HAL Id: jpa-00228859
https://hal.archives-ouvertes.fr/jpa-00228859
Submitted on 1 Jan 1988
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of
sci-entific research documents, whether they are
pub-lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
QUANTUM SPIN CHAINS WITH COMPOSITE SPIN
J. Sólyom, J. Timonen
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, dkcembre 1988
QUANTUM SPIN CHAINS WITH COMPOSITE SPIN
J. S6lyom ') and J. Timonen (3)
(I) Institut Laue-Langevin, f 56X, F-38042 Grenoble Cedex, Fmnce
1')
G e n t r ~ f Research Institute for Physics, P.O. Boz 49, H-1525 Budapest, Hungary( 3 ) Department of Physics, University of Jyviiskylii, SF-40100 Jyviiskylii, Finland
Abstract. - The ground state of quantum spin chains with two spin-112 operators per site is determined from finite chain calculations and compared to predictions from the continuum limit. As particular cases, results for the spin-1 Heisenberg chain, the spin-1 model with bilinear and biquadratic exchange and the extended Hubbard model are analysed.
We have studied the ground-state properties of one-dimensional composite spin models [I], where at each lattice site there are two spin operators, oi and
Ti.
First we considered the Hamiltonian
H = H O + H ' , (1) where
and
The unperturbed Hamiltonian
Ho
describes two un- coupled anisotropic S = 1/2 Heisenberg chains and H' gives the coupling between the two spin species.Using the Jordan-Wigner transformation this model can be transformed into a fermion model [2] and in the case J:, = 0, J: = J, this is the half-filled extended Hubbard model, where t = Jxy is the hopping,
U
= 2 0 is the on-site Coulomb coupling and V = -Jz is the intersite Coulomb interactjon.More generally, when Jxy = 0, the continuum limit of the fermion problem will be identical to the gen- eral interacting one-dimensional fermion problem. The backward, forward and umklapp scattering terms in the g-ology language [3] can be identified with the couplings between the spins as
gill = 45, - 25:
+
20, g i ~ = 2 (J:+
0),
g111-2g2=4J,+2~:-20, g 3 = 2
(4)
The phase diagram of the g-ology model and that of the extended Hubbard-model being well known, we
have checked whether finite size scaling from numeri- cal calculations on finite chains up to N = 10 lattice sites reproduces reasonably well these results. We have found that the lattice calculations and the continuum limit are in very good agreement for the phase diagram of the model.
Next we have taken the couplings between the a and
T spins to be J:, = X J x y , J: = XJ, and D = 0 and
studied the phase diagram in the (A, Jz/Jxy) plane. Relying on the analysis of a similar model by den Nijs
[4], Timonen and Luther [5] and Schulz [6] the con- tinuum theory predicts the opening of the gap with
X if -1
<
Jz/Jx,<
0. On the other hand, at X = 1the model has the same ground state properties as the S = 1 anisotropic Heisenberg chain [I], for which Hal- dane's conjecture [7] and subsequent numerical calcu- lations [8] predict a singlet, massive phase for -1.18
<
Jz/JSy<
-0.1. Knowing that all the energy levels of the model satisfy the self-duality relation E (A) =XE (l/X), the expected phase diagram is shown in fig-
ure 1. Notice that except for the ferromagnetic phase, the phase boundaries are not straight lines, they con- nect smoothly Jz/JzY = 0 at X = 0 and Jz/Jxy
=
-0.1 at X = 1 on one side of the singlet phase and Jz/Jxy =C8 - 1378 JOURNAL DE PHYSIQUE
-1 a t X = 0 and J , / J , , cz -1.18 a t X = 1 on the other side.
We calculated numerically exactly the gap for finite chains up to N = 10 sites. The extrapolation to in- finite chain length is not unambiguous, we can rely, however, on general trends in the X dependence of the gap. The numerical calculations indicate that for any fixed anisotropy the gap will either vanish for all X
or be a monotonously increasing function of X open- ing linearly with A. This would lead to X-independent phase boundaries for the singlet phase. This would in turn imply that these boundaries are determined by symmetry. It would follow that the antiferromagnetic phase extends to J,/J,, = -1 for any A, contrary to the expectation. One cannot exclude, however, that for longer chains a crossover to a different behaviour will appear which is not present in chains with N
5
10sites. Haldane's conjecture could then be valid. As a next problem: we have considered a general- ized Hamiltonian, adding four-spin coupling terms, re- stricting ourselves, however, to isotropic terms in spin space: (uzuz+i
+
r z r z t l )+
3 + g N sin 0 ( 5 ) and 1 H' = X cos(
0-
-sin 0 )SJ
( ~ i r i + l + r i u ; + ~ )+
2 + X ~ N sine
+
2 6sin 0C
iThe couplings are parametrized by 0 and X in such a way that the model satisfies self-duality and at X = 1
we recover
H = cos 0
x
( ~ i ~ i + l ) +sin 0x
( ~ i s i + l ) ~ ( 7 )with S; = a;
+
.ri. Moreover, it can be shown that the ground state and low lying excited states of this modelferro- magnet I -7c singlet
L
4 2 --
1I
singletL
0 ferro- magnetI
n/20
7cFig. 2. - Expected phase diagram for model given in (5) and (6).
are identical to that of the S = 1 bilinear-biquadratic exchange model, so the ground state properties of the latter model can be studied using the Hamiltonian in
( 5 ) and ( 6 ) .
At X = 0 the model consists of 1;wo uncoupled spin-
112 chains. The first term in H' is the isotropic version
of the coupling we studied above. IIaldanels conjecture would tell that it is a relevant perturbation opening a gap, whenever cos 8
-
;sin 0>
0 . The last term in H''4
is also, according to Affleck and H~tldane [9], a relevant perturbation and only for a special choice of 0 could the two mass-generating terms conlpensate each other. This should happen at 0 = f n / 4 , since it is known [lo]
that at B = f7r/4 and X = 1 the model is massless, furthermore the numerical calculatdons indicate that if the gap vanishes for X = 1 it will vanish for all A. The expected phase diagram is shown In figure 2.
Since the two supposedly relevimt terms have dif- ferent X dependence it is not easy to understand how they can compensate each other at 0 = *7r/4 for all A.
Moreover it has been proposed in [ l l ] that the spin-1 model, i.e. the present model at A = 1 is gapless for
-37r/4
5
0 <_ - a / 4 . There are ir~dications, [12].and [13] that this gapless phase extends beyond 0 = -7r/4.If this is true, it would follow from the numerical cal- culations that for all these values of 8 the gap vanishes for all A. This would then imply that the perturbations
@ [6] cannot be relevant and the gapless phase should exist for all X for 0
5
0 . The gap could appear for 6'>
0only, where the last term in H' changes sign. Whether
this is the case or the finite chain calculations could not reveal the true assumptotic beliaviour necessitates further studies.
[ l ] S6lyom, J. and Timonen, J., Phys. Rev. B 34 (1986) 487.
[2] Shiba, H., Prog. Theor. Phys. 48 (1972) 2171. [3] S6lyom, J., Adv. Phys. 28 '(1979) 201. [4] Den Nijs, M., Physica A 111 (1982) 273.
[5] Timonen, J. and Luther, A., J. Phys. C 18 (1985) 1439.
[6] Schulz, H. J., Phys. Rev. B 3.4 (1986) 6372. [7] Haldane, F . D. M., Phys. Rev. Lett. 50 (1983)
1153.
[8] Botet, R. and Jullien, R., Phys. Rev. B 27 (1983) 613.
[9] Affleck, I . and Haldane, F. I). M., Phys. Rev. B 36 (1987) 5291.
[lo] Takhtajan, L. A., Phys. Lett. A 87 (1982) 479. [ l l ] S6lyom, J . , Phys. 'Rev. B 36 (1987) 8642. [12] Oitmaa, J., Parkinson, J. B. and Bonner, J. C.,
J. Phys. C 19 (1986) L595.
