Variational Monte Carlo study of a spin one quantum antiferromagnet on a
fractal lattice
Dipan K. Ghosh and Latha S. Warrier
Citation: J. Appl. Phys. 95, 6992 (2004); doi: 10.1063/1.1689131
View online: http://dx.doi.org/10.1063/1.1689131
View Table of Contents: http://jap.aip.org/resource/1/JAPIAU/v95/i11
Published by the American Institute of Physics.
Related Articles
Anomalous low-temperature magnetic and magnetotransport properties in Ru-deficient SrRuO3
J. Appl. Phys. 111, 07E121 (2012)
Geometric properties of the magnetic Laplacian on the Euclidean 4-space
J. Math. Phys. 51, 123510 (2010)
Atomistic spin model simulation of magnetic reversal modes near the Curie point
Appl. Phys. Lett. 97, 192504 (2010)
Determination of spin Hamiltonians from projected single reference configuration interaction calculations. I. Spin
1/2 systems
J. Chem. Phys. 133, 044106 (2010)
The influence of magnetic anisotropy on magnetoelectric behavior in conical spin ordered multiferroic state
J. Appl. Phys. 107, 093908 (2010)
Additional information on J. Appl. Phys.
Journal Homepage: http://jap.aip.org/
Journal Information: http://jap.aip.org/about/about_the_journal
Top downloads: http://jap.aip.org/features/most_downloaded
Information for Authors: http://jap.aip.org/authors
Downloaded 29 Feb 2012 to 14.139.97.73. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
JOURNAL OF APPLIED PHYSICS
VOLUME 95, NUMBER 11
1 JUNE 2004
Variational Monte Carlo study of a spin one quantum antiferromagnet
on a fractal lattice
Dipan K. Ghosha) and Latha S. Warrierb)
Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai 400 076, India
共Presented on 8 January 2004兲
A variational Monte Carlo calculation is made on a spin one Heisenberg antiferromagnet on a fractal
lattice. Ground state energy and spin correlation functions are computed for a Sierpiński gasket
containing up to 123 sites. The results indicate approximate validity of Marshall sign rule for a
tripartite lattice. Correlation functions decay fast with distance, suggesting a disordered ground
state. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1689131兴
INTRODUCTION
According to this rule, the spin space of one of the sublattices 共B兲 is rotated about one of the spin axes by an angle ␲ .
If one considers the basis function in the rotated representation, all the coefficients in the expansion of the wave function in this basis are positive. The Sierpiński gasket considered here is a tripartite lattice. The classical ground state of
the lattice is one in which the spin vectors at the vertices of
every triangle makes an angle of 120° with one another, as
shown in Fig. 1.
In analogy with the two sublattice case, we rotate the
spins on the A sublattice by 2␲/3 and those on the B sublattice by 4␲/3 leaving the spins on the C sublattice unchanged.
Taking the rotation about the z axis, the transformed Hamiltonian may be written (J⫽1)
Study of ground state properties and low lying excitations of quantum antiferromagnets have attracted a great deal
of interest. Hulthén’s1 solution of one dimensional antiferromagnetic chain using Bethe ansatz shows that S⫽1/2 spin
chain has a gapless spectrum. The spin correlations decay
slowly with distance following a power law behavior. Contrasted to this, spin 1 systems show gapped excitation2 and
the spin correlations decay exponentially. In addition to the
difference between integer and half integer spins, significant
differences are observed between the properties in one and
two dimensions. While one dimensional spin-half systems
show no long range order 共LRO兲, most two dimensional systems show LRO, though with reduced magnetization. In
higher dimensions, one can have disordered ground states
due to frustration. With the above in view, we have studied
the ground state and correlation functions of a Sierpiński
gasket, which has a fractal dimension of ln 3/ln 2⫽1.58. As
the basic lattice is a triangular lattice, frustration effect is
very pronounced. Voigt et al.3 have carried out an exact diagonalization study of this system using a 15 site gasket and
have calculated the ground state energy and correlation functions. In view of the small number of sites considered, finite
size effect is expected to be significant. As exact diagonalization becomes unfeasible for large N, we have
used a variational Monte Carlo method to calculate various
properties.
H⫽H AB⫹H BC⫹H AC ,
where
H AB⫽
冋
1
兺 共 e ⫺ 2 ␲ i/3S ⫹i S ⫺j ⫹h.c. 兲 ⫹S zi S zj
i苸A 2
j苸B
册
and similar expressions for H BC and H AC .
VARIATIONAL MONTE CARLO METHOD AND
MARSHALL SIGN RULE
The variational Monte Carlo method is based on a
proper choice of the wave function in the position representation. Metropolis algorithm is used to sample the configuration space and at each of the points ‘‘local energy’’ is calculated. The variational energy is then given by the arithmetic
average of such local energies. Use of this procedure depends crucially on the applicability of Marshall–Pierls4 sign
rule, which has been proved exactly for bipartite lattices.
a兲
Author to whom correspondence should be addressed; electronic mail:
[email protected]
b兲
Electronic mail: [email protected]
0021-8979/2004/95(11)/6992/3/$22.00
FIG. 1. Classical ground state of a Sierpiński gasket with 15 sites.
6992
© 2004 American Institute of Physics
Downloaded 29 Feb 2012 to 14.139.97.73. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004
D. K. Ghosh and L. S. Warrier
6993
Following the procedure of Ref. 5 for obtaining the form
of the coefficients of the wave function, we take the trial
state to be
␾ ␣ ⫽ 共 S A⫹ 兲 S⫹m A 共 S B⫹ 兲 S⫹m B 共 S C⫹ 兲 S⫹m C . . . . . . 兩 0 典
⫽C 兩 m A ,m B ,m C , . . . 典 ,
共1兲
where the constant C, whose value depends on S and m is
given by standard angular momentum algebra.
The state 兩0典 is taken to be the vacuum state with values
of m to be ⫺S at every site. The ground state 兩⌿典 of the
system can be expanded in a complete set of this basis:
⌿⫽
⫽
兺␣
m1
f ␣␾ ␣
兺
,m ,...,m
2
f m 1 ,m 2 ,...,m n 兩 m 1 ,m 2 ,...,m n 典 .
共2兲
n
Substituting this and using spin commutation relations,
the expectation value of the off diagonal part H 1 of the
Hamiltonian is seen to satisfy
具 ⌿ 兩 H 1 兩 ⌿ 典 ⫽ 兺 共 S⫹m i 兲 ! 共 S⫹m j 兲 !
具 i, j 典
兿
k⫽i, j
共 S⫹m k 兲 !⫻2S 共 2S
⫺1 兲 . . . . . . . . . 共 S⫺m i ⫹1 兲
⫻2S 共 2S⫺1 兲 . . . . . . . . . 共 S⫺m j ⫹1 兲
⫻
兿
k⫽i, j
2S 共 2S⫺1 兲 . . . . . . . . . 共 S⫺m k ⫹1 兲
⫻ 兵 关 2e ⫺i 2 ␲ /3⫹e ⫺i 4 ␲ /3兴共 S⫹m j ⫹1 兲
⫻ 共 S⫺m j 兲 f m 1 ,m 2 ,m 3 , . . . .m i ⫺1,m j ⫹1, . . . .m N
⫹ 共 2e i2 ␲ /3⫹e i4 ␲ /3兲共 S⫹m i ⫹1 兲共 S⫺m i 兲
⫻ f m 1 ,m 2 ,m 3 , . . . .m i ⫹1,m j ⫺1, . . . .m N 其 .
共3兲
Writing
具 ⌿ 兩 H 1 兩 ⌿ 典 ⫽ 具 ␾ ␤ 兩 H 1 兩 ␾ ␣ 典 ⫽K ␤␣ .
共4兲
The Schrödinger equation can be written as
K ␤␣ f .. ⫹e 0 f .. ⫽E M f .. ,
共5兲
FIG. 2. Ground state energy per site as a function of 1/N. The optimum
values of the parameter are f 1 ⫽0.35, f 2 ⫽0.55, and f 3 ⫽0.758.
denotes the amplitude contribution from nearest neighbor
bonds between two sites, one with S z ⫽0 and the other with
S z ⫽1, while the latter gives the contribution from the bond
between two sites each with S z ⫽0. The remaining contributions are fixed by symmetry and probability conservation.
The initial configuration is taken to be a randomized S z
⫽0 state. A third of the total number of sites are randomly
selected and are assigned s z values 0 and ⫾1. Monte Carlo
moves are made so as to stay within this subspace. Initial
10 000 configurations are discarded so that the starting configuration is randomized. For N⫽6, 15, and 42, a million
Monte Carlo sweeps are used to calculate the ground state
energy while for N⫽123, because of computational time
constraint, only 100 000 sweeps were taken. Sample sizes in
each case were varied in units of 10 000 and the ground state
energy per site was found to vary less than one part in 104 ,
indicating the error in calculation to be systematic in nature
rather than of statistical origin. The ground state energy E,
obtained by minimizing the energy with respect to the variational parameters is shown in Fig. 2.
It is found that the ground state energy, as a function of
1/N can be fitted to a straight line
E⫽⫺1.272 19⫹1.283 79/N
so that the energy of the fractal lattice is ⫺1.272 19 per site.
The error in fit is about 2% in slope and less than 0.2% in the
where e 0 is the contribution of the diagonal part of the
Hamiltonian to the energy. For S⫽1, it can be seen that when
m i and m j are both ⫹1, the contribution from the off diagonal part is zero as the term in the curly bracket of Eq. 共3兲
vanishes. When m i and m j are ⫹1(⫺1) and ⫺1(⫹1) respectively, the coefficients K ␤␣ are complex while when they
are both equal to either zero or ⫺1, they are real. However,
the total energy obtained from Eq. 共5兲 is real. This is confirmed by numerical calculation, where the imaginary part of
the energy is found to be negligibly small.
RESULTS AND DISCUSSION
As the ground state is nondegenerate and is known to
have total S⫽0, we confine ourselves to those configuration
for which the total S z ⫽0. The variational wave function is
characterized by two parameters f 1 and f 2 . The former
FIG. 3. Spin correlation function vs distance.
Downloaded 29 Feb 2012 to 14.139.97.73. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions
6994
J. Appl. Phys., Vol. 95, No. 11, Part 2, 1 June 2004
intercept. Voigt et al.’s exact calculation3 for 15 sites gives
the energy per site to be ⫺1.319. The ground state energy
calculated by us is somewhat higher, which may be attributed partly to the nature of variational calculation and partly
to the failure of Marshall’s rule for the tripartite lattice.
Clearly, however, Marshall’s sign rule remains approximately valid.
We have also investigated the spin correlation function
z z
S
S
具 i j 典 as a function of the separation 兩 ri ⫺r j 兩 共Fig. 3兲. The
short range order, which exists for the first few neighbors,
decays rapidly thereafter, indicating a disordered ground
state. This is also confirmed by the behavior of the transverse
correlation function which decays to a nonzero constant
value after a few neighbors.6 However, we have not been
able to conclusively establish this by fitting the correlation
function within a sublattice as a function of the logarithm of
D. K. Ghosh and L. S. Warrier
the distance, as done by Voigt et al. It may be mentioned that
we had performed a unitary transformation on the original
Hamiltonian and had calculated the ground state energy assuming that all the states contribute to the energy with the
same sign. It is likely that the configurations which contribute to the minimum energy state in our variational calculation do not reproduce the correlations correctly.
The authors than Vinod Kumar for his help and advice in
numerical computation.
L. Hulthén, Ark. Mat., Astron. Fys. 26A, 11 共1938兲.
F. D. M. Haldane, Phys. Rev. Lett. 50, 1153 共1983兲.
A. Voigt, J. Richter, and P. Tomczak, Physica A 299, 107 共2001兲
4
W. Marshall, Proc. R. Soc. London, Ser. A 232, 48 共1955兲.
5
E. Lieb and D. C. Mattis, J. Math. Phys. 3, 749 共1962兲.
6
N. Trivedi and D. M. Ceperley, Phys. Rev. B 41, 4552 共1990兲.
1
2
3
Downloaded 29 Feb 2012 to 14.139.97.73. Redistribution subject to AIP license or copyright; see http://jap.aip.org/about/rights_and_permissions