In a recent letter [Phys. Rev. Lett. 89, 156603 (2002); cond-mat/0201300],
Alvarez and Gros have numerically analyzed the Drude weight for thermal
transport in spin ladders and frustrated chains of up to 14 sites and have
proposed that it remains finite in the thermodynamic limit. In this comment, we
argue that this conclusion cannot be sustained if the finite-size analysis is
taken to larger system sizes.Comment: One page REVTeX4, 1 figure. Published version (minor changes

Brenig, W.

Cabra, D. C.

Heidrich-Meisner, F.

Honecker, A.

Physical Review Letters

English

arXiv

A Comment on the Letter by J.V. Alvarez and Claudius Gros, Phys. Rev. Lett. 89, 156603 (2002). The authors of the Letter offer a Reply.Instituto de Física La Plat

Cabra, Daniel Carlos

SEDICI - Repositorio de la UNLP

ar
X
iv
:c
on
d-
m
at
/0
30
23
58
v2
  [
co
nd
-m
at
.st
r-
el
]  
12
 F
eb
 2
00
4
1
Comment on “Anomalous Thermal Conduc-
tivity of Frustrated Heisenberg Spin Chains and
Ladders”
In a recent Letter [1], Alvarez and Gros have presented
a numerical study of the thermal conductivity of frus-
trated chains and spin ladders with spin 1/2. Using exact
diagonalization of finite systems with N ≤ 14 sites, they
have computed the zero-frequency weight κ(th)(T,N),
i.e., the Drude weight, of the thermal conductivity where
T is the temperature and N the number of sites. One
of their main conclusions is that the numerical data in-
dicate a finite value of κ(th) in the thermodynamic limit
(N → ∞) for spin ladders and frustrated chains. In
the latter case, this conclusion is based on a finite-size
analysis of the high-temperature residue C(N), given by
limT→∞[T
2κ(th)(T,N)] = C(N), for α = 0.1, 0.24, 0.35
(see Refs. [1, 2] for definitions; also note [3]).
In this Comment, we argue that, from the systems
investigated in [1], no conclusions of a finite thermal
Drude weight in the gapped regime of frustrated chains
for N → ∞ can be drawn. This will be corroborated
by supplementary data for systems up to N = 18 sites.
In Figs. 1(a) and 1(b), we show the size dependence
of C(N) for 8 ≤ N ≤ 18 and α = 0.35, 0.5, 1 where
C(N)/C(N = 8) is plotted versus 1/N . Figure 1(b) is
a log-log display of Fig. 1(a). An overall and monotonic
decrease of C(N) is evident, consistent with a vanishing
thermal Drude weight for T ≫ J and N → ∞. In-
deed, the curvature of the curves in the log-log plot of
C(N)/C(N = 8) versus 1/N suggests that C(N) van-
ishes more rapidly than any power of 1/N as N → ∞.
Most important, for the case of α = 0.35 and 8 ≤ N ≤ 14
studied in [1], the overall decrease of C(N) remains
clearly observable, except for minor finite-size oscilla-
tions, which, however, justify no extrapolation to a finite
value for N →∞.
In their Letter, Alvarez and Gros have also argued that
the behavior of κ(th)(T,N) for α = 0.35 at low temper-
atures supports the conclusion of a finite thermal Drude
weight in the thermodynamic limit. Indeed, there is a
crossover temperature T ∗(N∗) [4] where the monotonic
decrease of κ(th)(T,N) with increasing system size ob-
served at high temperatures changes to a monotonic in-
crease of κ(th)(T,N) with system size (see Fig. 3 in Ref.
[2]). However, as shown in Fig. 1(c), T ∗(N∗) decreases
with increasing system size already for N∗ ≥ 11, i.e., in-
cluding systems studied in [1], and could well extrapolate
to zero for N∗ → ∞. In any case, finite-size effects for
T . 0.1J are far too large even at N = 18 to allow for
any reliable predictions regarding κ(th)(T ) in the ther-
modynamic limit.
Summarizing our numerical analysis of gapped, frus-
trated chains, both in the high- and the low-temperature
regime we find no evidence in favor of a finite thermal
Drude weight in the thermodynamic limit in contrast to
Ref. [1].
Finally, for spin ladders Alvarez and Gros have used
only the size dependence of κ(th)(T,N) at low tempera-
tures to conjecture a finite κ(th)(N →∞). In view of the
results for the frustrated chain presented in this Com-
ment, we suggest to perform a finite-size analysis of the
high-temperature residue C(N) to substantiate this con-
jecture. We believe that this is important for the analysis
of the thermal conductivity in the spin ladder material
La5Ca9Cu24O41[5].
We acknowledge financial support by the DFG and a
DAAD-ANTORCHAS exchange program.
0 0.05 0.1
1/N
0
0.2
0.4
0.6
0.8
1
C
(N
)/C
(N
=8
)
α=0.35
α=0.5
α=1
0.06 0.08 0.1 0.12
1/N
1
0.2
0.1
0.6
0.4
C
(N
)/C
(N
=8
)
0 0.02 0.04 0.06 0.08 0.1 0.12
1/N
0.05
0.1
0.15
0.2
0.25
0.30
T*
[J
]α=0.35
(a) (b)
(c)
N=14
FIG. 1: (a), (b) Size-dependence (N = 8, 10, 12, 14, 16, 18) of
the high-temperature residue C(N) for different values of the
frustration α in the gapped regime. (b) Both axes are scaled
logarithmically. (c) Size dependence of T ∗(N∗) (see text for
details) for α = 0.35. Solid lines are guides to the eye.
F. Heidrich-Meisner1, A. Honecker1, D.C. Cabra2, W.
Brenig1
1Institut fu¨r Theoretische Physik, TU Braunschweig,
38106 Braunschweig, Germany
2Laboratoire de Physique, ENS-Lyon, 69364 Lyon,
France
[1] J. V. Alvarez and C. Gros, Phys. Rev. Lett. 89, 156603
(2002).
[2] F. Heidrich-Meisner et al., Phys. Rev. B 66, 140406(R)
(2002).
[3] For N ≤ 14 and α = 0.35 the data in Ref. [1] are identical
to that published in [2] which, however, includes systems
up to N = 18. The definitions of κ(th) in Refs. [1, 2] differ
by a trivial factor of pi.
[4] We define T ∗(N∗) by κ(th)(T ∗, N+2) = κ(th)(T ∗, N) with
N∗ = (N + 1) and even N .
[5] C. Hess et al., Phys. Rev. B 64, 184305 (2001).


Comment on “Anomalous Thermal Conductivity of Frustrated Heisenberg Spin Chains and Ladders

A Comment on the Letter by J.V. Alvarez and Claudius Gros, Phys. Rev. Lett. 89, 156603 (2002). The authors of the Letter offer a Reply. © 2004 The American Physical Society.Fil: Heidrich-Meisner, F.. Institut für Theoretische Physik; AlemaniaFil: Honecker, A.. Institut für Theoretische Physik; 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. Centre National de la Recherche Scientifique. Laboratoire de Physique Et Chimie de I´environnement Et de I´espace; FranciaFil: Brenig, W.. Institut für Theoretische Physik; Alemani

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

arXiv:cond-mat/0302358v2  [cond-mat.str-el]  12 Feb 20041Comment on “Anomalous Thermal Conduc-tivity of Frustrated Heisenberg Spin Chains andLadders”In a recent Letter [1], Alvarez and Gros have presenteda numerical study of the thermal conductivity of frus-trated chains and spin ladders with spin 1/2. Using exactdiagonalization of finite systems with N ≤ 14 sites, theyhave computed the zero-frequency weight κ(th)(T, N),i.e., the Drude weight, of the thermal conductivity whereT is the temperature and N the number of sites. Oneof their main conclusions is that the numerical data in-dicate a finite value of κ(th) in the thermodynamic limit(N → ∞) for spin ladders and frustrated chains. Inthe latter case, this conclusion is based on a finite-sizeanalysis of the high-temperature residue C(N), given bylimT→∞[T2κ(th)(T, N)] = C(N), for α = 0.1, 0.24, 0.35(see Refs. [1, 2] for definitions; also note [3]).In this Comment, we argue that, from the systemsinvestigated in [1], no conclusions of a finite thermalDrude weight in the gapped regime of frustrated chainsfor N → ∞ can be drawn. This will be corroboratedby supplementary data for systems up to N = 18 sites.In Figs. 1(a) and 1(b), we show the size dependenceof C(N) for 8 ≤ N ≤ 18 and α = 0.35, 0.5, 1 whereC(N)/C(N = 8) is plotted versus 1/N . Figure 1(b) isa log-log display of Fig. 1(a). An overall and monotonicdecrease of C(N) is evident, consistent with a vanishingthermal Drude weight for T ≫ J and N → ∞. In-deed, the curvature of the curves in the log-log plot ofC(N)/C(N = 8) versus 1/N suggests that C(N) van-ishes more rapidly than any power of 1/N as N → ∞.Most important, for the case of α = 0.35 and 8 ≤ N ≤ 14studied in [1], the overall decrease of C(N) remainsclearly observable, except for minor finite-size oscilla-tions, which, however, justify no extrapolation to a finitevalue for N → ∞.In their Letter, Alvarez and Gros have also argued thatthe behavior of κ(th)(T, N) for α = 0.35 at low temper-atures supports the conclusion of a finite thermal Drudeweight in the thermodynamic limit. Indeed, there is acrossover temperature T ∗(N∗) [4] where the monotonicdecrease of κ(th)(T, N) with increasing system size ob-served at high temperatures changes to a monotonic in-crease of κ(th)(T, N) with system size (see Fig. 3 in Ref.[2]). However, as shown in Fig. 1(c), T ∗(N∗) decreaseswith increasing system size already for N∗ ≥ 11, i.e., in-cluding systems studied in [1], and could well extrapolateto zero for N∗ → ∞. In any case, finite-size effects forT . 0.1J are far too large even at N = 18 to allow forany reliable predictions regarding κ(th)(T ) in the ther-modynamic limit.Summarizing our numerical analysis of gapped, frus-trated chains, both in the high- and the low-temperatureregime we find no evidence in favor of a finite thermalDrude weight in the thermodynamic limit in contrast toRef. [1].Finally, for spin ladders Alvarez and Gros have usedonly the size dependence of κ(th)(T, N) at low tempera-tures to conjecture a finite κ(th)(N → ∞). In view of theresults for the frustrated chain presented in this Com-ment, we suggest to perform a finite-size analysis of thehigh-temperature residue C(N) to substantiate this con-jecture. We believe that this is important for the analysisof the thermal conductivity in the spin ladder materialLa5Ca9Cu24O41[5].We acknowledge financial support by the DFG and aDAAD-ANTORCHAS exchange program.0 0.05 0.11/N00.20.40.60.81C(N)/C(N=8)α=0.35α=0.5α=10.06 0.08 0.1 0.121/N10.20.10.60.4C(N)/C(N=8)0 0.02 0.04 0.06 0.08 0.1 0.121/N0.050.10.150.20.250.30T*[J]α=0.35(a) (b)(c)N=14FIG. 1: (a), (b) Size-dependence (N = 8, 10, 12, 14, 16, 18) ofthe high-temperature residue C(N) for different values of thefrustration α in the gapped regime. (b) Both axes are scaledlogarithmically. (c) Size dependence of T ∗(N∗) (see text fordetails) for α = 0.35. Solid lines are guides to the eye.F. Heidrich-Meisner1, A. Honecker1, D.C. Cabra2, W.Brenig11Institut für Theoretische Physik, TU Braunschweig,38106 Braunschweig, Germany2Laboratoire de Physique, ENS-Lyon, 69364 Lyon,France[1] J. V. Alvarez and C. Gros, Phys. Rev. Lett. 89, 156603(2002).[2] F. Heidrich-Meisner et al., Phys. Rev. B 66, 140406(R)(2002).[3] For N ≤ 14 and α = 0.35 the data in Ref. [1] are identicalto that published in [2] which, however, includes systemsup to N = 18. The definitions of κ(th) in Refs. [1, 2] differby a trivial factor of π.[4] We define T ∗(N∗) by κ(th)(T ∗, N +2) = κ(th)(T ∗, N) withN∗ = (N + 1) and even N .[5] C. Hess et al., Phys. Rev. B 64, 184305 (2001).

CONICET Digital

Servicio de Difusión de la Creación Intelectual

arXiv:cond-mat/0302358v2  [cond-mat.str-el]  12 Feb 20041Comment on “Anomalous Thermal Conduc-tivity of Frustrated Heisenberg Spin Chains andLadders”In a recent Letter [1], Alvarez and Gros have presenteda numerical study of the thermal conductivity of frus-trated chains and spin ladders with spin 1/2. Using exactdiagonalization of finite systems with N ≤ 14 sites, theyhave computed the zero-frequency weight κ(th)(T,N),i.e., the Drude weight, of the thermal conductivity whereT is the temperature and N the number of sites. Oneof their main conclusions is that the numerical data in-dicate a finite value of κ(th) in the thermodynamic limit(N → ∞) for spin ladders and frustrated chains. Inthe latter case, this conclusion is based on a finite-sizeanalysis of the high-temperature residue C(N), given bylimT→∞[T2κ(th)(T,N)] = C(N), for α = 0.1, 0.24, 0.35(see Refs. [1, 2] for definitions; also note [3]).In this Comment, we argue that, from the systemsinvestigated in [1], no conclusions of a finite thermalDrude weight in the gapped regime of frustrated chainsfor N → ∞ can be drawn. This will be corroboratedby supplementary data for systems up to N = 18 sites.In Figs. 1(a) and 1(b), we show the size dependenceof C(N) for 8 ≤ N ≤ 18 and α = 0.35, 0.5, 1 whereC(N)/C(N = 8) is plotted versus 1/N . Figure 1(b) isa log-log display of Fig. 1(a). An overall and monotonicdecrease of C(N) is evident, consistent with a vanishingthermal Drude weight for T ≫ J and N → ∞. In-deed, the curvature of the curves in the log-log plot ofC(N)/C(N = 8) versus 1/N suggests that C(N) van-ishes more rapidly than any power of 1/N as N → ∞.Most important, for the case of α = 0.35 and 8 ≤ N ≤ 14studied in [1], the overall decrease of C(N) remainsclearly observable, except for minor finite-size oscilla-tions, which, however, justify no extrapolation to a finitevalue for N →∞.In their Letter, Alvarez and Gros have also argued thatthe behavior of κ(th)(T,N) for α = 0.35 at low temper-atures supports the conclusion of a finite thermal Drudeweight in the thermodynamic limit. Indeed, there is acrossover temperature T ∗(N∗) [4] where the monotonicdecrease of κ(th)(T,N) with increasing system size ob-served at high temperatures changes to a monotonic in-crease of κ(th)(T,N) with system size (see Fig. 3 in Ref.[2]). However, as shown in Fig. 1(c), T ∗(N∗) decreaseswith increasing system size already for N∗ ≥ 11, i.e., in-cluding systems studied in [1], and could well extrapolateto zero for N∗ → ∞. In any case, finite-size effects forT . 0.1J are far too large even at N = 18 to allow forany reliable predictions regarding κ(th)(T ) in the ther-modynamic limit.Summarizing our numerical analysis of gapped, frus-trated chains, both in the high- and the low-temperatureregime we find no evidence in favor of a finite thermalDrude weight in the thermodynamic limit in contrast toRef. [1].Finally, for spin ladders Alvarez and Gros have usedonly the size dependence of κ(th)(T,N) at low tempera-tures to conjecture a finite κ(th)(N →∞). In view of theresults for the frustrated chain presented in this Com-ment, we suggest to perform a finite-size analysis of thehigh-temperature residue C(N) to substantiate this con-jecture. We believe that this is important for the analysisof the thermal conductivity in the spin ladder materialLa5Ca9Cu24O41[5].We acknowledge financial support by the DFG and aDAAD-ANTORCHAS exchange program.0 0.05 0.11/N00.20.40.60.81C(N)/C(N=8)α=0.35α=0.5α=10.06 0.08 0.1 0.121/N10.20.10.60.4C(N)/C(N=8)0 0.02 0.04 0.06 0.08 0.1 0.121/N0.050.10.150.20.250.30T*[J]α=0.35(a) (b)(c)N=14FIG. 1: (a), (b) Size-dependence (N = 8, 10, 12, 14, 16, 18) ofthe high-temperature residue C(N) for different values of thefrustration α in the gapped regime. (b) Both axes are scaledlogarithmically. (c) Size dependence of T ∗(N∗) (see text fordetails) for α = 0.35. Solid lines are guides to the eye.F. Heidrich-Meisner1, A. Honecker1, D.C. Cabra2, W.Brenig11Institut fu¨r Theoretische Physik, TU Braunschweig,38106 Braunschweig, Germany2Laboratoire de Physique, ENS-Lyon, 69364 Lyon,France[1] J. V. Alvarez and C. Gros, Phys. Rev. Lett. 89, 156603(2002).[2] F. Heidrich-Meisner et al., Phys. Rev. B 66, 140406(R)(2002).[3] For N ≤ 14 and α = 0.35 the data in Ref. [1] are identicalto that published in [2] which, however, includes systemsup to N = 18. The definitions of κ(th) in Refs. [1, 2] differby a trivial factor of pi.[4] We define T ∗(N∗) by κ(th)(T ∗, N+2) = κ(th)(T ∗, N) withN∗ = (N + 1) and even N .[5] C. Hess et al., Phys. Rev. B 64, 184305 (2001).

F. Heidrich-Meisner

A. Honecker

D. C. Cabra

W. Brenig

Crossref

Comment on “Anomalous Thermal Conductivity of Frustrated Heisenberg Spin Chains and Ladders”

https://ri.conicet.gov.ar/bitstream/11336/72084/2/CONICET_Digital_Nro.0936b629-db70-49d0-b2e0-b74c9fc6d316_A.pdf

Comment on "Anomalous Thermal Conductivity of Frustrated Heisenberg Spin Chains and Ladders"

Abstract

Similar works

Full text

Available Versions

SEDICI - Repositorio de la UNLP

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

CONICET Digital

Servicio de Difusión de la Creación Intelectual

Crossref