Scaling approach to related disordered stochastic and free-fermion models

0.26 MB·English

by  R. J. Harris

Scaling approach to related disordered stochastic and free-fermion models
R. J. Harris1,∗ and R. B. Stinchcombe2,†
1Institut fu¨r Festk¨orperforschung, Forschungszentrum Ju¨lich, 52425 Ju¨lich, Germany
2Rudolf Peierls Centre for Theoretical Physics, University of Oxford, 1 Keble Road, Oxford, OX1 3NP,UK
(Dated: January 2, 2007)
7
0 Motivated by mapping from a stochastic system with spatially random rates, we consider disor-
0 dered non-conserving free-fermion systems using a scaling procedure for the equations of motion.
2 Thisapproachdemonstratesdisorder-inducedlocalization actingincompetitionwiththeasymmet-
ric driving. Wediscuss theresulting implications for theoriginal stochastic system.
n
a
J PACSnumbers: 02.50.Ey,05.50.+q,05.40.-a,75.10.Jm

3

I. INTRODUCTION ance and therefore one can not utilize the usual Fourier
]
h andBogoliubovtransformations. However,workbyMerz
c and Chalker involving doubling the number of fermion
The role of disorder in stochastic non-equilibrium sys-
e
operators [21], offers the possibility of using a type of a
m tems is a topic of much recent interest (see e.g., [1, 2, 3,
localizedBogoliubovtransformationfollowedby numeri-
4,5]andreferencestherein). Thisisexemplifiedbystud-
- cal calculation. Another potential way to make progress
t ies of the asymmetric simple exclusion process (ASEP),
a is to derive and work with the linear dynamic equations
a one-dimensional lattice gas model exhibiting a phase
t
s transition for open boundary conditions. The effect of and this is the approachpursued in the present work.
.
t quenchedparticle-disorderedhopratesinthissystemwas Our scaling approach is based on a method developed
a
m investigated in, e.g., [6, 7, 8] and the case of quenched by Pimentel and Stinchcombe [22] to treat the equation
spatial disorder has been clarified by a number of works ofmotionofa1DMattis-transformedEdwards-Anderson
-
d (see,e.g.,[2,9,10,11])includingviaalinearizationtrans- Heisenberg spin glass and later applied by the present
n formation and scaling approach[12]. authors to the Cole-Hopf transformed ASEP [12]. Es-
o sentially, one looks at the evolution of parameters after
Inthispaperwestudyarelatednon-equilibriummodel
c a b = 2 decimation of the equation of motion. For ex-
with applications in catalysis—the partially asymmetric
[
tended states the parameters are found to evolve chaot-
exclusion process combined with deposition and evapo-

2 ration of dimers (or equivalently pair annihilation and ically under the scaling whereas for localized states they
v decayexponentiallyallowingtheidentificationofalocal-
creation). Studies of the pure model e.g., [13, 14, 15]
5 izationlength. Applicationofthismethodtothepresent
suggest that, unlike the ASEP, for adsorption and des-
9 problemcontributes to understandingofdisorderedfree-
orptionrateswhich do not vanishin the thermodynamic
5
fermions as well as illustrating a potential way to make
6 limit, there is no steady-state phase transitionevenwith
progress with the disordered stochastic model.
0 open boundary conditions [16, 17]. However, the effect
6 of disorder on both steady-state properties (such as the Theplanoftherestofthepaperisasfollows. InSec.II
0 density) and dynamics is still of interest. wedefineourmodel,introducethequantumHamiltonian
/ formalismanddemonstratethefree-fermionreductionfor
t Here we demonstrate that, for spatially disordered
a aparticularchoiceofdisorder. Wealsodiscusstheknown
rates obeying a particular condition, one can approach
m
pureresults. InSec.III,we derivethe coupledequations
this problemviaapowerfulmappingtofree-fermionsys-
- ofmotions forthe disorderedfree-fermioncase. Then, in
tems. Inthe pure case,the resultingfree-fermionsystem
d
Sec.IVweexploitthefactthattheseequationsarelinear
n can be treated by Fourier and Bogoliubov transforma-
to apply scaling techniques based on those in [12, 22].
o tions [13, 18]. The simplest disorderedscenario,the case
c of a single defect, is known to produce the stochastic As in those works, we find that disorder induces local-
: ization effects and in Sec. V we tentatively discuss the
v analogue of localized modes [19]. One might also expect
consequences for the original stochastic system. Finally,
i to make progress for the case of dilution, by breaking
X Sec. VI contains conclusions and suggestions for future
the chain up into finite uniform sections (compare the
r treatment of the diffusion-only problemin [20]; a similar work.
a
break-up into effective pure segments also occurs in [5]).
In this paper we are chiefly interested in more general
disordered cases where there is no translational invari- II. MODEL AND MAPPINGS
Letusstartbyconsideringthepurepartiallyasymmet-
ric exclusion process with dimer evaporation and depo-
∗Present address: Fachrichtung Theoretische Physik, Universita¨t
sition, shown schematically in Fig. 1. Subject to the ex-
des Saarlandes, 66041 Saarbru¨cken, Germany; Electronic address:
clusion constraints, particles hop to the right (left) with
[e-mail]
†Electronicaddress: [e-mail] rate p (q), and pairs of particles are deposited (evapo-
2
q p ability imposes the condition
ε ε′
hs|H =0. (3)
FIG. 1: Partially asymmetric exclusion process with dimer
Note that the general four-parameter model defined
evaporation and deposition (pure model). Filled circles indi-
by (2) contains many other important models as sub-
cate particles; open circles denotevacancies.
cases, e.g., the ASEP (q = ε = ε′ = 0), symmetric
l l l
diffusion (p = q, ε = ε′ = 0), etc. Further progress
l l l l
depends on what boundary terms are imposed and how
rated) with rate ε (ε′). This provides a simple model tractabletheresultantquantumspinmodelis. Inpartic-
ular, a Jordan-Wigner transformation to fermion opera-
foradsorption-desorptionprocessesandcatalyticsurface
reactions. tors c†l, cl gives both an easily treatable quadratic part
In the special case p + q = ε + ε′, the system can (arising from σ±σ± and σz terms) and a more difficult
quartic part (from the σzσz terms). The quartic terms
be mappedvia aJordan-Wignertransformation[23]and
will clearly be zero for the special case
a Bogoliubov-Valatin transformation [24, 25] to a free-
fermionproblemandhencesolvedexactly[13,14,18,26].
p +q =ε +ε′ (4)
The results are ofmoregeneralsignificance since,by du- l l l l
ality transformations and spin rotations, the quantum and this gives the disordered free-fermion model which
Hamiltoniancanalsobe mappedto otherstochasticsys- will be explored in more detail in the remainder of this
tems [15, 27], including some which are experimentally paper.
realizable [28]. A subcase is p = q = ε = ε′ which maps For this particular case, we now outline the details of
to the Ising model and so is well understood [18]. the transformationtofermionoperators(compareGryn-
The aim of the present work is to study the spatially- berg et al. [18] for the pure model). Writing σz in terms
disordered version of this model,with rates pl, ql, εl and of σ+σ− and imposing periodic boundary conditions,
ε′l for processes on the bond between sites l and l +1. Eq. (2) becomes
Inthis sectionwe demonstratethatthe mapping to free-
fermions is still valid in the case where pl+ql = εl +ε′l H =− [p σ−σ+ +q σ+σ− +ε σ+σ+ +ε′σ−σ−
for each l. FF l l l+1 l l l+1 l l l+1 l l l+1
l
Thegeneralizedpairprocessparameterizedbyp ,q ,ε X
and ε′ can be conveniently represented using thelqulan-l +(pl−1−pl+εl−ε′l−1)σl+σl−−εl]. (5)
l
tum Hamiltonian formalism [29]. In this approach one
Then applying the usual Jordan-Wigner transforma-
defines a probability vector |Pi = P |ni with |ni
n n tion [23]
the basis vector associated with the particle configura-
P
tion n = (n ,n ,...,n ) and P the probability mea-
1 2 L n
sure on the set of all such configurations. |Pi obeys the σ+ =c†exp iπ c†c (6)
normalization condition hs|Pi = 1 where hs| = hn| l l  j j
n j<l
and hn|n′i = δn,n′. Within this formalism the mPaster  X 
equation for the time evolution resembles a Schr¨odinger
equation σl− =exp−iπ c†jcjcl, (7)
j<l
X
d  
|P(t)i=−H|P(t)i (1) yields the Hamiltonian in terms of fermion operators1
dt
with H = [p c c† −qc†c −ε (c†c† −1)
FF l l l+1 l l l+1 l l l+1
l
X
H = +ε′c c −(p −p +ε −ε′ )c†c ]. (8)
l l l+1 l−1 l l l−1 l l
− p σ−σ+ +q σ+σ− +ε σ+σ+ +ε′σ−σ−
l l l+1 l l l+1 l l l+1 l l l+1 Thestudyofthedisorderedfree-fermionHamiltonian(8)
l
X(cid:2) is the central aim of this paper; we now digress to sum-
1 1
− (p +q +ε +ε′)+ (−p +q +ε −ε′)σz marize the known pure results.
4 l l l l 4 l l l l l
In the pure case one can exploit translational in-
1 1
+ (p −q +ε −ε′)σz + (p +q −ε −ε′)σzσz . variance via a Fourier transformation and then use a
4 l l l l l+1 4 l l l l l l+1
(2)
(cid:3)
Here the “spin-flip” operators σ± correspond to particle
creation and annihilation processes while the projection 1 Ifthetotalnumberoffermions,Plσl+σl− =Plc†lcl,iseventhen
the fermionoperators must obeycyclic boundaryconditions; in
operators σ are needed to account for the probability
z the odd subspace, anti-cyclic boundary conditions are required.
thatconfigurationsdonotchange. Conservationofprob- See[18].
3
Bogoliubov-type similarity transformation [24, 25] (well representation):
defined only for ε and ε′ non-zero) to diagonalize the
Hamiltonian. The details are given in [13, 18]; here we ∂cj =p c +q c −ε c† +ε c†
simply quote the resulting Hamiltonian: ∂t j−1 j−1 j j+1 j−1 j−1 j j+1
+(p −p +ε −ε′ )c , (14)
j−1 j j j−1 j
HFF = λkξk+ξk, (9) ∂c†
Xk ∂tj =−pjc†j+1−qj−1c†j−1+ε′j−1cj−1−ε′jcj+1
with spectrum −(p −p +ε −ε′ )c†. (15)
j−1 j j j−1 j
λ =ε+ε′+(ε−ε′)cosk+i(p−q)sink. (10) Notethatduetothenon-HermicityofH theseequations
k
are not simply Hermitian conjugates of each other. The
The imaginary part of λ implies ballistic motion while equations are linear thus raising the possibility of using
k
the real part indicates that excitations decay with time scaling techniques like those in [12, 22]—this will be the
constant [ε + ε′ + (ε − ε′)cosk]−1. For ε and ε′ non- subject of the next section.
zero, the spectrum is gapped giving exponentially fast As a preliminary to this scaling approach, we first as-
kinetics. Bymappingbackfromthis free-fermionmodel, sume that c and c† can be written as superpositions of
j j
one can obtain the steady-state density profile for the operators with exponential time dependence:
original pure stochastic problem. The result [18]
c = Γ (ω)eωt, (16)
j j
1
ρ = , (11) ω
l X
1+ ε′/ε c† = γ (ω)eωt. (17)
j j
p ω
isinagreementwithmean-fieldcalculation(onecanshow X
that the system is spatially uncorrelated). Other works Substituting into Eqs. (14) and (15) and equating com-
have calculated dynamic correlation functions [13, 14], ponents with the same time dependence yields
shock evolution [30], persistence probability, etc. Note
thatnon-steady-statepropertiesofthestochasticsystem (E −ω)Γ =−p Γ −q Γ +ε γ −ε γ ,
j j j−1 j−1 j j+1 j−1 j−1 j j+1
canbeunderstoodbymappingtotheGlaubermodel[31] (18)
and treating the asymmetric hopping as biased diffusion
(E +ω)γ =−p γ −q γ +ε′ Γ −ε′Γ ,
of domain walls [32]. j j j j+1 j−1 j−1 j−1 j−1 j j+1
(19)
In the disordered case where there is no translational
invariance a different approach is required—in the next
with E =(p −p +ε −ε′ ).
section, we demonstrate how to derive the equations of j j−1 j j j−1
Note that for the pure casewe can carryouta Fourier
motion starting from the free-fermion Hamiltonian (8).
transformation by writing Γ (ω) = e−ikjΓ , γ (ω) =
j k j
e−ikjγ . As expected, we then find that for eachFourier
k
component, Eqs. (18) and (19) have solutions with ω =
III. EQUATIONS OF MOTION
λ or ω =−λ , where λ is the pure spectrum (10). In
k −k k
other words the operators c and c† can be written as
j j
Let us first considerthe expectation value of a general
observable X, i.e., c = e−ikj(Γ eλkt+Γ′e−λ−kt), (20)
j k k
k
hXi(t)=hs|X|Pi. (12) X
c† = e−ikj(γ eλkt+γ′e−λ−kt), (21)
j k k
where we use angular brackets to denote this stochas- k
X
tic average over histories. Using the “Schr¨odinger equa-
where
tion” (1) oneseesthat the time evolutionofhXiis given
by ε′ k
γ =iΓ cot , (22)
k k
ε 2
∂hXi
∂t =h[H,X]i (13) γk′ =−iΓ′ktank2. (23)
with [H,X] the usual commutator.
The Bogoliubov angle can be obtained from the rela-
For the free-fermion Hamiltonian (8) we can use the tions (22) and (23); in the pure case this approach thus
fermion anti-commutation relations, together with the provides an alternative route to direct transformationof
identity [A,BC]={A,B}C−B{A,C}, to calculate the the Hamiltonian (8).
commutators[cj,H]and[c†j,H]andhenceobtaincoupled As an aside, we remark that Eqs. (18) and (19) can
dynamic equations for the operators (in the Heisenberg also be cast in the form of a transfer matrix mapping.
4
Analysing products of disordered transfer matrices then 3.0
p,q
providesawaytoinvestigatepossiblelocalizationeffects. p˜,q˜
′ε,ε˜′
Thisisaparticularlyattractiveapproachforsimplemod- 2.5 ε,ε˜E
E˜
els such as binary disorder. In the next section we de-
s
r
velop instead, a general numerical scaling approach for e
t2.0
e
arbitrary distributions of disorder. m
a
r
pa1.5
d
IV. SCALING e
cal1.0
s
e
A. Procedure r
0.5
In the generaldisorderedcase it is not obvious how to
solve Eqs. (18) and (19) explicitly but their form is rem- 0.0
0 2 4 6 8 10 12 14 16 18 20
iniscent of an equation which appears in the disordered I
ASEP (in a mean-field description) after a linearizing
FIG. 2: Evolution of parameters in Eqs. (24) and (25) under
(Cole-Hopf) transformation so we can develop a scaling
b = 2 decimation. Pure case with ω = 0.41 and starting
methodsimilartotheoneusedinthatcase[12](compare
parameter values p = p˜ = q = q˜ = 0.5, ε = ε˜= 0.8, ε′ =
alsotheoriginalworkofPimentelandStinchcombe[22]). ε˜′ = 0.2, E = E˜ = 0.6. Graph shows absolute magnitude of
Forfull generalitywe mustalloweachcoefficientto scale
parametersagainst numberofiterations I;linesareprovided
differently so we perform a b = 2 dilation of the system as an aid to the eye. Chaotic behaviour is typical of scaling
using decimation on the following equations in the allowed band.
(E −ω)Γ =−p Γ −q Γ +ε γ
j j j,j−1 j−1 j,j+1 j+1 j,j−1 j−1
−ε˜ γ , (24)
j,j+1 j+1
parameters. We find two distinct types of scaling be-
(E˜j +ω)γj =−p˜j,j+1γj+1−q˜j,j−1γj−1+ε′j,j−1Γj−1 haviour.
−ε˜′ Γ . (25) For 2ε′ ≤ ω ≤ 2ε the scaling parameters all evolve
j,j+1 j+1
chaotically under scaling (see Fig. 2). In contrast for
where we have extended the notation for clarity. For ω <2ε′ or ω >2ε, the “energies” E and E˜ tend to con-
example, pj,j−1 denotes the coefficient of Γj−1 in the ex- stant values whereas the rescaled rates r (i.e, p, q, etc.)
pression for Γj. Even in the pure case, this may scale tendrapidlyto zeroinside anexponentiallydecayingen-
differently from p˜j,j+1 which is the coefficent of γj,j+1 velope (see Fig. 3), i.e., r(l) ∼ f(l)e−l/ξ where l = 2I is
in the expression for γj. In the rescaled equation for Γj thedistancebetweensites(I isthe numberofiterations)
(γj) the coefficient of Γj−2 (γj+2) is pj,j−2 (p˜j,j+2). The and ξ is a localization length. We also see this localized
resulting 10-parameter scaling equations are rather in- behaviour for all non-real ω.
volvedandaredisplayedforconvenienceinAppendix A. Just as in the scaling of the Cole-Hopf-transformed
We now test this scaling procedure on the pure model mean-fieldASEP[12]thelocalizationlengthcanbemea-
(Sec. IVB) before applying it to the disordered problem sured numerically from the exponential decay of the pa-
(Sec. IVC). rameters. In Fig. 4 we plot the localization length ob-
tained from the scaling of p, as a function of real fre-
quency ω (localization lengths obtained from the other
B. Pure model rates are found to be identical). These numerical results
can be simply explained by reference to the pure spec-
We here demonstrate how the scaling of the equations trum for the symmetric case. The “allowed” band (real
of motion (24) and (25) reflects the known spectrum λ k) corresponds to the region 2ε′ ≤ ω ≤ 2ε; here states
k
of the pure model (10). For definiteness we consider the are extended and the localization length is infinite. Fre-
case ε > ε′, p ≥ q throughout; the extension to other quenciesoutsidethisbandcorrespondtoacomplexwave-
cases should be straightforward, since they are related vectork+iκ,i.e.,localizedstateswithlocalizationlength
by duality or space reflection. ξ =1/κ. Indeed, fromω =ε+ε′+(ε−ε′)cos(k+iκ)we
can obtain an analytical expression for the localization
length on the real-ω axis (k =0 or k =π):
1. Non-biased case, p=q
ε′+ε−ω −1
ξ = arccosh . (26)
For the pure case, if one starts with p = p˜ = q = q˜, ε′−ε
ε =ε˜, ε′ =ε˜′, and by implication E =E˜, then the rela- (cid:18) (cid:12)(cid:12) (cid:12)(cid:12)(cid:19)
tionships p = q, p˜= q˜, ε = ε˜and ε′ = ε˜′ are preserved This expression is shown a(cid:12)(cid:12)s a dashed(cid:12)(cid:12)line in Fig. 4 and
under scaling so one only needs to scale six independent agreement with the numerics is excellent.
5
Imλ
2.5 k
p,q localized
′pε˜,,qε˜˜′ (p−q)
ε,ε˜
E
2.0 E˜
ballistic
s
r
te 2ε′ ε+ε′ 2ε
e
m1.5 Reλk
a
r
a
p
d
ale1.0 −(p−q)
c
s
e
r
0.5
FIG. 5: Imλ versus Reλ for pure model (case p > q, ε >
k k
ε′). Solid ellipse represents allowed spectrum given by (10).
0.0 Inside the ellipse equations of motion scale to ballistic limit
0 2 4 6 8 10 12 14 16 18 20
I p/q→∞; outside of ellipse is localized region.
FIG. 3: Evolution of parameters in Eqs. (24) and (25) under
b = 2 decimation. Pure case with ω = 0.39 and starting
parameter values p = p˜ = q = q˜ = 0.5, ε = ε˜ = 0.8, ε′ = 50 p
ε˜′ = 0.2, E = E˜ = 0.6. Graph shows absolute magnitude of 45 pq˜
q˜
parametersagainst numberof iterationsI;linesareprovided ε
40 ε˜′
asanaid totheeye. Rateparametersdecayexponentially as s εε˜′
a function of 2I, as expected outside theallowed band. ter35 EE˜
e
m30
a
r
pa25
5
numericalscalingvalueofξ d
analyticalpredictionforξ e20
al
c
s15
4 re
10
5
3
0
0 1 2 3 4 5
ξ
I
2
FIG. 6: Evolution of parameters in Eqs. (24) and (25) under
b = 2 decimation (first few iterations). Pure case with ω =
1 0.41 and starting parameter values p = p˜ = 0.55, q = q˜ =
0.45, ε = ε˜ = 0.8, ε′ = ε˜′ = 0.2, E = E˜ = 0.6. Graph
shows absolute magnitude of parameters against number of
0 iterations I; lines are provided as an aid to the eye. Rates p,
0.0 0.5 1.0 1.5 2.0
ω q˜, ε, ε′ scale exponentially to infinity while p˜, q, ε˜, ε˜′ decay
exponentially and “energies” E, E˜ tend to constant values.
FIG. 4: Localization length ξ versus (real) frequency ω for Thisistypicalofballisticbehaviourinsidetheallowed ellipse
pure case with p=q =0.5, ε=0.8, ε′ =0.2. ξ is effectively (for case p>q).
infinite in allowed band 0.4<ω <1.6. Crosses indicate data
from numerical scaling of rate p; dashed line is analytical
prediction of Eq. (26).
and E˜ again tend to constant values, some of the rates
scale exponentially to zero, whereas others grow expo-
nentially to infinity (see Fig. 6). We argue that in this
2. Biased case, p6=q region the system is dominated by the ballistic motion
and scales to the totally asymmetric case. So if we start
In the pure biased case the “allowed” values of ω are with p > q, then p scales to infinity while q decays to
givenbyλ ofEq.(10)withkreal—anellipseincomplex zero.
k
λk-space, see Fig. 5. For these values of ω we againfind, In Figs. 7 and 8 we plot the localization lengths for
asexpected,thatthescalingparametersallevolvechaot- p and q along the principal axes of the ω-space ellipse
ically under scaling corresponding to extended states. (again for the case p > q). As is evident from the dis-
Outside this allowed ellipse we see localized states once cussion above, the localization length for q is small and
more. positiveinallregions—qisanirrelevantparameterunder
Inside the ellipse we see a new type of behaviour—E scaling. Thelocalizationlengthforpispositiveinthelo-
6
25 16
ξp
ξq
20
14 p
p˜
15 q
q˜
s12 ε
105 ameter10 εε˜EE˜ε˜′′
r
ξ 0 pa 8
d
-5 ale 6
c
-10 s
re 4
-15
2
-20
-25 0
0.0 0.5 1.0 1.5 2.0 0 2 4 6 8 10 12
ωRe I
FIG. 7: Numerical localization lengths ξp, ξq as a function FIG. 9: Evolution of parameters in Eqs. (24)and (25) under
of real frequency ωRe for pure case with p = 0.55, q = 0.45, b=2decimation. Disorderedcasewithω=1.41andstarting
ε = 0.8, ε′ = 0.2. Note difference in scaling behaviour for p parameter values pj,j−1 = p˜j−1,j = qj−1,j = q˜j,j−1 = 0.5,
and q. Lines are provided as an aid to theeye. εj,j−1 =ε˜j−1,j =z,ε′j,j−1 =ε˜′j−1,j =1−z,E =E˜ =pj,j−1−
pj+1,j −εj+1,j −ε′j,j−1 where z is a random variable drawn
fromauniformdistributionbetween0.6and1.0. Graphshows
150 absolute magnitude of parameters at site j0 (averaged over
ξp 100 realizations of disorder) against number of iterations I;
ξq
linesareprovidedasanaidtotheeye. Rateparametersdecay
100 exponentially as a function of 2I showing a clear localization
effect.
50
ξ 0 the case where the disorder is in ε and ε′ with the rates
chosen to maintain the free-fermion condition for each
-50 bond (specifically ε′ = 1−ε , p =p and q = q, where
j j j j
p+q =1). Onceagain,weconsiderinturnthesymmetric
-100 (p=q) and asymmetric (p6=q) cases.
-150
-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4
ωIm 1. Non-biased case, p=q
FIG. 8: Numerical localization lengths ξp, ξq as a function In Fig.9 we show anexample of the scalingbehaviour
of complex frequency ω = 1+iωIm for pure case with p =
forthe casewith p =q =0.5,ε drawnfromauniform
0.55, q = 0.45, ε = 0.8, ε′ = 0.2. Note difference in scaling j j j
distributionandε′ =1−ε . The value ofω corresponds
behaviour for p and q. Lines are provided as an aid to the j j
eye. to being in the allowed band for the pure case (compare
Fig.2). Wefindthattheratesevolvechaoticallytowards
zerowithinanexponentialenvelopeandsowecandefine
a localization length via r(l) ∼ f(l)e−l/ξ. In Fig. 10 we
calized region outside the ellipse, infinite on the allowed plot the localization length for the decay of p, averaged
ellipse, and negative in the ballistic region inside (refer over realizations of disorder. Localization is clearly seen
again to Fig. 5). The value of this relevant localization for all values of ω in the pure band.
lengthcanagainbe understoodanalyticallyby consider-
ing a complex wavevectorin the dispersion relation.
2. Biased case, p6=q
C. Disordered model In Fig. 11 we plot the localization lengths for p and
q along the real frequency axis for a sample disordered
Armed with this understanding of the pure case, we case, viz., p =0.55, q =0.45, ε drawn from a uniform
j j j
move on to look at the effect of disorder on the free- distribution and ε′ = 1−ε . The pure results are also
j j
fermion system. There are obviously various different shownforreference(comparealsoFig.7). Inthiscasewe
possiblewaystoadddisorder;herewemainlyinvestigate see that disorder reduces the size of the allowed ellipse
7
250 1000
disorderedcase p=0.6,q=0.4
purecase p=0.7,q=0.3
800 pp==00..89,,qq==00..21
p=1.0,q=0.0
200
600
400
150
200
ξ ξp
0
100
-200
-400
50
-600
0 -800
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50
ω ω
FIG. 10: Localization length for p scaling versus (real) fre- FIG.12: Numericallocalizationlengthξpasafunctionofreal
quencyfordisorderedcasewithp=q=0.5,εj =z,ε′j =1−z frequency ωRe for disordered cases with εj = z, ε′j = 1−z
wherezisdrawnfromauniformdistributionbetween0.6and where z is drawn from a uniform distribution between 0.6
1.0. Pureresultsoutsidethebandarealsoshownforcompar- and1.0. Resultsareshownforarangeofvaluesofpandq as
ison (dashed). In the disordered case, thelocalization length indicated by the legend. Lines are provided as an aid to the
is finite everywhereexcept ω=1. eye.
100
disordered,ξp bosonic problem arises from the study of columnar de-
disordered,ξq
ppuurree,,ξξpq fects in the superconductor.
Forallcaseswithdisorderonlyinεandε′,wefindthat
50 thelocalizationlengthisalteredalongthereal-ωaxisbut
unchanged along the axis in the imaginary direction.2
However,the change in the position of the allowed band
ξ 0 (where the localizationlengthis infinite) depends on the
strength of the driving p−q, see Fig. 12. We shall now
considersometentativeanalyticalideastoquantifythese
disorder effects.
-50
Recall that the pure spectrum is given by
λ =b −a cosk+i(p−q)sink, (27)
k 0 0
-100
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
ωRe with b0 = ε′ +ε and a0 = ε′ −ε. We seek to modify
this to representthe effective spectrum ofthe disordered
FIG.11: Numericallocalizationlengthsξp,ξq asafunctionof model. Forthecaseofdisorderedεandε′ (withε +ε′ =
realfrequencyωRefordisorderedcasewithp=0.55,q=0.45, p+q =1) then a naive suggestion is j j
εj =z,ε′j =1−zwherezisdrawnfromauniformdistribution
between 0.6 and1.0. Linesare providedasan aid totheeye.
λ =b −acosk+i(p−q)sink (28)
Pureresultsareshownforcomparison. Notethereductionof k 0
theballistic region in thedisordered case.
where a can, in principle, have both real and imaginary
parts, i.e.,
a=a˜+i2δ (29)
alongthereal-ω axis,i.e.,itactstoincreasethelocalized
region at the expense of the ballistic region. The results
witha˜, δ real. The imaginarypartofa causes arotation
for ω = 1+iω are essentially the same as in the pure
Im of the allowed ellipse (see Fig. 13); if a˜ 6= a , then there
0
case (Fig. 8).
is also stretching or compression. In agreement with the
This competition between disorder-induced localiza- numerical localization length data, this form leaves the
tion and driving is analogous to that arising in another
non-Hermitian Hamiltonian studied by Hatano and Nel-
son [33]. Their work was motivated by the mapping be-
tween flux lines in a (d+1)-dimensional superconductor 2 Obviously disorder in p and q affects the imaginary-ω direction
and d-dimensional bosons. A random potential in the instead.
8
Imλ
k 0.6690
p=0.5,q=0.5
p=0.55,q=0.45
replaceme(npt−s q) p=0.7,q=0.3
p=1.0,q=0.0
ρ0+bw2
0.6685
b0+2a0 b0 b0−2a0
Reλk 0.6680
ρ
0.6675
−(p−q)
0.6670
FIG.13: Schematicofeffectivedisordereddispersionrelation
in ω-spacefor λk =b0+(a0+i2δ)cosk+i(p−q)sink (solid
line). Pure case λk = b0+a0cosk+i(p−q)sink is shown 0.66605.00 0.05 0.10 0.15 0.20
for comparison (dashed line). Inside the ellipse equations of w
motion scale to ballistic limit p/q → ∞; outside of ellipse is
localized region. FIG. 14: Simulation results showing density dependence on
widthofε,ε′ distributionforrangeofp,qvalues(seelegend)
inasystemofsize1000. εj isdrawnfromauniformdistribu-
tionbetween0.8−wand0.8+w;ε′j givenby1−εj. Dashed
centre of the ellipse and the intersection with the ω = line is a squared fit given by ρ=ρ0+bw2 with b=0.05.
b axis unchanged, It also explains why any amount of
0
disorder causes localization in the p=q case (where the
ellipse is a line along the real axis) for all values of real
ofthe Bogoliubovtransformation[cf. Eqs.(22)and(23)]
frequency except ω =1.
andbycomparisonswithsimulation,wesuggestthatthe
There is no a priori reasonwhy the imaginary part of ratio ε′/ε is the important quantity and hence define
a should have a positive sign so we naturally expect the
possibility ofboth ±2δ. Furthermore,we anticipate that ε′ 1 ε′+iδ ε′−iδ
= + . (30)
a˜ is relatedtothe meanofε′−εandδ isproportionalto ε 2 ε−iδ ε+iδ
(cid:18) (cid:19) (cid:18) (cid:19)
thewidthofthedistribution. Infact,comparisonsofthe
This then yields a disorder-averageddensity
obtained localization lengths (e.g., see Fig. 12) with the
prediction from the dispersion relation (28) suggest that 1
ρ = , (31)
the scalingresultscannotbe describedbyunique values l
ofa˜,δ. Itappearsthatwhatwemeasureinthenumerical 1+ (ε′/ε)
scaling procedure, is the average of 1/ξ for some range q
whichis increasedfromthe pure case by anamountpro-
of values. Further work is needed to check and quantify
portionaltoδ2 (forsmallδ). Simulationsonthe stochas-
this hypothesis.
tic model (see Fig. 14) confirm a density shift propor-
tional to the square of the width of the distribution, i.e.,
thelocalizationinthedisorderedfree-fermionmodeldoes
V. DISORDER EFFECTS IN STOCHASTIC appear to map back to a density shift in the equivalent
SYSTEM
disorderedstochasticsystem. Furtherworkisstillneeded
torigorouslyestablishthispictureandtorelatethevalue
In this section we attempt to infer disorder effects of δ both to the details of the disorder distribution and
in the original stochastic hopping process by mapping- to the numerical localization data.
back the results for the equivalent quantum free-fermion Similarly, from knowledge of the effective dispersion
model. In particular, we are interested in disorder- relation for the disordered case, it should be possible to
induced changes to the pure steady-state density (11). adaptthepuretreatmentof[18]forbothsteady-statedy-
We argue from the previous section that one can namiccorrelationfunctionsandnon-steady-stateproper-
crudely characterize the effects of disorder in the ties. Thesequantitiestypicallycontaintime-dependences
evaporation-deposition rates by the replacements ε′ → ofthe forme−Reλkt soany decreaseinthe effective spec-
ε′ ±iδ, ε → ε∓iδ (a → a ±i2δ) where δ is related tral gap would be expected to lead to a slowing-downof
0 0
to the width of the disorder distribution. This gives two the dynamics (as was found for the ASEP in [12]).
different effective Bogoliubov angles (possibly connected
to the doubling of fermionic degrees of freedom in the
work of Merz and Chalker [21]). Stochastic observables VI. DISCUSSION
are presumably related to the average of these two pos-
sibilities but it is not yet clear what is the appropriate In this paper we have studied the effect of disor-
function of ε, ε′ to average over. Motivated by the form der on a model consisting of the partially asymmet-
9
ric exclusion process combined with dimer evaporation- depositionratesscalewithsystemsize,i.e.,theyarepro-
deposition. We presentedanexactmapping to anequiv- portional to 1/L where L is the number of sites. In this
alent quantum system and then employed a numerical case the interplay between boundary and bulk effects is
scaling technique on the quantum equations of motion. expected to give a steady-state phase transition in the
This provided a clear demonstration of disorder-induced pure model—compare recent work for a similar dimer
localization acting in competition with the asymmet- model [34] and earlier results for the case with adsorp-
ric driving. We discussed tentatively how these local- tion/desorption of monomers [35, 36]. The addition of
ization effects are related to the observed steady-state disorder to the latter pure monomer model was recently
density shift in the original stochastic system; exact de- consideredin the contextof minimal models for intercel-
tails of the inverse transformation remain to be clari- lular transport [37].
fied. The stochastic system is relatively robust to dis-
order since particle pairs can be evaporated/deposited
along the whole length of the chain.
Acknowledgments
Thereismuchscopeforfurtherworkonthisdisordered
model. In particular, a more rigorous development of
WewouldliketothankJohnChalkerforhelpfuldiscus-
the mapping from the disordered quantum model back
sions. ThisworkwassupportedbyEPSRCunderOxford
to the stochastic system is needed. This would permit
Condensed Matter Theory Grants GR/R83712/01 and
the translation of numerical scaling results for different
GR/M04426/01together with Grant No. 309261.
formsofdisorder(including, forexample,the casewhere
all rates are disorderedand ε +ε′ =p +q is position-
j j j j
dependent) into quantitativestatements aboutthe effect
ondensityandcorrelationfunctions. Asindicatedatthe APPENDIX A: DISORDERED FREE-FERMION
SCALING
endofSec.Vweshouldalsoliketoextendthediscussion
to dynamics.
Another open question relates to the effect of disor- Here we present scaling relations for the coupled free-
derintheopen-boundarycasewhentheevaporationand fermion equations of motion (24) and (25):
(E −ω)Γ =−p Γ −q Γ +ε γ −ε˜ γ , (A1)
j j j,j−1 j−1 j,j+1 j+1 j,j−1 j−1 j,j+1 j+1
(E˜ +ω)γ =−p˜ γ −q˜ γ +ε′ Γ −ε˜′ Γ . (A2)
j j j,j+1 j+1 j,j−1 j−1 j,j−1 j−1 j,j+1 j+1
A b=2 decimation of these equations leads to
(E′ −ω)Γ =−p′ Γ −q′ Γ +(ε)′ γ −(ε˜)′ γ , (A3)
j j j,j−2 j−2 j,j+2 j+2 j,j−2 j−2 j,j+2 j+2
(E˜′ +ω)γ =−p˜′ γ −q˜′ γ +(ε′)′ Γ −(ε˜′)′ Γ , (A4)
j j j,j+2 j+2 j,j−2 j−2 j,j−2 j−2 j,j+2 j+2
with rescaled “energy” parameters
E′ =E − pj,j−1qj−1,j − qj,j+1pj+1,j + εj,j−1ε˜′j−1,j + ε˜j,j+1ε′j+1,j, (A5)
j j Ej−1−ω Ej+1−ω E˜j−1+ω E˜j+1+ω
E˜′ =E˜ − q˜j,j−1p˜j−1,j − p˜j,j+1q˜j+1,j + ε′j,j−1ε˜j−1,j + ε˜′j,j+1εj+1,j, (A6)
j j E˜j−1+ω E˜j+1+ω Ej−1−ω Ej+1−ω
10
and the rescaled rates
− pj,j−1pj−1,j−2 + εj,j−1ε′j−1,j−2 + Aj ε′j,j−1pj−1,j−2 + q˜j,j−1ε′j−1,j−2
p′ = Ej−1−ω E˜j−1+ω E˜j′+ω Ej−1−ω E˜j−1+ω , (A7)
j,j−2 (cid:16) 1−(cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− qj,j+1qj+1,j+2 + ε˜j,j+1ε˜′j+1,j+2 − Aj ε˜′j,j+1qj+1,j+2 + p˜j,j+1ε˜′j+1,j+2
q′ = Ej+1−ω E˜j+1+ω E˜j′+ω Ej+1−ω E˜j+1+ω , (A8)
j,j+2 (cid:16) 1−(cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− pj,j−1εj−1,j−2 + εj,j−1q˜j−1,j−2 + Aj ε′j,j−1εj−1,j−2 + q˜j,j−1q˜j−1,j−2
(ε)′ = Ej−1−ω E˜j−1+ω E˜j′+ω Ej−1−ω E˜j−1+ω , (A9)
j,j−2 (cid:16) 1−(cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− qj,j+1ε˜j+1,j+2 + ε˜j,j+1p˜j+1,j+2 − Aj ε˜′j,j+1ε˜j+1,j+2 + p˜j,j+1p˜j+1,j+2
(ε˜)′ = Ej+1−ω E˜j+1+ω E˜j′+ω Ej+1−ω E˜j+1+ω , (A10)
j,j+2 (cid:16) 1−(cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− ε˜′j,j+1ε˜j+1,j+2 + p˜j,j+1p˜j+1,j+2 − Bj qj,j+1ε˜j+1,j+2 + ε˜j,j+1p˜j+1,j+2
p˜′ = Ej+1−ω E˜j+1+ω Ej′+ω Ej+1−ω E˜j+1+ω , (A11)
j,j+2 (cid:16) 1− (cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− ε′j,j−1εj−1,j−2 + q˜j,j−1q˜j−1,j−2 + Bj pj,j−1εj−1,j−2 + εj,j−1q˜j−1,j−2
q˜′ = Ej−1−ω E˜j−1+ω Ej′+ω Ej−1−ω E˜j−1+ω , (A12)
j,j−2 (cid:16) 1−(cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− ε′j,j−1pj−1,j−2 + q˜j,j−1ε′j−1,j−2 + Bj pj,j−1pj−1,j−2 + εj,j−1ε′j−1,j−2
(ε′)′ = Ej−1−ω E˜j−1+ω Ej′+ω Ej−1−ω E˜j−1+ω , (A13)
j,j−2 (cid:16) 1−(cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
− ε˜′j,j+1qj+1,j+2 + p˜j,j+1ε˜′j+1,j+2 − Bj qj,j+1qj+1,j+2 + ε˜j,j+1ε˜′j+1,j+2
(ε˜′)′ = Ej+1−ω E˜j+1+ω Ej′+ω Ej+1−ω E˜j+1+ω (A14)
j,j+2 (cid:16) 1− (cid:17) AjBj (cid:16) (cid:17)
(E′−ω)(E˜′+ω)
j j
where
p ε˜ q ε ε p˜ ε˜ q˜
j,j−1 j−1,j j,j+1 j+1,j j,j−1 j−1,j j,j+1 j+1,j
A = − − + , (A15)
j Ej−1−ω Ej+1−ω E˜j−1+ω E˜j+1+ω !
B = −ε′j,j−1qj−1,j + ε˜′j,j+1pj+1,j + q˜j,j−1ε˜′j−1,j − p˜j,j+1ε′j+1,j . (A16)
j Ej−1−ω Ej+1−ω E˜j−1+ω E˜j+1+ω !
Note the difference in meaning between ε′ (the evaporation rate) and (ε)′ (the scaled deposition rate).
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