a r X i v :c o n d -m a t /0201300v 3 [c o n d -m a t .s t r -e l ] 9 S e p 2002
Anomalous thermal conductivity of frustrated Heisenberg spin-chains and ladders
J.V.Alvarez and Claudius Gros
Fakult¨a t 7,Theoretische Physik,University of the Saarland,66041Saarbr¨u cken,Germany.
We study the thermal transport properties of several quan-tum spin chains and ladders.We ?nd indications for a diverg-ing thermal conductivity at ?nite temperatures for the models
examined.The temperature at which the non-diverging pref-actor κ(th )(T )peaks is in general substantially lower than the temperature at which the corresponding speci?c heat c V (T )is maximal.We show that this result of the microscopic ap-proach leads to a substantial reduction for estimates of the magnetic mean-free path λextracted by analyzing recent ex-periments,as compared to similar analyses by phenomenolog-ical theories.
PACS numbers:
Introduction -The nature of thermal transport in
systems with reduced dimensions is a long standing prob-lem and has been studied intensively in classical 1sys-tems either by direct numerical out-of-equilibrium simu-lations 2or by investigation of soliton-soliton scattering processes 3.There has been,on the other hand,con-siderably less progress for quantum systems.Huber,in one of the ?rsts works on the subject 4,evaluated the thermal conductivity κ(T )for the Heisenberg-chain with an equation-of-motion approximation and found a ?nite κ(T ),a result which is,by now,known to be wrong.It has been shown recently 5,that the energy-current op-erator commutes with the hamiltonian for the spin-1/2Heisenberg chain.The thermal conductivity is conse-quently in?nite for this model.The intriguing ques-tion,“under which circumstances does an interacting quantum-system show an in?nite thermal conductivity”,is to-date completely open,the analogous question for classical systems being intensively studied 1.
A second motivation to study κ(T )for quantum spin-systems comes from experiment.An anomalous large magnetic contribution to κhas been observed 6for the hole-doped spin-ladder system Sr 14?x Ca x Cu 24O 41.For Ca 9La 5Cu 24O 41,which has no holes in the ladders,an even larger thermal conductivity has been measured 7raising the possibility of ballistic magnetic transport lim-ited only by residual spin-phonon and impurity scatter-ing.There is,however,up-to-date no microscopic calcu-lation for the thermal conductivity of spin-ladders.
In this Letter we present a ?nite size analysis of the thermal conductance in Heisenberg J 1?J 2chains and ladders suggesting ballistic thermal transport in these two families of spin models.Following this result we propose a possible scenario that would account a num-ber of anomalies found in the thermal conductivity in spin systems.We apply these ideas to the case of Ca 9La 5Cu 24O 41,obtaining good agreement with recent
experimental measurements.
Models -We consider quasi one-dimensional systems
for which the hamiltonian takes the form H =
L x=1H x ,where L is the number of units along the chain.We will consider two models,the isotropic Heisenberg chain with dimerized nearest and homogeneous next-nearest neigh-bor exchange couplings,
H (ch )x =J
(1+δ(?1)x )S x ·S x+1+αS x ·S x+2 ,and the two-leg Heisenberg-ladder:
H (lad )x =J (S 1,x ·S 1,x+1+S 2,x ·S 2,x+1)+J ⊥S 1,x ·S 2,x .
H (ch )has been proposed to model the magnetic prop-erties of the spin-Peierls compound CuGeO 3.The non-frustrated dimerized Heisenberg chain (α=0)models the magnetic behavior 8of (VO)2P 2O 7.Ca 9La 5Cu 24O 41contains doped spin-chains and undoped spin-ladders 7,described by H (lad ).
Method -Our goal is to make connection between the microscopic transport-properties of the low dimensional models presented above and the experimental measure-ments of thermal conductivity mentioned in the intro-duction.
The thermal conductivity is de?ned as the response of the energy-current density j E x to a thermal gradient j E
x =?κ?T and it has units of [κ]=W
c
j E
x ,
where c is the lattice constant along the chain-direction.
In the absence of applied magnetic ?elds spin-inversion symmetry holds and the Kubo formula for κ=
lim ω→0κ(ω)reduces to
4
κ=lim
ω→0
β2
α=0=δand a spin-anisotropy).The total thermal current commutes with the hamiltonian in this case5and the current-current correlation function is therefore time-independent.Eq.(1)reduces then to the static expecta-tion valueκ(ω=0)=β2 (J E)2 /(Ls),a quantity which can be evaluated by Bethe-Ansatz10.
In general we have[J E,H]=0,but the thermal trans-port will be ballistic if
β2
κ(th)=
(3)
v s
to hold for the thermal conductivityκ.
Let us discuss the relation of Eq.(3)to the usual phe-nomenological formula13(here in one dimension)
κ(ph)=c V v sλ,(4) where c V is the speci?c heat.When applied in order to analyze experimental data,the microscopic equation (3)will in general yield di?erent values for the magnetic mean-free pathλ.On the other hand,we might expect Eq.(4)to hold at low temperature for the Heisenberg-chain in the gapless phase,when the Luttinger-liquid quasiparticles are well de?ned and a Boltzmann approach is justi?ed.Consequently we expectκ(th)λ/v s≡c V v sλin this case,i.e.we expectκ(th)/c V≡v2s in the limit T→0.All three quantities in this equation(κ(th),c V and v s)can be computed by Bethe-Ansatz for the XXZ-chain.One?nds that this equation is exact10in the limit T→0.
Numerics-The computation ofκ(th)(T)demands the whole spectrum of the hamiltonian,restricting the max-imum lattice size and a careful?nite-size analysis is re-quired.In general one?nds,e.g.for the XXZ-model for which the exact Bethe-Ansatz result is known10,that the κ(th)in chains with odd(even)number of sites chains is a upper(lower)bound of the exact result.The value of κ(th)(T)decreases(increases)for chains with odd(even) number of sites when the lattice is enlarged.This obser-vation has led us to consider the average
L0κ(th)(L0)+L1κ(th)(L1)
κ(th)(L e?)=
κ(exp )=
k B J
abc
λ
ˉ
h v s
?κ(th )
(6)
where the quantities in the brackets are dimensionless.
?κ(th )is the dimensionless thermal conductance (2)which we will evaluate by exact diagonalization.a ,b and c are the lattice constants (the chains run along the c direction)and N c is the number of chains per unit cell.Note that λ
ˉh v s
= JτJ ?7?
725
T
2
.The assumption of a weak spin-environment coupling entering Eq.(3)is there-fore justi?ed in the experimental relevant temperature regime.The lifetime is,to give an example,132times larger than the coupling constant J at T =150K,lead-ing to an external broading of the energy levels of only 1/132=0.76%,in units of J .
To estimate the e?ective mean-free path we use λ(T )=ˉv s (T )τ(T ),were we have used for ˉv s the ther-mal expectation value of the magnon-velocity within the independent-triplet model:
ˉv s =1Z ?2
?
dεn (ε)
where Z =
dk n (ε(k ))is the partition function and n (ε(k ))=3/(exp(βε(k )+3).We approximated the one-magnon dispersion ε(k )by 2ε2(k )=(?2+?22)+(?2
2??2)cos(k ).For J ⊥=J we have 16that ?2≈3.8?.
In the low-temperature limit ˉv s (T )≈
√
1
For an overview see F.Bonetto,J.L.Lebowitz and L.Rey-Bellet,arXiv:math-ph/0002052(2000).
2
C.Giardin′a ,R.Livi,A.Politi and M.Vassalli,Phys.Rev.Lett.84,2144(2000).3
R.Khomeriki,arXiv:cond-mat/0112067(2001).4
D.L.Huber,Prog.Theo.Phys.39,1170(1968);D.L.Huber and J.S.Semura,Phys.Rev.182,602(1969).5
X.Zotos,F.Naef,and P.Prelovˇs ek,Phys.Rev.B 55,11029(1997).6
A.V.Sologubenko,K.Giann′o ,H.R.Ott,U.Ammerahl and A.Revcholevschi,Phys.Rev.Lett.84,2714(2000).7
C.Hess, C.Baumann,U.Ammeral, B.B¨u chner,F.Heidrich-Meissner,W.Brenig and A.Revcholevschi,Phys.Rev.B 64184305(2001).8
A.W.Garrett,S.E.Nagler,D.A.Tennant,
B.
C.Sales2and T.Barnes,Phys.Rev.Lett.79,745(1997).9
F.Heidrich-Meisner, A.Honecker, D.C.Cabra and W.Brenig,cond-mat/0208282.10
A.Kl¨u mper and K.Sakai,J.Phys.A35,2173(2002).11
C.Gros,Z.Phys.B 86,359(1992).
12J.V.Alvarez and C.Gros.Eur.Phys.J.B,15,641(2000)13
H.W.Ashcroft and N.D.Mermin,“Solid State Physics”,Holt-Saunders (1976).14
M.Matsuda,K.Katsumata,R.S.Eccleston,S.Brehmer and H.-J.Mikeska,Phys.Rev.B 62,8903(2000)15
M.Windt,M.Gr¨u ninger,T.Nunner, C.Knetter,K.P.Schmidt,G.S.Uhrig,T.Kopp,A.Freimuth,U.Am-merahl,B.B¨u chner,and A.Revcolevschi,Phys.Rev.Lett.87,127002(2001).16
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00.20.40.6
0.81 1.2 1.4
T [K]
0.1
0.2
0.3
0.4
0.5
κ
(th)
L eff =5.5L eff =7.5L eff =9.5L eff =11.5L eff =13.5
Bethe Ansatz
0.5
1
1.5
0.25
0.5
L=13,11,9,7,5
L=14,12,10,8,6
FIG.1.κ(th ),as de?ned by Eq.(2),for the Heisen-berg-chain as a function of temperature.Inset:Data for L =5...14,in comparison with the Bethe-Ansatz re-sult 10(full line).Main panel:Results using Eq.(5),with L ef f =(L 0+L 1)/2.We note,that the position of the maxi-mum in κ(th )can be determined con?dently with the averag-ing procedure.
00.20.40.6
0.81 1.2 1.4
T[J ||]
0.1
0.2
0.3
0.4
0.5
c V a n
d κ
(t h )
C v QMC L eff =3.5L eff =4.5L eff =5.5 L eff =6.5
κ
(th)
c V
J perp =2J ||?=1.2J ||
0.3
0.5
3
574
6
FIG.2.Temperature dependence of κ(th )and the dimen-sionless speci?c heat for ladders with twisted boundary con-ditions and J ⊥=2J .Both magnitudes are computed using the average procedure (5)described in the text.The speci?c heat for a 2×100-ladder computed with QMC is plotted for comparison (squares).Inset κ(th )for L=3,5,7(from bottom up)and L =4,6(from top down)in the low-temperature region.
sizes L=8,10,12,14,the arrows indicate increasing L .Inset :Finite size analysis (L=6...14)of the high temperature residue C (L )de?ned as κ(th )=C (L )/T 2at T ?J for di?erent val-ues of α.
50
150
250
350
T [K]
0500
1000
m e a n ?f r e e p a t h [A ]
La 5Ca 9Cu 24O 41
1.0*10
?12
0.5*10
?12
τ[s]
τ
λ
100
200300
T[K]050
100
150
κ
(e x p )
[w /m K ]
FIG. 4.τ(T )(right-scale)and λ(T )(left-scale)for Ca 9La 5Cu 24O 41extracted using Eq.(6).The experimental data by Hess et al.7is shown in the inset (?lled circles)to-gether with an analytic ?t (solid line)discussed in the text.
统计热力学基础 一、选择题 1. 下面有关统计热力学的描述，正确的是：( ) A. 统计热力学研究的是大量分子的微观平衡体系 B. 统计热力学研究的是大量分子的宏观平衡体系 C. 统计热力学是热力学的理论基础 D. 统计热力学和热力学是相互独立互不相关的两门学科B 2. 在研究N、V、U有确定值的粒子体系的统计分布时，令刀n i = N,刀n i & i = U , 这是因为所研究的体系是：( ) A. 体系是封闭的，粒子是独立的 B 体系是孤立的，粒子是相依的 C. 体系是孤立的，粒子是独立的 D. 体系是封闭的，粒子是相依的C 3. 假定某种分子的许可能级是0、&、2 ￡和3 &,简并度分别为1、1、2、3四个这样的分子构成的定域体系，其总能量为3￡时，体系的微观状态数为：() A. 40 B. 24 C. 20 D. 28 A 4. 使用麦克斯韦-波尔兹曼分布定律，要求粒子数N 很大，这是因为在推出该定律时：( ) . 假定粒子是可别的 B. 应用了斯特林近似公式 C. 忽略了粒子之间的相互作用 D. 应用拉氏待定乘因子法A 5. 对于玻尔兹曼分布定律n i =(N/q) ? g i ? exp( - ￡ i/kT)的说法：(1) n i是第i能级上的粒子分布数; (2) 随着能级升高，￡ i 增大，n i 总是减少的; (3) 它只适用于可区分的独立粒子体系; (4) 它适用于任何的大量粒子体系其中正确的是：( ) A. (1)(3) B. (3)(4) C. (1)(2) D. (2)(4) C 6. 对于分布在某一能级￡ i上的粒子数n i，下列说法中正确是：() A. n i 与能级的简并度无关 B. ￡ i 值越小，n i 值就越大 C. n i 称为一种分布 D. 任何分布的n i 都可以用波尔兹曼分布公式求出B 7. 15?在已知温度T时，某种粒子的能级￡ j = 2 ￡ i，简并度g i = 2g j，则「和￡ i上 分布的粒子数之比为：( ) A. 0.5exp( j/2￡kT) B. 2exp(- ￡j/2kT) C. 0.5exp( -￡j/kT) D. 2exp( 2 j/k￡T) C 8. I2的振动特征温度? v= 307K，相邻两振动能级上粒子数之n(v + 1)/n(v) = 1/2的温度是：( ) A. 306 K B. 443 K C. 760 K D. 556 K B 9. 下面哪组热力学性质的配分函数表达式与体系中粒子的可别与否无关：( ) A. S、G、F、C v B. U、H、P、C v C. G、F、H、U D. S、U、H、G B 10. 分子运动的振动特征温度?v是物质的重要性质之一，下列正确的说法是： ( ) A. ? v越高，表示温度越高 B. ?v越高，表示分子振动能越小 C. ?越高，表示分子处于激发态的百分数越小 D. ?越高，表示分子处于基态的百分数越小 C 11. 下列几种运动中哪些运动对热力学函数G与
统计热力学基础 题 择 一、选 1. 下面有关统计热力学的描述，正确的是：( ) A. 统计热力学研究的是大量分子的微观平衡体系 B. 统计热力学研究的是大量分子的宏观平衡体系 C. 统计热力学是热力学的理论基础 D. 统计热力学和热力学是相互独立互不相关的两门学科B 2.在研究N、V、U 有确定值的粒子体系的统计分布时，令∑n i = N，∑n iεi = U， 3.这是因为所研究的体系是：( ) A. 体系是封闭的，粒子是独立的 B 体系是孤立的，粒子是相依的 C. 体系是孤立的，粒子是独立的 D. 体系是封闭的，粒子是相依的 C 4.假定某种分子的许可能级是0、ε、2ε和3ε，简并度分别为1、1、2、3 四个这样的分子构成的定域体系，其总能量为3ε时，体系的微观状态数为：( ) A. 40 B. 24 C. 20 D. 28 A 5. 使用麦克斯韦-波尔兹曼分布定律，要求粒子数N 很大，这是因为在推出该定律 6.时：( ) . 假定粒子是可别的 B. 应用了斯特林近似公式 C. 忽略了粒子之间的相互作用 D. 应用拉氏待定乘因子法 A 7.对于玻尔兹曼分布定律n i =(N/q) ·g i·exp( -εi/kT)的说法：(1) n i 是第i 能级上的 粒子分布数; (2) 随着能级升高，εi 增大，n i 总是减少的; (3) 它只适用于可区分的独 8.立粒子体系; (4) 它适用于任何的大量粒子体系其中正确的是：( ) A. (1)(3) B. (3)(4) C. (1)(2) D. (2)(4) C 9.对于分布在某一能级εi 上的粒子数n i ，下列说法中正确是：( ) 10.A. n i 与能级的简并度无关 B. εi 值越小，n i 值就越大 C. n i 称为一种分布 D.任何分布的n i 都可以用波尔兹曼分布公式求出 B 11. 15．在已知温度T 时，某种粒子的能级εj = 2εi，简并度g i = 2g j，则εj 和εi 上分布的粒子数之比为：( ) A. 0.5exp( j/2εk T) B. 2exp(- εj/2kT) C. 0.5exp( -εj/kT) D. 2exp( 2 j/kεT) C 12. I2 的振动特征温度Θv= 307K，相邻两振动能级上粒子数之n(v + 1)/n(v) = 1/2 的温度 13.是：( ) A. 306 K B. 443 K C. 760 K D. 556 K B 14.下面哪组热力学性质的配分函数表达式与体系中粒子的可别与否无关：( ) A. S、G、F、C v B. U、H、P、C v C. G、F、H、U D. S、U、H、G B 15. 分子运动的振动特征温度Θv 是物质的重要性质之一，下列正确的说法是： ( ) A.Θv 越高，表示温度越高 B.Θv 越高，表示分子振动能越小 C. Θv 越高，表示分子处于激发态的百分数越小 D. Θv 越高，表示分子处于基态的百分数越小 C 16.下列几种运动中哪些运动对热力学函数G 与A 贡献是不同的：( ) A. 转动运动 B. 电子运动 C. 振动运动 D. 平动运动 D 17.三维平动子的平动能为εt = 7h 2 /(4mV2/ 3 )，能级的简并度为：( )
第七章统计热力学基础 一、单选题 1．统计热力学主要研究（）。 (A) 平衡体系(B) 近平衡体系(C) 非平衡体系 (D) 耗散结构(E) 单个粒子的行为 2．体系的微观性质和宏观性质是通过（）联系起来的。 (A) 热力学(B) 化学动力学(C) 统计力学(D) 经典力学(E) 量子力学 3．统计热力学研究的主要对象是：（） (A) 微观粒子的各种变化规律(B) 宏观体系的各种性质 (C) 微观粒子的运动规律(D) 宏观系统的平衡性质 (E) 体系的宏观性质与微观结构的关系 4．下述诸体系中，属独粒子体系的是：（） (A) 纯液体(B) 理想液态溶液(C) 理想的原子晶体 (D) 理想气体(E) 真实气体 5．对于一个U，N，V确定的体系，其微观状态数最大的分布就是最可几分布，得出这一结论的理论依据是：（） (A) 玻兹曼分布定律(B) 等几率假设(C) 分子运动论 (D) 统计学原理(E) 能量均分原理
6．在台称上有7个砝码，质量分别为1g、2g、5g、10g、50g、100g，则能够称量的质量共有：（） (A) 5040 种(B) 127 种(C) 106 种(D) 126 种 7．在节目单上共有20个节目序号，只知其中独唱节目和独舞节目各占10个，每人可以在节目单上任意挑选两个不同的节目序号，则两次都选上独唱节目的几率是：（） (A) 9/38 (B) 1/4 (C) 1/180 (D) 10/38 8．以0到9这十个数字组成不重复的三位数共有（） (A) 648个(B) 720个(C) 504个(D) 495个 9．各种不同运动状态的能级间隔是不同的，对于同一种气体分子，其平动、转动、振动和电子运动的能级间隔的大小顺序是：（） (A)△e t >△e r >△e v >△e e(B)△e t <△e r <△e v <△e e (C) △e e >△e v >△e t >△e r(D)△e v >△e e >△e t >△e r (E)△e r >△e t >△e e >△e v 10．在统计热力学中，对物系的分类按其组成的粒子能否被分辨来进行，按此原则：（） (A) 气体和晶体皆属定域子体系(C) 气体属离域子体系而晶体属定域子体系 (B) 气体和晶体皆属离域子体系(D) 气体属定域子体系而晶体属离域子体系 11．对于定位系统分布X所拥有的微观状态t x为：（B） (A)(B)