Resistor-network anomalies in the heat transport of random harmonic chain

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by  Isaac Weinberg

Resistor-network anomalies in the heat transport of random harmonic chain
Isaac Weinberg1, Yaron de Leeuw1, Tsampikos Kottos2,3, Doron Cohen1
1Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel
2Department of Physics, Wesleyan University, Middletown, Connecticut 06459 and
3Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
We consider thermal transport in low-dimensional disordered harmonic networks of coupled
masses. Utilizing known results regarding Anderson localization, we derive the actual dependence
of the thermal conductance G on the length L of the sample. This is required by nanotechnology
6
implementations because for such networks Fourier’s law G 1/Lα with α=1 is violated. In par-
1 ∝
ticularweconsider“glassy”disorderinthecouplingconstants,andfindananomalywhichisrelated
0
byduality tothe Lifshitz-tail regime in thestandard Anderson model.
2

n PACSnumbers: 76.50.+g,11.30.Er,05.45.Xt
u
J

3 I. INTRODUCTION II. SCOPE
2

Considering heat transport for low-dimensional disor-
]
n The theory of phononic heat conduction in disordered dered networks of coupled harmonic masses, we utilize
n low-dimensional networks is a central theme of research known results from the field of mesoscopic electronic
-
s in recent years [1–3]. The interest in this theme is not physics, in order to derive the actual L dependence of
i onlypurely academic,but itis alsomotivatedby the on- G for regular as well as for “glassy” type of disorder.
d
. goingdevelopmentsinnanotechnology. Inspiteofthere- The information about the latter is encoded in the de-
t
a centresearchefforts,theunderstandingofthermaltrans- pendence of the inverse localization length γ on the vi-
m port is still at its infancy. This becomes more obvious if bration frequency ω. Our results explain the transition
- one compares with the achievements that have been ex- fromoptimaltosuper-optimalscalingbehaviorandeven-
d periencedduringthelastfiftyyearsinunderstandingand tually to exponential dependence on L. We address the
n managingelectrontransport. Inthisrespecteventhemi- implications of the percolation threshold, and the geo-
o
croscopiclawsthatgovernheatconductioninlowdimen- metrical bandwidth. Along the way we highlight a sur-
c
[ sional systems have only recently start being scrutinized prisinganomalythatisrelatedbydualitytotheLifshitz-
via both theoretical,numericalandexperimentalstudies tailregimeinthestandardAndersonmodel,andtestthe
3 [1–8]. These studies unveil many surprising results, the borders of the one-parameter scaling hypothesis.
v
most dramatic of which is the violation of the naive ex- The outline of this paper is as follows: Sections III-V
7
0 pectation (Fourier’s law) which states that the thermal definethegeneralmodelofinterest,emphasizingthatfor
2 conductanceGisinverseproportionaltothesizeLofthe “glassydisorder”a resistor-networkperspective is essen-
2 system, namely, G 1/Lα with α=1. tial. SectionVIclarifiesthattheanalysisofheatconduc-
∝
0 tionofquasionedimensionalnetworkseffectivelyreduces
.
1 Currentlyitiswellestablishedthatinlow-dimensional to the analysis of a single-channel problem. Section VII
0 disordered systems, in the absence of non-linearity, explains how we use the transfer matrix method in the
6 Fourier’s law is violated. The underlying physics is re- numerical analysis: we highlight the procedure for the
1
lated to the theory of Anderson localization of the vi- determination of the optimal leads, and the significance
:
v brationalmodes [2, 9–16]. On the basis of the prevailing of the percolation parameter s in this context. Section
i theory [2, 9] it has been claimed that for samples with VIII use the Born approximation in order to provide an
X
“optimal” contacts α=1/2, while in generalα might be explanation for the numerical findings of [17]. These re-
r
a larger,sayα=3/2forsampleswith“fixedboundarycon- sults had been obtained for weak disorder.
ditions”. Recentlythe“optimal”valueα=1/2hasbeen Subsequently we focus on the single-channel model.
challengedbythenumericalstudyof[17]. Theseauthors Our main interest is to explore the implications of
found a super-optimal value α 1/4 for moderate sys- “glassy” disorder, and to highlight the resistor-network
∼
tem sizes L, while asymptotically, in the presence of a aspect. In Sections IX and X we go beyond the born
pinning potential, G decays exponentially as exp( γL). approximation by establishing a duality between glassy
−
off-diagonal disorder and weak diagonal disorder. Con-
Itisobviousthatifthe finalgoalistoachievethecon- sequently we deduce that the Lifshitz-tail anomaly is re-
trolofheatflowonthenanoscale,firstwehavetounder- flected in the frequency dependence of the inverse local-
stand the fundamental mechanisms of heat conduction, ization length. This prediction is verified numerically.
and provide an adequate description of its scaling with The remaining sections XI to XIII clarify how scaling-
the system size for any L, including the experimentally theoryoflocalizationcanbeusedinordertocalculatethe
relevant cases of intermediate lengths. heat conductance. Here no further surprises are found.
2
Infactweverifynumericallythatastraightforwardappli- in the latter case is
cation of the weak-disorder analytical approach is quite
s
satisfactory. In spite of the “glassy” disorder the devia- P(w) = ws−1 (w <w ) (3)
ws c
tions from one-parameter scaling are not alarming. c
where s is the normalized density of the sites. Large s is
like regular weak disorder, while small s implies glassy
III. THE MODEL
disorder that features log-wide distribution (couplings
distributed over several orders of magnitude). The case
We consideraone-dimensionalnetworkofLharmonic
s=0withanaddedlowercutoffformallycorrespondsto
oscillators of equal masses. The system is described by
“random barriers”.
the Hamiltonian
1 1
= PTP + QTWQ (1)
H 2 2
V. RESISTOR-NETWORK PERSPECTIVE
where QT (q ,q , ,q ), and PT (p ,p , ,p )
1 2 N 1 2 N
≡ ··· ≡ ···
are the displacement coordinates and the conjugate
Itisusefultonoticethattheproblemofphononicheat
momenta. The real symmetric matrix W is deter-
conduction in the absence of a pinning potential is for-
mined by the spring constants. Its off-diagonal ele-
mallyequivalenttotheanalysisofarateequation,where
mentsW = w originatefromthecouplingpotential
nm nm the spring-constantsareinterpretedasthe ratesw for
(1/2) w− (q q )2, while its diagonal elements nm
m,n nm n − m transitions between sites n and m. Optionally it can be
containanadditionaloptionaltermthatoriginatefroma
regarded as a resistor-network problem where w rep-
pinninPg potential(1/2) v q2 that couples the masses nm
n n n resent connectors. We define w0 as the effective hopping
tothesubstrate. AccordinglyW =v + w . For
P nn n m nm ratebetweensites. We later justify thatitshouldbe for-
achainwithnear-neighbortransitionsweusethesimpli-
mally identified with the conductivity of the correspond-
P
fied notation w w .
n+1,n n ing resistor-network.
≡
Ingeneralthe interestis inquasione-dimensionalnet-
The detailednumericalanalysisin the subsequentsec-
works, for which W is a banded matrix with 1+2b di-
tions concerns the b=1 chain, for which the “serial ad-
agonals. For b=1 the near-neighbor hopping implies a
dition”ruleimpliesthatw equalstheharmonicaverage.
single-channel system. For b>1 the dispersion relation 0
For the “random distance” disorder of Eq.(3) we get
(see sectionVI below)hasseveralbranches,whichis like
having a multi-channel system. The heat conduction of
−1
1 s 1
such networks has been investigated numerically in [17], w = = (s>1) − w (4)
0 c
with puzzling findings that have not been explainedthe- w s
(cid:20)(cid:28) (cid:29)(cid:21) (cid:20) (cid:21)
oretically. We shall see that the essential physics can
be reduced to single channel (b=1) analysis. On top we For s<1 the network is no longer percolating, namely
would like to consider not only weakly disordered net- w0 =0. In the present context w0 determines the speed
work, but also the implications of “glassy” disorder as of sound (see below).
defined below. Forthe lateranalysisweneedalsothe secondmoment
of the couplings. For s>2 one obtains
IV. THE DISORDER 1 2 s 1 2
= (5)
w s 2 w
Both the w and the v are assumed to be ran- *(cid:18) (cid:19) +s>2 (cid:20) − (cid:21)(cid:18) c(cid:19)
nm n
dom variables. The diagonal-disorder due to the pin-
Hence the variance is [s/((s 1)2(s 2))]w−2. For s<2
ning potential is formally like that of the standard An- − − c
derson model with some variance σ2 Var(v). The off- the second moment diverges. But for a particular real-
k ≡ ization the sample-specific result is finite, and depends
diagonal disorder of the couplings might be weak with
on the effective lower cutoff δw of the distribution. The
some variance σ2 =Var(w), but more generally it can
⊥ number w /δw reflects the finite size ofthe sample,
reflecttheglassinessofthenetwork. By“glassydisorder” N ≡ c
and we get the sample-size dependent result
wemeanthatthecouplingwhasanexponentialsensitiv-
ity to physical parameters. For random barrier statistics
2 2
w e−B, where B is uniformly distributed within [0,σ], 1 = s 2−s 1 1 (6)
∝
accordingly *(cid:18)w(cid:19) +s<2 2−s N − (cid:18)wc(cid:19)
(cid:2) (cid:3)
1 w
P(w) e−σ < <1 (2) Aswegofroms>2tos<2the dependence ofthevari-
∝ w w
(cid:18) c (cid:19) anceons hasacrossoverfrompower-lawtoexponential.
For random distance statistics w e−R, where R is im- We shall see later that this crossover is reflected in the
∝
plied by Poisson statistics. The probability distribution localization-length of the eigenstates.
3
103
0.1
0.5
102 1.0
N
P
101
100
0 5 10 15 20
λ
FIG. 1. The participation number (PN) [a] of the eigenstates of a conservative banded matrix are plotted against their
eigenvalues λ. The 0< m n b elements of W are random numbers w [1 ς,1+ς] (box distribution). The values of ς
| − |≤ ∈ −
areindicatedin thelegend(notethatthenumericsof[17]correspondstoς=0.5). Thenumberofbandsisb=5,andthelength
of the sample is N=1000 with periodic boundary conditions. The support of the clean-ring channels is indicted by the lower
horizontallines. Weobservethegradualblurringofthenon-disorderedbandstructure. Alltheeigenstates thathavelargePN
reside in thelower part of thespectrum and belong to a single channel.
VI. THE SPECTRUM VII. LOCALIZATION
The eigenvaluesλ aredeterminedviadiagonalization The disorder significantly affects the eigenmodes:
k
WQ=λQ,fromwhichonededucestheeigen-frequencies rather than being extended as assumed by Debye, they
via λ =ω2. In the absence of disorder the eigenmodes become exponentially localized. We use the standard
k k
are Bloch states with notation γ(ω) for the inverse localization length. Con-
b sidering a single-channel (b=1) system it is defined via
λ = 2w [1 cos(rk)] ω2 (7) the asymptotic dependence of the transmission g on the
k 0 − ≡ k
length L of the sample. Namely,
r=1
X
where k is the associated wavenumber. For a single- 1 ln(g)
ω
channel λ = 2w (1 cosk) w k2, where the small-k γ(ω) = lim h i (11)
0 − ≈ 0 −L→∞2 L
approximationholdscloseto the bandfloor. With disor-
where indicates an averaging over disorder realiza-
deredcouplings,butintheabsenceofapinningpotential h···i
tions. The notionoftransmissionis physicallyappealing
thelowesteigenvalueisstillλ =0,whichcorrespondsto
the trivialextended state Q=0 (1,1,...,1)T, that is inter- here,becausewecanregardW astheHamiltonianofan
electron in a tight binding model. The transmission can
preted as the ergodic state in the context of rate equa-
be calculated from the transfer matrix T of the sample:
tions. All higher eigenstates are exponentially localized,
and are characterized by a spectral density ̺(ω). 4 sin(k)2
g = | | (12)
In Fig.1 we provide a numerical example considering T T +T exp(ik) T exp( ik)2
21 12 22 11
a b=5 quasi-one dimensional sample. The dispersion | − − − |
where
relation Eq.(7) has 5 branches. The support of the 1st,
3rdand5thascendingbranchesisindicatedinthefigure. n=L λ−(vn+wn+wn+1) wn
Itis importanttoobservethatat thebottomof thebanda T = wn+1 −wn+1 (13)
singlechannel-approximation ismostappropriate. Hence n=1 1 0 !
Y
within the framework of the Debye approximation the Above it is assumed that the sample is attached to
dispersion at the bottom of the band is always two non-disorderedleads. Optimal coupling requires the
ω ck [Debye] (8) hopping-rates there to be all equal to the “conductiv-
≈ ity” w , meaning same speed of sound. This observation
0
For b=1 the speed of sound is c=√w0, while for b≫1 has been verified numerically, see Fig.2. We see that
it is easily found that
it is the resistor-network harmonic-average and not the
c [(1/3)b3]1/2√w (9) algebraic-averagethat determines the optimal coupling.
0
≈
In Fig.3 we display an example for the calculation of
Eitherwaythelow-frequencyspectraldensityisconstant,
γ versus s. Well-defined results are obtained for s>2
namely
where the second moment Eq.(5) is finite. In the next
L
sectionweshallderiveanaiveBornapproximationforγ.
ρ(ω) (10)
≈ πc This is displayed in Fig.3 too as a dashed line. The esti-
The effect of weak disorder on this result is negligible. mateisbasedonanalyticalensemble-averageofVar(1/w)
4
1 101
0.8 0
10
0.6
−1
10
i
g
h γ
0.4
−2
10
0.2
−3
10
0
−5 0
10 10 10−4
wlead 0 1 2 3 s 4 5 6 7
FIG. 2. The average transmision g as a function of the
h i FIG. 3. The inverse localization length γ for the random
hopping rate wlead within the leads. The calculation is done
distancedisorderofEq.(3)versuss. Solidlinewithdiamonds
for L=50 disordered sample, where the w are distributed
accordingtoEq.(2)withσ=10atk=0.028nπ. Thealgebraic isforthenon-optimized(wlead =wc)results. Thicksolidline
and the harmonic mean values of the w are indicated by with squares is for the optimized (wlead =w0) results in the
vertical dotted and dashed lines respectivenly. range s>1, where w0 is finite. The tangent thin solid line
has been fitted in the range 1<s<2 where Eq.(6) is ex-
pected to hold. The naive Born approximation Eq.(15) is
illustrated by dashed line in the range s>2. The numerical
andthereforedivergesass=2isapproachedfromabove.
results here and in the next figures are based on thetransfer
In the range 1<s<2 the second moment Eq.(6) is ill- matrix method (symbols), with several hundreds of realiza-
defined(sample-specific). Givenanindividualsamplethe tions up to L 104.
∼
Born approximation can be used with sample-variance
(which is always finite) and provide a rough estimate.
Thetypicalresultinthisrangeisexpectedtodependex- The Fermi-Golden-Rule (FGR) picture implies that
ponentially ons as implied by Eq.(6). This expected de- the scattering rate is τ−1 =2π̺(ω)Wk′,k 2. The Born
pendenceisindeedobserved. Fors<1thesdependence approximation for the mean free pa|th is|ℓ=[dλ/dk]τ,
ofγ iscompletelyill-defined: thechainisnon-percolating
where the expression in the square brackets is the group
inthe L limit, andthe contactoptimizationproce- velocityintheelectronicsense(λislikeenergy). TheDe-
→∞
dures becomes meaningless. bye approximation implies dλ/dk 2[c2]k. The inverse
localization length is γ = (2ℓ)−1.≈From here (without
taking the small k approximation) it follows that
VIII. BORN APPROXIMATION
1 9 σ 2 1 2
k
γ(ω) (14)
In the absence of disorder W describes hopping with ≈ 8(cid:20)b6(cid:21)(cid:18)w0(cid:19) (cid:18)sin(k)(cid:19)
some rate w0, and the eigenstates are free waves labeled 1 9 σ 2 k 2
⊥
by k. With disorderthe w ofthe unperturbed Hamilto- + 2tan (15)
0 8 5b w 2
nian is loosely defined as the average w. Later we shall (cid:20) (cid:21)(cid:18) 0(cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
go beyond the Born approximation and will show that wheretheprefactorsinthesquarebracketsassumeb 1,
≫
it should be the harmonic average (as already defined andshouldbereplacedbyunityforb=1. Intheabsence
previously). The disorder couples states that have dif- of pinning the localization length diverges (γ k2) at
∝
ferentk. Fordiagonaldisorder(”pinning”)thecouplings the band floor, as assumed by Debye. This behavior is
areproportionaltothevarianceofthediagonalelements, demonstrated in Fig.4 and Fig.5 for two types of glassy
namely Wk,k′ 2 =(1/L)Var(v). Foroff-diagonaldisorder disorder. The deviations from Eq.(15) will be explained
| |
(randomspringconstants)thecouplingsareproportional in the next paragraphs.
tothevarianceoftheoff-diagonalelements,anddepends
on b and on k too:
IX. BEYOND BORN
b
Var(w)
Wk′,k 2 = [2sin(rk′/2)2sin(rk/2)]2
| | L The Born approximation has assumed weak disor-
r=1
X der. Here we would like to consider the more gen-
It follow that for small k we have |Wk,k′|2 ∝b5σ⊥2k4. eral case of glassy disorder. For this purpose, as
5
0.3 0.015
Numerics Numerics
dual map with R=1 dual map with R=1
0.25 naive Born naive Born
dual Born dual Born
0.2 beyond Born 0.01 beyond Born
γ0.15 γ
0.1 0.005
0.05
00 0.05 0.1 0.15 0.2 0.25 0.3 00 0.05 0.1 0.15 0.2 0.25 0.3
k/π k/π
0.3
Numerics FIG. 5. The inverse localization length γ versus k for the
dual map with R=1 random distance disorder of Eq.(3) with s=4. The symbols
0.25 naive Born
andlinesarethesameasinFig.4. Notethatthe“dualBorn”
dual Born
beyond Born approximation cannot be resolved from its improved version
0.2
Eq.(24). HereweconsiderthedistributionofEq.(3)forwhich
theduality implies k dependencethat does not coincide with
γ0.15
that of thenaiveBorn approximation.
0.1
in [18], we write the equation Wψ =λψ for the eigen-
0.05
states as a map of a single variable r =ψ /ψ ,
n n n−1
namely r = R /r A , where R =w /w ,
n+1 n n n n n−1 n
00 0.05 0.1 0.15 0.2 0.25 0.3 and An =(λ v−n wn −wn−1)/wn. In the case of di-
k/π − − −
agonal disorder it takes the from
1
0.3 r = ǫ+f (16)
Numerics n+1 −r − n
dual map with R=1 n
0.25 naive Born where f =v /w is the scaled disorder and
dual Born n n 0
beyond Born
0.2 λ
ǫ = 2 2cos(k) (17)
w − ≡ −
0
γ0.15
isthescaledenergymeasuredfromthecenteroftheband.
Without the random term f this map has a fixed-point
0.1 n
that is determined by the equationr2+ǫr+1=0, with
0.05 elliptic solution for ǫ [ 2,2]. The random term is re-
∈ −
sponsibleforhavinganonzeroinverselocalizationlength
0 γ = ln(r) . The Born approximation Eq.(14) is writ-
0 0.05 0.1 0.15 0.2 0.25 0.3 −h i
k/π ten as
1 Var(f) 1 Var(f)
γ = (18)
≈ 8 [1 (ǫ/2)2] 8 [sin(k)]2
FIG.4. Theinverselocalizationlengthγversuskfortheran- −
dom barrier disorder of Eq.(2) with σ = 5,10,15 from up to This standard estimate does not hold close to the band-
down. Pure off-diagonal disorder is assumed. The numerical
edge ǫ = 2, which can be regarded as an anomaly [18].
0
resultsbasedonthetransfermatrixmethod(circles)arecom- −
Closeness to the band-edge means that ǫ ǫ becomes
paredwiththosethataregeneratedbythemapEq.(20)with comparable with the kinetic energy γ|2.− 0H|ence the
the approximation R =1. The naive Born estimate Eq.(15)
n
so called Lifshitz tail region is ǫ ǫ <ǫ with ǫ =
isillustratedbydashedredline,whilethebluedashed-dotted 0 c c
| − |
lineisbasedontheimprovedestimatewithEq.(23). Herewe [Var(f)]2/3. Optionally this energy scale can be deduced
consider the distribution of Eq.(2) for which both estimates by dimensional analysis. In the Lifshitz tail region the
coincide identically, therefore a small shift has been inserted inverse localization length has finite value γ √ǫc. An
∼
artificially so the two lines could be discern. The inverse lo- analytical expression can be derived using white-noise
calization length γ is over-estimated as k becomes larger due approximation(see Appendix A for details):
to the Lifshitz tail anomaly. The solid blue line is based on
Eq.(24) with no fittingparameters. γ = 1Var(f) 13 2Var(f)2 −31 k2 (19)
2 K
(cid:18) (cid:19) (cid:20)(cid:16) (cid:17) (cid:21)
6
OutsideoftheLifshitz tailregionthisexpressionreduces same result. But for random distance disorder the two
back to Eq.(18). To be more precise, if we want to take prescriptions differ enormously. This is demonstrate in
the exact dispersion into account an add-hock improve- Fig.4 and Fig.5, were we present our numerical results
ment of Eq.(19) would be to to replace k2 by [sin(k)]2. together with the theoretical predictions.
But our interest is in small k values, for which this im- Having adoptedthe reviseddefinition Eq.(23), we still
provementisnot requiredinpractice: this has beencon- seeinFig.4andFig.5thattheinverselocalizationlength
firmed numerically (not displayed). γ isover-estimatedaskbecomeslarger. Wecantracethe
origin of this discrepancy to the Lifshitz anomaly in the
Anderson model. The condition ǫ ǫ <ǫ translates
0 c
X. DUALITY into λ>w−3[Var(1/w)]−2. Thus|th−e an|omaly develops
0
not at the band floor but as we go up in ω, where the
Wenowturntoconsidertheglassydisorderduetothe inverse localization length becomes γ ω4/3 instead of
∝
dispersion of the w . Here we cannot trust the Born ap- γ ω2. To verify that this is indeed the explanation for
n
∝
proximation because a small parameter is absent. How- thedeviationwebaseourcalculationonEq.(19),namely
ever, without any approximation we can write the map
in the form 1 −1
1 σ 2 3 σ 4 3
1 λ γ ⊥ k4 2 ⊥ k2 (24)
rn+1 = Rn 1− r +1− w (20) ≈ 2(cid:18)w0(cid:19) ! K (cid:18)w0(cid:19) ! 
(cid:18) n(cid:19) n
 
For λ=0 the zero momentum state is a solution as ex-
The anomalyappearswheneverthe argumentof (E) is
pected, irrespective of the disorder: the randomness K
small, meaning large k ratherthan small k. The validity
in R is not effective in destroying the r =1 fixed
n of this formula is numerically established in Fig.4 and
point. Therefore, for small λ, we can set without
Fig.5withnofitting parameters. We note thata slightly
much error R =1. The formal argument that jus-
n better version of Eq.(24) can be obtained by replacing
tifies this approximation is based on the linearization
the ks by appropriatetrigonometricfunctions as implied
(r 1)=R (r 1), and the observation that the
n+1− n n− bytheremarkafterEq.(19)andEq.(22). Butthenumer-
product R R R ... remains of order unity. In Fig.4 and
1 2 3 ical accuracy is barely affected by such an improvement.
Fig.5 we verify numerically that setting R =1 does not
n
affect the determination of γ.
HavingestablishedthatEq.(20)withR =1inavalid
n
approximation,werealizethatitreducestoEq.(16),with XI. THE AVERAGE TRANSMISSION
zero-averagerandom term
1 1 For the calculation of the heat transport we have to
f = λ (21)
n knowwhatis g(ω) g . Givenγ the commonapprox-
− w − w ω
(cid:18) n 0(cid:19) imation is g e−≡(1/h2)iγL. But in-fact this asymptotic
This random term corresponds to the diagonal-disorder approximathioin≈canbetrustedonlyforverylongsamples.
of the standard Anderson model. Consequently, the im- Moregenerally,assumingweakdisorder,thefollowingre-
plieddefinitionofǫviaEq.(17)justifiestheidentification sult can be derived [19–21]:
of the harmonic averagew as the effective coupling.
0
naWmeelyo,bstehreveditshpaetrstihoenreoifsfa,nwemhiechrgeisntpsrmopaollrtpiaornaaml etoterλ, g = ∞du 2πutanh(πu) e−[(41+u2)γL] (25)
h i cosh(πu)
irrespective of the glassiness. Thus we have deduced a Z0
dualitybetween“strong”glassyoff-diagonaldisorderand
The question arises whether this relation can be trusted
the “weak”diagonaldisorder. In the context of the dual
also in the case of a glassy disorder, where the one-
problemwecanusetheBornapproximationEq.(18)with
parameter scaling assumption cannot be justified. This
2 is tested in Fig.6. For weak disorder (large s) the ex-
k 1
Var(f) = 2sin w2Var (22) pected relation between the first and second moments
2 0 w
(cid:20) (cid:18) (cid:19)(cid:21) (cid:18) (cid:19) of x = ln(g) is confirmed, namely Var(x)=2 x . For
− h i
leading to Eq.(15) but with twoimportantmodifications strongglassy disorder(small s) clear deviationfrom this
withrespecttoFGR-basedderivation: (i)werealizethat relation is observed.
w0 shouldbetheharmonicaverage,asconjecturedinthe Still we see in Fig.6 (lower panel) that the failure
introduction; (ii) we realize that the dispersion for off- of one-parameter scaling is not alarming for g =
diagonal disorder should be re-defined as follows: exp[ x] . The exact calculation of the integralhisithe
h − i
solid line, while the asymptotic result exp[ (1/2) x ] is
1
σ2 := w4 Var (23) indicated by dashed line. Note that the l−atter imh pilies
⊥ 0 w
(cid:18) (cid:19) g =exp[(1/4) ln(g) ]. We realize that the asymptotic
Forlog-boxdistributiontheFGRdefinitionσ2 =Var(w) ahpiproximation mh ightibe poor, but the exact calculation
⊥
and the revised definition Eq.(23) provide exactly the using Eq.(25) is quite satisfactory.
7
50 1
Actual
Debye
0.8
40 G=1
1/4
L
g)) 30 0.6 L1/2
( G
n
(l 0.4
r 20
a
V
0.2
10
0
0 0 20 40 60 80 100
0 5 10 15 20 L
−
hln(g)i
4
12
3
10
G
8
) 2 2
gi / Actual
(h 6 1L Debye
n
l
− G=1
4 1
1/4
L
2 L1/2
0
0 2 4 6
0
0 5 10 15 20 25 L1/4
−hln(g)i
FIG. 7. The heat conductance G as a function of L is cal-
FIG. 6. Testing one parameter scaling for glassy disorder. culated using Eq.(28) with Eq.(25). For the black solid line
Thevarianceofln(g)(upperpanel)andthelogoftheaverage with symbols we use γ(k) that is based on the numerical re-
ln g (lowerpanel)areplottedagainstthescalingparameter sultsthathavebeenobtainedforarandombarrierdisorderas
− h i
ln(g) . The calculation is done for the random distance in Fig.4 with σ=1. Forthebluesolid line weusetheDebye
−h i
disorder of Eq.(3) with s=15 (blue diamonds) and s=1.2 approximation for the density of states, and extrapolate the
(red squares). The solid line is the standard one parameter initial γ k2 dependence up to the cutoff k = π. On the
scaling prediction for weak disorder, and the dashed line is lower pan∝el we plot √LG as a function of L1/4 in order to
itsasymptoticapproximation. Oneobservesthatananomaly highlight the α=1/2 (dotted red) and the α=1/4 (dashed
develops as thedisorder becomes glassy. red) asymptotic dependence for long and short samples re-
spectively. The black dotted line is G=1.
XII. HEAT CONDUCTANCE
theleftandrightleads. Inanalogytomesoscopicsstudies
[2, 9] one can argue that
Following [2, 9, 11] the expression for the rate of heat
flow from a lead that has temperature T to a lead that (ω) g(ω) (0)(ω) (27)
H
T ≈ T
has temperature T is
C where (0)(ω) refers to a non-disordered sample, and
T
T T ∞ dω g(ω) is the disordered averaged transmission. For “fixed
Q˙ = H− C (ω) G(TH TC) (26) boundary conditions” (0)(ω) η2ω2, where the damp-
2 Z0 π T ≡ − ing rate η characterizTes the co∼nta0ct point. In contrast,
0
Here (ω) is a complicated expression that reflects the for “free boundary condition” one obtains (0)(ω) 1,
T T ≈
transmissionofthesample. Ifweweredealingwithinco- which is the most optimal possibility. In the latter case
herent or non-linear transport [22], it would be possible c π dk
to justify the Ohmic expression (ω)=ℓ/L, where ℓ is G = g(ω ) (28)
k
T 2 π
the inelastic mean free path. But we are dealing with an Z0
isolatedharmonicchain,therefore (ω)isdeterminedby The standard approach is to use two incompatible ap-
T
the couplings of the eigenmodes to the heat reservoirsat proximations: On the one hand one use the asymptotic
8
estimate g(ω) e−(1/2)γ(ω)L which holds for long sam- XIV. CONCLUSIONS
∼
ples for which γL 1. On the other hand one extends
≫
theupperlimitoftheintegrationtoinfinity,arguingthat We have considered in this work the problem of heat
the major contribution to the integral comes from small conduction of quasi one-dimensional (b 1) as well as
ω values. In the absence of pinning γ(ω) ω2, hence single channel harmonic chains; addres≫sing the effects
∝
by rescalingof the dummy integrationvariable it follows of both glassy disorder (couplings) and pinning (diag-
that the result of the integral is precisely 1/√L. We onal disorder). We were able to provide a theory for the
∝
shalldiscussinthenextparagraphthelimitationsofthis asymptoticexponentiallength(L)andbandwidth(b)de-
prediction. Going on with the same logic we can ask pendence; as well as for the algebraic L dependence in
what happens in the presence of a weak pinning poten- the absence of a pinning potential. A major observation
tial. Using a saddle-point estimate we get along the way was the duality between glassy disorder
and weak Anderson disorder. That helped us to figure
out what is the effective hopping w , hence establish-
1 0
G exp[ (1/2)γ L] (29) ing a relation to percolation theory. It also helped us
0
∼ √L − to go beyond the naive Born approximationthat cannot
be justified for glassy disorder, and to identify a (dual)
Lifshitz anomaly in the ω dependence of the transmis-
where γ is the minimal value of γ(ω). From Eq.(15)
0 sion. Finally, we have established that known results
withEq.(14)wededuceγ b−η,withη =7/2. Thisex-
0 ∝ from one-parameter scaling theory can be utilized in or-
plains the leading exponential dependence on the length
der to derive the non-asymptotic L dependence of the
that has been observed in [17]. However the above cal-
heat conductance.
culationfailsinexplainingthesub-leadingLdependence
that survives in the absence of pinning. Namely it has
Note added after acceptance.– It has been ob-
been observed that instead of 1/Lα with α = 1/2 the
servedin[25]thattheone-dimensionallocalizationprob-
numerical results are characterizedby the super-optimal
lem with the distribution Eq.(3) describes in a univer-
value α 1/4.
≈ sal way the phononic excitations of a one-dimensional
Bose-Einsteincondensate in a random potential: the su-
perfluid regime is percolating (s>1), while the Mott-
insulator regime corresponds to the “s=0” random-
barrierdistributionEq.(2). Thelocalizationlengthinthe
XIII. BEYOND THE ASYMPTOTIC ESTIMATE
anomalous regime 1<s<2 had been worked out in [26],
leading to γ ωs, while γ ω2 applies if s>2.
∝ ∝
We now focus on the L dependence that survives in
Acknowledgements.– We thank BorisShapiro(Tech-
the absence of a pinning potential. As already note that
deviation from the 1/√L law is related to two incom- nion) for helpful comments. This researchhas been sup-
ported by by the Israel Science Foundation (grant No.
patible approximations regarding the γ dependence of g
29/11),and by the NSF Grant No. DMR-1306984.
and the upper limit of of the integration in Eq.(28). We
can of course do better by using the analytical expres-
sion Eq.(25). For the density of states we can use ei- Appendix A: Localization in white noise potential
thernumericalresultsoroptionallywecanusetheDebye
approximation. The latter may affect the results quan-
Thereisananalyticalexpressionforthecountingfunc-
titatively but not qualitatively. Within the framework
tion of the energy-levels for a particle in a white-noise
of the Debye approximation we assume idealized depen-
disordered potential. We cite Eq(1.62) of [23]:
dence γ(ω ) σ2k2 in accordance with Eq.(15), up to
the cutoffkat∝k =⊥π. The result of the calculation is pre- 1 −1
(E) = [Ai( 2E)]2+[Bi( 2E)]2 (A1)
sentedinFig.7. Onthelowerpanelthereweplot√LGas N π2 − −
a function of L1/4 in order to highlight the α=1/2 and (cid:16) (cid:17)
In this expressionthe levels are counted per unit length;
the α=1/4 asymptotic dependence for long and short
AiandBiaretheAiryfunctions; andtheenergyE isex-
samples respectively. Note that in the latter case, as in pressedinscaledunits. Itreducesto (E)=(2E)1/2/π,
0
[17], a small offset has been included in the fitting pro- N
as expected, in the limit of zero disorder. Here we use
cedure. We do not think that the α=1/4 dependence
Eq.(A1) as an approximationthat applies at the bottom
is “fundamental”. The important message here is that
of the band, where the scaled energy is
a straightforward application of an analytical approach
canexplainthefailureofthe1/√Llaw. Wealsoseethat [Var(v )]2 −1/3
n
the numericalprefactor ofthe 1/Lα dependence is sensi- E = 2 λ (A2)
w
tivetotheline-shapeofthe largek cutoff,henceitisnot (cid:20) 0 (cid:21)
the same for the numerical spectrum and for its Debye The inverselocalizationlength is relatedto the count-
approximation. ing function by the Thouless relation, namely Eq(19) of
9
0.4 [24]. Note that the subsequent formulas there are con-
fused as far as units are concerned. In order to remedy
this confusion we define
0.3
(z) (z)
0
(E) = N −N dz (A3)
) K E z
E Z −
0.2 and write the Thouless relation as follows:
(
K
Var(v ) 1/3 [Var(v )]2 −1/3
n n
γ = 2 λ (A4)
0.1 (cid:18) 2w02 (cid:19) K"(cid:18) w0 (cid:19) #
For large E one obtains (E) 1/(8E) and the Born
K ≈
approximationis recovered:
0
0 1 2 3 4
Var(v ) Var(v )
E γ = n = n (A5)
8w λ 8w2k2
0 0
FIG. 8. The function (E) calculated numerically (sym- For practical purpose we find that an excellent approxi-
K
bols). The solid and dashed lines are our approximation mation for any E >0 is provided by
Eq.(A6) and theasymptotic 1/(8E) respectively.
E 1
(E) 1 exp (A6)
K ≈ − −E 8E
(cid:20) (cid:18) 0(cid:19)(cid:21)
where E =0.35. See Fig.8 for demonstration.
0
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