Diffusive transport of waves in a periodic waveguide

1.8 MB·English

by  Felipe Barra

The diffusive transport of waves in a periodic waveguide
Felipe Barra and Jaime Zun˜iga
Departamento de F´ısica, Facultad de Ciencias F´ısicas y Matem´aticas,
Universidad de Chile, Casilla 487-3, Santiago Chile
Vincent Pagneux
Laboratoire d’Acoustique de l’Universit´e de Maine, UMR CNRS 6613
Avenue Olivier Messiaen, 72085 Le Mans Cedex 9, France
(Dated: January 11, 2013)
Westudythepropagationofwavesinquasi-one-dimensionalfiniteperiodicsystemswhoseclassical
(ray)dynamicsisdiffusive. ByconsideringarandommatrixmodelforachainofLidenticalchaotic
2
cavities, we show that its average conductance as a function of L displays an ohmic behavior even
1
thoughthesystemhasnodisorder. Thisbehavior,withanaverageconductancedecayN/L,whereN
0 √
2 is the number of propagating modes in the leads that connect the caviti√es, holds for 1(cid:28)L<∼ N.
After this regime, the average conductance saturates at a value of O( N) given by the average
n
numberofpropagatingBlochmodes(cid:104)N (cid:105)oftheinfinitechain. Wealsostudytheweaklocalization
a B
correction and conductance distribution, and characterize its behavior as the system undergoes
J
the transition from diffusive to Bloch-ballistic. These predictions are tested in a periodic cosine
3
waveguide.
2

PACSnumbers:
]
n
n
I. INTRODUCTION phononiccrystals[7]manyinterestingpropertiesofwaves
-
s in periodic media have been found [8]. In this work we
i
d Wave propagation in periodic media has been stud- will consider a new one, namely the existence of a diffu-
. sive regime for waves in periodic media, a property usu-
t ied quantitatively at least from the advent of quantum
a ally associated to disordered systems. In fact, it is not
mechanics and the quantum theory of solids. Bloch-
m
always appreciated that the diffusive regime is a semi-
Floquet theorem, the underlying theoretical tool, allows
- classical property of some chaotic systems and disorder
to identify the propagating and non-propagating states
d
is not essential for its appearance.
n that form bands as functions of the quasi-momentum.
o The group velocity of the propagating waves is given by
There exists a wealth of literature where the classical
c the derivatives of the energy bands with respect to the
(ray) dynamics of particles in periodic billiards is stud-
[
quasi-momentum, thus explaining the ballistic character
iedinrelationtotransportprocesses. TheLorentzchan-

2 of the Bloch states [1]. Another well studied subject is nel[9]isprobablythebestknownexamplebecauseisone
v wave propagation in disordered systems [2]. Here, three
of the few cases where hyperbolicity, the mathematical
5 regimes are usually recognized depending on the system
expression of hard chaos, has been proved. The Lorentz
3 size L. When L is smaller than the mean free path (cid:96)ˆ
channel consists in a quasi-one-dimensional region pop-
3
0 the propagation is in the ballistic regime, if (cid:96)ˆ(cid:28) L (cid:28) ξ, ulated by hard wall disks placed regularly in a lattice.
. with ξ the localization length, the system is in the dif- Particlestravelfreelyexceptfortheelasticcollisionswith
2
fusive regime and if ξ (cid:28) L it is in the localized regime. the obstacles. If the geometry of the lattice is such that
0
1 In three dimensional systems ξ could be either finite or therearenotrajectoriesallowedtotravelinfinitelywith-
1 infinite and the transition between these two regimes is out collisions, then an initial density of trajectories will
: called the Anderson transition. In waveguides (quasi- spread such that its variance grows proportional to time
v
one-dimensionalsystems)[3]ξ ∼N(cid:96)ˆwithN thenumber t, i.e. it will exhibit normal diffusion. Otherwise, diffu-
i
X of modes in the scattering leads, so the diffusive regime sion is anomalous [10] and the particle density variance
r can be observed in the semiclassical limit. The conduc- grows as tlogt. In this work we always assume normal
a
tance is a natural quantity [4] to study these wave prop- diffusive dynamics. The essential ingredient for this dif-
erties in electronic, optical and acoustical systems. In fusive behavior is the chaotic dynamics of the particle in
disordered systems, its scaling with L is such that, in the billiard unit cell. Hence, a natural question to ask
the ballistic regime it is independent of L, in the diffu- istowhatextentthisclassicaldiffusivebehaviorappears
siveregimeitscalesas1/Landinthelocalizedregimeit in the wave transport properties of periodic systems, es-
decreases exponentially with L. While the ballistic and pecially when we think in the contrast between the well
diffusive regimes have been observed in electronic sys- knownballisticwavepropagationofBlochstatesandthe
tems (for a review see [5]), localization has been more diffusive character of the classical dynamics. One ap-
elusivebutrecently[6]hasbeenexperimentallyobserved proach to this problem is the one considered in [11] were
with acoustic waves. they studied the dependence of spectral statistics on the
Since the experimental realization of photonic and diffusion coefficient of a ring of L identical chaotic cells.
2
Another approach is to study the time evolution of a The number of cells in the slab is L and we choose the x
wave-packet[12]. Unfortunately,thisisnumericallydiffi- direction as the waveguide axis.
cult if we are interested in the semiclassical regime. One A natural way to describe the wavefunction φ(x,y) in
can simplify this task by considering model systems like the waveguide is to project it on the local transverse ba-
the spatially extended multibacker map [12], where the sis, this is, writing the wavefunction as
diffusivetoballistictransitionwasexhibitedinthemean
∞
square displacement. φ(x,y)= (cid:88)(cid:0)c+(x)+c−(x)(cid:1) ρ (x,y), (2)
In this work, we consider a time-independent ap- n n n
proachbasedonthescatteringmatrixandfocusonfinite n=1
quasi-one-dimensional periodic systems. Besides solv- where ρ (x,y) are the local transverse modes which
n
ing the wave equation in a particular waveguide system, satisfy the boundary conditions on each x, and c+(x)
n
namely the cosine billiard, we use Random Matrix The- (c−(x)) is the right-going (left-going) longitudinal mode.
n
ory (RMT), which has been successfully used to model In the particular case of a hard-wall waveguide,
the wave properties of chaotic cavities [13] and of dis-
ordered wires, and also to study spectral properties of (cid:115) (cid:18) (cid:19)
2 y−h (x)
extended systems [11]. Here, we employ RMT to study ρ (x,y)= sin nπ 1 , (3)
n h(x) h (x)−h (x)
scattering in chaotic periodic waveguides. Since the unit 2 1
cell of the periodic billiard is an open chaotic cavity, its
n = 1,...,∞, where h (x) < h (x) are the walls height
scattering matrix can be modeled by elements of the 1 2
as a function of the longitudinal coordinate x. This
Dyson circular ensembles [13]. Using this matrix, the
set of functions satisfies the null boundary conditions
scattering matrix of the slab with L identical cells is
ρ (x,h (x))=ρ (x,h (x))=0 everywhere in the guide.
constructed,fromwherephysicalproperties,likeconduc- n 1 n 2
The longitudinal modes are obtained by inserting (2)
tance,canbecomputed[14]. Averagingthisconductance
in (1), which transforms the original partial differential
overtheappropriateDysonensembleofrandommatrices
equation into a system of coupled ordinary differential
we can obtain a prediction for the average conductance
equationswhichcanbeefficientlysolvednumerically[16].
ofaperiodicwaveguidecomposedofgenericchaoticcav-
Inaplanelead, h (x)andh (x)areconstant, so(1)is
ities, which we verify in the cosine billiard. 1 2
separable since ρ (x,y) = ρ (y) is independent of x. In
The plan of the paper is the following. In Sec. II we n n
this region, the longitudinal modes c±(x) are given by
review the basic tools to analyze scattering in waveg-
uides, introduce the so called cosine periodic billiard
e±iknx
and a random matrix model for periodic waveguides. e±(x)= √ , (4)
In Sec. III, we show numerically in the random matrix n kn
model and in the cosine billiard that for system length
1 (cid:28) L <∼ √N the average conductance behaves diffu- where kn2 =k2−(nπ/W)2 is the longitudinal wavenum-
ber,W =h −h istheleadwidthandthenormalization
sively, i.e. as N/(L + 1) and at a length of the order 2 1
√
is to impose unit flux. In this region, there are
N, the so-called diffusive-Bloch ballistic transition oc-
cursandtheaverageconductancesaturatestoaconstant (cid:22) (cid:23)
Wk
value. In two different subsections, we analyze conduc- N = (5)
π
tancefluctuationsandweaklocalizationcorrectionasthe
systems undergoes the transition from diffusive to Bloch
propagating modes because for n > N the longitudinal
ballistic. Finally, in Sec. IV we offer some conclusions.
wavenumber k is imaginary, implying null flux. These
n
arecalledevanescentmodesanddecayexponentiallywith
x. The far field wavefunction in the leads can be de-
II. WAVEGUIDE SYSTEM scribedwitha2N dimensionalcomplexvectorcomposed
of coefficients A and B for n = 1,...,N and we can
n n
Inthispaper,thephysicalproblemweaddressregards write the wave function as
scattering in a waveguide composed of a slab made of a
N
finite periodic chain of two dimensional chaotic cavities φ(x,y)= (cid:88)(cid:0)A e+(x)+B e−(x)(cid:1) ρ (x,y). (6)
connected by leads [see Figs. 1 and 2]. The leftmost and n n n n n
n=1
rightmost leads extend to x going to minus and plus in-
finity. ThewavefunctionφisgovernedbytheHelmholtz Let Ar (Br) be the N dimensional complex vector of
equation right-going (left-going) amplitudes in the right lead and
Al (Bl) the same on the left lead. We denote the in-
∇2φ+k2φ=0, (1) comingandoutgoing(orincidentandscattered)fieldsin
vector notation as
with Dirichlet boundary conditions at the walls, and
(cid:18) (cid:19) (cid:18) (cid:19)
where k is the wavenumber. Experimental realizations Al Bl
ψ = and ψ = . (7)
of this system can be built with microwave cavities [15]. in Br out Ar
3
The scattering matrix S is defined as the linear transfor-
mation that maps incoming to outgoing fields,
ψ =Sψ , (8)
out in
and has a block structure,
(cid:18)r t(cid:48) (cid:19)
S = (9)
t r(cid:48)
with r the left (r(cid:48) the right) reflection matrix and t the FIG. 1: Cosine billiard chain with five unit cells connected
to two plane leads. The unit cell boundaries are defined in
left to right (t(cid:48) the right to left) transmission matrix,
(12)–(13) as a function of the amplitudes A and A shown
eachofthemofdimensionN×N. Thescatteringmatrix 1 2
in the figure. The width W of the leads is A
1
of a chain of L identical cavities (such as Fig. 2) can be
obtained from a standard concatenation rule [17], start-
ing from the knowledge of the unit cell scattering matrix
speed v is constant. Note that our cosine billiard al-
i.e. S =f (S )withf (S )=S ,whereS andS
L L uc 1 uc uc uc L ways has finite horizon, i.e. it does not allow unbounded
are the unit cell and L-cells scattering matrices, respec-
collision-free trajectories for any values of A > 0 and
1
tively. It is also useful to consider the transfer matrix
A >0, and is chaotic choosing these parameters appro-
2
M , which maps the field on the left lead ψ to the field
L l priately [20]. We always consider configurations display-
in the right lead ψ =M ψ, where
r L l ing strongly chaotic dynamics such that classical parti-
(cid:18) Ar (cid:19) (cid:18)Bl (cid:19) cles in the cavity follow a normal diffusion process, i.e.
ψr = Br and ψl = Al . (10) x2 ∼Dt,wheretheaverage(·)iscomputedforeachtime
t over an initially spatially-bounded ensemble of initial
The concatenation rule for transfer matrices is a simple conditions [20] with random velocity.
multiplication, so if Muc is the transfer matrix of the Wenotethattheunitcellmirrorsymmetryx→−xis
unit cell, then ML = (Muc)L and if λi are the eigen- not relevant for the classical transport properties of the
values of Muc then λLi are the eigenvalues of ML. The billiard but makes the numerical solution of the quan-
consequence of this for the infinite 1D periodic systems tum scattering problem faster (the transmission and re-
arewellknown[18]. Sincethewavefunctionoftheinfinite flexion matrices t and r are the same in both direction).
periodic system must remain bounded along the chain, However, this induces an anti-unitary symmetry in the
the only allowed states are those associated to the eigen- quantumHamiltonianwhichplaysaroleinthestatistical
values that satisfies |λi|=1. The number of these states andtransportpropertiesofthewaveguideaswediscussed
iscalledthenumberofpropagatingBlochstates[19]and in [20].
will be denoted by 2N (k). Since in this case we can
B We will solve numerically the scattering wave problem
write λ = eiθ(k), we can invert the relation and obtain
i in this billiard as a function of L and compute the con-
the allowed energy bands k =k (θ).
n ductance (11). In general, all quantities dependent on
As we have already mentioned, in this paper we are
the M spectrum (in particular g (L)) are highly fluc-
uc k
interested in the transport properties of finite periodic
tuating as a function of k over variations of the order of
systems, in particular in the dimensionless conductance
the(unitcell)meanlevelspacing[11]. Averagingoverak
of a chain with L cells, which can be obtained directly
interval several mean level spacings wide gives a smooth
from thetransmission part ofthe S matrix bythe Lan-
L function that changes on much larger k scales. We call
dauer formula [14],
the ensemble of wavenumbers (or energies) realizations
g (L)=tr[t t †]. (11) in this interval the semiclassical ensemble. In the fol-
k L L
lowing sections, we will compute these averages for the
cosine billiard and the results will be compared with the
A. Cosine waveguide predictions that follows from the RMT periodic waveg-
uide model that we discuss below. If the typical size of
the cavity is much larger than the leads width W, then
As our particular model we will employ the periodic
it is possible to define the semiclassical ensemble over
cosine billiard. We define the unit cell as the region en-
a wavenumber interval such that N is constant. This
closed by h (x)<y <h (x) for each x∈[−1,1], where
1 2 assumption is important to make the connection to our
h (x) = A1 [1+cos(πx)] and (12) RMT model.
1 2
A
h (x) = A + 2 [1+cos(πx)]. (13)
2 1 2 B. RMT periodic waveguide model
The classical limit of (1) corresponds to noninteracting
freeparticleswithinthesystem;collisionsagainstthebil- The RMT model is constructed by taking a 2N ×2N
liard boundaries h and h are elastic, thus the particle random matrix from the Dyson Circular Orthogonal
1 2
4
y Ti
otaFlw[aiWannnaIindGedrakcdtu.re/hrxseni.π2tgalW:h]heATestmAsmcnhsowueouasdemsccnelthhdessbepaleemtttderahhhcsadaeaoitfsttftiylhcecepiniaefxtrpglttoltlotesophhctntLaehodogneoiwfarntfswtotaefieahvcisxnsetepeinaut.metulenrhoFimi(inenoothubgdreccseithretcrhealhalecneoiLishmdtsaai=nkca.pianlwc4tlTuyha)oshhs,vefiieiritscntcehhihocfieeafaanwsvlorcieeetithoftyitnsayc,Nmevdathbeuonatisc=ldoast-- 00001.....24680 (cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:224)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)
(cid:224)
we deal with a scattering system. (cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224)(cid:224) L
10 20 30 40 50 60
(Unitary)EnsembleCOE(CUE)andusingitasthescat- FIG. 3: The first five transmission eigenvalues T (L) in a
i
tering matrix Suc of a chaotic unit cell with (without) cosine billiard with A1 =0.5 and A2 =4.5 for k=30.2157π.
time reversal invariance. Modeling the scattering matrix Forthisenergy,NB=4andtheassociatedTi areplottedwith
of a chaotic cavity by elements of these ensembles is a circles. In addition, the slowest to decay evanescent mode is
also plotted with squares. It can be seen that the ballistic
common method [21] known to be accurate in the pre-
transmission modes tend to spread in the [0,1] interval and
diction of statistical properties. The connection to the
therefore (cid:104)g(L)(cid:105)<N in the generic case.
physical system is made by replacing k dependent quan- B
tities by RMT realization dependent quantities. The av-
eragesovertheRMTensembletaketheplaceofaverages
They are related to the eigenvalues {Λ (L),Λ (L)−1}N
overthesemiclassicalensemble. UsingtheSmatrixcom- i i i=1
of the 2N ×2N matrix M M† by
position rule SL =fL(Suc), the scattering matrix of the L L
L cells connected by leads with N modes is obtained.
4
Thus, we have a RMT ensemble for periodic chains of L T (L)= , i=1,...,N, (16)
chaotic cavities. i 2+Λi(L)+Λ−i 1(L)
If we denote by µ the Dyson measure of the RMT en-
sembleandnotethattheconductance(11)isafunctional arelationthatfollowsfromthepolardecomposition[23].
of S , we can express the averaged conductance of the Tosimplifynotation,wedroptheexplicitlykdependence
L
RMT model as in all quantities below; we note that N =[Wk/π] is also
(cid:90) k dependent.
(cid:104)g (L)(cid:105)= dµ(S )g[S (S )] (14) Theconductance(15)–andingeneralanyothertrans-
N uc L uc
port property dependent of the transmission eigenvalues
We perform this computation numerically. T – is a function of M M† eigenvalues. In view of the
i L L
Toendthissectionwewouldliketoremarkthatmod- simpledescriptionoftheinfiniteperiodicsysteminterms
eling a chaotic cavity by RMT is justified if the particle ofallowedandforbiddenstatesitisinterestingtolinkthe
stay trapped inside the cavity for a long time [21]. To eigenvalues {Λ (L),Λ (L)−1}N to the eigenvalues λ of
i i i=1 i
bemoreprecise,theRMTmodelassumesthattheparti- M . Oseledets theorem [22] provides us such relation,
uc
cle escape time is much longer than its correlation decay given by [23]
characteristic time, so the particle effectively undergoes
a random walk between unit cells. Therefore, the case Λ (L) −→ a (L)e−2Llog|λi|, (17)
i i
of anomalous diffusion mentioned in the introduction is L→∞
excluded from our analysis. In order to study this case,
wherea (L)isapositiveand(generically)boundedfunc-
the Dyson ensembles should be properly modified. i
tion of L. Then, using relation (16), we can decompose
g(L) [Eq.(15)] in two terms, one with the sum of the N
B
III. CONDUCTANCE OF A FINITE PERIODIC non-decayingtransmissionmodesTi relatedbyEqs.(16)
CHAIN OF CHAOTIC CAVITIES and (17) to the 2NB propagating Bloch modes |λi| = 1
(which we choose to have indices i = 1,...,N ), and
B
another with the sum of the transmission modes related
Landauer’s formula can be written more explicitly as
to evanescent Bloch states |λ | (cid:54)= 1 (which we choose to
i
g (L)=tr[t t †]=(cid:88)N T (L), (15) have indices i=NB+1,...,N). The second term has a
k L L i decay length
i=1
where {T (L)}N are the N eigenvalues of the N ×N (cid:18) (cid:19)−1
i i=1 (cid:96)= min {log|λ |} , (18)
matrixt t † whichareboundedintherealinterval[0,1]. i
L L |λi|>1
5
determined by the slowest to decay non-propagating From P(T ,...,T ;L) it is possible to obtain all the de-
1 N
state. From (17) and (16) we deduce that the N trans- sired statistical information of the conductance, which
B
(cid:82)
mission eigenvalues T associated to Bloch modes is of can be written as gˆ(L) = dTP (T)T with P (T) =
i L L
order one, so for chains of length L>∼(cid:96), N (cid:82) dT2...dTNP(T,T2,...,TN;L)(withN anormaliza-
tion constant ). From the solution of the DMPK equa-
g(L)<∼NB +4am(L)e−2L/(cid:96), (19) tion, it is possible to show that the conductance behaves
diffusivelyi.e. asgˆ(L)=N(cid:96)ˆ/(L+1)for(cid:96)ˆ<L<(cid:96)ˆN,and
where am is the Oseledet function ai associated to the then, for L > (cid:96)ˆN, as gˆ(L) ∝ exp(−L/2βN(cid:96)ˆ) with β = 1
mode with decay length (cid:96). The equality in (19) is non-
( β =2) for systems with (without) time reversal invari-
generic and occurs if Muc is a normal matrix (as we ance and (cid:96)ˆthe mean free path in the disordered medium
discuss below), in which case T (L) = 1 for all Bloch
i uptoanumericalconstant[23]. Fromthisanalysisitalso
modes. More generally, for chains of length L (cid:28) (cid:96), all
followsthatthelocalizationlengthforthedisorderedwire
transmission modes contribute to the conductance,
is (cid:96)ˆN. It is also possible to characterize the fluctuations
forinstancetogiveanexplicitexpressionforP (T). For
g(L)=NBP(L)+ (cid:88)N 2+Λ (L)4+Λ−1(L) (20) instancePL(T)= N2L(cid:96)ˆT√11−T,thesocalledDoroLkhovdis-
i=NB+1 i i tribution,inthediffusiveregime(cid:96)ˆ<L<(cid:96)ˆN. Thisresult
is linked to the distribution of the localization lengths
with 0 < P(L) < 1 an O(1) quasi-periodic function of
spectrum(cid:96)ˆ =ln|λˆ |definedfromtheeigenvaluesofthe
L. The function P(L) takes into account the fact that n n
there exists a repulsion between the eigenvalues Ti(L) disordered M matrix satisfying |λˆn| > 1. Acording to
associated to propagative Bloch modes and that these Dorokhov [24] (cid:96)ˆ (N)−1 ∼ n/N(cid:96)ˆfor n = 1,...,N. It is
n
quantities fluctuate quasi-periodically as a function of L interestingtonote,thatasaconsequenceof(cid:96)ˆ−1linearde-
n
[see Fig. 3]. This effect makes the conductance strictly pendenceonn,(16)impliesthatgˆ(L)∼1/Lintheinter-
lower than NB in the generic case, and can be thought val (cid:96)ˆ(cid:28)L(cid:28)(cid:96)ˆN. There are other two important results
asageometricconsequenceofM beingnotnormal,i.e.
uc that follow from the DMPK equation. Firstly, there is a
not diagonalizable in a orthogonal base. The transmis-
weaklocalizationcorrection(WLC)thathastobeadded
sion eigenvalues repulsion is also observed in disordered
to gˆ(L) in the metallic (ohmic) regime in time reversal
chains [23].
invariant systems, which for large N is δgˆ(L) = −1/3
We remark that P(L), N and the spectrum λ are
B i independent of L. Secondly, the conductance distribu-
k dependent. Intervals where N (k)=0 corresponds to
B tion in the metallic regime turn out to be Gaussian with
forbiddenbandsandEq.(20)shows,asexpected,thatfor
O(1) variance, independent of the system length and of
kvaluesinaforbiddenband,g(L)decreaseexponentially
the disorder properties (universal conductance fluctua-
with L, while in allowed bands, where N (k) ≥ 1, the
B tions), and then changes from this Gaussian form to a
conductance remains finite as L → ∞. As k increases,
log-normal distribution as the size of the wire reaches
the probability of finding an energy gap decreases, nev-
the localization length.
ertheless gaps play an important role in some statistical
Otherapproacheshavebeenusedtostudytheconduc-
measures [see Fig. 5].
tance of a disordered wire, for instance using a random
matrix theory for the description of the hamiltonian in
thescatteringregion[26],butintheappropriatelimitsof
A. Digression: the Disordered chain
theweaklydisorderedwirethesedescriptionswereshown
tobeequivalent. ItisexpectedthatthedisorderedRMT
WewillcontrastourresultsfortheperiodicRMTchain chain is well described by this model, except perhaps in
with a disordered RMT chain, where the composition of thetransitionbetweenthedifferentregimes. Ournumer-
the scattering matrix is performed each time with a dif- ical results [see Fig.4 (lower panel)] confirm this expec-
ferentrealizationoftheCUEorCOEensemble. Thesta- tation. For instance, a transition from the diffusive to
tistical properties of the conductance of this disordered the localized regime when the system reaches a length
chain are expected to be well described by the statistical L∼O(N) corresponding to the localization length.
properties of the disordered wire which allow an analyti-
calstudy[23]. Oneofthetheoreticalframeworkstostudy
this model is through a Fokker-Planck equation for the
B. Average conductance of the periodic waveguide
probability P(T ,...,T ;L). In this approach, the sys-
1 N
tem size L is a continuous variable and the transmission
coefficients are random variables that evolves stochas- In this section we focus on the average conductance
tically as L increase. There are important hypothesis (cid:104)g(L)(cid:105) and resistance (cid:104)1/g(L)(cid:105), that we compute in the
in the derivation of this Fokker-Plank equation (called semiclassical and RMT ensembles for a periodic waveg-
DMPK equation in this context [24, 25]), for instance, uide. From the decomposition (20) of g(L) as a sum
the disorder is spatially homogeneous and the scattering of propagating and decaying modes, it is clear that for
producedbyashortslabofthewireisisotropicandweak. small values of L we have (cid:104)g(L)(cid:105)∼O(N), as in the dis-
6
orderedcase[23]and,forLlargeenough,(cid:104)g(L)(cid:105)<∼(cid:104)NB(cid:105) (cid:88)g(cid:144)N(cid:92)
is independent of L. The average number of propa- 0.5(cid:231)(cid:225)(cid:168)(cid:243)(cid:245) (cid:88)N(cid:144)g(cid:92)
gating Bloch modes in the infinite periodic waveguide, 50 (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)
(cid:244) (cid:236)
t(cid:104)taNrhsaaBn(cid:104)tNs(cid:105)i,ftBoiwro(cid:105)ant∼shfrseot√umdNdiffite.uhdseTiivbnheae[lrw1lie9saft]ovirceaegnp,udriaod[sp2eas0Lg],waiwtneicohcrneeoran(cid:104)esgseii(dstL,ew)rt(cid:105)ahi∼setrseOshco(iasNwlenas) 0.25 (cid:231)(cid:225)(cid:168)(cid:243)(cid:245) (cid:231)(cid:225)(cid:168)(cid:243)(cid:245) (cid:231)(cid:225)(cid:168)(cid:243)(cid:245) 1234000000(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)5(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)(cid:230)(cid:224)(cid:242)(cid:244)(cid:236)1(cid:224)(cid:242)(cid:244)(cid:236)(cid:224)(cid:242)(cid:244)0(cid:236)(cid:224)(cid:242)(cid:244)(cid:236)(cid:224)(cid:242)(cid:244)(cid:236)(cid:224)(cid:242)(cid:244)(cid:236)1(cid:224)(cid:242)(cid:244)(cid:236)(cid:242)(cid:244)5(cid:236)(cid:242)(cid:244)(cid:236)(cid:242)(cid:244)(cid:236)(cid:242)(cid:244)(cid:236)2(cid:242)(cid:244)(cid:236)(cid:244)0(cid:236)(cid:244)(cid:236)(cid:244)(cid:236)(cid:244)(cid:236)2(cid:236)5(cid:236)(cid:236)(cid:236)30L
through short systems√to the Bloch-ballistic propagation (cid:231)(cid:225)(cid:168)(cid:243)(cid:245)
(cid:104)sgy(sLte)m(cid:105)s<∼. I(cid:104)nNBor(cid:105)de∼r tOo(stNud)ytthhriosugtrhanlosintgionfinwiteecpoemripoudtice 0.1 (cid:231)(cid:225)(cid:168)(cid:243)(cid:245) (cid:231)(cid:225)(cid:168)(cid:243)(cid:245) (cid:231)(cid:225)(cid:168)(cid:243)(cid:245) (cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)(cid:231)
(cid:104)uWgid(eLef)oa(cid:105)cnuadss faionrfuptnahcrettiiocRunMlaoTrf iLmnfotodhreelthlioemfciatosopifnereliaoprdgeiercioNwd,iacvi.eweg.auvitedhgee-. (cid:225)(cid:168)(cid:243)(cid:245) (cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245) L
semiclassical limit. 1 2 5 10 20 50
In Fig. 4 (upper panel) we present the scaled averaged (cid:88)g(cid:144)N(cid:92)
conductance (cid:104)g(L)/N(cid:105) as a function of L for our RMT 0.50 (cid:225)(cid:168)(cid:243)(cid:245)
model with S taken from the COE ensemble and com- (cid:225)(cid:168)(cid:243)(cid:245)
uc
pare it with the characteristic 1/(L + 1) decay of the 0.20 (cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)
gofwoihivltmleohnwic(cid:104)tslgheb(enLegho)ta(cid:105)hhCvmiOOoiErc(√=[b3Ne].h)aWv(cid:104)cNlie(ooLBsroe,NbP+(cid:104)st(geo1Lr()v(cid:104)L)Ne(cid:105))(cid:105)Bt1h(cid:105)=a(cid:28)√wtNNhtLhe/re((cid:28)<∼eLna√Lu+mtNr1ae)rn,iscuiatpliodt(no2a1ttaao) 0000....00011250 (cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)(cid:225)(cid:168)(cid:243)(cid:245)
L
anLindependentregimeisobserved. Thisisthediffusive 0 10 20 30 40 50
to Bloch-ballistic transition. The ohmic regime manifest
itself also in the resistance, FIG. 4: Upper panel: For the periodic RMT chain, plots of
(cid:104)g(L)/N(cid:105) as a function of L for N =10 (dots), 20 (squares),
(cid:28) (cid:29)
1 30 (triangles up), 40 (triangles down) and 50 (diamonds) for
(cid:104)R(cid:105)= , (22)
g(L) the COE case. As N increase, convergence to the linear de-
pendence on 1/(L+1) (line) and the scaling with N of the
which is plotted in the inset of Fig. 4 (upper panel) for diffusiveregimeareclearlyobserved. Thedeparturefromthis
the RMT model and in Fig. 5 for a cosine waveguide. lawtoanLindependentregimesignalsthediffusivetoBloch
Qualitatively, we found Ohm’s law (cid:104)R(cid:105) = (L + 1)/N ballistic transition. In the inset we depict (cid:104)N/g(L)(cid:105) for the
samevaluesofN andthelineL+1predictedbyOhm’slaw.
holds for small L, before (cid:104)g(L)(cid:105) reaches its asymptotic
The divergence of this quantity that follows after the ohmic
value. This regime is followed by localization-like ex-
behaviorisdiscussedinthetext. Lowerpanel: Forthedisor-
ponential grow of the resistance. The latter may seem
dered RMT chain (see sect. IIIA), plots of (cid:104)g(L)/N(cid:105). Sym-
surprising but is a consequence of the non-null proba-
bols correspond to the same matrix dimension as in upper
bility of N = 0. In fact, although this probability de-
B panel (N = 10 not shown). The continuous line for L < 20
cays to zero as N →∞ [27], it dominates in the average representsOhm’slawandforL>20anexponentialdecayas
(cid:104)1/g(L)(cid:105) for long chains. However, in the limit of large expected in the localized regime.
N,forparticularrealizationsof1/g(L)themostprobable
is 1/g(L)=1/(N P(L))<∞ with N (cid:54)=0.
B B
The existence of the ohmic regime is a consequence |λn| of Muc = Muc(Suc) with Suc taken from COE has
of the full MLML† matrix spectrum and it is observed a(cid:104)Nsim(cid:105)i≤larnp(cid:28)ropNer,tyf,ronmamwehliych(cid:96)−nw1e=conlcolgu|dλen|th≈e enx/isNtenfocer
for for lengths L (cid:28) (cid:96), where several non-propagating B
of an ohmic regime, in periodic diffusive waveguides [see
T still contribute to the sum in the right hand side of
i
Fig. 7]. This is a statistical characteristic of the transfer
(20). This is clearly observed in Fig. 6 where the aver-
matrix spectrum, which can be recasted stating that
age conductance of the cosine billiard (filled squares) is
compared to the average of (20) eliminating the second 1 (cid:104)N (cid:105)
termoftherighthandside(filledcircles). Inordertoun- p(|λ|)∼ , for B <log|λ|(cid:28)1 (23)
|λ| N
derstand this ohmic regime we remember (see Sec.IIIA)
that, for the disordered wire, it can be explained by the with p(|λ|) the marginal pdf of the spectrum absolute
scaling of Dorokhov localization lengths spectrum [24] values |λn|.√From this follows that 1/g ∼ L for L <∼
(cid:96)ˆ (N)−1 ∼ n/N(cid:96)ˆ, n = 0,...,N −1. Now, it turns out N/(cid:104)N (cid:105) ∼ N, which can also be understood from the
n B
that,inourmodelforperiodicwaveguides,thespectrum two following arguments.
7
1(cid:144)g(cid:72)L(cid:76) (cid:227)(cid:227)(cid:227) (cid:227) (cid:227) N(cid:123)n(cid:45)1 (cid:245) (cid:168)
3.5 (cid:227) 140 P(cid:72)N(cid:123)(cid:45)1(cid:76) (cid:248) (cid:168)
0.05
3.0 (cid:227) (cid:230) 120 00..0012 (cid:243) (cid:245) (cid:248)(cid:248) (cid:168)(cid:168)(cid:168)
2.5 (cid:227) 100 00..000025 (cid:245) (cid:248)(cid:248) (cid:168)(cid:168)(cid:168)(cid:168)
00112.....050500 (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)5(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) 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(cid:230)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) (cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227) 1(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)5 (cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)(cid:227)L 24680000(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)1(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)1(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)0(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)2(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)3(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)20(cid:248)(cid:168)(cid:231)(cid:225)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)4(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)5(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)6(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)30(cid:248)(cid:168)(cid:225)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:243)(cid:245)(cid:248)(cid:168)7(cid:243)(cid:245)(cid:248)(cid:168)(cid:243)(cid:245)0(cid:248)(cid:168)(cid:243)(cid:245)N(cid:248)(cid:168)(cid:243)(cid:245)(cid:248)(cid:168)(cid:243)(cid:245)(cid:123)(cid:248)(cid:168)(cid:243)(cid:245)(cid:45)(cid:248)(cid:168)(cid:243)(cid:245)(cid:248)(cid:168)(cid:243)(cid:245)14(cid:248)(cid:168)(cid:245)0(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:245)(cid:248)(cid:168)(cid:248)(cid:168)5(cid:248)(cid:168)0(cid:248)(cid:168)(cid:248)(cid:168)(cid:248)(cid:168)(cid:248)(cid:168)(cid:168)(cid:168)(cid:168)(cid:168)(cid:168)60 70 n
FIG. 5: Average resistance (cid:104)1/g(L)(cid:105)k (black dots) for the FIG. 7: Average inverse decay lengths spectrum (cid:96)−n1 =
cosine billiard with A1 = 0.5, A2 = 4.5 and k = 30.33π. log|λn| for the COE periodic chain model. The plot shows
The full line shows the ohmic regime holding for L < 10. N(cid:96)−1 forN =10,20,30,40,50,60,70(circles,squares,uptri-
n
Also, several realizations of 1/g(L) are displayed in gray: a angles,downtriangles,stars,diamonds). Theintegernisthe
fewcaseswithNB =0growexponentiallyandtherestreach absolute-value-sorted λn eigenvalue index. For this system
an asymptotic oscillating value 1/g∞(L) ∼ O(1/NB). The (cid:104)NB(cid:105)=4.7. As can be seen in the plot N(cid:96)−n1 is close to zero
exponentialgrowoftheaverageresistance(cid:104)R(cid:105)k forL>10is for n <∼ (cid:104)NB(cid:105) and is followed by a range with linear growth
explained by the presence of the rare cases with NB =0. N(cid:96)−1 ∼ n (dashed line) which explains the existence of the
n
ohmic regime in the periodic chain for L < (cid:104)N (cid:105)/N. In the
B
inset,partofthepdfP(N(cid:96)−1)forN =70isplotted,showing
7.0(cid:224) a constant range for 10 <∼ N(cid:96)−1 <∼ 30 which is equivalent to
P(|λ|)∼|λ|−1 [see (23)].
5.0 (cid:88)g(cid:72)L(cid:76)(cid:92)
(cid:224)
gate ballistically.
(cid:224)
3.0
(cid:224)
C. Weak localization
(cid:224)
(cid:230)
(cid:224)
(cid:230) (cid:224)
2.0 (cid:88)NBP(cid:72)L(cid:76)(cid:92) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224) weHtauvrinngtoestthaebilsisshueedotfhweeeaxkisltoecnacleizoaftitohne.dSiffoufsairvewreeghiamvee
focused only on time-reversal symmetric systems, which
1.5
1 2 5 10 20 50 L are described by the COE ensemble. On the other hand,
the CUE ensemble models chaotic cavities where time-
FIG. 6: average conductance of the cosine billiard (filled cir- reversal invariance has been broken, for instance, a two-
cles)iscomparedtotheaverageof(20)eliminatingthesecond dimensional degenerate electron gas in a diffusive peri-
term of the right hand side (filled squares). odic structure subject to a perpendicular magnetic field.
For a chaotic quantum dot, i.e. a chain of length L=1,
it is well known that the conductance is given by [5]
First, if we consider that the conductance in the diffu-
sive regime (cid:104)g(L)(cid:105) ≈ N/(1+L) must match its asymp- N (cid:18) 2(cid:19)1
(cid:104)g(L=1)(cid:105)= + 1− , (24)
totic average value (cid:104)g (L)(cid:105)∼(cid:104)N (cid:105), we obtain that this
∞ B 2 β 4
happens at L∼N/(cid:104)N (cid:105).
B
Second, if we consider a wave-packet representing an where β = 1 for the COE case and β = 2 for the CUE
electron in a periodic system, we expect from semiclassi- case. The difference δg(1) = (cid:104)g(1)(cid:105) −(cid:104)g(1)(cid:105) =
COE CUE
calargumentstoobservechaoticdiffusionifmanyenergy −1/4iscalledWeakLocalizationCorrection(WLC)and
bands contribute to the superposition. This is the case can be explained semiclassically by the enhancement of
if the diffusion time tD =L2/D is smaller than the unit- thereflectionprobabilityduetoconstructiveinterference
cell Heisenberg time tH = mAc/(cid:126), where we recall that of time-reversed trajectories in time-reversal invariant
D = D1v, with v the particle’s speed and m the elec- systems [28]. In the disordered wire, the WLC is ob-
tron mas√s. Then, we√obtain that tD < tH if and only server in the metallic (ohmic) regime and is of slightly
if L <∼ AcD1k ∼ N which is the √expected result. larger magnitude with δgˆ(L)=−1/3 independent of L.
For longer times (correspondingly L >∼ N) the energy Now we consider the WLC in a periodic chain. It can
bands are resolved and the wave-packet starts to propa- be assumed that a weak magnetic field will not change
8
∆g 4 P (cid:72)T(cid:76)
L L
5 10 15 20 25 30
4 PL(cid:72)T(cid:76)
(cid:45)(cid:45)00..2250 (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224) (cid:230)(cid:224)(cid:242) (cid:230)(cid:224)(cid:242)(cid:244) (cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) 3 23
(cid:45)0.30 (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:230)(cid:224)(cid:242)(cid:244) (cid:236)(cid:224)(cid:242)(cid:244) (cid:236)(cid:224)(cid:242)(cid:244) (cid:236)(cid:224)(cid:242)(cid:244) (cid:236)(cid:242)(cid:244) (cid:236)(cid:242)(cid:244) (cid:236)(cid:242)(cid:244) (cid:236)(cid:244) (cid:236) (cid:236) Va01r..80(cid:64)g(cid:68)COE 2 1
0.6 N
0.4 1 T
(cid:45)0.35 0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0 L
10 20 30 40 50
T
FIG. 8: For the RMT model, plots of the WLC δg(L) for 0.0 0.2 0.4 0.6 0.8 1.0
N = 10,20,30,40,50 using the same symbols as in Fig. 4.
In the inset we plot the conductance variance Var[g(L)] for
N =1 to N =50 for the COE case.
FIG. 9: Transmission pdf P (T) of a cosine periodic waveg-
L
uide chain model with N = 50 for L = 5 in the diffusive
regime. The continuous line represents Dorokhov’s distribu-
the diffusion coefficient of a chain of strongly chaotic
systems, thus we can extract the WLC in the periodic tion for a disordered wire PL(T) = 2NLT√11−T. Histogram is
computed with the same parameters than Fig. 3. The inset
chain as a function of L analogously to the quantum
showsthehistogramofP (T)forL=25, deepintheBloch-
dot as δg(L) = (cid:104)g(L)(cid:105) − (cid:104)g(L)(cid:105) . In Fig. 8 we L
COE CUE ballistic regime and the continuous line is the corresponding
plotδg(L),whichforanN-dependentL-rangeiscloseto
Dorokhov’s distribution.
theweaklocalizationcorrectionofadisorderedwire[23].
Although weak localization is usually associated to the
metallic regime, here we see that in periodic systems it consider the second order expansion
extends for large L with a correction δg(L) ≈ −0.2 that
persists during the Bloch-ballistic regime. (cid:104)g−1(L)(cid:105) =1−Var[g(L)](cid:104)g(L)(cid:105)−2+O(cid:0)(cid:104)g(L)(cid:105)−3(cid:1). (25)
(cid:104)g(L)(cid:105)−1
Recently, the WLC for a periodic system was stud-
ied [29] assuming that the Ehrenfest time of the cavity
Since (cid:104)g(L)(cid:105) = N/(L + 1) we have that, deep in the
√
islargerthantheergodictime[30]ofthe(closed)cavity.
ohmic regime, where L (cid:28) N < N, Var[g(L)] (cid:28) L
We have in mind the opposite case, where the Ehrenfest
and Var[g(L)](cid:104)g(L)(cid:105)−2 (cid:28) 1. Hence, keeping the domi-
timeissmallerthantheergodictimeensuringthatRMT
nant term in (25) we obtain (cid:104)1/g(L)(cid:105) ∼ 1/(cid:104)g(L)(cid:105). Inset
is a good model for the periodic waveguide.
in Fig. 8 shows that the variance grows linearly with L
√
for L <∼ N an√d then reaches a constant value with
Var[g(L)] ∼ O( N), signaling the diffusive to Bloch-
ballistic transition. From the RMT quantum dot de-
D. Conductance Fluctuations
scription we know that Var[g(L = 1)] = 1/8. On the
other hand, the limit value Var[g(L → ∞)] can be un-
In the previous discussion we noticed that finite pe- derstood from the fluctuations of N which are of order
√ B
riodic waveguides with chaotic cells possess a diffusive O( N)[27]. Thesimilaritiesanddifferencesbetweenthe
regime where (cid:104)g(L)(cid:105), including its WLC, are similar to diffusive regime of periodic systems and disordered sys-
those of the metallic regime in a disordered wire. We tems is also illustrated in Fig. 9 where we compare his-
now address conductance fluctuations. In a disordered togram of the probability distribution function P (T) of
L
wireinthemetallicregime,theconductancehasaGaus- transmissioneigenvaluesT fortheperiodiccosinewaveg-
i
sian distribution with O(1) variance, independent of the uide with the Dorokhov’s distribution for a disordered
system length and of the disorder properties. The con- wire [23, 24]. We can observe that qualitatively the dis-
ductancedistributionchangesfromthisGaussianformto tributions resembles each other but they differ quantita-
a log-normal distribution as the size of the wire reaches tively, for instance for the periodic waveguide there is a
the localization length. In our periodic chain, we also slightly increase in the population of transmission eigen-
found that in the diffusive regime the conductance has values larger than 0.8 and a corresponding decrease in
Gaussian fluctuations [see Fig. 10], however its variance the eigenvalues smaller than that. On the other hand,
Var[g(L)]=βLdependslinearlyonL[seeinsetinFig.8] asillustratetheinsetinFig.9deepintheBloch-ballistic
butwithaslopeβ ≈0.05sufficientlysmalltoensurethat regimelargetransmissionprobabilitiesoccursmuchmore
the average is representative of a typical value, e.g. we often than in the diffusive regime as expected.
check that 1/(cid:104)g(L)(cid:105) ≈ (cid:104)1/g(L)(cid:105) in this regime. Let’s A remarkable feature that characterize the Bloch-
9
1P.21(cid:72)g(cid:76) 1P.29(cid:72)g(cid:76) transmission coefficients which we have discussed in sec-
1.0 1.0 tionIII.Owingtothisrepulsion,onlyoneTi ∼1andthe
0.8 0.8 restarenecessarilysmaller,spreadinginthe[0,1]interval
0.6 0.6 as shown in Fig.3, thus g(L)=(cid:80)NB T <N .
i=1 i B
0.4 0.4
0.2 0.2 IV. CONCLUSIONS
g g
23.023.524.024.525.025.526.0 2 3 4 5 6 7
1P.127(cid:72)g(cid:76) 1P.225(cid:72)g(cid:76) In this work we have studied the propagation of waves
in diffusive periodic quasi-one-dimensional systems, by
1.0 1.0
numerically computing the conductance of a cosine-
0.8 0.8
shaped waveguide and by employing a RMT model to
0.6 0.6
0.4 0.4 describe the system. We have shown that wave propaga-
0.2 0.2 tion in such systems displays a diffusive√regime for sys-
0 1 2 3 4 5 6 7 g 0 1 2 3 4 5 6 7 g tems of length L in the range 1 (cid:28) L√(cid:28) N, where the
conductance varies from O(N) to O( N). This should
1P.323(cid:72)g(cid:76) 1P.520(cid:72)g(cid:76) be compared with the diffusive regime of disordered sys-
1.0 1.0 temswhichholdsfor1(cid:28)L(cid:28)N,wheretheconductance
0.8 0.8 varies from O(N) to O(1). The ohmic behavior of bulk
0.6 0.6
disordered wires has been studied for a long time [31]
0.4 0.4
and more recently was reported in surface disordered
0.2 0.2
wires [3]. Here we have shown that this regime is also
g g
0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 observedinperiodicchainsofcavitieswithdiffusiveclas-
sical dynamics.
FIG. 10: Conductance pdf P (g) of a COE periodic chain
L Ontheotherhand,forperiodicwaveguideswithlength
model with N = 50 for several values of L. We see that up √
L (cid:29) N, wave propagation acquires the ballistic char-
to L=9 the distribution is clearly gaussian (black line) and
acter of the Bloch states of the associated unfolded in-
at L = 17 peaks start to develop. At L = 50 the distri-
bution has reached its long-chain stationary form consisting finite periodic system, with a constant average conduc-
in a multimodal structure which arises from the asymptotic tance (cid:104)g(L)(cid:105) which is close (and bounded by) (cid:104)NB(cid:105). We
quasi-periodicbehaviorofg arounditsaverage∼N . Given foundthatthereisaWLCintheconductancebothinthe
B
an integer n the peak just below g =n is due to realizations ohmic and Bloch-ballistic regimes. In the former, we ob-
with NB =n. Note that for NB =1 the peak is very narrow serveavaluesimilartodisorderedwireswithδg(L)≈0.3,
and its center close to one but for bigger NB the peaks start whereasinthelatterthecorrectionsissomewhatsmaller
to spread and locate further away from its respective integer
with δg(L)≈0.2.
upper boundary. This issue is addressed in the main text.
A difference we have observed between the ohmic
Note that the x-axis range covered changes in each plot.
regimes in disordered and periodic systems is in the con-
ductance fluctuations. While the conductance variance
Var[g(L)] is O(1) for the disordered wire, in diffusive pe-
ballistic regime is the conductance distribution multi- √
riodicwaveguidesitgrowslinearlywithLuptoL∼ N
modal shape observed in Fig. 10. In fact, as we have al-
and then reaches a constant asymptotic value. The pas-
ready mentioned, the conductance in the Bloch-ballistic
sage between these two regions signals the diffusive to
regime is dominated by the first term in Eq. (20) that
representthesum(cid:80)NB T overtheN non-decayingT . Bloch-ballistic transition.
i=1 i B i
Thereforethefluctuationsofg(L)arerelatedtothefluc-
tuations of these T and also to the fluctuations in their
i
number N which take only integer values. The sharp Acknowledgments
B
peaks of the conductance distribution P (g) at g = 0
L
and g = 1 corresponds to realizations with N = 0 and FB and JZ thank ACT 127 and Conicyt for support.
B
N = 1, respectively. We see that peaks at larger value F.B.thanksFondecytProjectNo. 1110144. F.BandV.P
B
of g becomes less sharp and also are not located at in- thank ECOS project C09E07 and Anr-Conicyt project
teger values of g. This is due to the repulsion between 38.
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