Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder
Using continued fractions we examined the density of states and thermodynamic properties of a few periodic nonuniform spin-1/2 XX chains in a transverse field. We considered the transverse spin dynamics in spin-1/2 XX chain with correlated Lorentzian disorder.
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Інститут фізики конденсованих систем НАН України
1999
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| Schriftenreihe: | Condensed Matter Physics |
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| Zitieren: | Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder / O. Derzhko, T. Krokhmalskii, O. Zaburannyi // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 339-344. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1203952017-06-13T03:03:18Z Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder Derzhko, O. Krokhmalskii, T. Zaburannyi, O. Using continued fractions we examined the density of states and thermodynamic properties of a few periodic nonuniform spin-1/2 XX chains in a transverse field. We considered the transverse spin dynamics in spin-1/2 XX chain with correlated Lorentzian disorder. Використовуючи неперервні дроби, ми дослідили щільність станів і термодинамічні властивості кількох періодично неоднорідних спін- 1/2 XX ланцюжків у поперечному полі. Ми розглянули поперечну спінову динаміку в спін-1/2 XX ланцюжку із скорельованим лоренцовим безладом. 1999 Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder / O. Derzhko, T. Krokhmalskii, O. Zaburannyi // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 339-344. — Бібліогр.: 7 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.339 PACS: 75.10.-b http://dspace.nbuv.gov.ua/handle/123456789/120395 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| description |
Using continued fractions we examined the density of states and thermodynamic properties of a few periodic nonuniform spin-1/2 XX chains in a
transverse field. We considered the transverse spin dynamics in spin-1/2
XX chain with correlated Lorentzian disorder. |
| author |
Derzhko, O. Krokhmalskii, T. Zaburannyi, O. |
| spellingShingle |
Derzhko, O. Krokhmalskii, T. Zaburannyi, O. Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder Condensed Matter Physics |
| author_facet |
Derzhko, O. Krokhmalskii, T. Zaburannyi, O. |
| author_sort |
Derzhko, O. |
| title |
Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder |
| title_short |
Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder |
| title_full |
Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder |
| title_fullStr |
Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder |
| title_full_unstemmed |
Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder |
| title_sort |
spin-1/2 xx chains in a transverse field with regular nonuniformity or correlated lorentzian disorder |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120395 |
| citation_txt |
Spin-1/2 XX chains in a transverse field with regular nonuniformity or correlated Lorentzian disorder / O. Derzhko, T. Krokhmalskii, O. Zaburannyi // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 339-344. — Бібліогр.: 7 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT derzhkoo spin12xxchainsinatransversefieldwithregularnonuniformityorcorrelatedlorentziandisorder AT krokhmalskiit spin12xxchainsinatransversefieldwithregularnonuniformityorcorrelatedlorentziandisorder AT zaburannyio spin12xxchainsinatransversefieldwithregularnonuniformityorcorrelatedlorentziandisorder |
| first_indexed |
2025-07-08T17:48:10Z |
| last_indexed |
2025-07-08T17:48:10Z |
| _version_ |
1837101910014820352 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 339–344
Spin-1/2 XX chains in a transverse field
with regular nonuniformity or
correlated Lorentzian disorder
O.Derzhko, T.Krokhmalskii, O.Zaburannyi
Institute for Condensed Matter Physics
of the National Academy of Sciences of Ukraine,
1 Svientsitskii Str., 290011 Lviv, Ukraine
Received June 11, 1998, in final form June 24, 1998
Using continued fractions we examined the density of states and thermo-
dynamic properties of a few periodic nonuniform spin-1/2 XX chains in a
transverse field. We considered the transverse spin dynamics in spin-1/2
XX chain with correlated Lorentzian disorder.
Key words: periodic nonuniform spin-1/2 XX chain, correlated Lorentzian
disorder, density of states, magnetization, specific heat, dynamic structure
factor
PACS: 75.10.-b
Since the pioneering paper by Lieb, Schultz and Mattis [1] statistical mechanics
calculations for spin- 1
2
XY chains have been the subject of long-lasting interest both
from fundamental and applied viewpoints. Our communication deals with some new
results for thermodynamics and dynamics of isotropic transverse XY chains with
regular nonuniformity or correlated Lorentzian disorder. The former model gener-
alizes the XX chain with the alternating coupling constant that was investigated
by some authors as a toy model to consider spin-Peierls phase transition [2-4]. The
latter model was studied in [5]. However the treatment in that paper was restricted
to thermodynamics in contrast to the present study dealing with dynamics of trans-
verse spin correlations. The basic tools to study these models are the Jordan-Wigner
method, the continued fractions, and the numerical approach for examining the spin
correlation dynamics developed recently [6].
Hereinafter we investigate a nonuniform XX chain in a magnetic field along the
z-axis consisting of N spins 1
2
wherein Hamiltonian is defined by
H =
∑
n
Ωns
z
n +
∑
n
Jn
(
sxns
x
n+1 + syns
y
n+1
)
c© O.Derzhko, T.Krokhmalskii, O.Zaburannyi 339
O.Derzhko, T.Krokhmalskii, O.Zaburannyi
=
∑
n
Ωn
(
s+n s
−
n − 1
2
)
+
∑
n
In
(
s+n s
−
n+1 + s−n s
+
n+1
)
. (1)
By the Jordan-Wigner transformation the Hamiltonian (1) can be represented as
the Hamiltonian of non-interacting spinless fermions
H =
∑
n
Ωn
(
c+n cn −
1
2
)
+
∑
n
In
(
c+n cn+1 − cnc
+
n+1
)
. (2)
To examine the thermodynamics of the model one must find the density of magnon
states ρ(E) that is related to the temperature double-time fermion Green functions
G∓
nm ≡ G∓
nm(E ± iǫ) according to the formula
ρ(E) = ∓ 1
πN
N
∑
n=1
ImG∓
nn. (3)
Using the equation of motion for G∓
nm it is a simple matter to show that
G∓
nn =
1
E ± iǫ− Ωn −∆−
n −∆+
n
,
∆−
n =
I2n−1
E ± iǫ− Ωn−1 − I2
n−2
E±iǫ−Ωn−2−...
,
∆+
n =
I2n
E ± iǫ− Ωn+1 − I2
n+1
E±iǫ−Ωn+2−...
. (4)
The continued fraction representation for the diagonal Green functions (4) is ex-
tremely useful for examining thermodynamics of regularly nonuniform chains when
periodic continued fractions emerge.
Consider for example a regular alternating chain Ω1I1Ω2I2Ω1I1Ω2I2 . . . when pe-
riodic continued fractions with period 2 emerge. As a result
ρ(E) =
{
0, E < b4, b3 < E < b2, b1 < E,
1
2π
|2E−Ω1−Ω2|√
B(E)
, b4 < E < b3, b2 < E < b1.
(5)
Here
B(E) = 4I21I
2
2 −
[
(E − Ω1) (E − Ω2)− I21 − I22
]2
= − (E − b1) (E − b2) (E − b3) (E − b4) ,
{bi} =
{
1
2
[
Ω1 + Ω2 ±
√
(Ω1 − Ω2)
2 + 4 (I1 ± I2)
2
]}
, (6)
and in (5), (6) it is implied that b1 > b2 > b3 > b4. In principle the calculation
of ρ(E) can be done for an arbitrary periodic chain, however, the calculations in
340
Spin-1/2 XX chains . . .
Figure 1. Density of states (a, b), transverse magnetization versus transverse
field at zero temperature β = ∞ (c, d) and temperature dependence of specific
heat (e, f) for nonuniform chain Ω1I1 . . .Ω12I12Ω1I1 . . .Ω12I12 . . . . Ω1 = Ω12 =
0.5 + Ω0, Ω2 = . . . = Ω11 = 1 + Ω0, J1 = . . . = J11 = 1, J12 = 0.5 (a, c, e);
Ω1 = . . . = Ω12 = 1 + Ω0, J1 = . . . = J6 = 1, J7 = . . . = J12 = 0.5 (b, d, f);
dotted curves correspond to uniform case.
the case of large periods become cumbersome. In figures 1a, 1b we plotted ρ(E) for
two chains with the period 12. The main result of introducing the regular nonuni-
formity is a spliting of the magnon band into subbands (a number of subbands
is equal to or less than the period of nonuniformity; compare figure 1a and fig-
ure 1b) that has important consequences in the thermodynamic properties of spin
model. For instance the low-temperature dependence of transverse magnetization
mz = −1
2
∫∞
−∞
dEρ(E) tanh βE
2
on transverse field is made up of sharply increasing
parts and horizontal parts, their number being determined by the period of nonuni-
formity (figures 1c, 1d). In figures 1e, 1f we plotted the temperature dependence
of specific heat c =
∫∞
−∞
dEρ(E)
(
βE
2
)2
/ cosh2 βE
2
that due to the introduced peri-
odic nonuniformity exhibits a two-peak structure, i.e. it has low-temperature and
high-temperature peaks.
Let us consider spin model (1) assuming that the exchange couplings Jn are
independent Lorentzian variables with distribution
p(Jn) =
1
π
Γ
(Jn − J0)
2 + Γ2
(7)
(J0 is the mean value of exchange coupling and Γ is the width of distribution that
controls the strength of disorder) and the transverse fields are determined by the
341
O.Derzhko, T.Krokhmalskii, O.Zaburannyi
Figure 2. Frequency dependence of Szz(κ, ω) (10) at κ = π
4 and κ = π
2 . 1 –
correlated disorder with a = −1.01; 2 – correlated disorder with a = 1.01; 3 –
independent exchange couplings and transverse fields; the latter have distribution
(9) with |a|Γ = 0.101; 4 – non-random case Γ = 0 (dashed curves).
neighbouring exchange couplings according to the formula
Ωn − Ω0 =
a
2
(Jn−1 + Jn − 2J0) . (8)
It can readily be checked that the distribution for the random variable Ωn reads
p(Ωn) =
1
π
|a|Γ
(Ωn − Ω0)
2 + (|a|Γ)2
. (9)
We shall be interested in calculation of the random-averaged dynamic structure
factor
Szz(κ, ω) =
∑
n
eiκn
∫ ∞
−∞
dte−ǫ|t|eiωt
[
〈szj(t)szj+n〉 − 〈szj〉〈szj+n〉
]
(10)
using for this purpose numerical approach. As shown in [6], to achieve this goal
it is necessary to solve N × N eigenvalue and eigenvector problem for the matrix
Anm = Ωnδnm + Jn
2
δm,n+1 +
Jn−1
2
δm,n−1, i.e.
N
∑
j=1
gkjAjm = Λkgkm,
N
∑
j=1
gkjgqj = δkq,
N
∑
k=1
gkjgkm = δjm. (11)
In our numerical calculations we considered chains of N = 300 spins with J0 = −1,
Ω0 = 0.5 and Γ = 0.1 at low temperature β = 1000. We computed correlation
functions 〈sz150(t)sz150+n〉 − 〈sz150〉〈sz150+n〉 with n = 0,±1, . . . ,±100 for the times up
to t = 200, put ǫ = 0.01 and averaged the zz dynamic structure factor (10) over
3000 random realizations to obtain the results presented in figure 2. We carefully
analyzed the accuracy of our calculations studying finite-size effects, the effects of
finite number of terms in the sum in (10) and of finite time cut-off in the integral in
(10), and the effects of finite number of random realizations.
Let us comment the results we have obtained for Szz(κ, ω). Figure 2 nicely shows
the difference in frequency shapes of the dynamic structure factor for correlated
342
Spin-1/2 XX chains . . .
Figure 3. Density of states for model (1), (7). a) – correlated disorder with
a = −1.01; b) – correlated disorder with a = 1.01; c) – independent exchange
couplings and transverse fields with |a|Γ = 0.101. The density of states for non-
random case is depicted by dashed curves.
disorder (7), (8) with different signs of a and for the case of independent random
exchange couplings and transverse fields with distributions (7) and (9), respectively.
The transverse dynamic structure factor is determined by two magnon excitations
and these spectacular changes in the frequency dependence are caused by the changes
in the density of magnon states depicted in figure 3.
To summarize, we applied continued fractions to study rigorously the thermody-
namic properties of periodic nonuniform spin- 1
2
XX chain in a transverse field and
extended a previous analysis of the spin- 1
2
XX chain with correlated Lorentzian
disorder examining numerically the dynamics of transverse spin correlations. The
theoretical results observed in our study should prove valuable in understanding
the experimental data for XX chain materials the synthesis of which is becoming a
reality [7].
A study of periodic nonuniform spin- 1
2
XX chains was inspired by the papers of J.
Freericks and R. Lemański presented at the 22nd Seminar of the Middle European
Cooperation in Statistical Physics (Szklarska Porȩba, 1997). The authors thank
R. Lemański for useful correspondence. O. D. is indebted to Mr. Joseph Kocowsky
for continuous financial support.
References
1. Lieb E., Schultz T., Mattis D. Two soluble models of an antiferromagnetic chain. //
Ann. Phys. (N.Y.), 1961, vol. 16, p. 407–466.
2. Pincus P. Instability of the uniform antiferromagnetic chain. // Solid State Commun.,
1971, vol. 9, p. 1971–1973.
3. Lima R.A.T., Tsallis C. Magnetic field influence on the spin–Peierls instability in the
quasi–one–dimensional magnetostrictive XY model: Thermodynamical properties. //
Phys. Rev. B, 1983, vol. 27, No. 11, p. 6896–6915.
4. Okamoto K. Alternating s = 1
2 XY chain in the Lorentzian random field. // J. Phys.
Soc. Jpn., 1990, vol. 59, No. 12, p. 4286–4296.
343
O.Derzhko, T.Krokhmalskii, O.Zaburannyi
5. Derzhko O., Richter J. Solvable model of random spin- 12 XY chain. // Phys. Rev. B,
1997, vol. 55, No. 21, p. 14298–14310.
6. Derzhko O., Krokhmalskii T. Numerical approach for a study of the spin- 12 XY chains
dynamic properties. // Phys. Status Solidi B, 1998, vol. 208, No. 1, p. 221–248.
7. Collins M.F., Petrenko O.A. Triangular antiferromagnets. Preprint, cond-
mat/9706153, 1997.
Спін-1/2 XX ланцюжки в поперечному полі з
регулярною неоднорідністю або із скорельованим
лоренцовим безладом
О.Держко, Т.Крохмальський, О.Забуранний
Інститут фізики конденсованих систем НАН Укpаїни,
290011 Львів, вул. Свєнціцького, 1
Отримано 11 червня 1998 р., в остаточному вигляді –
24 червня 1998 р.
Використовуючи неперервні дроби, ми дослідили щільність станів і
термодинамічні властивості кількох періодично неоднорідних спін-
1/2 XX ланцюжків у поперечному полі. Ми розглянули поперечну
спінову динаміку в спін-1/2 XX ланцюжку із скорельованим лоренцо-
вим безладом.
Ключові слова: періодично неоднорідний спін-1/2 XX ланцюжок,
скорельований лоренців безлад, щільність станів, намагніченість,
теплоємність, динамічний структурний фактор
PACS: 75.10.-b
344
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