. Namely, the previous data at low temperatures were frozen in some metastable states, maybe due to the nonergodic nature of the Monte Carlo steps. We will report on the methological point elsewhere. Here we conclude that correlations for long separated spins are reduced because (N,‘)/N*, which is the summation of all correlations, is not enhanced although the short-range correlations are enhanced. In table 2, values of (Nz>/N’ are listed. We may expect that quantum fluctuations could reduce the classical percolation limit of dilution, which will be reported elsewhere.
Fig. 2. Positions of nn and nnn correlation functions which are enhanced by more than 2% are indicated by *.
References
qualitative nature does not change within it. We see that their nature is similar to the 4 x 4 lattice case. In order to see the nature of long separate correlations, we investigate the value of ( NZz)/N2 where N is the number of spins. In the previous report [l], large enhancement of this value was shown. We have then performed Monte Carlo simulation with a more elaborate algorithm which concerns the ergodicity problem for the winding number in the systems with nonmag-
[l] J. Behre, S. Miyashita and H.-J. Mikeska, J. Phys. A 23 (1990) L1175. [2] D.D. Betts and J. Oitmaa, private communication. N. Nagaosa, Y. Hatsugai and M. Imada, J. Phys. Sot. Jpn. 58 (1989) 978. R.E. Camley, W. von der Linden and V. Zevin, Phys. Rev. B 40 (1989) 119. N. Bulut, D. Hone, D.J. Scalapino and E.Y. Loh, Phys. Rev. Lett. 62 (1989) 2192. K.J.B. Lee and P. Schlottmann, Phys. Rev. B 42 (1990) 4426. [3] D.D. Betts and S. Miyashita, Can. J. Phys. 68 (1990) 1410.