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PHASE DIAGRAM OF A HEISENBERG
HEXAGONAL LATTICE WITH IN-PLANE
COMPETING INTERACTIONS: CLASSICAL AND
QUANTUM SCENARIOS
A. Harris, E. Rastelli, A. Tassi
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplbment au no 12, Tome 49, dbcembre 1988
PHASE DIAGRAM OF
AHEISENBERG HEXAGONAL LATTICE WITH IN-PLANE
COMPETING INTERACTIONS: CLASSICAL AND QUANTUM SCENARIOS
A. B. Harris (2), E. Rasteili (I) and A. Tassi (I)
( I ) Dipartimento di Fisica, Universita' di Parma, 43100 Parma, Italy
(2) Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, U.S.A.
Abstract. - Further-than-third neighbor exchange interactions and quantum fluctuations produce new interesting effects in the phase diagram of the Heisenberg magnet with competing interactions. These effects are considered for a simple hexagonal lattice.
Competing exchange interactions in a simple Heisenberg model lead to non simple spin configu- rations. In the classical approximation (S = m) in- plane competing interactions up to third neighbors (3N model) give a phase diagram at T = 0 with [I] ferromagnet (F), antiferromagnet (AF), and two he- lix (H) phases which we label H1 and Hz, respectively. The 3 N models studied in reference [I] are the tetrag- onal and the h e x a ~ n a l
-
lattices with the following in--
plane exchange interactions: N N ferromagnetic inter- action J1>
0, second NNJ2 and third N N J 3 with arbitrary sign. Here we consider a ferromagnetic out- of-plane coupling J'.The classical phase diagram at T = 0 for the hexag- onal lattice is shown in figure 1 where continuous curves correspond to continuous phase transitions and broken curves to first order phase transitions. The
phase booundary between HI and H2 is an "infinite
degeneration" line [2], along which the classical ground state energy is minimized by helices whose wavevectors Q span a continous range between the two helical con- figurations HI and Hz.
Quantum fluctuations in helimagnets are not easy to treat consistently: indeed standard perturbative meth- ods work well for F and AF configurations but fail dramatically for helimagnets, because near the phase boundary the elementary excitations in the classical approximation are not well defined: imaginary fie
quencies are obtained from perturbation theory 131. This fact indicates that the choice of the ground state is not self-consistent: the helix wavevector, Q, is surely modified by quantum fluctuations but any standard perturbative approach is unable t o account for this.
The problem is still open even though partial answers have been obtained for the ground state energy. A cal- culation to leading order in 1/S has been performed for a tetragonal lattice [4] while, recently, a formulation exact to all orders in 1/S has been given to discuss the F-H transition line [5]. In this formulation the ground state energy, EG, is expanded in powers of the helix wavevector Q which lies in the basal plane. To order
Q~ one has the exact result,
where
Fig. 1.
-
Phase diagram of the 3N model for classical is the classical ground state energy (a = 1, 2, 3 and(S = m) and quantum systems. For S = m : solid lines AE
and AB are continuous phase transitions between AF - Sff is a vector joining a spin with its neighbours of the and F-H phases, respectively. The broken line AD is the a- th shell) and
first order AF-F transition line and the chain curve BB'
is the "infinite degenerationn HI -H2 transition line. For A k = 2 J. S (1
-
cOS k -6,)+
j j
-
-
0.1 and S = 512 quantum effects give the A"E' andQ
A"D' transition lines and at A' the F-H transition becomes 6, first order. Dotted lines near A" are conjectured. Quan-
tum effects on the HI-H2 phase boundary have yet to be +2 J'S (1
-
cos k .6') (3)studied. 6'
C8
-
1396 JOURNAL DE PHYSIQUEBk =
-
x
J ~ Sx
(1/2) ( 4 s,)? cos k .6. (4)0 6,
We use the T-matrix [6] of equation (5) to sum contri- butions t o all orders in 1/S, where Vk,, is the interac- tion potential in the Dyson-Maleev representation [7]. As one can see directly from equations (1)-(6) quan- tum effects first appear at order Q4. The Q2 contri- bution is zero along the classical F-H transition line (1
+
3j2+
4j3 = 0, with jp = Jp/J1) in which case equation (1) is of the formEG (Q) = 6 JI N S ~ [eo
+
e4 ( a ~ ) ~ ] (.7) where a is the in-plane N N distance. The classical result gives e42
0 along the whole F-H transition line and leads t o a continuous phase transition. Quantum fluctuations, on the contrary, lead to e4<
0 over a non negligible part of the F-H line in the neighbor- hood of the triple point A, giving a first order phase transition between F and H phases. In figure 1A' is the point where e4 changes its sign for j' = J'/J~ =0.1 and S = 512. The "quantum region" AA' increases as S and/or j' decreases. Before concluding that the H-F tramition becomes discontinuous at A', we need to show that the H-AF-F triple point perturbed by quantum fluctuations, A"
,
remains inside the interval A-
A'. To see whether this is the case we compared the zero point energy of the AF phase to order 1/S with that of the quantum fluctuating phase. For S2
1 we found that the AF phase does not extend as far as the point A', in which case there is a finite region of quantum-fluctuation-induced first order behavior.A similar approach is used to calculate the new AF
-
H1 transition line (E'A" in Fig. 1) because in this case a T-matrix calculation is not feasible. An ex- pansion around the AF wavevector QAF is performed up to (Q-
Q A F ) ~ contributions. The E'A" line is the locus where the coefficient of (Q-
Q A F ) ~ van- ishes. Notice that both classical and quantum effects contribute at this order. Along the E'A" line the co- efficient of the fourth order term is negative so that a wfirst order phase transition occurs in the region where H1 phase is metastable.
We now consider the classical 4N model adding a fourth NNJ4 to the 3 N model.
The phase diagram is shown in figure 2 for j 4 =
f 0.025. The more striking results are:
Fig. 2. - Classical phase diagram of the 4N model: (+)
labels refer to j 4 = 0.025, (-) labels to j 4 = -0.025. Con-
tinuous and broken curves are continuous and first order phase transitions, respectively. The inserts are sketch of the ground state energy as function of Q in F, AF, H ~ , - H ~ regions.
degeneration" line in the 3N model [Z], is shifted and becomes first order.
The ground state configurations of the 4N model are qualitatively the same as those of 3N model. We think that our finding a first order H1 -Ha t~ansition line ow- ing to the presence of J4, is not generic: we expect that appropriate further neighbor couplings would lead to a new "intermediate" phase characterized by a helix wavevector swinging between the HI and H2 configu- rations.
In summary, we have shown that both, quantum fluctuations and additional exchange couplings pro- duce significant and interesting new qualitative fea- tures in the phase diagram of the Heisenberg model with competing interactions.
[l] Rastelli, E., Tassi, A., Reatto, L., Physica B 97
(1979) 1.
[2] Rastelli, E., Reatto, L., Tassi, A., J. Phys. C 16
(1983) L331.
[3] Rastelli, E., Tassi, A., J. Phys. C 19 (1986) 1993. [4] Rastelli, E., Reatto, L., Tassi, A., J. Phys. C 19
(1986) 6623.
[5] Harris, A. B., Raktelli, E., J. Phys.
C
20 (1987) i) for ' j 4>
0 a part of F-H transition line becomes L741; J. Appl. Phys. 63 (1988) 3083.first order (curve PA+ in Fig. 2); [6] Silberglitt, R., Harris, A. B., Phy:r. Rev. Lett. 19 ii) for j 4
