CC BY
5Scientific Notes of Taurida National V. I. Vemadsky University
Series : Physics and Mathematics Sciences. Volume 26 (65). 2013. No. 2. P. 60-68
UDK 537.612
SPECTRA OF ELEMENTARY EXCITATIONS AND PHASE DIAGRAM OF NON-HEISENBERG SPIN-2 MAGNETIC Kosmachev O. A.
Taurida National V. I. Vernadsky University, 4 Vernadsky Ave., Simferopol 95007, Ukraine
Е-mail: lkosma@tnu. crimea. ua
The spectra of elementary excitations have been investigated for spin-2 non-Heisenberg magnetic with the account of all spin invariants. Analysis of the spectra of elementary excitations allowed to construct the phase diagram of the magnet at various relationship between the exchange integrals. In case of single-sublattice magnet, there is equivalence with the phase states and excitation spectra behavior of spin-2 Bose-gas of ultracold atoms. Keywords: non-Heisenberg magnet; phase transition, Hubbard operators; nematic phase; tetrahedral phase.
PACS: 75.10. ± b
INTRODUCTION
Recently, the investigation of magnetically ordered systems with the account of highorder spin invariants has drawn great attention [1-4], because of the fact that such systems are equivalent to the Bose-condensate of "cold" atoms. This condensate can be obtained with the help of various "atom traps" [5, 6]. One can make one or another type of interaction corresponding to spin invariant prevailing by varying trap's parameters. As it was noted in [1], the investigation of such systems can be carried out within the frameworks of the exchange Hamiltonian. Research of the model Hamiltonian with the account of high-order spin invariants [7, 8] allows to find the phase states of the system and also to determine the behavior of excitations spectra near the phase transition lines. Besides, it is possible to determine the relationship between the exchange constants and the parameters of scattering lengths of the corresponding spin systems. It was found during investigations [9, 10] that the increase of the spin of a magnetic ion results in the emergence of the new quantum effects, in particular, to the realization of new nematic phases: tetrahedral and antitetrahedral. It was shown that the geometrical image is a biaxial ellipsoid, while in spin-1 case [11] the geometrical image of the nematic phase is single-axis ellipsoid. It should be also noted that there appears additional parameter in tetrahedral (antitetrahedral) phase - the pseudospin which transforms like the real spin vector at time reflection t ^ -t. The next stage in the investigation of this system is the investigation of the behavior of the spectra of elementary excitations in the vicinity of the phase transitions lines.
1. PHASE STATES
The aim of the present work is to investigate the phase states and spectra of elementary excitations in the vicinity of the phase transitions lines of the isotropic spin-2 ferromagnetic with the account of the complete set of spin invariants. The Hamiltonian of such a system has the following form:
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X Ynn'{ßrßn') +Knn'{ßrßn') + A»j'(4Ä') +Fnn'(snsn'} I 2 „W v /
2^« +Knn'{SnSn') +D„„'(SnSn') +Fnn'(SnSn') } (1)
2 n*n'
where J, K, D, F are the exchange integrals corresponding to various spin invariants. It is supposed that the system considered is at low temperature (T << TC where TC is the Curie temperature), as the quantum properties of the system are more evident in this case.
Before we proceed to the investigation of the spectra, we want to remind which phases realize in the system at various relationships between the constants of exchange interactions [10].
1. If the relationship between exchange integrals is such that J0 > K0, D0, F0, then the
wave-function of the ground state is given by = | 2) . Therefore, the averages on this
state are ^S2} = 2, ^(Sz)2 ^ = 4, ^(Sx)2 ^ = Sy)2 ^ = 1. This state is ferromagnetic (FM).
2. At K0 > J0, D0, F), the wave-function of the ground state is l^g,.^ = -^cos^(|2) + ) + sinP\0). This state is characterized by the
"V2
quadrupolar order parameters: qf = 3^(Sz )2^ — 6 = 6cos2^,
q| = ^(Sx— ^(Sy)2^ = 2>/3sin2^, because (S) = 0. This spin state is "spin
nematic" and will be denoted as N.
3. And there is one more case: D0 > K0, F0. In this case, the wave-function of the
ground state looks like = (| 2) + V2| — l)) . This phase is characterized with
order parameters of higher order in spin operators S': q° = 10, q3 =1I ((S+ )J + ((S— = 4^2, because = 0, q0 = qf = 0. This phase state
21......
will be denoted as Tig-state.
In case of negative constant of Heisenberg exchange interaction J0 < 0, the two-sublattice magnetically ordered structures realize in the system:
4. Antiferromagnetic (AFM) state. In this state, only axial moments are differ from
zero: (S^ = 2 q2 = 6, q3 = 6, q4 = 12. The order parameters of the second sublattice
are (Sz) = —2, q2 = 6, q30 = —6, $ = 12 .
5. And finally, the ATQ-antitetrahedral phase, characterized with tensor component of the higher order: q° = 10 , q3 = WI, q4 = —28, q4 = 2*Jl in the first sublattice,
and = —10, q33 = —•W2, q4 = —28, q4 = 2^2 in the second sublattice.
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2. SPECTRA OF ELEMENTARY EXCITATIONS OF SINGLE-SUBLATTICE NON-HEISENBERG SPIN-2 MAGNETIC
The spectra of elementary excitations are determined by the poles of the Green function [1]: Gaa' (n,r; n\z') = -(T~: (r) X",' (r')), (2)
where X"(t) = exp(Ht)X" exp(-7/r ) are the Hubbard operators in the Heisenberg
representation; t is the Wick operator; Jf = }f:) +://rint [12, 13]. The further evaluations will be carried out within the mean-field approximations; therefore we can restrict ourselves with the dynamic part of the exchange Hamiltonian which can be presented as follows:
Hmt = -- £ £ {c(a)An c(P)\^X„ ■ AXpn,, (3)
2 n^n' a,p
where AXa = Xa -(Xa)Q , and the components of the 24-dimensional vector c(a)
are determined from the relationship between the spin and tensor operators with the Hubbard operators; Ann, matrix is given by
Ann' =-n-(2E 0 I)®-^n-(6E 0 I 0 /10 -nn-(l6^ 0 3I 0 301 0 20/10 2 v ' 2X ' 160 v '
a (4) /.....V
0_nnL(2E: 0 10I 0 5I 0 70I 0 35/), 560 v '
. . f0 1^ where E is unit matrix, I = .
11 °J
The derivation of the dispersion equation determining the spectra of elementary excitations is given in Refs. [12, 13]. The equation is valid at arbitrary spin value, arbitrary symmetry of the single-ion anisotropy, and arbitrary temperature (except the region of fluctuation). The dispersion equation can be presented in the following form:
det ||l + Xy | = 0, z, j = 1,.. ,24. (4)
Solutions of equation (4) determine magnon spectra in different phases.
Let us proceed with the analysis of the spectra of elementary excitations for each phase. 1. Using the explicit form of the order parameters in the FM-phase, we can obtain the magnon spectra in the long wave-length limit (at k ^ 0):
sl(k) = A[j0+98k2)-, (5)
s2(k) = 4\2(jQ+3KQ) + 3{A + y + \3S)k2^ (6)
£3 (k) = 2 (a + 41 + 16/ + 64S) k2; (7) s4 (k) = 6
~2(j,+3K,) + 3{r-A5)k2\ (8) where J0 = 2 J0 - K0 + 41D0 - 79F0 and K0 = K0 - 5D0 + 43F0; J0 - Jk = ak2,
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(11)
K0 — Kk = Ak2, D0 — Dk = /k2,F0 — Fk = 5k2.
As it is seen from Eq. (5) and (8), the magnon spectra soften at J0 = 0 and J0 + 3K0 = 0, i.e., the FM-phase becomes unstable. Thus, the phase transition from the FM-phase occurs along the magnon branches £1 (k) and S4 (k). It should be noted that
nevertheless the gap in the spectrum £2 (k) has the same form as in the spectrum £4 (k),
the phase transition occurs along the branch (8), as the magnon "velocity" in this branch is higher, than in the branch (6).
2. Now consider the spectra of elementary excitations in the N-phase in the long wave-length limit (with the account of the notations, introduced above):
s2 (k) = 48 [(A — 2/+ 285) k2 sin2 / + 3 -5k2 cos2 /] x x |j3K0 sin2 J3 — J0 cos2 /?J;
s2(k) = U4K0(Z-2y + 2SS)k2-, (10)
s324 (k) = 12 [(A — 2/ + 315) k2 + (A — 2/ + 255) k2 cos (2/ ± n/ 3)
x 3KQ - JQ + (3KQ + JQ)cos(2/3 ± tt/3)
It is seen that the magnon spectra (9) - (11) in the N-phase are linear in the wave-vector far away from the stability points. The branch £1 (k) softens at line J0 = 0 when parameter / = 0 (parameter /3 is re-determined exactly), and takes the form s1 (k) = 365k2. Consequently, the line J0 = 0 is the line of the phase transition N - FM
phase, at this, the spin nematic tends to single-axis nematic near this line. Besides, the
•—- n
magnon spectrum (9) becomes unstable at K0 = 0, and at 3 = ^" takes the form:
s1 (k) = 36 (/ — 55) k2, and the order parameters of the N-phase tend to the parameters of the "flat" nematic near this line. It should be noted that the spectrum (10) also loses stability at line K0 = 0, and is proportional to k squared: s2 (k) = 36(/ — 55)k2. Besides, the line of excitations (11) is degenerated which is related with the degeneracy of the energy levels of a magnetic ion E1 = E 1 in the N-phase at / = 0, and / = .
3. Now consider the spectra of elementary excitations in the TQ-phase. As it was mentioned above, the three-fold degeneracy of the excited energy levels of a magnetic ion is observed in this phase which results in coincidence of three branches of elementary excitations. In the long wave-length limit these spectra have the following form:
(*) = 144(J0 + 3Xq)(7 " 4S)k2; (12)
£2 (k) = 144(-^0 +(A-2y + 28S)k2)2 . (13)
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As it follows from Eq. (12), the spectra s"i 2 3 (k) are linear in the wave-vector k far from the line J0 + 3K0 = 0, but are square in k near the line J0 + 3K0 = 0 :
^1,2,3 (k) = 18(y-40)k2,
i.e., the magnon spectra soften on this line, and the line J0 + 3K0 = 0 is the line of the
phase transition TQ - FM phase. The spectrum (13) is unstable at the line K0 = 0 and becomes square in k:
( k) = 36 (y- 53) k2.
Consequently, the line K0 = 0
is the line of the phase transition TQ- N - phase.
Thus, the analysis of the magnon spectra allows to construct the phase diagram m of the system considered (see Fig. 1). The coincidence of the spectra at the lines of the phase transitions testifies that these phase transitions are of the second kind. It should be noted that this phase diagram completely coincides with the phase diagram, obtained for ultracold neutral atoms with S = 2 [5, 6].
3. SPECTRA OF ELEMENTARY EXCITATIONS OF TWO-SUBLATTICE NONHEISENBERG SPIN-2 MAGNETIC
As before, the spectra of elementary excitations are determined by the poles of the Green function. Using order parameters, determined above, we can obtain the spectra of elementary excitations in the corresponding phase states.
1. AFM-phase. The magnon spectra in the AFM-phase can be presented as the difference of two squares, consequently, their behavior in the center of the Brillouin zone (k = 0) and on the boundary ( k = n ), are almost equivalent. The magnon spectra in the AFM-phase in the long wave-length limit (k ^ 0) look like:
L FM
TQ N
Fig. 1. Phase diagram of non-Heisenberg spin-2 ferromagnetic on the (./,,. A',, ) -plane.
ef {k) = \6(J0 -18F0 + 90k2)(J0 -90k2) ; (14)
s2 (k) = 4[J0 - K0 - 3 (A - 5y + 435)k2 ] [J0 - 3K0 - 6(K0 - 5D0 + 43F0 )] ; (15) e\ (k) = 4 [J0 - 3K0 -6(K0- 5D0 + 43F0)](cr -51 + 34/ -179S)k2 ; (16) s24(k) = 9[j0-3K0 + 6(r-6ö)k2){j0-3K0 +12(£>0-6F0)). (17)
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The spectrum (14) softens at the line J0 = 2J0 — K0 + 41D0 — 61F0 = 0 . This line
describes the phase transition from the AFM-phase into the N-state.
The energy gaps in spectra (15) and (17) decrease while approaching to the line of the phase transition into the ATQ-phase. Both branches are unstable at the line of the
phase transition the AFM-ATQ phase ~0 — 3KT0 = 2J0 — 4K0 + 38D0 —100F0 = 0;
however, the velocities of the "spin" waves are different: the phase transition occurs along
s4 (k), because its velocity coincides with the velocity of the wave in the ATQ-phase at
line J'0-3K'0=0.
2. Consider the spectra of elementary excitations in the N-phase. In the long wavelength limit at k ^ 0:
si (k) = 98 [(Aq - 2D0 + 28E0)sin2 J3 + 3F0 cos2 /?] x [3^ sin2 J3 + +J'0 cos2 j3-(2a-A + 4ly-70£)k2 cos2 j3-9(y-5S)k2 sin2 /?]. (18)
e\ (k) = 144(K0 - 5D0 + 43F0- 2y + 28S)k2;
^3,4
( k ) = 48
K0 -2D0 + 31^0 +(K0 -2D0 + 25F0)cos| 2ß±^ \ + a'k2
J0 + K0 + 22D0 — 11F0 — (J0 — 2K0 + 19D0 — 50F0)cos 2/ ± — I + a"k2
v 3 J
where a' and a" are the combinations of a,A,y,S.
As it is seen from Eq. (18), the spectrum softens at the line of the phase transition into the AFM-phase (at / = 0): ~0 = 2 J 0 — K0 + 41D0 — 61F0 = 0. On the other hand, this spectrum also softens at the line of the phase transition into the ATQ-phase (at 3 = —2): K~0 = K0 + D0 + 13F0 = 0. The branches s2 (k) — s4 (k), as it easy to notice, do not soften in the vicinity of the phase transitions lines ~0 = 2J0 — K0 + 41D0 — 61F0 = 0 and K0 = K0 + D0 + 13F0 = 0.
3. Now consider the spectra of elementary excitations in the ATQ-phase. There observed a three-fold degeneracy of the excited energy levels of a magnetic ion E1 = E 1 = E0 which
results in coincidence of three branches of elementary excitations e1 (k) = s2 (k) = s3 (k). In the center of Brillouin zone (k ^ 0), the spectra are given by
£22 3(>) = 72[-J0 +3Kq +(2cc-4j3 + 47y-\54S)k2^(DQ -6E0); (21)
e\ (k) = 144[K0 -(P -2y + 285)k2][6(D0 -5F0)-K0], (22)
(19)
, (20)
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The gap in spectrum 2 3 (k) vanishes at the line of the phase transition ATQ-AFM
phase J 0 — 3K'0 = 2 J 0 — 4K0 + 38D0 — 100F0 = 0, and the spectrum becomes linear in k.
The spectrum s4 (k) becomes unstable at the line of the phase transition ATQ-N phase
K 0 = K0 + D0 + 13F0 = 0, and becomes linear in k. At the boundary of the Brillouin zone (k ^ —), the spectra look like:
£L23 (k) = 36[J0 -3r0 +18(D0-6F0)-(2a-4/3 + 47y-\545)k2^(y-65)k2;
e\ (k) = 144-(P-2y + 285)] x[(/?-2y + 285)k2 -KQ+6(D0 -5F0)] . Spectrum s12 3 (k) is linear in the wave-vector k; however, it is stable at the phase
boundaries. Behavior of s4 (k) at the boundary of the zone is equivalent to the behavior in
the center of the zone. Analysis of the spectra if elementary excitations and of the free energy density allows to construct the phase diagram of the two-sublattice nonHeisenberg magnetic. This phase diagram on the (J'.K' )-planc is given in Fig. 2.
ATQ
J'
AFM
N
CONCLUSIONS
The carried out investigations of the spin-2 non-Heisenberg magnetic allow to stay that the account of high-order spin invariants is essential and leads to the realization of the magnetically ordered states with more complex structure, than ferro- or antiferromagnetic. The nematic phase together with the tetrahedral
and the antitetrahedral phases Fig- 2- Phase diagram
belongs to such more complex Heisenberg spin-2 magnetic on the (./,',./<,',)-plane. states. These phases are characterized with that their
magnetization (per site) equals zero, while the states, realized in them, are magnetically ordered, and the order parameters are the components of the tensor of quadrupolar moments. The states with zero magnetization per site, but with finite multipole order parameters are purely quantum effect [14]. Nevertheless the fact that magnetization equals zero in these phases, these phases are different, because they have different ground states, different topology in the spin space, and, consequently, different symmetry. The key feature of these phases for S = 2 is their more complex structure (the geometrical images
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in the spin space), in comparison with the structure of the nematic phases, investigated previously for S = 1 and S = 3/2 [9-11]. Thus, the nematic phase is a single-axis ellipsoid in spin-1 magnetic; while its geometrical image in the case considered is "goffered" biaxial ellipsoid which loses its "goffering" and becomes single-axis only at the lines of the phase transitions. Besides, the antinamatic phase is absent in spin-2 magnetic, while it is observed in magnetic with the spin of a magnetic ion S = 3/2. However, the tetrahedral/antitetrahedral phase can realize in the system under consideration which, in some way, is analogues of the antinematic phase. However, the tetrahedral/antitetrahedral phase has more complex geometrical structure (in the spin space) in comparison with the antinematic phase in spin-3/2 magnetic; however, similar to the magnetic with S = 3/2 the tetrahedral/antitetrahedral phase has additional order parameter - the pseudospin a which is described by non-zero averages from expressions cubic in spin operators. It should be noted that the appearance of the states with the pseudospin order parameter is possible only in non-Heisenberg magnets with S > 1, because this parameter describes with non-zero averages from expressions cubic in spin operators.
Thus, the mean-field analysis of the non-Heisenberg magnetic with spin-2 allowed us to describe both, the dynamic, and the static properties of the system, to reveal formation peculiarities of the phases with multipole order parameters, and to construct the phase diagram of the system.
This work was supported by the State Fund for Fundamental Research of Ukraine.
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У робота дослщжено спектри елементарних збуджень негейзенберговського магнетика зi стном магттного юна 2, при облжу вах спгнових iнварiантiв. Аналiз спекав елементарних збуджень дозволив побудувати фазову дiаграму магнетика при рiзних сшввщношеннях обмгнних iнтегралiв. У разi однотдграткового магнетика е вщтаждтсть фазових сташв i поведшки спектргв збудження бозе-газу ультрахолодних атомгв з S=2.
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Received 11 February 2013
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