Dynamics of alternating spin chains∗)
Oleg Derzhko, Taras Krokhmalskii
Institute for Condensed Matter Physics, 1 Svientsitskii Street, L’viv–11, 79011, Ukraine
Institut f¨ ur Physik, Universit¨ at Dortmund, 44221 Dortmund, Germany
Received 8 October 2001
We consider the dimerised spin-1
the xx and zz dynamic structure factors for different values of the Hamiltonian parameters
2isotropic XY chain in a transverse field and study
Key words: dimerised XY chain, dynamic structure factor
Because of progress in material sciences a relatively large number of magnetic
compounds which can be modeled by regularly alternating quantum spin chains
has become available. Neutron scattering, EPR, NMR etc. are the basic experi-
mental techniques used to study the properties of such materials and the interpre-
tation of experimental data requires corresponding theoretical predictions for the
dynamic properties of regularly alternating quantum spin chains. Although usu-
ally the Heisenberg model is used to model the magnetic degrees of freedom for
most of the compounds, a simpler case to study is the spin-1
for which a rigorous analysis of many statistical mechanics properties is feasible.
In this paper we present the results for the xx dynamic structure factor obtained
numerically as well as for the zz dynamic structure factor obtained analytically
for the dimerised spin-1
results may be viewed as a further extension of the recent study on uniform chains
, whereas the analytical ones extend those reported in  to the case of nonzero
We consider N (→ ∞) spins1
in which two different exchange interactions J(1 + δ) and J(1 − δ) (0 ≤ δ ≤ 1 is
the dimerisation parameter) come into play alternatively. Besides, Ω is the value
of the (transverse) magnetic field directed along the z axis. Our goal is to calculate
the dynamic structure factors
2isotropic XY model
2isotropic XY chain in a transverse field. The numerical
2described by the Hamiltonian
H = Ω
J (1 − (−1)nδ)?sx
∗) Presented at 11th Czech and Slovak Conference on Magnetism, Koˇ sice, 20–23 August 2001
Czechoslovak Journal of Physics, Vol. 52 (2002), No. 2
O. Derzhko, T. Krokhmalskii, and J. Stolze
For this purpose we first calculate the time-dependent αα spin correlation functions.
The (numerical) calculation of the correlation function ?sx
in detail in . It is based on the Jordan–Wigner fermionization and on the numer-
ical computation of Pfaffians. In the present study we consider chains of 400 spins
with J = −1, δ = 0,...,0.2 and Ω = 0,...,1 at temperatures β = 20 and higher.
We put j = 32(33), j = 41(42) and take n up to 50 for times t up to 200. We do the
integral over t in (2) multiplying the integrand by e−?|t|typically with ? = 0.001.
We performed many test calculations similar to those described in  and compared
our numerical findings with the exact results available at infinite temperature β = 0
 in order to make sure that our numerical results for Sxx(κ,ω) pertain to infinite
The time-dependent zz spin correlation function can be found analytically for
any set of values of the Hamiltonian parameters and temperature following the
standard derivation of [2–4]. First we perform in the Hamiltonian (1) the Jordan–
Wigner transformation from spin to Fermi operators. Then we perform the Fourier
and the Bogolyubov transformations bringing the fermionic Hamiltonian into di-
agonal form. Using the Wick–Bloch–de Dominicis theorem we obtain the explicit
expression for the time-dependent zz spin correlation function. Finally we perform
the Fourier transformations with respect to time and space (see (2)) to end up with
the following result:
j+n? is explained
+(uκ?vκ?−κ−uκ?−κvκ?)2nκ? (1−nκ?−κ+π)δ (ω+λκ?−λκ?−κ+π)
(thermodynamic limit is implied). Here
(uκ?uκ?−κ+vκ?vκ?−κ)2nκ? (1 − nκ?−κ)δ (ω+λκ?−λκ?−κ)
cos2κ + δ2sin2κ, nκ=
eβΛκ+ 1, Λκ= Ω + λκ, λκ= sgn(cosκ)J?κ.
Formulas (3), (4) generalize the zero-temperature result given in .
In Figs. 1, 2 and Fig. 3 we present some of our findings for the zz and xx dy-
namic structure factors for dimerisation δ = 0.1 at different values of field and
temperature. We start our discussion recalling what is long known , i.e., consid-
ering Szz(κ,ω) at zero temperature (Figs. 1a, 2a). Generally speaking, Szz(κ,ω) is
determined by two-fermion excitations (modes) which form two (upper and lower)
continua. If Ω < δ only the upper continuum contributes to Szz(κ,ω) (Fig. 1a),
if Ω > δ both continua contribute to Szz(κ,ω) (Fig. 2a). The two-particle char-
acter of Szz(κ,ω) leads to a high-frequency cutoff in the (κ,ω)-plane. Apart from
boundary singularities shown in  for β = ∞, Szz(κ,ω) is almost structureless.
Figures 1a,b,c (Ω = 0) show how the intensity of Szz(κ,ω) in the upper continuum
redistributes as the temperature increases. Moreover, the lower continuum which is
absent at zero temperature arises as the temperature increases. In Figs. 2a,b,c the
Czech. J. Phys. 52 (2002)
Dynamics of alternating spin chains
Fig. 1. The zz dynamic structure factor of the dimerised XY chain (J = −1, δ = 0.1)
without field (Ω = 0) at different temperatures β = ∞ (a), β = 20 (b), β = 1 (c).
Fig. 2. The same as in Fig. 1 at presence of the field Ω = 0.5.
temperature effects may be followed for Ω > δ. As the temperature increases some
sharp continuum boundaries disappear and become washed-out, whereas the upper
and the central lower boundaries of the upper continuum and the upper boundary
of the lower continuum persist. In the high-temperature limit Szz(κ,ω) becomes
field independent (compare Figs. 1c and 2c). The observed behaviour of the zz
dynamic structure factor can be understood on the basis of the exact analytical
result for Szz(κ,ω) (3), (4).
Let us pass to the xx dynamic structure factor. Comparing Figs. 3b and 2b one
immediately notes that similarly to the uniform case δ = 0  Sxx(κ,ω) at low
temperatures exhibits washed-out excitation branches which follow the boundaries
of two-fermion excitation continua. Thus, the two-fermion modes give an impor-
tant contribution to Sxx(κ,ω). Obviously, like in the uniform case  Sxx(κ,ω) at
low temperatures exhibits also nonzero values outside the two-fermion continua,
and, due to its many-particle character, no high-frequency cutoff. In the high-
temperature limit only the autocorrelation functions (with known time dependence
Czech. J. Phys. 52 (2002)
O. Derzhko et al.: Dynamics of alternating spin chains
Fig. 3. The xx dynamic structure factor of the dimerised XY chain (J = −1, δ = 0.1),
Ω = 0.1, β = 20 (a), Ω = 0.5, β = 20 (b), Ω = 0.5, β = 1 (c).
) contribute to Sxx(κ,ω). That explains the almost κ-independent pattern seen
in Fig. 3c.
Finally, let us comment to what extent the results obtained may refer to a
more realistic spin-1
fermionization with the Hartree–Fock approximation for interacting fermions one
can analyze in a similar manner the dynamic properties of the dimerised Heisenberg
chain. Note, however, that the two-fermion bound state outside the continuum
observed in the latter model  is probably difficult to obtain within such a kind
of simplified consideration .
To summarize, we have reported the first results for xx and zz dynamic structure
factors at finite temperatures for the dimerised spin-1
transverse field. We have compared those dynamic structure factors emphasizing
similarities and differences.
2isotropic Heisenberg chain. Combining the Jordan–Wigner
2isotropic XY chain in a
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