Download Comment on “Quantum Monte Carlo Approach to Elementary

VOLUME 77, NUMBER 13
PHYSICAL REVIEW LETTERS
Comment on “Quantum Monte Carlo Approach to
Elementary Excitations of Antiferromagnetic
Heisenberg Chains”
In his recent Letter [1], Yamamoto investigates the lowlying excitation spectrum of antiferromagnetic Heisenberg
1
chains with S . 2 using an improved quantum Monte
Carlo (QMC) method. Among other questions, he addresses the question of the size of the finite gap in the S ­
2 isotropic antiferromagnetic Heisenberg chain predicted
by Haldane [2]. Using chain lengths up to L ­ 128, he
estimates the gap to be 0.049(18). This result is compared
to the findings of other authors [3] and disagrees substantially from our result [4] 0.085(5).
In this Comment, we would like to point out why we
feel this result to be inconsistent with other findings, and
that it must be considered as a lower bound of the actual
gap. The two main issues seem to be system size and
the extrapolation law: His results up to L ­ 128 are
extrapolated in a 1yL plot without further assumptions
about an extrapolation law.
Using the DMRG algorithm, we found the correlation
length to be j ­ 49s1d [4], to be compared to j ­
6.03s2d [5] for S ­ 1. This implies that system sizes
of L ­ 128 are not yet big enough to see directly the
L ! ` correct scaling behavior. In fact, in our gap
analysis going up to L ­ 270 in an open chain, we did
not yet see the crossover from a 1yL to a 1yL2 scaling
law, which has been extremely well established for the
S ­ 1 chain [5,6] and is due to the massive relativistic
low-energy spectrum the two systems share. Consistency
arguments showed, furthermore, that this crossover should
happen for substantially larger system sizes L ø 450 only
[4]. This scaling law holds for open boundary conditions,
S ­ 1 findings for periodic boundary conditions as used
by Yamamoto imply exponential convergence of the gap
to its thermodynamic limit [7], which set in for smaller L,
as there are no additional end effects. So far, all authors
agree that at the Heisenberg point the physics of S ­ 1
and S ­ 2 chains is fundamentally the same, so S ­ 1
results should provide a useful guideline. This is why
we argue that 1yL fits are not correct, suggested by the
fact that for small system sizes the S ­ 2 system with its
small gap still looks critical, which implies a 1yL scaling.
The large error bars provided by the QMC method tend
to further complicate the issue. However, given the
scaling behavior observed for S ­ 1, one expects that
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23 SEPTEMBER 1996
Yamamoto’s result still gives a lower bound to the gap;
the DMRG estimate of the lower bound is consistent with
it, albeit more precise [0.080(1) [4]]. Further research by
Yamamoto [8] actually strongly supports the validity of
the above argument.
Let us also point out that we found the cited correlation
length of the S ­ 2 chain using an approach which has
been highly successful for the S ­ 1 chain [5]. Using,
furthermore, that semiclassical analysis yields a spin wave
velocity c ­ 2S ­ 4, which implies a slightly larger true
quantum spin wave velocity, we can deduce that the gap
should (at least) be of the order of 0.08, in consistency
with our findings and above arguments.
All in all, it seems that, at the moment, White’s DMRG
gives better gap estimates than QMC, as it allows to
treat longer systems, and gives, by far, smaller (about one
order of magnitude) error bars for the raw results used in
extrapolations.
Ulrich Schollwöck
Sektion Physik
Ludwig-Maximilians-Universität München
Theresienstrasse 37yIII, 80333 München, Germany
Thierry Jolicoeur
Service de Physique Théorique
CEA Saclay
91191 Gif-sur-Yvette CEDEX, France
Received 24 June 1996
[S0031-9007(96)01216-1]
PACS numbers: 75.10.Jm, 05.30.– d, 75.40.Mg
[1] S. Yamamoto, Phys. Rev. Lett. 75, 3348 (1995).
[2] F. D. M. Haldane, Phys. Lett. 93A, 464 (1983); Phys. Rev.
Lett. 50, 1153 (1983).
[3] J. Deisz, M. Jarrell, and D. L. Cox, Phys. Rev. B 42, 4869
(1990); 48, 10 227 (1993); Y. Nishiyama, K. Totsuka,
N. Hatano, and M. Suzuki, J. Phys. Soc. Jpn. 64, 414
(1995); G. Sun, Phys. Rev. B 51, 8370 (1995).
[4] U. Schollwöck and Th. Jolicoeur, Europhys. Lett. 30, 493
(1995).
[5] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); S. R. White
and D. A. Huse, Phys. Rev. B 48, 3844 (1993).
[6] E. S. Sørensen and I. Affleck, Phys. Rev. Lett. 71, 1633
(1993).
[7] O. Golinelli, Th. Jolicoeur, and R. Lacaze, Phys. Rev. B
50, 3037 (1994).
[8] S. Yamamoto, Phys. Lett. A 213, 102 (1996).
© 1996 The American Physical Society