Table Of ContentUnbinding transition in semi-infinite two-dimensional localized systems
A. M. Somoza1, P. Le Doussal2 and M. Ortun˜o1
1 Departamento de F´ısica-CIOyN, Universidad de Murcia, Murcia 30.071, Spain and
2 CNRS-Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure, 24 rue Lhomond,75231 Cedex 05, Paris France.
Weconsideratwo-dimensionalstronglylocalizedsystemdefinedinahalf-spaceandwhosetransfer
integral in the edge can be different than in the bulk. We predict an unbinding transition, as the
edge transfer integral is varied, from a phase where conduction paths are distributed across the
bulk to a bound phase where propagation is mainly along the edge. At criticality the logarithm of
the conductance follows the F Tracy-Widom distribution. We verify numerically these predictions
1
for both the Anderson and the Nguyen, Spivak and Shklovskii models. We also check that for a
5 half-space,i.e.,whentheedgetransferintegralisequaltothebulktransferintegral,thedistribution
1 of the conductance is the F Tracy-Widom distribution. These findings are strong indications that
4
0 random signs directed polymer models and their quantum extensions belong to the Kardar-Parisi-
2 Zhanguniversalityclass. Wehaveanalyzedfinite-sizecorrectionsatcriticalityandforahalf-plane.
n
a PACSnumbers: 72.20.-i,71.23.An,71.23.-k
J
5
The probability distribution function of the conduc- scatteringpaths. MedinaandKardar[9]studiedindetail
1
tance g of quantum random hopping models, such as the NSS model, focusing on its formulation as a model
] the Anderson model, has been much studied. In one of directed polymers (DP) with non-positive Boltzmann
n
dimension, it has been shown that all the cumulants of weights. They found that the variance of the tunneling
n
lng scale linearly with system size [1]. Thus, the dis- probability increases with distance as L2/3 for 2D sys-
-
s tribution function of lng approaches a Gaussian form tems. Thissuggeststhatin2DtheDPwithnon-positive
i
d for asymptotically long systems and is fully character- Boltzmann weights should be in the same universality
t. izedbytwoparameters,themean(cid:104)lng(cid:105)andthevariance class as the standard (i.e. positive weight) DP problem,
a σ2 =(cid:104)ln2g(cid:105)−(cid:104)lng(cid:105)2. Bothparametersarerelatedtoeach well known to exhibit a wandering exponent of 2/3. A
m
other, supporting the extension of the single parameter qualitative replica argument led to the same conclusion
-
scaling hypothesis [2] to the distribution function of the [9].
d
n conductance [3]. In the mean-time, much progress happened in the
o In two-dimensional (2D) systems, calculations of the
study of growth models in the Kardar-Parisi-Zhang
c conductance distribution function are possible in the
(KPZ) universality class [10]. The standard DP prob-
[
metallic regime, thanks to the non-linear sigma model lem belongs to this class (the KPZ height h maps onto
1 [4], but not until now in the strongly localized regime.
lnZ where Z is the DP partition sum). Remarkably, it
v
However, two of us have argued, and demonstrated nu-
was found that the TW distribution arises at large time
2
merically, that in that regime L (cid:29) ξ, ξ being the local-
1 scales(i.e. longpolymerlength)indiscretemodelsinthe
6 ization length, the conductance takes the form [5, 6] KPZ class. For example, the height in the polynuclear
3 growthmodel[11],theoptimalenergyinDPmodels[12],
0 2L (cid:18)L(cid:19)1/3
lng =− +α χ (1) and the length of the longest increasing subsequence in
1. ξ ξ a random permutation [13] all follow TW distributions.
0
where α is a constant and χ a random variable with This body of facts thus led to the conjecture (1) with
5
1 a Tracy-Widom (TW) cumulative distribution function the 1/3 exponent of the KPZ class, and to its numer-
: (CDF). The TW distributions are CDF for the largest ical verification. More recently the KPZ equation and
v
eigenvaluesoflargeGaussianrandommatrices[7]. Itwas the DP problem have been solved directly in the contin-
i
X foundthattherandomvariableχdependsonthegeome- uum, using integrability by the Bethe Ansatz of an as-
r try of the model. For narrow leads, the TW distribution sociated quantum boson model [14–17]. Various bound-
a
associated to the Gaussian unitary ensemble χ = χ is ary conditions where treated, suggesting new tests for
2
the CDF F (x), i.e., Prob(χ < x) = F (x) = F(x)2, the conjecture (1) with various lead geometries. In par-
2 2 2
where F(x) is defined as F(x) = exp{−1(cid:82)+∞(s − ticular the continuum DP problem in a half-space was
2 x
x)u(s)2ds} and u(x) is the solution of the Painlev´e II found [18] to lead to the F4 TW distribution, associated
equation u(cid:48)(cid:48)(x) = 2u(x)3 +xu(x), with u(x) → −Ai(x) to the Gaussian simplectic ensemble, in agreement with
as x→+∞, Ai(x) being the Airy function. earlier results for discrete models [19, 20]. This distri-
The chain of arguments leading to (1) goes as follows. bution verifies F4(x) = F(x)(E(x)−1 + E(x))/2 where
It was argued by Nguyen et al. (NSS) [8] that quan- E(x)=exp{1(cid:82)+∞u(s)ds}, F(x) and u(x) are the func-
2 x
tum interference effects in the localized regime are faith- tions defined above. It was thus conjectured [18] that in
fullydescribedbyretainingonlytheshortestorforward- a half-space the conductance near the edge should be of
2
the form (1) with χ=χ of CDF given by F . The Green function between two sites a and b can be
4 4
The aim of this paper is to study the logarithm of the written in terms of the locator expansion
conductance lng in 2D systems in the strongly localized
(cid:88)(cid:89) 1
regime in order to (i) verify numerically the conjecture (cid:104)a|G|b(cid:105)= , (3)
E−(cid:15)
that the CDF of for a half-plane is the F (x) function, i
4 Γ i∈Γ
and(ii)predictandanalyzethetransitionfrom2Dto1D
as the hopping amplitude t at the edge is increased, fa- where the sum runs over all possible paths connecting
voring propagation along the edge. Exactly at criticality the two sites a and b. The NSS model assumes that, in
we observe that the CDF of lng is the TW distribution the strongly localized regime, (3) is dominated by the
F (x) = F(x)E(x) of the Gaussian orthogonal ensem- forward-scatteringpathsandonlytakesintoaccountthe
1
ble. This transition is a generalization of the unbinding contributionsofsuchpathstothetransmissionamplitude
transition studied for positive weight DP [20, 21]. Re- betweenoppositecornersofasquarelattice. Aswewant
markably, the full crossover distribution obtained there to study a half-space, we consider a triangular sample as
fits our data for the whole parameter range, providing a represented in Fig. 1b. In order to mimic the Anderson
very delicate test that the random sign DP belongs to model, we choose the site disorder energy at random in
the KPZ universality class. This is all the more precious the interval (cid:15)i ∈[−W/2,W/2], but if |E−(cid:15)i|<1, we set
since at present no integrable system has been found to 1/|E−(cid:15)i|=1,partiallyincorporatingmultiplescattering
solve the NSS model. effects. We note that at least for the planar symmetry
this choice of disorder does no change the universality of
We focus on the Anderson model on a square sample
the problem [23].
of finite size L×L described by the Hamiltonian
The calculation of the quantum amplitude in the NSS
(cid:88) (cid:88) model is formally similar to the calculation of the parti-
H = (cid:15) a†a + t a†a +h.c., (2)
i i i i,j j i tion function of DP in a random potential
i (cid:104)i,j(cid:105)
(cid:40) (cid:41)
(cid:88) (cid:88)
where the operator a†i (ai) creates (destroys) an electron Z = exp −β hi , (4)
at site i of an square lattice and (cid:15)i is the energy of this Γ i∈Γ
site chosen at random (cid:15) ∈ [−W/2,W/2], with W the
i
where β = 1/kT, h is a random site energy and Γ runs
strengthofthedisorder. Thedoublesumrunsovernear- i
over all possible configurations of the DP. Equations (3)
est neighbors. The hopping matrix element t is taken
i,j
and(4)areequivalentprovidedthatwecanidentify−βh
equalto1everywhere,exceptbetweenthesitesalongone i
withln(E−(cid:15) ). WhileinDPthedisorderenergiesh are
edgewhereisequal t>1(seefigure1). Thevalue inthe i i
real, in the quantum case E − (cid:15) does not have to be
bulk sets the energy scale, while the lattice constant sets i
positive, implying a more general problem.
the length scale. The unit of conductance is 2e2/h.
It is interesting to study first the conductance distri-
bution of the NSS and Anderson models in a half-space
a b and check whether the conjecture of Ref. [18], based on
the exact result for the continuum DP, that the CDF of
χ is the TW function F is verified by our two models.
4
In Fig. 2 we plot histograms of lng for the NSS model
as a function of z = (lng −A)/B, where A and B are
chosen so that z has the same mean and variance as F .
4
The lateral dimensions are L = 2500 (blue dots), 5000
FIG. 1: Scheme of the geometrical arrangements of the a)
Anderson model, and the b) NSS model. Blue dots are site (red dots) and10000 (blackdots), and the number of re-
withrandomdisorderenergiesandreddotssitesontheleads, alizations is 4×106. The solid line corresponds to the
without disorder. Thin lines correspond to the bulk hopping TW function F(cid:48). We see a perfect agreement between
4
strength equal 1, while thick lines along the edge to t. our numerical results and the TW distribution for more
than four orders of magnitude. A similar agreement is
For the Anderson model we have calculated the con- found for the Anderson model (not shown), taking into
ductance through Landauer’s formula in terms of the account that it corresponds to smaller system sizes.
transmissionbetweenperfectleads,arrangedasschemat- The inset of Fig. 2 we plot the skewness of lng ver-
ically shown in Fig. 1a. This is obtained from the Green sus L−2/3, which according to the discussion below is
function, which can be calculated propagating layer by expectedtobetheleadingordercorrection. Theredline
layer [22]. It is convenient to propagate along the di- correspondstotheNSSmodel, whilethedotstotheAn-
rection perpendicular to the leads, starting from the op- derson model (W = 10 blue, 20 magenta). The dashed
posite edge, so that each calculation of the bulk Green line is Sk = 0.165509... The skewness of the numerical
4
function can be used for any value of t. results for both models tends to the predicted value.
3
100 transition point [20]. Our prediction is that lng always
behaves as in (1) with the random variable χ having
10-1 a CDF Prob(χ < x) = F(x,w = −cw˜) that depends
on the scaling variable w˜ = (t − t )L1/3, c > 0 being
c
10-2 an (unknown) proportionality constant. The universal
0.15 crossoverfunctionF(x,w)(correspondingtoF (x,w/2)
z) (cid:27)
F'(410-3 0.10 in Ref. 20) satisfies
Sk
0.05 F(x,w)= 1F(x)(cid:0)(f(x,w)−g(x,w))E(x)−1 (5)
10-4 2
0.000.00 0.04 0.08 0.12 +(f(x,w)+g(x,w))E(x)(cid:1)
L-2/3
10-5
-4 -2 0 2 4 where the functions f(x,w) and g(x,w) are the ones of
z
[24] and verify:
FIG. 2: Histogram of lng versus the scaled variable z for ∂ f(x,w) = u(x)2f(x,w)−(wu(x)+u(cid:48)(x))g(x,w)
w
three sizes and a disorder W = 10 of the NSS model. The
continuous line corresponds to F(cid:48)(z). Inset: skewness vs. ∂wg(x,w) = −(wu(x)−u(cid:48)(x))f(x,w)+
4
L−2/3 fortheNSSmodel(redcurve)fortheAndersonmodel (w2−x−u2(x))g(x,w) (6)
with W =10 (blue dots) and 20 (magenta dots); the dashed
line is Sk . and are subjected to the initial condition at criticality
4
f(x,0)=−g(x,0)=E(x)2. Then,theCDFatcriticality
is F(x,0) = F (x). In the limit w → +∞, one has
1
f → 1 and g → 0, hence F(x,∞) = F . In the opposite
We now argue theoretically that the NSS and the An- 4
(cid:112)
limit, w →−∞ [24] F(w2+y |w|,w)→erf(y) and one
derson models exhibit a phase transition between (i) a
recovers Gaussian fluctuations for lng.
2D phase where conduction paths wander unboundedly
To study the transition and to characterize the two
in the half-space and the fluctuations of lng at large L
phases,weanalyzetheskewnessofthedistributionoflng
are described by (1) with χ=χ , and (ii) a quasi-1D lo-
4
for the NSS model as we vary t. The skewness should
calized phase where the conduction paths have 1D char-
tend to 0 in the localized phase, to Sk = 0.293464...
acter at large scale, and lng has a log-normal distribu- 1
tion with fluctuations scaling as L1/2, i.e. (1) still holds at the transition and to Sk4 in the unbound phase. In
Fig. 3 we plot the skewness as a function of size on a
but with a size dependent variable χ→χ , scaling now
L
as χ ∼ L−1/6. Exactly at the transition t = t , the logarithmic scale for several values of the edge strength
L c
t. Theupperhorizontallinecorrespondstotheexpected
fluctuations of lng at large L are described by (1) with
value of the skewness at the critical point, Sk , and the
now χ = χ distributed according to F . Hence the 1/3 1
1 1
lower horizontal line to the limiting value in the bulk
KPZ exponent holds up to and at the transition. This
phase, Sk . The black curve represents the skewness at
transition can be described as an unbinding transition 4
the critical value t ≈ 1.613(1) and tends to the upper
of conduction paths. In the standard DP the unbinding c
horizontal line, Sk . Solid curves are for t>t and after
transition was predicted in [21] and worked out in detail 1 c
reachingamaximumstartdecreasing,eventuallytending
in the framework of symmetrized random permutations
to zero. Dashed lines correspond to t < t and tend to
in [20]. Before using these results below, let us first give c
Sk , middle horizontal line in Fig. 3.
the physical picture and the scaling arguments. 4
It is possible to scale the raw data for the skewness at
In the unbound phase, t > t , the paths wander in
c large values of L into a universal curve using as scaling
the 2D space, but come back from time to time to
variable w˜ = (t−t )L1/3. Looking at the behavior of
the boundary (they still feel the boundary hence the c
the skewness for the critical value t in Fig. 3, it is clear
change from F to F ). Their typical transverse wan- c
1 4 that finite size effects are important. To take them into
dering is L2/3. Ins the bound phase, the paths return
account we assume Sk(t,L) ≈ Sk(w˜)f(L), where f(L)
to the wall, with a typical length ξ which diverges at
u incorporates finite size effects in a simple way f(L) =
the transition as ξ ∼ (t−t )−3. Near the transition
u c Sk(t ,L)/Sk . In Fig. 4 we plot this renormalized values
there are thus L/ξ independent pieces of paths, each c 1
u Sk(w˜) for several values of t. All curves are plotted for
fluctuating as ξu1/3 hence the variance should behave as the range 200 < L < 104. The black dot represents the
(lng)2c =ξu2/3(L/ξu)=L/ξu1/3 ∼(t−tc)L. Farfromthe transition point and so is equal Sk1 and is placed at the
transition this picture predicts that the skewness should vertical axis, t = t . The dashed line is the theoretical
c
decay as Sk∼(L/ξu)−1/2, since in the 1D phase the cen- prediction for the scaling function, i.e., the solution of
tral limit theorem holds for lng and all cumulants scale Eqs. (6), using a value c = 0.9 for the multiplicative
c
as L i.e. (lng)n ∼L. constant. It presents a maximum in the localized phase
Let us now consider the critical scaling around the well reproduced by our simulations. The blue dotted
4
0.30 lng(here)andoftheinterfaceheighth(there,timebeing
2
1.8 denoted as L) and the scaled cumulants kn = Kn/Ln/3
0.25
1.75 which converge to constants. The finite size corrections
1.7
0.20 1.65 oftheknwherefound(thereinthebulk)toscaleasL−1/3
s 1.63 for n=1 and as at most L−2/3 for all n≥2 [25, 26]. We
es 1.614
wn 0.15 1.61 study these corrections for the NSS model. In the top
Ske 11..558 panels of Fig. 5 we plot k2 (left) and k3 (right) versus
0.10 1.3 L−2/3 for the half-space, t = 1. We note that k follows
3
1
a very good linear behavior, while k shows some curva-
2
0.05
ture. Based on the curvature of k it is difficult to de-
2
termine its leading finite size corrections exponent, but
0.00
101 102 103 104 the exponent −2/3 produces an extrapolated value for
L the skewness (Sk= 0.166) very close the the theoretical
expectation,indicatingthatfinitesizecorrectionsforthe
FIG.3: SkewnessversusLonalogarithmicscalefortheval- half-space are similar to those observed for bulk growth
uesoftshowninthefigure. Thereddashedlinecorresponds
modelsinKPZclass. InthelowerpanelsofFig.5weplot
to Sk and the dashed black horizontal line to Sk .
1 4 k (left)andk (right)atthecriticalpointversusL−1/3.
2 3
The smooth behavior of both curves indicate that finite
size corrections exhibit a distinct and novel behavior at
line on the right represents the limiting behavior in the
bound phase, bw˜−3/2, with the fitting parameter b = criticality. From the intercepts with the vertical axis of
linear fits at large L, we obtain Sk= 0.297, confirming
0.55. The overlap and the agreement with theoretical
that the CDF is the TW function F . The histograms
expectationsisquitegood,speciallynotingthattheonly 1
of lng at the critical point agree fairly well with these
free parameter to overlap all the curves is t .
c
theoretical predictions.
1.3 1.62 1.2
0.30 11..45 11..6635 4.0 Half-plane 1.0 Half-plane
1.55 1.7
0.25 1.58 1.75 3.8 0.8
1.6 1.8 k2 k30.6
1.612 1.9 3.6
ttSk() Sk/Sk()1c 00..1250 00..1250 1.614 2 53..044.00 0.03 L- 2/3 0.06 0.09 0.044.00 0.03 L- 2/3 0.06 0.09
0.10 Sk0.10 80 5.2 Criticality 3 Criticality
160
0.05 320
0.05 400 k25.0 k32
0.00
1.0 1.5 2.0 2.5 3.0
t 4.8
0.00 1
-3 -2 -1 0 1 2 3 4 5 0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3
(t-t) L1/3 L-1/3 L-1/3
c
FIG. 5: Leading order corrections of the second k (left) and
2
FIG.4: Skewnessscaledasafunctionofw˜=(t−tc)L1/3 for third k3 (right) cumulants in the NSS model for a half-space
several values of t. The black dot corresponds to Sk1 at the versus L−2/3 (top panels) and at the unbinding transition
transitionandthehorizontaldashedlinetoSk4. Thedashed versus L−1/3 (bottom panels).
curveisthetheoreticalpredictionforthescalingfunctionand
the dotted curve the predicted asymptotic behavior at large
w˜. Inset: skewnessasafunctionoftfortheAndersonmodel. We have shown that the NSS and the Anderson mod-
elsinahalf-planeundergoanunbindingtransitionasthe
TheresultsfortheAndersonmodelconfirmallpredic- hopping amplitude at the edge is varied. We show evi-
tions,althoughthemaximumsizethatcanbecalculated, dencethattheanalyticalexpressionsfortheconductance
L=400,isstillrelativelyfarfromtheasymptoticbehav- distribution functions and the scaling functions at the
ior. IntheinsetofFig.4,weplottheskewnessasafunc- unbindingtransitionsinlocalizedtwo-dimensionalquan-
tion of t for several system sizes of the Anderson model tum systems are obtained from Eqs. (5-6). We have also
and a disorder W =20. A peak around t≈1.8 develops studied the corresponding finite-size corrections. Similar
with size, but its maximum is still far from Sk . This unbindingtransitionscanoccurin3Dsystemswhencon-
1
results are fully consistent with NSS for similar sizes. duction through a surface or a line is favored. They may
Itisinterestingtocomparethefinitesizecorrectionsin be relevant to the behavior of edge states.
ourmodelswiththosefordiscretegrowthmodelsandex- We thank J. Baik and K. A. Takeuchi for a useful
perimentsintheKPZclass. ConsiderK thecumulantof exchange. AS and MO acknowledge financial support
n
5
from the Spanish DGI and FEDER Grant No. FIS2012- (2000);J.BaikandE.M.Rains,J.Stat.Phys.100,523
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