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QUANTIZED BREATHERS IN A DOUBLE
SINE-GORDON SYSTEM
Qing Xia, P. Riseborough
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, decembre 1988
QUANTIZED BREATHERS IN A DOUBLE SINE-GORDON SYSTEM
Qing Xia and P. S. RiseboroughDepartment of Physics, Polytechnic University, 333 Jay Street, Brooklyn, New York 11201, U.S.A.
Abstract. - We examine the excitations of a quasi-one dimensional easy plane antiferromagnet, in a symmetry breaking applied magnetic field. The antiferromagnetic exchange includes a small Dzyaloshinski Moriya component that produces the easy plane. In the classical continuum limit, this system is described by a double-sine-Gordon field theory, which supports soliton excitations and the multi spin wave bound states/resonances of the discrete spin system. The latter correspond to the quantized breather excitations of the double sine Gordon field theory.
1. Introduction
The double sine-Gordon field theory is not known to be integrable, however, many of the elementary exci- tations of this system show strong resemblence t o their counterparts in the integrable sine-Gordon field theory. Numerical studies have shown that the double sine- Gordon system does support long lived, large ampli- tude, breather excitations which like the corresponding sine-Gordon large amplitude breathers can be viewed as being composed of an interacting soliton/antisoliton pair. We investigate the dynamics of a quantum sys- tem, which in the classical continuum limit may be described by a double sine-Gordon field theory. We examine the quantized low amplitude breathers.
2. The model
lattices A and B. The spins on the A sublattice are described by a classical vector with components S: = S [sin (8,
+
8,) cos (9,+
cp,) ;sin (8,
+
8,) sin (9,+
~ n ) ; cos (Qn+
On)], (2.2a)9: =
-S
[sin (@,-On) ess(a,-cp,)
;sin (@,-On) sin (9,-cp,) ; cos (0,-Qn)]
.
(2.2b) The equation of motion for the four variables can calculated using the Poisson brackets1 sin 8, cos 8, {Q,, 9,) = {On, a n ) =
--
2 5 sin2 0,-
sin2 On'
(2.3a) and 1 cos 0, sin 8,{ e n ,
9,) = {en,an)
=--
2 s sin2 0, - sin2 0,'
(2:3b) We consider a one dimensional, nearest neigh- In the continuum limit, the purely inplane dynamics bour Heisenberg Antiferromagnet, which has a small is governed by the double sine-Gordon equation Dzyaloshinski-Moriya component t o the exchange in-teraction. In the ground state, the D.M. interaction
a2as
has the effect of introducing a canting between theantiparallel spins also constrains the spins t o orient themselves within a plane. The continuous invariance under a rotation of the spins in the easy plane, is bro- ken by the application of a small field within the easy plane. The system is described by the Hamiltonian,
H =
C
I JI
Si.Si+l+C
D. (Si A Si+l)-
where i labels the sites of the magnetic atoms on the chain. The convention which we use is that the direc- tion 6f the D.M. interaction D defines the Z axis, and
the direction of the field within the easy plane defines the X-axis.
The classical continuum limit of the model is de- rived, first by defining two interpenetrating sub-
T
where
as=
@+-.
2The small amplitude excitations of the Harniltonian (2.1) may be obtained by expressing the spin opera- tors in terms of an implane angle [5] &, and the Z component of the spin S;Z :
where
3
=d m .
The angle operator @; and theZ component of the spin S? are canonically conjugate.
operators,
[pi,
sf]
= i6ii. (2.6)C8 - 1588 JOURNAL DE PHYSIQUE
We shall express $; in terms of the deviations @; from the equilibrium values of the spins,
where a is the canting angle. The resulting Hamilto- nian is expanded in powers on the operators S: and
4i.
These operators are then expressed in terms of their Fourier Transforms, and in terms of boson creation and annihilation operatorswhere a~ is undetermined. The resulting Hamiltonian is then normal ordered. One finds that the terms in the Hamiltonian of first order in the deviation operators is
D
-2
I J I
6 4 .
[(sin a + ~ c o s a) xFig. I. - The singularity structure of the two spin wave
continuum, as evaluated in the harmonic approximation. The lower most line is the single spin wave energy, shown for comparison. The next line denotes the lower edge of the two spin wave continuum, and the topmost line represents the upper edge. The other lines represent structures occuring within the continuum.
x
((1-
cos ka) a$-
ak2)
113, one finds that the singularity at the lower band edge of the two spin wave continuum is washed out by the interaction, and the resulting spectral density + g p9
cos!? ~ ~ ~-
&
(.$
+ a;')}] (2.9) is transfered into the bound state of twospin waves2
K which has an energy below the bottom of the two spin
The canting angle a is choosen such that this term wave _continua. On retaining terms of higher order
vanishes. in 1/S, one obtains further two spin wave resonances
The resulting normal ordered quadratic form is di- bound states branches. In general, this procedure has
agonalized yielding to be performed numerically.
(2.10)
3. Discussion
where the spin wave dispersion relation contain cor- We have examined the low amplitude excitations of rections of all powers in 113. The form factor is deter- a discrete quantum spin chain that has a double sine- mined self consistently. To zeroth order, we have Gordon field theory as its classical continuum limit.
2
W K = The excitations have been calculated to lowest non- trivial order in the coupling constant. We find a quantum renormalization t o the energy of the n = 1
(I
Jl cos a-Ds-,)+
--
2 S breather, the single spin wave. We also examined the spectra corresponding to two spin waves. This is com- posed of a continuum of scattering states, and bound x { ( I J I cos a-Ds-,} ( 1
+
cos bo) states. It is these latter objects that correspond to the(2.11) n = 2 breathers of the double sine-Gordon system. The single spin wave self energy is calculated from the
equations of motion. Analogously to the case of the sine-Gordon systems, this spin wave is expected t o cor- respond t o the lowest amplitude breather excitation.
In the harmonic approximation, the two spin wave spectrum exhibits many singularities located a t the edge of the continua and lie within it, cf. figure 1. The effect of multiple interactions between spin waves is included by the Random Phase Approximation. The bound states and resonances can be obtained directly from the zero's of the determinant. To leading order in
[I] Condat, C. A., Guyer, R. A. and Miller, M. D.,
Phys.
Rev.
B 27 (1983) 474.[2] Bullough, R. K., Caudrey, P. J. and Gibbs, H. M., Solitons, Eds. R. K. Bullough and P. J. Caudrey (Springer, Berlin) 1980, p. 107.
[3] Xia, Q. and Riseborough, P. S., J. Appl. Phys. [4] Fairbairn, W . M., J. Phys. C 17 (1984) 2575.
[5] Villain, J., J . Phgs. Fmnce 35 (1974) 27.
[6] Maki, K. and Takayama, H., Phys.
