We analyze the Drude weight for both spin and thermal transport of
one-dimensional spin-1/2 systems by means of exact diagonalization at finite
temperatures. While the Drude weights are non-zero for finite systems, we find
indications of a vanishing of the Drude weights in the thermodynamic limit for
non-integrable models implying normal transport behavior.Comment: 2 pages, 1 figure. Proceedings of the ICM 2003, Rom

A. Honecker

Alvarez

Bonča

Castella

D.C. Cabra

F. Heidrich-Meisner

Fujimoto

Heidrich-Meisner

Klümper

Orignac

Rosch

Saito

Shastry

W. Brenig

Zotos

English

arXiv

We analyze the Drude weight for both spin and thermal transport of one-dimensional spin-12 systems by means of exact diagonalization at finite temperatures. While the Drude weights are non-zero for finite systems, we find indications of a vanishing of the Drude weights in the thermodynamic limit for non-integrable models implying normal transport behavior.Fil: Heidrich Meisner, F.. Technische Universitat Carolo Wilhelmina Zu Braunschweig.; AlemaniaFil: Honecker, A.. Technische Universitat Carolo Wilhelmina Zu Braunschweig.; AlemaniaFil: Cabra, Daniel Carlos. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Física La Plata. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Física La Plata; Argentina. Ecole Normale Supérieure de Lyon; FranciaFil: Brenig, W.. Technische Universitat Carolo Wilhelmina Zu Braunschweig.; Alemani

Heidrich Meisner, F.

Honecker, A.

Cabra, Daniel Carlos

Brenig, W.

CONICET Digital

Journal of Magnetism and Magnetic Materials

Transport in dimerized and frustrated spin systems

We analyze the Drude weight for both spin and thermal transport of one-dimensional spin-12 systems by means of exact diagonalization at finite temperatures. While the Drude weights are non-zero for finite systems, we find indications of a vanishing of the Drude weights in the thermodynamic limit for non-integrable models implying normal transport behavior.Instituto de Física La Plat

Heidrich-Meisner, F.

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Transport in dimerized and frustrated spin systems
F. Heidrich-Meisnera∗, A. Honeckera, D.C. Cabrab and W. Breniga
aInstitut fu¨r Theoretische Physik, Technische Universita¨t Braunschweig,
Mendelssohnstrasse 3, 38106 Braunschweig, Germany
bLaboratoire de Physique, Ecole Normale Superieure de Lyon,
46, Alle´e d’Italie, 69364 Lyon, France
We analyze the Drude weight for both spin and thermal transport of one-dimensional spin-1/2 systems by
means of exact diagonalization at finite temperatures. While the Drude weights are non-zero for finite systems,
we find indications of a vanishing of the Drude weights in the thermodynamic limit for non-integrable models
implying normal transport behavior.
Transport properties of one-dimensional spin-
1/2 systems have recently attracted strong in-
terest from the theoretical side (see e.g. [1,2,3,
4,5,6] and references therein). An intensively
studied issue is the question under which con-
ditions ballistic transport occurs at zero fre-
quency and finite temperature T > 0, charac-
terized by a non-zero Drude weight D. This
quantity is the zero-frequency weight of the
real part of the conductivity σ(ω), namely
Reσ(ω) = D(T )δ(ω) + regular part. While it
is known that the anisotropic Heisenberg chain
has a finite Drude weight for thermal transport
since the energy-current operator is a conserved
quantity[7], there is a controversial discussion[2,
3,4,5] whether transport remains ballistic if the
model is extended by dimerization or frustration.
In this paper, we will focus on two examples,
namely thermal transport of dimerized chains and
spin transport in frustrated chains. We consider
the Hamiltonian (S = 1/2)
H =
N∑
l=1
hl = J
N∑
l=1
[λl ~Sl · ~Sl+1 + α ~Sl · ~Sl+2]. (1)
Here, N is the number of sites and hl is the lo-
cal energy density. We set λl = 1 if l is even
and λl = λ otherwise. Note that we use periodic
∗Corresponding author. Tel.: +49-531-391-5184. E-mail:
f.heidrich-meisner@tu-bs.de.
boundary conditions.
The thermal Drude weight Dth(T ) can be ob-
tained from[7] (pn = e
−En/T /Z;Z =
∑
n e
−En/T )
Dth(T ) =
π
N T 2
∑
m,n
Em=En
pn |〈m|jth|n〉|
2 (2)
while the Drude weight Ds(T ) for spin transport
follows from[8,9]
Ds(T ) =
π
N

〈−Tˆ 〉 − 2
∑
m,n
Em 6=En
pn
|〈m|js|n〉|
2
Em − En

 . (3)
jth and js denote the energy- and spin-current
operator, respectively. They obey equations of
continuity: i
[
H,hl[S
z
l ]
]
= −(jth[s],l+1 − jth[s],l)
with jth[s] =
∑
l jth[s],l. The local energy den-
sity hl is defined in Eq. (1) and S
z
l is the local
magnetization density. We refer the reader to
Refs. [3,10] for full expressions of jth[s]. Both
current operators do not commute with H as
soon as λ 6= 1 or α 6= 0. The operator Tˆ =
J
∑
l[λl S
+
l S
−
l+1+4αS
+
l S
−
l+2+H.c.] is the kinetic
energy[8,10].
Now we turn to the discussion of our numeri-
cal results for the thermal Drude weight Dth(T )
of dimerized chains which we obtain by complete
diagonalization of H . The data are shown in
the main panel Fig. 1(a) for N = 8, . . . , 16 sites.
Since finite-size effects are strongest at low tem-
1
2peratures, we concentrate on T > J . Here, the
leading contributions to Dth(T ) take the form
Dth(T ) = Cth,1/T
2 + Cth,2/T
3 + . . . . (4)
While we compute Cth,1 directly by setting Z =
2N and e−βEn = 1 in Eq. (2), Cth,2 is extracted
by a fit of Eq. (4) to the numerical data at high
T . As can be seen in the inset of Fig. 1(a), both
coefficients exhibit a significant decrease with sys-
tem size though not monotonically. The system
sizes are too small to draw conclusions about the
actual finite-size dependence of Dth, but the ob-
served behavior does not support the notion of
a finite Drude weight for N → ∞ contrasting a
recent claim of Ref. [2].
The second example is spin transport of frus-
trated chains in the gapped regime. In the main
panel of Fig. 1(b), we show Ds(T ) for α = 0.35
and even N ≤ 18. The main features are a fi-
nite Drude weight at T = 0 and a monotonic
decrease of Ds(T ) for T > 0.5J . Ds(T = 0) > 0
has also been reported in Ref. [10] for α < 0.5
and N ≤ 20. Extension to larger systems at
T = 0 (e.g. with Lanczos techniques) should clar-
ify whether Ds(T = 0) > 0 is a property that
survives in the thermodynamic limit N →∞.
We further concentrate on the analysis at large
T , namely the high-temperature prefactor Cs =
limT→∞[T · Ds(T )]. Its finite-size dependence is
illustrated in the inset of Fig. 1(b) for α = 0.35, 1
and N = 8, 9, . . . , 18. Apart from odd-even
finite-size effects, which are more pronounced for
α = 0.35, both data sets exhibit a monotonic de-
crease of Cs with system size, as it is particu-
larly obvious in the case of α = 1. This provides
strong evidence for Ds → 0 for N → ∞ in the
gapped regime of frustrated chains which is con-
sistent with Ref. [11].
In summary, we have numerically studied the
thermal Drude weight of dimerized chains and the
spin Drude weight of frustrated chains at T > 0.
The finite-size analysis supports the conclusion of
a zero Drude weight in the thermodynamic limit
in these examples. We have found analogous re-
sults for both kinds of transport for the models
discussed here and spin ladders which will be re-
ported in detail elsewhere[12].
0 0.2 0.4 0.6 0.8 1 1.2
T [J]
0
0.1
0.2
0.3
0.4
D
s [
J]
0 0.05 0.11/N
0
0.5
1
C
s(N
)/C
s(8
) α=0.35
α=1
0
0.1
0.2
0.3
D
th
 [J
2 ]
N=8
N=10
N=12
N=14
N=16
0 0.05 0.1
1/N
0
0.02
0.04
C
th
,i [
J4
] Cth,1
Cth,2
N=8,10,...,18
(a)
λ=0.5,α=0
(b) λ=1,α=0.35
Figure 1. (a) Main panel: Thermal Drude weight
Dth(T ) of a dimerized chain with λ = 0.5 (see Eq.
(1)). Inset: Cth,i, i = 1, 2 vs. 1/N (see text). (b)
Main panel: Drude weight Ds(T ) of a frustrated
chain with α = 0.35. Arrows indicate increasing
N . Inset: Cs vs. 1/N for N = 8, 9, . . . , 18.
REFERENCES
1. A. Klu¨mper and K. Sakai, J. Phys. A 35
(2002) 2173.
2. J. V. Alvarez and C. Gros, Phys. Rev. Lett.
89 (2002) 156603.
3. F. Heidrich-Meisner et al., Phys. Rev. B 66
(2002) 140406(R); Phys. Rev. Lett 92 (2004)
069703.
4. K. Saito, Phys. Rev. B 67, (2003) 064410.
5. E. Orignac et al., Phys. Rev. B 67 (2003)
134426.
6. S. Fujimoto and N. Kawakami, Phys. Rev.
Lett. 90 (2003) 197202.
7. X. Zotos et al., Phys. Rev. B 55 (1997) 11029.
8. B.S. Shastry and B. Sutherland, Phys. Rev.
Lett.˙65 (1990) 243.
9. H. Castella et al., Phys. Rev. Lett.˙74 (1995)
972.
10. J. Boncˇa et al., Phys. Rev. B 50 (1994) 3415.
11. A. Rosch and N. Andrei, Phys. Rev. Lett. 85
(2000) 1092.
12. F. Heidrich-Meisner et al., Phys. Rev. B 68
(2003) 134436.


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arXiv:cond-mat/0406362v1  [cond-mat.str-el]  16 Jun 2004Transport in dimerized and frustrated spin systemsF. Heidrich-Meisnera∗, A. Honeckera, D.C. Cabrab and W. BrenigaaInstitut für Theoretische Physik, Technische Universität Braunschweig,Mendelssohnstrasse 3, 38106 Braunschweig, GermanybLaboratoire de Physique, Ecole Normale Superieure de Lyon,46, Allée d’Italie, 69364 Lyon, FranceWe analyze the Drude weight for both spin and thermal transport of one-dimensional spin-1/2 systems bymeans of exact diagonalization at finite temperatures. While the Drude weights are non-zero for finite systems,we find indications of a vanishing of the Drude weights in the thermodynamic limit for non-integrable modelsimplying normal transport behavior.Transport properties of one-dimensional spin-1/2 systems have recently attracted strong in-terest from the theoretical side (see e.g. [1,2,3,4,5,6] and references therein). An intensivelystudied issue is the question under which con-ditions ballistic transport occurs at zero fre-quency and finite temperature T > 0, charac-terized by a non-zero Drude weight D. Thisquantity is the zero-frequency weight of thereal part of the conductivity σ(ω), namelyReσ(ω) = D(T )δ(ω) + regular part. While itis known that the anisotropic Heisenberg chainhas a finite Drude weight for thermal transportsince the energy-current operator is a conservedquantity[7], there is a controversial discussion[2,3,4,5] whether transport remains ballistic if themodel is extended by dimerization or frustration.In this paper, we will focus on two examples,namely thermal transport of dimerized chains andspin transport in frustrated chains. We considerthe Hamiltonian (S = 1/2)H =N∑l=1hl = JN∑l=1[λl ~Sl · ~Sl+1 + α ~Sl · ~Sl+2]. (1)Here, N is the number of sites and hl is the lo-cal energy density. We set λl = 1 if l is evenand λl = λ otherwise. Note that we use periodic∗Corresponding author. Tel.: +49-531-391-5184. E-mail:f.heidrich-meisner@tu-bs.de.boundary conditions.The thermal Drude weight Dth(T ) can be ob-tained from[7] (pn = e−En/T /Z; Z =∑n e−En/T )Dth(T ) =πN T 2∑m,nEm=Enpn |〈m|jth|n〉|2 (2)while the Drude weight Ds(T ) for spin transportfollows from[8,9]Ds(T ) =πN〈−T̂ 〉 − 2∑m,nEm 6=Enpn|〈m|js|n〉|2Em − En . (3)jth and js denote the energy- and spin-currentoperator, respectively. They obey equations ofcontinuity: i[H, hl[Szl ]]= −(jth[s],l+1 − jth[s],l)with jth[s] =∑l jth[s],l. The local energy den-sity hl is defined in Eq. (1) and Szl is the localmagnetization density. We refer the reader toRefs. [3,10] for full expressions of jth[s]. Bothcurrent operators do not commute with H assoon as λ 6= 1 or α 6= 0. The operator T̂ =J∑l[λl S+l S−l+1 +4αS+l S−l+2 +H.c.] is the kineticenergy[8,10].Now we turn to the discussion of our numeri-cal results for the thermal Drude weight Dth(T )of dimerized chains which we obtain by completediagonalization of H . The data are shown inthe main panel Fig. 1(a) for N = 8, . . . , 16 sites.Since finite-size effects are strongest at low tem-12peratures, we concentrate on T > J . Here, theleading contributions to Dth(T ) take the formDth(T ) = Cth,1/T2 + Cth,2/T3 + . . . . (4)While we compute Cth,1 directly by setting Z =2N and e−βEn = 1 in Eq. (2), Cth,2 is extractedby a fit of Eq. (4) to the numerical data at highT . As can be seen in the inset of Fig. 1(a), bothcoefficients exhibit a significant decrease with sys-tem size though not monotonically. The systemsizes are too small to draw conclusions about theactual finite-size dependence of Dth, but the ob-served behavior does not support the notion ofa finite Drude weight for N → ∞ contrasting arecent claim of Ref. [2].The second example is spin transport of frus-trated chains in the gapped regime. In the mainpanel of Fig. 1(b), we show Ds(T ) for α = 0.35and even N ≤ 18. The main features are a fi-nite Drude weight at T = 0 and a monotonicdecrease of Ds(T ) for T > 0.5J . Ds(T = 0) > 0has also been reported in Ref. [10] for α < 0.5and N ≤ 20. Extension to larger systems atT = 0 (e.g. with Lanczos techniques) should clar-ify whether Ds(T = 0) > 0 is a property thatsurvives in the thermodynamic limit N → ∞.We further concentrate on the analysis at largeT , namely the high-temperature prefactor Cs =limT→∞[T · Ds(T )]. Its finite-size dependence isillustrated in the inset of Fig. 1(b) for α = 0.35, 1and N = 8, 9, . . . , 18. Apart from odd-evenfinite-size effects, which are more pronounced forα = 0.35, both data sets exhibit a monotonic de-crease of Cs with system size, as it is particu-larly obvious in the case of α = 1. This providesstrong evidence for Ds → 0 for N → ∞ in thegapped regime of frustrated chains which is con-sistent with Ref. [11].In summary, we have numerically studied thethermal Drude weight of dimerized chains and thespin Drude weight of frustrated chains at T > 0.The finite-size analysis supports the conclusion ofa zero Drude weight in the thermodynamic limitin these examples. We have found analogous re-sults for both kinds of transport for the modelsdiscussed here and spin ladders which will be re-ported in detail elsewhere[12].0 0.2 0.4 0.6 0.8 1 1.2T [J]00.10.20.30.4Ds [J]0 0.05 0.11/N00.51Cs(N)/Cs(8)α=0.35α=100.10.20.3Dth [J2]N=8N=10N=12N=14N=160 0.05 0.11/N00.020.04Cth,i [J4 ]Cth,1Cth,2N=8,10,...,18(a)λ=0.5,α=0(b) λ=1,α=0.35Figure 1. (a) Main panel: Thermal Drude weightDth(T ) of a dimerized chain with λ = 0.5 (see Eq.(1)). Inset: Cth,i, i = 1, 2 vs. 1/N (see text). (b)Main panel: Drude weight Ds(T ) of a frustratedchain with α = 0.35. Arrows indicate increasingN . Inset: Cs vs. 1/N for N = 8, 9, . . . , 18.REFERENCES1. A. Klümper and K. Sakai, J. Phys. A 35(2002) 2173.2. J. V. Alvarez and C. Gros, Phys. Rev. Lett.89 (2002) 156603.3. F. Heidrich-Meisner et al., Phys. Rev. B 66(2002) 140406(R); Phys. Rev. Lett 92 (2004)069703.4. K. Saito, Phys. Rev. B 67, (2003) 064410.5. E. Orignac et al., Phys. Rev. B 67 (2003)134426.6. S. Fujimoto and N. Kawakami, Phys. Rev.Lett. 90 (2003) 197202.7. X. Zotos et al., Phys. Rev. B 55 (1997) 11029.8. B.S. Shastry and B. Sutherland, Phys. Rev.Lett.˙65 (1990) 243.9. H. Castella et al., Phys. Rev. Lett.˙74 (1995)972.10. J. Bonča et al., Phys. Rev. B 50 (1994) 3415.11. A. Rosch and N. Andrei, Phys. Rev. Lett. 85(2000) 1092.12. F. Heidrich-Meisner et al., Phys. Rev. B 68(2003) 134436.

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