Page 1

Dynamics of alternating spin chains∗)

Oleg Derzhko, Taras Krokhmalskii

Institute for Condensed Matter Physics, 1 Svientsitskii Street, L’viv–11, 79011, Ukraine

Joachim Stolze

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

and temperature.

2isotropic XY chain in a transverse field and study

PACS: 75.10.-b

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

[1], whereas the analytical ones extend those reported in [2] to the case of nonzero

temperatures.

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

n

sz

n+

?

n

J (1 − (−1)nδ)?sx

nsx

n+1+ sy

nsy

n+1

?

(1)

Sαα(κ,ω) =1

N

?

j

?

n

eiκn

?∞

−∞

dteiωt?sα

j(t)sα

j+n?.

(2)

∗) Presented at 11th Czech and Slovak Conference on Magnetism, Koˇ sice, 20–23 August 2001

Czechoslovak Journal of Physics, Vol. 52 (2002), No. 2

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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 [1]. 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 [1] and compared

our numerical findings with the exact results available at infinite temperature β = 0

[3] in order to make sure that our numerical results for Sxx(κ,ω) pertain to infinite

systems.

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(t)sx

j+n? is explained

Szz(κ,ω) =

?π

−π

dκ??

+(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)

?

(3)

uκ=

1

√2

?

1 +|cosκ|

?κ

,vκ= sgn(sin(2κ))1

√2

?

1 −|cosκ|

?κ

,

?κ=

?

cos2κ + δ2sin2κ, nκ=

1

eβΛκ+ 1, Λκ= Ω + λκ, λκ= sgn(cosκ)J?κ.

(4)

Formulas (3), (4) generalize the zero-temperature result given in [2].

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 [2], 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 [2] 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

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Czech. J. Phys. 52 (2002)

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Dynamics of alternating spin chains

0123

0

1

2

κ

ω

123

κ

123

κ

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).

0123

0

1

2

κ

ω

123

κ

123

κ

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 [1] 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 [1] 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)

323

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O. Derzhko et al.: Dynamics of alternating spin chains

0123

0

1

2

κ

ω

123

κ

123

κ

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).

[3]) 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 [5] is probably difficult to obtain within such a kind

of simplified consideration [6].

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

References

[1] O. Derzhko, T. Krokhmalskii, and J. Stolze: J. Phys. A 33 (2000) 3063.

[2] J. H. Taylor and G. M¨ uller: Physica A 130 (1985) 1.

[3] J. H. H. Perk and H. W. Capel: Physica A 100 (1980) 1 (and references therein).

[4] K. Kawasaki, N. Maya, A. Kouzuki, and K. Nakamura: J. Phys. Soc. Jpn. 66 (1997)

839.

[5] G. S. Uhrig and H. J. Schulz: Phys. Rev B 54 (1996) R9624; 58 (1998) 2900.

W. Yu and S. Haas: Phys. Rev. B 62 (2000) 344.

[6] A. Kouzuki, K. Kawasaki, and K. Nakamura: Phys. Rev. B 60 (1999) 12874.

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Czech. J. Phys. 52 (2002)