Table Of ContentEigenstate Structure in Graphs and Disordered Lattices
L. Kaplan∗
Institute for Nuclear Theory and Department of Physics
University of Washington, Seattle, Washington 98195
A quantum graph consists of V vertices connected by
We study wave function structure for quantum graphs in
B bonds, each of which is modeled as a one-dimensional
the chaotic and disordered regime, using measures such as
wire of length L (j = 1...B) along which the wave
j
the wave function intensity distribution and the inverse par-
propagates freely with zero potential. At each of the
ticipation ratio. The result is much less ergodicity than ex-
i = 1...V vertices, v 2 bonds will meet (note that
1 pected from random matrix theory,eventhough thespectral i ≥
0 statistics are in agreement with random matrix predictions. ivi =2B), and one must impose wave function conti-
0 nPuity and the current conservation condition
Instead,analyticalcalculationsbasedonshort-timesemiclas-
2
sical behavior correctly describe the eigenstate structure.
d
n Ψ (x)=λ Ψ (0), (1)
i,j i i,j
a Quantumgraphs,knownalsoas networkmodels, have Xj dx
J been used successfully for many years as simple dynam-
6 ical systems in which to study complex wave behavior. where the sum is over all v bonds j that meet at ver-
i
2 For appropriate parameter values, graphs can be made tex i and Ψ (x) is the wave function in the bond j,
i,j
] to display generic chaotic, disordered, or integrable mo- with x = 0 corresponding to the vertex i. λi is a free
D tion, and at the same time the quantum mechanics of parameter associatedwith the height of the effective po-
these systems has the simplifying advantage of being tentialatvertexi,andallowsone tointerpolatebetween
C
semiclassically exact. Originally, graphs were developed Neumann (λ = 0) and Dirichlet (λ ) boundary
n. as models for electrons moving between atoms in an or- conditions. Oi ne sees easily that theifo→rm∞of Eq. (1) is
i ganic molecule [1]. Recently, graph models have been the only one consistent with an s-wave pointlike scat-
l
n used to study issues as diverse as Anderson localization terer at the vertex; it can be obtained formally using
[ withinthecontextofperiodicorbittheory[2],thespatial self-adjoint extension theory. Because of time reversal
1 distribution and transport properties of persistent cur- invariance, the wave function in each bond can be writ-
v rents [3], Aharonov-Bohm conductance modulations in ten as Ψ (x) = a(k)eikx +a(k)∗e−ikx, for an eigenstate
8 GaAs/GaAlAs networks [4], spectral statistics and the i,j i,j i,j
atenergyk2/2m. Thus,thedistributionofwavefunction
4 trace formula in chaotic systems [5], spectral determi-
intensitiesinthesystemcanbecompletelycharacterized,
0 nants, with applications to thermodynamic and trans-
1 inthelimitkL ,bythedistributionofthequantities
port properties of mesoscopic networks [6], and chaotic →∞
0 a(k) 2. In this limit, the simple normalization condition
1 scattering and resonance behavior [7]. A discussion of | i,j|
0 the earlier history of quantum graphs can be found in
n/ Rtheef.m[8o]d,ewl.hich also provides an extensive discussion of Xi,j Lj|a(i,kj)|2 =Xj Lj (2)
i
l Though substantial work now exists on the spectral
n andscatteringpropertiesofquantumgraphs,andalsoon ensures that the mean bond intensity a(k) 2 is set to
v: their large-scale localization behavior, surprisingly little unity. h| i,j| i
i
X attention has been paid so far to the detailed wavefunc- We begin our analysis with a simple one-dimensional
tion structure of this paradigmatic quantum system. In “ring graph” [8] where the V vertices are arranged in a
r
a this paper we begin to address questions relating to the circle and each vertex i is connected by bonds to neigh-
statisticsofwavefunctionsongraphsandtheirrelationto boring vertices i v/2...i 1,i+1...i+v/2. v, the
− −
the underlying classical structures in the system. In the numberofbondsmeetingateveryvertex,isknownasthe
process,we examine the relationshipbetween short-time valency(heretakentobeconstantovertheentiregraph).
andlong-timeeffectsonstationarybehavior,makingcon- Setting allthe vertexpotentials λ =0 leadstothe max-
i
nectionstorecentworkonothersimplequantumchaotic imum possible delocalization in the graph (the opposite
problems,includingquantummaps,Sinaibilliards,Buni- limit, λ would instead produce eigenstates local-
i
→ ∞
movich stadia, tunneling in double wells, conductance ized on individual bonds), while non-integrability is en-
through chaotic quantum dots, and many-body systems sured through randomness in the bond lengths Lj. In
with random two-body interactions [9]. ourcalculationwetakethe bondlengthstobeuniformly
distributedinaninterval[1 δL,1+δL];becausethescat-
−
tering matrices depend only on kL mod 2π, all choices
j
of δL are equivalent as long as δL k−1. The eigen-
≫
∗ states of the system may be obtained by examining a
[email protected]
1
V V secular matrix h(k) [8]; k is an eigenvalue when- deviation from ergodicity in the local wave function be-
×
ever det h(k) = 0, and the associated null vector cor- havior.
responds to an eigenvector of the graph. In fact, small The key theoretical idea discussed and applied in sev-
but non-zero singular values ǫ of h(k) can easily be seen eralrecentworks[9]isthatwavefunctionintensitiesina
to correspond to eigenvectors of the same system with complexsystemcanoftenbeconvenientlyseparatedinto
slightlyperturbed potentialsλ λ kǫ,sosufficiently a product of short-time and long-time parts:
i i
→ −
small singular values (ǫ < k−1λ) can also be used to
produce eigenstates. T|h|is method allows the collection a(k) 2 =ρ (E(k)) r . (4)
| j | j × jk
of several independent wave functions at a given value
of k for any given realization of the disorder ensemble. Here ρ (E) is a smooth local density of states (known
j
One easily checks that collecting one or more wavefunc- alternatively as the strength function) on the bond j at
tionsatagivenvalueofk hasnodiscernibleeffectonthe energyE,obtainedastheFouriertransformoftheshort-
resulting wave function statistics. time part of the autocorrelationfunction
Comparing the Heisenberg time (h¯ divided by the
mean level spacing, at which individual levels are re- Ashort(t)= j e−iHte−t/2Tcutoff j , (5)
j h | | i
solved) with the time required classically to diffuse over
while r is obtained (formally) by Fourier transforming
the entire system, or alternatively making an analogy jk
the long-time behavior, at times t T and larger.
with banded random-matrix behavior, we see that the cutoff
∼
condition for avoiding localizationin our system is v2 The decomposition is useful because in many situations
≫ the short-time return amplitude Ashort(t) has a known
V. We note that the localization condition is k and j
− approximate analytical expression, which can be trans-
¯h independent and depends only on the classical graph
− formed to obtain ρ (E). On the other hand the long-
geometry. Increasingthevalencyvforafixedsystemsize j
time return amplitude in a chaotic or disordered system
V, one easily observes a transition from localized to de-
isgivenbyaconvolutionofthe short-timebehaviorwith
localizedbehavior,whichcanbe detectedeitherbylook-
a sumof exponentiallymany contributions,and thusr
ingatthechangeinlevelspacingstatistics(fromPoisson jk
may be regardedas random variable:
to GOE) or at the change in the wave function intensity
correlation(fromstronglynegativetonearzero)between
r =1
distantpoints onthe graph. Either method confirmsthe h jki
expected scaling behavior for the transition. One can rjkrj′k′ =1+(F 1)δjj′δkk′, (6)
h i −
thereforetakethe large-volumelimit V ,wheresta-
→∞ where the statistical average is performed over an ap-
tistical behavior is expected, while easily satisfying the
delocalization condition √V <v V 1. propriate ensemble. (If rjk is the square of a complex
≤ − Gaussian random variable, then F = 2.) Because the
In the delocalized regime, full information about wave
smooth localspectraldensity ρ (E) and the fluctuations
function statisticalbehavior is contained in the distribu- j
tionofbondintensities a(k) 2andtheircorrelations(note rjk are associated with distinct time scales (before and
| j | after the mixing time, respectively, in a chaotic system),
that we may freely drop the i index as it is immaterial
the two quantities are regarded as statistically indepen-
which of the two endpoints we take to be the beginning
dent. Thus, for example, the local IPR can be written
of bond j). It is convenient to introduce a simple one-
as
number measure of wave function ergodicity, the inverse
participation ratio (IPR):
= ρ2 r2 = ρ2 F , (7)
Ij h jih jki h ji
= a(k) 4 , (3)
I h| j | i where ρ2 , the second moment of the smooth local den-
h ji
where the averaging is performed over the B = Vv/2 sity function, is proportional to the sum of short-time
bonds j andovera disorderensemble,at a fixedvalue of return probabilities Ashort(t)2. This formalism has suc-
| j |
k. Of course, averagingover nearby values of k may also cessfully been used to quantitatively study scars of un-
be done. It is often useful to introduce a local versionof stableperiodicorbitsandrelatedphenomenainbilliards,
the IPR, , where the bond j is fixed; then = , in smooth potential wells, and in many-body interacting
j j
I I hI i
averaging over all bonds. The IPR is the first nontrivial systems.
momentoftheintensitydistribution (a(k) 2)(werecall To apply these ideas to the ring graphs, we focus on
that the mean intensity has been noPrm|ajlize|d to unity), one(arbitrary)bondjconnectingvertices1,2. Aninitial
andcanrangefrom1inthemaximallyergodiccasewhere wavepacketlaunchedinthisbondmovingfrom1towards
all intensities are equal up to a maximum value of B in 2 will have a probability
the case of perfect wavefunction localizationon individ-
ual bonds. Random matrix theory or the random vector P =v−2 1+e−2itan−1(λ2/vk) 2 (8)
trans
hypothesis would predict Gaussian random fluctuations (cid:12) (cid:12)
(cid:12) (cid:12)
inthecomplexcoefficientsa(k),thus =2. Anenhance- of being transmitted i(cid:12)nto one of the othe(cid:12)r v 1 bonds
j I −
ment of the IPR above this baseline value indicates a meeting at vertex 2; the remaining part of the wave
2
packetthengetsreflectedbackintotheoriginalbond[8]. thatthelong-timefluctuationsaresuper-Gaussian). Nu-
To begin with, we set λ = 0 at all vertices for sim- mericaldatacanthenbeusedtofitthecoefficientbofthe
plicity, and find that the reflected probability is P = subleading semiclassical correction; the fit is quite good
refl
1 (v 1)P = 1 4(v 1)/v2. The process is re- as can be seen in Fig. 1. As expected, the IPR is pri-
trans
− − − −
peated at vertex 1, and the remaining probability P2 marily a function of valency v, with volume-dependence
refl
travelsagainthe pathtakenby the originalwavepacket, being a higher-order effect as long as the delocalization
leading to a nontrivial contribution to the return prob- condition v2 > V is satisfied. The wave function statis-
ability. We may iterate this process until almost all of ticsclearlydeviatestronglyfromrandommatrixexpecta-
the initial probability to be in the bond j has decayed tionsasv becomeslarge. Thisisdespitethefactthatthe
(after O(1/v) bounces for v 1), and find that the sum level spacing statistics of this system are well-predicted
≫
of return probabilities behaves as byrandommatrixtheory,indicatinganabsenceofstrong
localization.
∞ 1+P2
dt A(t)2 (P2 )|t| = refl (9) In a one-dimensional system it is of course impossible
Z | | ∼ refl 1 P2 to take the semiclassical (large-volume) limit for fixed v
t=X−∞ − refl
while staying in the delocalized regime. It is therefore of
(note that we always sum return probabilities over both
interesttoconsiderhigher-dimensionalsystems,suchasa
positive and negative times). To leading order in v,
d-dimensional cubic lattice, with v =2d. In the absence
we therefore obtain ρ2 = v/4 O(1) for the short-
h ji − of vertex potentials (λ = 0) the above analysis still ap-
time factor. Taking into account intermediate-time re-
plies. Whathappenswhenweintroducedisorderintothe
currences(wherethewaveistransmittedateithervertex
system via the potentials λ in addition to the disorder
i
into an adjoining bond and is subsequently transmitted
already presentin the bond lengths? Let the λ be inde-
i
back into the bond j) cancels the O(1) term, leading to
pendentanddistributedforlargeλ inaccordancewitha
i
hρ2ji=v/4+O(v−1), and thus power law P(λi) ∼ λ−i α, for λ0 < λi < ∞ (with α > 1).
WenowclaimthatthetailoftheIPRdistributionwillbe
v b
= = 1 F +O(v−1). (10) strongly modified by the rare events where a strong po-
j
I I 4(cid:18) − V (cid:19)
tential λ is present on both sides of a given bond j with
endpoints 1 and 2. Indeed, we easily see that Eq. (8)
for transmission probability reduces to P λ−2 for
trans
∼
25 strong λ. Clearly the weaker of the two potentials λ
1
and λ will dominate the escape rate. The short-time
2
20 enhancement factor for the local IPR is proportional to
IPR 15 Sphroerdti-cttiimoen tthhuesinwveerhsaevoeftthheepersecdaipcetiorante, i.e. Ij ∼min(λ21,λ22), and
d
e
r
asu 10 P(Ij)∼λ02(α−1)Ij−α (11)
e
M for 1 V, modifying the exponential falloff pre-
j
5 RMT prediction ≪ I ≪
dicted by random matrix theory. Similarly, the tail of
the intensity distribution becomes
0
8 10 12 14 16 18 20 22 24 26 (a2) λ2(α−1)(a2)−α−1 (12)
Short-time predicted IPR P | | ∼ 0 | |
for 1 a2 V, in contrast with the exponential
FIG.1. ObservedIPRvs. thatpredictedbytheshort-time ≪ | | ≪
Porter-Thomas prediction valid for a system satisfying
theory, Eq. (10), for ring graphs with parameter values
7 ≤ v ≤ 15 and 15 ≤ V ≤ 33. Data points represented random matrix statistics.
by + signs for v = 7, crosses for v = 9, stars for v = 11,
empty squares for v=13 and full squares for v=15. 0.01
n
o 0.001
i
We note that in Eq.(10) we have included the leading t
O(h¯) correction to the semiclassical answer (for graph bu 0.0001
i
systems V−1 plays the role of an effective h¯, and b is an tr 1e-05
s
undetermined constant). The long-time factor F can be di 1e-06
obtaineddirectlyby measuringthe meansquarevalueof y
a(k) 2/ρ (E(k)),i.e. ofthebondintensitynormalizedby lit 1e-07
|thje a|nalyjtically computed short-time spectral envelope. abi 1e-08 RpMrTediction
b
For a wide range of values for the system size V and the o 1e-09
r
valency v, the result 4 F 5 is obtained, supporting P
≤ ≤ 1e-10
the conjecture of independence of short-time and long- 10 100 1000 10000
time fluctuations in the spectrum (we note F >2 means Intensity
3
FIG.2. Thetailofthewavefunctionintensitydistribution In conclusion, we have seen that various aspects of
isplotted for athree-dimensionallattice with random poten-
wave function structure in chaotic and disordered quan-
tials λ (see text). From top to bottom, the three solid data
tum graphs are adequately described using the analyti-
curves are for exponent α = 1.2, 1.5, and 1.8. The corre-
cally known short-time behavior of the system, specifi-
sponding theoretical power laws from Eq. (12) are plotted as
cally the return amplitude atshort times andits Fourier
dashedlines,withtheRMTpredictionappearingasadotted
transform. On the other hand, random matrix theory
curvefor comparison.
completely fails to describe the wave function structure,
even though its predictions are good for the spectral
The result of Eq. (12) is confirmed in Fig. 2, where
statistics. This failure can be understood as resulting
we have used an ensemble of 373 lattices with one out of
fromtheomnipresenceofshortperiodicorbitsingraphs,
every D = 11 sites randomly chosen to contain a scat-
in contrast with the situation prevailing in most other
terer (the particle executes free motion while traveling
chaotic systems.
between the scatterer sites, so that the classical motion
Several very valuable discussions with T. Kottos are
isdiffusivewithmeanfreepathD). The totalnumberof
gratefully acknowledged. This work was supported by
scatterers is then V = 4604, and we indeed see that the
the DOE under grant No. DE-FG03-00-ER41132.
data curves tend quickly to zero for a2 V. We have
| | ≥
checked that with increasing number of scatterers V at
fixedα,theintensitydistributioncurveskeepfollowinga
given power law behavior for larger and larger intensity,
beforeeventuallydroppingofftozero. Similarly,wehave
checkedthatvaryingthediffusionconstantD byafactor
of2doesnotsignificantlyaffecttheintensitydistribution
[1] M. J. Richardson and N. L. Balazs, Ann. Phys. (N. Y.)
for fixed V, as would have been expected for a diffusion- 73, 308 (1972).
dominated log-normal tail, logP ∼ −Dlog2(|a|2) [10]. [2] H. Schanz and U. Smilansky, Phys. Rev. Lett. 84, 1427
We see instead that the tail of the intensity distribution
(2000).
isdominatedentirelybythe short-timesystembehavior. [3] M. Pascaud and G. Montambaux, Phys. Rev. Lett. 82,
Finally, we return to the volume-averaged IPR , the 4512 (1999).
I
simplest overall measure of the degree of wave function [4] J. Vidal, G. Montambaux,and B. Doucot, Phys. Rev. E
localization. FromEq.(11),weeasilyobtaininthelarge- 62, R16294 (2000).
volume limit [5] H. Schanzand U. Smilansky,arXiv:chao-dyn/9904007.
[6] E. Akkermans,A. Comtet, J. Desbois, G. Montambaux,
I =Cαλ02(α−1)V2−α. (13) and C. Texier, J. Phys. A 33, L63 (2000).
[7] T. Kottos and U. Smilansky, Phys. Rev. Lett. 85, 968
for 1 < α < 2. As we can see in Table I, this result (2000).
compares favorably to the numerically computed value [8] T.KottosandU.Smilansky,Ann. Phys.274,76(1999).
overarangeofvolumesV andfordifferentclassicalmean [9] L.KaplanandT.Papenbrock,Phys. Rev. Lett.84,4553
free paths D and minimum potentials λ . None of the (2000); L.KaplanandE.J.Heller,Phys. Rev. E62,409
0
IPR’s are close to the random matrix prediction =2. (2000); L. Kaplan, Phys. Rev. E 62, 3476 (2000); W.
I
E. Bies, L. Kaplan, M. R. Haggerty, and E. J. Heller,
arXiv:nlin.CD/0004024; W. E. Bies, L. Kaplan, and E.
V D λ I I
0 predicted actual J.Heller,arXiv:nlin.CD/0007037; L.Kaplan,Phys. Rev.
549 4 0.3 21 23 Lett. 80, 2582 (1998).
549 4 1.0 70 72 [10] V. I Fal’ko and K. B. Efetov, Phys. Rev. B 52, 17413
2197 8 1.0 141 137
(1995);I.E.SmolyarenkoandB.L.Altshuler,Phys.Rev.
2315 4 1.0 144 131 B 55, 10451 (1997).
4604 11 1.0 204 210
TABLEI. Numericallyobtainedinverseparticipationratio
I for a disordered three-dimensional lattice with exponent
α=1.5,comparedwiththepredictionofEq.(13)forvarious
numbers of scatterers V, mean free paths D, and minimum
potential values λ0. The constant used is Cα=3.0.
4
