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Submitted on 1 Jan 1988
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RENORMALIZATION GROUP STUDY OF THE
ANISOTROPIC AND ALTERNATING HEISENBERG
ANTIFERROMAGNETS
H. Lin, C. Pan
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplhment au no 12, Tome 49, dkembre 1988
RENORMALIZATION GROUP STUDY OF THE ANISOTROPIC AND
ALTERNATING HEISENBERG ANTIFERROMAGNETS
H. Q. Lin (I) and C. Y. Pan (2)
(I) Department of Physics, Brookhaven National Laboratory, Upton, New York 11973, U.S. A.
(2) International Institute of Theoretical Physics, Department of Physics, Utah State University, Logan, Utah 84322, U.S.A.
Abstract. - A real space renormalization group method is applied to study the properties of the anisotropic and alternating Heisenberg antiferrornagnets with spin 112, 1 and 312. To improve the accuracy of the numerical estimates larger spin cluster is used. The ground state energy and the energy gap are obtained. For the case of S = 1 at Heisenberg point, the ground state energy per site is E (0) / N = -1.4021 f 0.0005 and the energy gap is 0.4097 f 0.0005, in excellent agreement with a recent Monte Carlo calculation. For the cases of spin 112 and 312 the energy gap is absent. Therefore our results are shown to support Haldane's conjecture. We briefly discuss the properties of the novel phase.
In recent years, studies of quantum spin chains continue t o be active and fruitful. This was partly motivated by the prediction made by Haldane [I] that there exists a radical different between the class of integer-spin and the class of half-integer-spin of the antiferromagnetic XXZ chains. Haldane has conjec- tured that the energy spectrum has no gap for the half- integer-spin chains but has a finite gap for the integer- spin chains. This surprising feature was investigated quite extensively [2-61. Although most numerical work on finite chains are in favor of the Haldane's conjecture there is, however no conclusive answer up to date.
There are two aspects of the problem need to be con- cerned. One is to investigate the validity of the Hal- dane's conjecture. The other one is to identify the crit- ical point where a gapless phase turns into the phase of finite energy gap or vice versa. In order to do that it is not sufficient to study the properties only at the isotropic point (J,, = J + )
,
a global picture of the en- ergy gap as a function of anisotropy and alternating parameters is needed. Encouraging by the partial suc- cess in the previous study on this problem by using a real space renormalization group (RSRG) study with the smallest spin cluster mapping [6], we use larger spin cluster in our RSRG mapping to study the prop- erties of the anisotropic and alternating antiferromag- netic Heisenberg (AFH) model. It enables us to obtain a clear global picture of the energy gap as function of anisotropy and alternating parameters. For the case of spin-1 chains the ground state energy and the singlet- triplet gap at Heisenberg point are in excellent agree- ment with Monte Carlo simulation results [3]. The two critical value of A, which enclose the novel phase, are also estimated. The energy gap obtained by the same approach vanishes at the Heisenberg point for both spin-l/2 and spin-3/2 cases. Therefore, our results are consistent with the Haldane's conjecture.We consider the anisotropic and alternating AFH linear chains:
where
s+,
S- and Sz are spin operators with spin S =1/2, 1, and 3/2. J1and J2 are coupling constants, A = J,,
/
Jz is the anisotropy parameter and 7 = Jz/ JI is the alternating parameter. In this work we subdivide the lattice into blocks of six sites and truncate the ba- sis of each block to two lowest levels, a singlet and a triplet. Symmetries of the Hamiltonian are preserved at each step of iteration. We write the Hamiltonian in the same form as the original one so to obtain recursion relations which relate the parameters of the renormal- ized Hamiltonian to those of the unrenormalized one. The ground state energy per spin is obtained bywhere rn numbers iteration step. The energy gap is defined as the energy difference between the ground state and the first excited state. It is well known that the accuracy of the RG results should be improved sys- tematically by increasing block size and retaining more energy levels. We have also performed few RG calcula- tions with block sizes more than six and the results are shown to be slightly better, so we only present results with block size six in this report.
Figure 1 shows the global picture of the energy gap
versus the anisotropy (A) parameter for the case of
C8 - 1416 JOURNAL DE PHYSIQUE
Fig. 1.
-
Energy gap us. anisotropy X for the spin 1 AFH Fig. 2. - Energy gap us. alternating parameter -y for the model, where X = 1.175 is the critical point. spin 1 AFH model.S = 1. It clearly shows a finite energy gap a t the Heisenberg point. The ground state energy per site and the energy gap are E (0) / N =
-
1.4021 f 0.0005 and 0.4097 f 0.0005, respectively. Comparing with re- sults obtained by using Monte Carlo simulation tech- nique: E (0) / N =-
1.4015 f 0.0005 and energy gap =0.41 [3], the agreement is excellent.
We did a similar calculation as described above for both spin 1/2 and spin 3/2 cases. The energy gap is indeed zero within the error bar of our RG calcula- tions for anisotropy paranieter X varies from zero (XY point) to one (Heisenberg point). This is not new for S = 1/2 chains since exact results are already known but for S = 312 there exists no exact solution and our results are consistent with the Haldane's conjecture. For lack of space the relevant figures are not shown here.
From figure 1 we also see that when the anisotropy parameter changes from the X Y point (A = 0) to the Heisenberg point (A = 1) the energy gap varies from zero to a very small value around 0
<
X5
0.5 then increases dramatically at X e 0.5 to the maximum at X = 1. From this observation we are hardly to say that there exists a c r i t f d point a t X = 0.49 [7]. It may well be that XI = 0 is a critical point, which is a stable fixed point. However, when X changes from the Heisenberg point (A = 1) t o Ising-like region (A>
1) the energy gap sharply drops to zero at A2 = 1.175 then remainszero in all region X
> 1.175. Hence we deduce that
X = 1.175 is a critical point where a finite gap phase turns into gapless phase. We noticed that a finite-size scaling calculation also predicates this critical point t o be X = 1.18 [4]. Considering errors in numerical calculations both results agree with each other very well.Figure 2 shows a possible novel property for the spin 1 case. The energy gap decreases from a finite value t o a very small value (N 0) when the alternating pa- rameter y vaies from the dimer limit (y = 0) to the neighborhood of the uniform limit (y < 1
-
E ),
whichis the same behavior as in the spin 1/2 case, however, right at the uniform limit (y = 1) there is an energy jump (0.4097) instead of zero as in the spin 112 case. Although this surprising feature dj~serves further in- vestigation, however, if we tend t o believe that the integer-spin and the half-integer spin systems are rad- ically different, it might not be so siurprised.
In conclusion, the RSRG method we applied to study the anisotropic and alternating AFH model gives reliable results as compared with tht! exact solution for S = 112 and with other numerical work for S = 1. The spin-1 AFH chains seem t o have a novel phase as pred- icated by Haldane and critical values of anisotropy pa- rameter are estimated. A systematic studies of quan- tum spin models using RSRG method are in progress. Acknowledgement
The authors thank V. Emery and D. Matti's for many helpful discussions. H. Q. Lin is supported by U.S.D.O.E. under contract DE-AC02-76CH00016. C. Y. Pan gratefully acknowledge the support from V. G. Lind and a grant from IBM (20.
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