Directed polymer in a random medium of dimension 1+3 : multifractal properties at the localization/delocalization transition

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by  Cecile Monthus

Directed polymer in a random medium of dimension 1+3 :
multifractal properties at the localization/delocalization transition
C´ecile Monthus and Thomas Garel
Service de Physique Th´eorique, CEA/DSM/SPhT
Unit´e de recherche associ´ee au CNRS
91191 Gif-sur-Yvette cedex, France
7
0
We consider the model of the directed polymer in a random medium of dimension 1+3, and
0
investigate its multifractal properties at thelocalization/delocalization transition. In close analogy
2
with models of the quantum Anderson localization transition, where the multifractality of critical

n wavefunctionsiswellestablished,weanalysethestatistics ofthepositionweightsw (~r)oftheend-
L
a pointofthepolymeroflengthLviathemomentsY (L)=P [w (~r)]q. Wemeasurethegeneralized
J q ~r L
9 exponentsτ(q)and τ˜(q)governingthedecayofthetypicalvaluesYqtyp(L)=elnYq(L) ∼L−τ(q) and
disorder-averaged values Y (L) ∼ L−τ˜(q) respectively. To understand the difference between these
2 q
exponents,τ(q)6=τ˜(q)abovesomethresholdq>q ∼2,wecomputetheprobabilitydistributionsof
c
] y=Yq(L)/Yqtyp(L)overthesamples: wefindthatthesedistributionsbecomesscaleinvariantwith
n a power-law tail 1/y1+xq. These results thus correspond to the Ever-Mirlin scenario [Phys. Rev.
n Lett. 84,3690(2000)]forthestatisticsofInverseParticipation RatiosattheAndersonlocalization
- transitions. Finally,thefinite-sizescalinganalysisinthecriticalregionyieldsthecorrelationlength
s
i exponentν ∼2.
d
.
t
a
I. INTRODUCTION
m
-
d The notion of multifractals is now important in various areas of physics (see for instance [1, 2, 3, 4, 5, 6, 7]
n and references therein). For classical systems with frozen disorder of interest here, the idea that multifractality is
o present at criticality has been mostly investigated for correlation functions in two-dimensional diluted ferromagnets
c
[8, 9, 10, 11, 12], spin-glasses and random field spin systems [13, 14, 15]. For disordered quantum spin-chains, the
[
statistics of critical correlation functions is described by “multiscaling”, which is even stronger than multifractality

1 [16]. Forquantumlocalizationmodels,the multifractality ofthe criticalwavefunctionshowsupthroughthe statistics
v ofinverseparticipationratios(I.P.R.s)[17,18]. ManyresultsarenowavailablefortheAndersonlocalizationtransition
9
in d=3 [19, 20, 21, 22], the integer quantum Hall transition [23, 24] Dirac fermions in a random magnetic field [25],
9
and power-law random banded matrices [26]. Connections have been also established with the scaling properties of
6
the correlation functions [27] and with the time evolution of wave packets [28]. More recently, it was realized that
1
0 typical and disorder-averaged I.P.R.s can actually lead to two different multifractal spectra as a consequence of the
7 broadness of their probability distributions [21, 26, 29].
0 In this paper, we consider the localization/delocalization transition of the directed polymer in a random medium
/ of dimension 1+3 (see the review [30] and references therein), to investigate whether some multifractality in present
t
a at criticality, in analogy with the quantum localization models quoted above.
m
The paper is organized as follows. In Section II, we introduce the directed polymer model and the observables
- displaying multifractal behavior at criticality. We then describe our numerical results concerning the generalized
d
dimensions D(q) and D˜(q) that govern typical and averaged values (Section III), the singularity spectrum (Section
n
o IV), and the probability distributions over the samples ( Section V). Finally, we present in Section VI the finite-size
c scaling analysis in the critical region. Section VII contains our conclusions.
:
v
i
X II. MODELS AND OBSERVABLES
r
a
A. Model definition
Therandombonddirectedpolymermodel[30]isdefinedbythefollowingrecursionrelationforthepartitionfunction
on the cubic lattice in d=3
2d
Z (~r)= e−βǫt(~r+~ej,~r)Z (~r+~e ) (1)
t+1 t j
Xj=1
2
The bond energies ǫ (~r+~e ,~r) are random independent variables drawn from the Gaussian distribution
t j
1 −ǫ2
ρ(ǫ)= e 2 (2)
√2π
In this paper, we consider the following boundary conditions. The first monomer is fixed at ~r = ~0, i.e. the initial
condition of the recurrence of Eq. (1) reads
Z (~r)=δ (3)
t=0 ~r,~0
The lastmonomeris free,i.e. the fullpartitionfunctionofthe polymer oflengthL is thenobtainedbysumming over
all possible positions~r at t=L
Ztot = Z (~r) (4)
L L
X~r
The phase diagram of this directed polymer model as a function of space dimension d is the following [30]. In
dimension d 2, there is no free phase, i.e. any initial disorder drives the polymer into a strong disorder phase,
≤
where the order parameter is an ‘overlap’ [31, 32, 33, 34, 35]. In dimension, d > 2, there exists a phase transition
betweenthe low temperaturedisorderdominatedphase anda free phaseathightemperature [36, 37, 38]. This phase
transition has been studied exactly on a Cayley tree [31]. In finite dimensions, bounds on the critical temperature T
c
have been derived [37, 39, 40] : T (d) T T (d). The upper bound T (d) corresponds to the temperature above
0 c 2 2
≤ ≤
whichthe ratioZ2/(Z )2 remainsfinite asL . ThelowerboundT correspondstothe temperaturebelowwhich
L L →∞ 0
the annealed entropy becomes negative. On the Cayley tree, the critical temperature T coincides with T [31]. In
c 0
finite dimensionshowever,we havearguedin[41]thatT coincideswithT , andourrecentnumericalsimulations[42]
c 2
are in agreement with the numerical value given in [39] for T (d=3) :
2
T =T (d=3)=0.790.. (5)
c 2
The numerical results given below have been obtained using polymers of various lengths L, with corresponding
numbers n (L) of disordered samples with
s
L =6,12,18,24,36,48,60,72,84,96 (6)
n (L) =108,107,2.106,8.105,2.105,5.104,3.104,2.1044.104,2.104 (7)
s
In the following, A denotes the averageof A over the disorder samples. We also define (∆A)2 =A2 A2.
−
B. Notion of multifractal statistics at criticality
In this paper, we will focus on the statistical properties of the weights
Z (~r)
L
w (~r)= (8)
L Ztot
L
normalized to (Eq. 4)
w (~r)=1 (9)
L
X~r
and study whether they present some multifractal properties at criticality. We thus consider the following moments
of arbitrary order q
Y (L)= wq(~r) (10)
q L
X~r
that are the dynamical analogs of the Inverse ParticipationRatios (I.P.R.s) in quantum localization models [26]
P (L)= dd~rψ(~r)2q (11)
q
Z | |
Ld
3
As a consequence of the normalization of Eq. 9, one has the identity Y (L)=1. The localization/delocalization
q=1
transitioncanbecharacterizedbytheasymptoticbehaviorinthelimitL oftheY (L)forq >1. Inthelocalized
q
→∞
phase T <T these moments Y (L) converge to finite values
c q
Y (L= )>0 for T <T (12)
q c
∞
In the delocalized phase, the spreading of the polymer involves the Brownian exponent ζ = 1/2, and, space being of
dimension d=3, the decay of the moments follows the scaling
Yq(L) L−(q−1)dζ =L−(q−1)23 for T >Tc (13)
≃
Exactly at criticality, the typical decay of the Y (L) defines a series of generalized exponents τ(q)=(q 1)D(q)
q
−
Ytyp(L) elnYq(L) L−τ(q) =L−(q−1)D(q) (14)
q ≡ |T=Tc ≃
The notion of multifractality corresponds to the case where D(q) depends on q, whereas monofractality corresponds
toD(q)=cstasinEq. (13). TheexponentsD(q)representgeneralizeddimensions[1]: D(0)representthedimension
of the support of the measure, here it is simply given by the space dimension D(0) = d = 3; D(1) is usually called
the information dimension [1] , since it describes the behavior of the ’information’ entropy
s w (~r)lnw (~r)= ∂ Y (L) D(1)lnL (15)
L L L q q q=1
≡− − | ≃
X~r
Finally D(2) is called the correlation dimension [1] and describes the decay of
Ytyp(L) elnY2(L) L−D(2) (16)
2 ≡ |T=Tc ≃
In the multifractal formalism, the singularity spectrum f(α) is given by the Legendre transform of τ(q) [1] via the
standard formulas
′
q =f (α) (17)
τ(q) =αq f(α) (18)
−
The physical meaning of f(α) is that the number (α) of points ~r where the weight w (~r) scales as L−α typically
L L
N
behaves as
(α) Lf(α) (19)
L
N ∝
So the Legendre transform of Eq. (18) corresponds to the saddle-point calculus in α of the following expression
Ytyp(L) dα Lf(α) L−qα (20)
q ∼Z
The general properties of the singularity spectrum f(α) are as follows [1] : it is positive f(α) 0 on an interval
≥
[α ,α ] where α =D(q =+ )is the minimal singularityexponent and α =D(q = ) is the maximal
min max min max
∞′′ −∞
singularity exponent. It is concave f (α) < 0. It has a single maximum at some value α where f(α ) = D(q = 0)
0 0
(so here f(α )=3), and contains the point α =f(α )=D(1).
0 1 1
Following[1],many authorsconsiderthatthe singularityspectrum hasa meaning onlyfor f(α) 0 [19, 20,23, 24,
≥
25]. However,when multifractality arises in random systems, disorder-averagedvalues may involve other generalized
exponents [43, 44, 45, 46] than the typical values (see Eq. 14). In quantum localization transitions, these exponents
were denoted by τ˜(q)=(q 1)D˜(q) in [26, 29]
−
Y (L) L−τ˜(q) =L−(q−1)D˜(q) (21)
q |T=Tc ≃
For these disorder averaged values, the corresponding singularity spectrum f˜(α) may become negative f˜(α) < 0
[26, 29, 43, 44, 45, 46] to describe rare events (cf Eq. 19). The difference between the two generalized exponents sets
D(q) and D˜(q) associated to typical and averaged values has for origin the broad distributions at criticality [21, 26]
as we now describe.
4
C. Probability distributions of the Y (L)
q
The scenario proposed in [21, 26] in the context of quantum localization models is as follows : the probability
distribution of the logarithm of the Inverse Participation Ratios of Eq. 11 becomes scale invariant around its typical
value [21, 26], i.e.
lnP (L)=lnP (L)+u (22)
q q
where u remains a random variable of order O(1) in the limit L . According to [26] the probability distribution
→∞
G (u) generically develops an exponential tail
L
G∞(u) e−xqu (23)
u→≃∞
As a consequence, the ratio y = Pq(L)/Pqtyp(L) = eu with respect to the typical value Pqtyp(L) = elnPq(L) presents
the power-law decay
P (L) 1
q
Π y (24)
(cid:18) ≡ Ptyp(L)(cid:19)y→∝∞y1+xq
q
The conclusions of [21, 26] are then as follows : for q small enough q < q , the exponent satisfies x > 1, and the
c q
corresponding generalized dimensions coincide D˜(q) = D(q). However, for larger values q > q , the exponent x(q)
c
may become smaller x(q) < 1, and then the corresponding generalized dimensions differ D˜(q)= D(q) : the decay of
6
the averagedvalue P (L) is then determined by the finite-size cut-off of the power-lawtail. In this case,the averaged
q
values are not representative but are governedby rare events.
In this paper, we show that this scenario for the I.P.R.’s statistics at Anderson transitions describes well our data
for the directed polymer at criticality.
III. RESULTS FOR THE GENERALIZED EXPONENTS D(q) AND D˜(q)
3 3
2 2
~
D(q)
D(q)
1 1
~
D(q)
(b)
(a)
0 0
0 2 4 6 8 10 0 2 4 6 8 10
q q
FIG.1: (Color online) (a) Multifractality at criticality (T =0.79) : generalized dimensionsD(q) ((cid:13))andD˜(q)((cid:3))associated
to typical and disorder averaged values (Eq. 14 and 21) (b) Monofractality in the high-temperature phase (T = 2.): D(q) =
D˜(q)= 3 (see Eq. 13)
2
We show on Fig. 1 (a) our results for the generalized exponents D(q) and D˜(q) governing the decay of typical
and disorder averaged values (Eqs. 14 and 21). In agreement with the scenario proposed in [21, 26] for Anderson
transitions, we find that there exists a threshold q , of order q 2 here, such that
c c
∼
D(q) =D˜(q) for q <q (25)
c
D(q) >D˜(q) for q >q (26)
c
5
In particular, for q =1, the information dimension of Eq.(15) is
D(1)=D˜(1) 1.5 (27)
∼
and corresponds to the monofractal dimension D (q)=3/2 of the delocalized phase (see Eq. 13), as numerically
T>Tc
checked on Fig. 1 (b).
For q =2, the correlation dimension D(2) defined in Eq. (16) is found to be of order
D(2) D˜(2) 1.3 (28)
∼ ∼
For q 3, the values for D(q) and D˜(q) are clearly different, in particular for q =3
≥
D(3) 1.1 (29)
∼
D˜(3) 0.9 (30)
∼
and for q =10
D(10) 0.9 (31)
∼
D˜(10) 0.3 (32)
∼
The limit q will be discussed more precisely below (see Eq. 39)
→∞
Finally, for q <0, we find that Ytyp(L) and Y (L) divergemore rapidly than the power-lawsof Eqs. 14 and 21, i.e.
q q
D(q <0) =+ (33)
∞
This can be understood in the delocalized phase T >T where it is also true, since for the free Gaussian probability
c
wT=∞(~r) e−(~r)2/L/L3/2 at T = leads to exponential divergence of Yq in the negative region q <0. This finding
∼ ∞
for our ’dynamical’ model is thus very natural, but is a major difference with the Anderson localization cases where
the generalized exponents D(q) are finite for q <0.
IV. RESULTS FOR THE SINGULARITY SPECTRUM f(α)
3 6
f(α) α
2.5 5
2 4
1.5 3
1 2
0.5 1
(a) (b)
0 0
0 1 2 3 4 5 6 0 1 2 3 4 5
α q
FIG. 2: (a) Singularity spectrum f(α) : starting at some α = D(q = +∞) ∼ 0.77 where f(α ) = 0, it is tangent to
min min
the diagonal α = f(α) at α = D(1) ∼ 1.5 and asymptotically goes to f(+∞) = D(0) = 3. (b) Corresponding curve α(q) :
1
diverging at q→0, it goes through thepoint α(q=1)=D(1)∼1.5 and tendsto α =D(q=+∞) as q→∞.
min
To measure the singularity spectrum f(α), we have followed the method of the q-measures proposed in [47]. As
explained above,in the negative regionq <0, our model does not lead to power-law (see Eq. 33). As a consequence,
the maximum (α ,f(α )) of the curve f(α) which is at finite distance in Anderson localization models, is rejected
0 0
towards infinity in our case
α =+ (34)
0
∞
f(α ) =D(q =0)=3 (35)
0
6
Our result for the curve f(α) are shown on Fig. 2 (a) : it begins at some α =D(q =+ ) where f(α )=0,
min min
∞
it is tangent to the diagonal at α =D(1) 1.5 and asymptotically goes to f(+ )=D(0)=3.
1
∼ ∞
The q-values used in our computations, and the corresponding α(q) (see Eq. 20) are shown on Fig. 2 (b).
To measure better the minimal exponent α = D(q = + ) where f(α) vanishes f(α = 0) = 0, we have
min min
∞
studied the statistics of the maximal weight in each sample (Eq. 8)
wmax =max [w (~r)] (36)
L ~r L
The disorder-averagedvalue of its logarithm gives
lnwmax 0.77lnL (37)
L ∼−
The first moment involves a similar value
w L−0.75 (38)
L
∼
so our conclusion is that the minimal exponent in a typical sample is around
α =D(q =+ ) 0.77 (39)
min
∞ ∼
V. RESULTS FOR PROBABILITY DISTRIBUTIONS
A. Probability distributions of the Y (L)
q
0
P (ln Y ) ln G (u)
L 2 L
1 −2
−4
−6
0.5
−8
(b)
−10
0
−8 −6 −4 −2 −12
−5 −2.5 0 2.5 5 7.5 10
ln Y
2 u
FIG. 3: (Color online) Histogram of lnY (L) at criticality (T = 0.79) (a) Probability distribution P (lnY ) for L =
2 c L 2
6,12,18,24,36,48,60,72,84,96 (b) Rescaled distributions lnG (u = lnY (L)−lnY (L)) : the exponential tail of Eq.(41)
L 2 2
is clearly visible, the corresponding slope being x ∼1.
2
Tounderstandthedifferencebetweenthegeneralizedexponentsassociatedtotypicalandaveragedvalues(Eq. 26),
we now consider the probability distributions of Y (L) over the samples. Our results for the histograms of lnY (L)
q 2
for variousL atcriticality areshownonFig. 3 (a). Remarkably,asL grows,this distributionsimply shifts alongthe
x-axiswithafixedshape,asalsofoundin[21,26]forI.P.R.satAndersontransitions. As inEq. 22, wemaytherefore
write
lnY (L)=lnY (L)+u (40)
q q
where u is a randomvariable oforderO(1) in the limit L . The probabilitydistribution G (u) ofu=lnY (L)
2 2
→∞ −
lnY (L) is shown on Fig. 3 (b) for various L. It clearly develops an exponential tail as u
2
→∞
GL→∞(u)u→≃∞e−xqu (41)
7
10 ln G (u)
L
P (ln Y ) −2
L 2
5
−7
(a) (b)
0 −12
−8 −7 −6 −5 −4 −3 −6 −4 −2 0 2 4 6
ln Y
2 u
FIG.4: (Color online) Histogram of lnY (L)in thehigh-temperaturephase( T =2)(a) Probability distributionof P (lnY )
2 L 2
pour L = 6,12,18,24,36,48,60 (b) The distributions G (u) of the rescaled variable u = (lnY (L)−lnY (L))/[∆lnY (L)]
L 2 2 2
become Gaussian.
As stressed in [21, 26], the ratio y =Y (L)/Ytyp(L)=eu then presents the power-law decay
q q
Y (L) 1
q
Π y (42)
(cid:18) ≡ Ytyp(L)(cid:19)y→∝∞y1+xq
q
Whenever x <1, the corresponding generalized dimensions differ D˜(q)=D(q) (Eq. 26) : the decay of the averaged
q
6
value Y (L)is then determinedby the finite-size cut-offofthe power-lawtail. Our resultsfor the histogramsofY (L)
q q
for q =3,4,.. are similar to the results shown for q =2 on Fig. 3.
For comparison, we show on Fig. 4 (a) the histogram of lnY (L) in the delocalized phase at T = 2 : as L grows,
2
the width shrinks around the averaged value lnY (L) (3/2)lnL. The corresponding rescaled distribution shown
2
∼ −
on Fig. 4 (b) tends towards the Gaussian distribution.
B. Probability distributions of the last-monomer entropy s =−P w (~r)lnw (~r)
L ~r L L
Q (s) 0.6
L
H (z)
L
2 0.5
0.4
0.3
1
0.2
(a) 0.1 (b)
0 0
2 3 4 5 6 7 8 −6 −5 −4 −3 −2 −1 0 1 2 3
s z
FIG.5: (Coloronline)Histogramoftheentropys =−P w (~r)lnw (~r)atcriticality(T =0.79)(a)Probabilitydistribution
L ~r L L
Q (s) for L=6,12,24,36,48,60 (b)Thedistributions H (z) of therescaled variable z= sl−sL are clearly asymmetric.
L L ∆sL
Since the last-monomer entropy s is closely related to the Y (L) (Eq. 15), we have also computed its histogram
L q
over the samples both at criticality (Fig 5 ) and in the delocalized phase at T =2>T (Fig. 6). Again, the rescaled
c
distribution is Gaussian for T >T (Fig. 6 b), and strongly asymmetric at criticality (Fig. 5 b).
c
8
15 0.5
Q (s) (a) H (z) (b)
L L
0.4
10
0.3
0.2
5
0.1
0 0
4 5 6 7 8 −5 −4 −3 −2 −1 0 1 2 3 4 5
s z
FIG.6: (Coloronline)Histogramoftheentropys =−P w (~r)lnw (~r)inthehightemperaturephase(T =2)(a)Probability
L ~r L L
distributionQ (s)forL=6,12,24,36,48,60(b)ThedistributionsH (z)oftherescaledvariablez= sl−sL becomeGaussian.
L L ∆sL
VI. FINITE-SIZE SCALING IN THE CRITICAL REGION
For Andersontransitions,finite-size scaling involvesthe multifractalspectrumatcriticality but asingle correlation
length exponent ν (see the reviews [23, 24]). In this section, we thus try the following finite-size scaling form in the
critical region
Y (L,T)=L−τ˜(q) Φ (T T )L1/ν (43)
q c
(cid:16) − (cid:17)
For T <T , the convergence to finite-values Y (L= ,T) in the L limit yields
c q
∞ →∞
Y (L= ,T)=(T T)β˜(q) with β˜(q)=ντ˜(q) (44)
q c
∞ −
This relation between the multifractal exponents τ˜(q) and the critical exponents β˜(q) and ν is well known for the
Anderson transitions (see the reviews [23, 24]). Our results for T < T are shown on Fig. 7 for q = 2 (a) and q = 3
c
(b) with the value ν = 2. This value is one of the two values ν 2 and ν 4 found previously in the literature for
∼ ∼
other observables [38, 39, 42].
35 150
−
v −
2 v
30 3
25
100
20
15
50
10
5 (a) (b)
0
−3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0
−3.5 −2.5 −1.5 −0.5
x x
FIG. 7: (Color online) Finite-size scaling (T < T ) of v = Lτ˜(q)Y as a function of x = (T −T )L1/ν, see Eq. 43, with the
c q q c
valueν =2and for the sizes L=12((cid:13)),24((cid:3)),36(♦),48(△),60(⊲) (a) q=2 (b) q=3
For T >T , the results of the finite-size scaling form of Eq. 43 are shown on Fig. 8 for q =2 with the value ν =2.
c
Note that the matching between the finite-size scaling form of Eq. 43 and the asymptotic behavior in the delocalized
9
0.5
−
v
2
0.4
0.3
0.2
0.1
0
0 2 4 6 8 10
x
FIG.8: (Color online) Finite-size scaling (T >T )of v =Lτ˜(2)Y as afunction of x=(T −T )L1/ν with thevalueν =2and
c 2 2 c
for the sizes L=12((cid:13)),18(∗),24((cid:3)),36(♦),48(△),60(⊲), see Eq. 43.
phase (Eq. 13) yields a diverging amplitude A(T) in Eq. 13 for q >1
Yq(L,T >Tc)∼ LA23((qT−)1) with A(T)∼(T −Tc)−ν(q−1)(23−D˜(q)) (45)
Our data thus points towards a correlation length exponent ν 2 above and below T , i.e. towards a value very
c
∼
close to the general lower bound ν 2/d=2 of disordered systems [48].
≥
VII. CONCLUSION
In this paper, we have found that the directed polymer in a random medium of dimension 1+3 exhibits mul-
tifractal properties at the critical localization/delocalization transition. We have numerically studied the statistics
of the Y (L) (see Eqs.8 and 10), which are the dynamical analogs of the Inverse Participation Ratios of Anderson
q
localization quantum models [23, 24, 26]. Our results are very close to the Evers-Mirlin scenario [21, 26] for the
Anderson transitions case. In particular, we have found that the generalized dimensions D(q) and D˜(q) for typical
and disorder averaged values coincide for q < q 2 but differ for q > q , and that the probability distributions of
c c
y =Yq(L)/Yqtyp(L) over the samples becomes sca∼le invariant with a power-law tail 1/y1+xq. We have also measured
the corresponding typical singularity spectrum f(α), which starts at the value α = D(+ ) 0.77, and ends at
min
∞ ∼
α =+ . Off-criticalresultslead, througha finite size scalinganalysis,to a value ν 2 forthe correlationlength
max
∞ ∼
exponent on both sides of the transition.
Finally, our numerical results, in particular the scale invariant shape of the histogram of lnY (L) shown on Fig. 3
2
(a), strongly support the equality T =T (d=3) (see Eq. 5 and the corresponding discussion).
c 2
Since the directed polymer in a random medium can be mapped onto a growth model in the Kardar-Parisi-Zhang
universality class[30], multifractality is also expected to show up at the critical point of these growth models. More
generally, the present study confirms that it may be interesting to characterize the critical points of quenched disor-
dered models by their multifractal properties.
Acknowledgements
We thank F. Igl´oi for drawing our attention to many relevant references.
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10
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