Localization-delocalization transition in disordered systems with a direction

4 Pages·0.13 MB·English

by  A. V. Kolesnikov

Localization-delocalization transition in disordered systems with a direction
A. V. Kolesnikov1 and K. B. Efetov1,2
1Fakult¨at fu¨r Physik und Astronomie, Ruhr-Universit¨at
Bochum, Universit¨atsstr. 150, Bochum, Germany
2L.D. Landau Institute for Theoretical Physics, Moscow, Russia
(February 1, 2008)
0
0
0
2 gauge transformationapplies only if the functions φk(r)
Usingthesupersymmetrytechnique,westudythelocaliza- do not grow at infinity. For eigenfunctions of the form
n
tion-delocalization transition in quasi-one-dimensional non-
a Hermitian systems with a direction. In contrast to chains, φ0 ∼exp(−|r|/lc), this is possible if h<hc. For h>hc,
J one should take extended wave functions, which leads to
our model captures the diffusive character of carriers’ mo-
9 complex eigenenergies.
tion at short distances. We calculate the joint probability
1 Qualitative arguments of Ref. [1] have been confirmed
of complex eigenvalues and some other correlation functions.

We find that the transition is abrupt and occurs as a result for the disordered chains numerically as well as analyti-
]
n of an interplay between two saddle-points in the free energy cally[4–6]. At the same time, they arequite generaland
n functional. one might expect that such a transition occurs in more
- complex systems provided all states at h=0 are local-
s
i 72.15.Rn, 73.20.Fz, 74.60.Ge ized. For example, one can think of disordered quasi-
d
. one-dimensional wires and films. Using the mapping of
t
a Non-Hermitianmodelswithdisorderhaveattractedre- Ref. [1], one expects such models to be relevant for de-
m scription of the vortices in slabs and 3D bulk supercon-
cently a considerable attention [1–11]. Non-Hermitian
- Hamiltoniansappearincontextoffluxlinesinsupercon- ductors. The wires and films are richer than the chains
d
becauseatdistanceslargerthanthemeanfreepathl but
ductors [1], in transfer phenomena in lossy media [8], in
n
smallerthanthe localizationlengthL the motionis dif-
o hydrodynamics [9], in QCD [10], in quantum mechanics c
c of open systems [11]. fusive,whereasindisorderedchainsonlyballisticmotion
[ orlocalizationarepossible. Asaresult,propertiesofthe
The most important property of the non-Hermitian
localization-delocalization transition in wires and films
1 Hamiltonians is that their eigenenergies can be complex
v independently of their origin. Very interesting physical can differ. The problem has not been addressed yet and
3 systemsaremodelswithadirection. ThesimplestHamil- our paper is aimed to provide its basic understanding.
6 Westudythedistributionfunctionofcomplexeigenen-
tonian of such models is written as
2 H ergies P (ε,y), where ε and y are the real and imaginary
1 (p+ih)2 parts of the energy, and correlations of eigenfunctions of
0 = +V(r), (1)
0 H 2m the Hamiltonian , Eq. (1), at different points for wires
H
0 using the supersymmetry technique [12]. A proper non-
where p is the momentum operator, V (r) is a random
/ linear supermatrix σ-model has been derived in Ref. [3].
t potential and h is a constant vector. One comes to the
a Its free energy functional F [Q] reads
m Hamiltonian , Eq. (1), reducing the d+1 dimensional h
H
d- ccloalsusmicnaalrpdroebfelcetms toofavodr-tdicimeseninsioanaslupqeuracnotnudmucptororbwleimth. Fh[Q]= πνSTr [D(∂Q)2 4(γΛ+yΛ1τ3)Q]dr (3)
8 Z −
n Then, the vector h is proportional to the component of
o the magnetic field perpendicular to the line defects [1]. whereD istheclassicaldiffusioncoefficient,ν istheden-
c
Considering the one-dimensional (1D) version of the sity of states, and the standard notations for the super-
:
v Hamiltonian , Eq. (1), the authors of Ref. [1] pre- trace Str and supermatrices Q, τ3, Λ, and Λ1 are used
i H
X dictedalocalization-delocalizationtransitionthatoccurs [12,3]. The free energy functional Fh[Q], Eq. (3), differs
r at hc = lc−1, where lc is the localization length at h=0. from the conventional one F0[Q] (describing the Hermi-
a At h l−1 the “imaginary vector-potential” h can be tian problem) by the presence of the “gauge-invariant”
≤ c
removed by the “gauge transformation” combination ∂Q= Q+h[Q,Λ] instead of Q.
∇ ∇
The zero-dimensional (0D) version of the σ-model is
φ (r)=φ (r)exp(hr), φ¯ (r)=φ (r)exp( hr)
k 0k k 0k equivalent to models of random non-Hermitian matrices
−
(2) [3],wherenotransitionoccurs. Buttheσ-model,Eq. (3),
holdsinanydimensionandonemightexpectthatalready
In Eq. (2), φ (r) and φ¯ (r) are the right and left eigen- the 1D version would manifest the transition. So, one
k k
functions, φ (r) are those at h=0. Carrying out the could describe the transition using the transfer-matrix
0k
transformation, Eq. (2), we see that the eigenvalues of technique developed for the σ-models [13,14].
the localizedstatesǫ donotdependonh. However,the Surprisingly,the1Dσ-modelwithF [Q],Eq.(3),does
k h
1
nothaveanytransitionandphysicalquantitiesofinterest integrate over ψ(r). As a result, we obtain a functional
are smooth functions of h giving always a finite proba- integral over Q with a free energy functional
bility of complex eigenenergies. At first glance, this re-
1 πνQ2(r) Q(r)
sult is in an evident contradiction with the arguments F[Q]= STr ln[ i + ] dr
of Ref. [1]. The resolution of this paradox is quite in- 2Z (cid:18) 4τ − − H 2τ (cid:19)
teresting. It turns out that the σ-model with F [Q],
h
Eq.(3),describesproperlyonlytheregionofthedelocal- =(p2 h2)/2m ǫ+iγΛ+iΛ ′ (8)
1
ized states. The functional F [Q] should be used in the H − − H
0
localizedregime,foranyfiniteh<h ,whereh isacriti- where ′ = ih /m+yτ and τ is the mean free time.
c c H − ∇ 3
calvector-potential. Thisreplacementofthe freeenergy, ThenextstandardstepistofindtheminimumofF[Q]
′
resembling a first order transitions, results in an abrupt neglecting . The minimum is reached at
H
change of the distribution of eigenvalues. At h < h , all
c
Q=VΛV (9)
eigenenergiesarerealwhereasath>h onegetsabroad
c
distribution in the complex plane. In the limit h h ,
≫ c VV = 1, V+ = KVK. (The notations are the same as
our results areuniversalfor any dimension. All this is in
in Ref. [3,12]). Expanding the functional F[Q] near the
acontrastwiththeresultsobtainedforthechains,where minimum in the gradients of Q and in ′ one comes to
the spectrum changessmoothly, showingthe appearance H
thefunctionalF [Q],Eq.(3). Thisisexactlythewayhow
h
of“arcs”whenincreasingh [1,4],or“wings”whenstart-
the σ-model, Eq. (3), was derived in Ref. [3]. However,
ing from the regime of the strong non-Hermiticity [6].
it has been mentioned that this σ-model is not valid in
Having formulated the rough picture of what happens
the localized regime. What is wrong in the derivation?
when changing the parameter h, let us present some de-
ItnotdifficulttofindthatthefunctionalF[Q]hasalso
tails of calculations. We first introduce the quantities
another minimum. Performing the transformation
which are to be studied. Since eigenvalues of the Hamil-
tonian(1)canbe complex, itis convenientto double the Q=exp(Λ rh)Qexp( Λ rh) (10)
1 1
−
sizeofrelevantmatrices[3],thus“hermitizing”theprob-
e
lem. In suchan approach,the roleof the Greenfunction we rewrite the functional F [Q], Eq. (8) in terms of Q.
is played by the function As a result, the imaginary vector-potentialh is removed
e
from F and the minimum is achieved at
γφ (r)φ (r′)
B(r,r′)= k k (4)
(ǫ ǫ′)2+(y ǫ′′)2+γ2 Q=VΛV (11)
Xk − k − k
which corresponds toeQ varying in space.
with an eigenenergy ǫ =ǫ′ +iǫ′′. The “density-density
k k k Which of these two saddle points should be cho-
correlator”in the present context is given by
sen? The answer depends on the value of h. To clar-
1 ify this question we consider the correlation functions
Y(r,r′;ǫ,y)= lim B(r,r′)B(r′,r) , (5) Y (r,r′;ǫ,y) and X(r,r′;ǫ,y), Eqs. (5,7). They can be
π γ→0h i
written as functional integrals over the supermatrices Q
where brackets imply impurity averaging. The limit
γ 0, understood in all correlators,becomes important Y (X)= πν2 Q1±Q′1± Q2±Q′2± (12)
as→soon as h 0. The function Y(r,r′;ǫ,y), Eq. (5), 4 42 24 − 42 24 Q
→ (cid:10) (cid:11)
establishesalinkbetweenthelocalizationpropertiesand
where the sign + ( ) corresponds to the correlator Y
the joint probability density of complex eigenenergies (X),Q1± =Q11(r) −Q22(r),Q2± =Q12(r) Q21(r),and
thematricesQ′ are±takenatr′. Thesymbol±... stands
P(ǫ,y)= 1 drdr′Y(r,r′;ǫ,y) (6) for averaging with the functional F[Q], Eq.h(8)i.QTo cal-
V Z
culate the integral in Eq. (12) let us use the transforma-
whereV isthevolume. Weintroducealsoanotherimpor- tion,Eq. (10),andtakethesaddle-point,Eq.(11). Since
tantcorrelatorX(r,r′;ǫ,y)=C(r,r′;ǫ,y)+C(r′,r;ǫ,y), thecombinationofthesupermatricesQenteringEq.(12)
forthe function Y is invariantunder the transformation,
1
C(r,r′;ǫ,y)= lim B(r,r′)B∗(r,r′) (7) Eq.(10),hisgaugedoutinthisfunction. Therefore,one
2π γ→0h i can use the standard results of the transfer-matrix ap-
The correlator Y(r,r′;ǫ,y), Eq. (5), is invariant under proach [12] developed for the Hermitian case. The final
the transformation, Eq. (2), but X(r,r′;ǫ,y) is not. result for the correlator Y (r,r′;ǫ,y)
Wefurtherexpressasusual[3,12]thecorrelationfunc- Y (r,r′;ǫ,y)=p∞(r)δ(y) (13)
tions in terms of integrals over eight-component super-
field ψ(r), average over the white-noise disorder poten- where r = r r′ , contains the function p∞(r) charac-
| − |
tial V(r), decouple ψ4 term by 8 8 matrix Q(r) and terizing localization properties
×
2
p∞(r)= φ0k(r)2 φ0k(r′)2δ(ǫ ǫk) (14) ϕˆ= ϕ 0 , θˆ= θ 0 (19)
Xk | | | | − (cid:18) 0 iχ(cid:19) (cid:18)0 iθ1 (cid:19)
Indisorderedwires,thefunctionp∞(r)describesthede- InEqs.(19),thesupermatricesT(θˆ)containnotonlyreal
cay of the wave functions at r Lc variables θˆbut also Grassmann variables. The “angles”
≫
vary in the following intervals π/2 < ϕ < π/2, <
p∞(r)≈ 4√1πL (cid:18)π82(cid:19)2(cid:18)4Lrc(cid:19)3/2exp(cid:18)−4Lr (cid:19) (15) χ<Th∞e,va−rπiab<leθs<θˆaπn,d−ϕˆ∞ar<enθo1t−<eq∞ui.valent. Forex−am∞ple,
c c
neglectingthegradienttermsinEq.(3)onecomestothe
where Lc is the localization length. For the unitary en- 0D free energy F(0), containing the variables ϕˆ only
semble it equals L =2πνSD, S is the cross-section.
c
Incontrast,thevector-potentialhentersexplicitlythe F(0)[ϕˆ]=h˜2(λ iy˜/2h˜2)2+h˜2(λ+y˜/2h˜2)2, (20)
1
−
pre-exponentialofthefunctionX aftermakingthetrans-
where λ = sinhχ, λ = sinϕ, h˜2 = 2h2L2 and y˜ =
formation, Eq. (10). However, the dependence on h is 1 c
simpleandcalculationsforX(r,r′;ǫ,y)canbeperformed 2yL2c/D. In contrast to real magnetic field H, gradually
in the same way as for Y (r,r′;ǫ,y) yielding suppressingandfinally freezingoutsome degreesoffree-
dom with increasing H, the imaginary magnetic field h
X(r,r′;ǫ,y)=cosh(2hr)p∞(r)δ(y) (16) shifts the saddle point as a whole. Noticeable changesin
behavior occur only at y˜&2h˜2, where sinϕ 1.
± ≈
Eqs. (13,16) demonstrate that, provided one may per- CalculationsperformedinRef.[3]forthe0Dcaseshow
form the transformation, Eq. (10), all eigenenergies re- thatthevariablesθˆplayaminorpartin0D.Theydonot
main real. However, the validity of this procedure de- enter F(0)[ϕˆ] but their role is even less pronounced due
pends on the value of h. The function X(r,r′;ǫ,y) does to the singularity of the Jacobian at θ, θ 0. A de-
1
→
not grow at infinity only if h<hc where tailed discussion of Ref. [3] leads to the conclusion that
one should replace the θˆ-dependent part of the Jacobian
hc =(8Lc)−1 (17) by a constant and put everywhere else θˆ= 0. It is also
interesting to note that the free energy F [Q], Eq. (3),
Ath>h Eqs.(13,16)cannotbeusedbecausethiswould h
c is notinvariantagainstthe replacement,Eq.(10). Using
correspond to growing wave functions, which are forbid-
the parametrization, Eqs. (19), one obtains that this re-
den for a closed geometry. This agrees with the argu-
placement leads to the shift θ =θ 2irh, θ =θ 2rh.
mentsofRef.[1]. AccordingtoRef.[1]oneshoulduse at − 1 1−
This shift changes the contour of the integration over
h>hc extended states having complex eigenenergies. e e
π < θ < π in a complicated way, thus demonstrating
In the present formalism, the other saddle-point, −
the violation of the “gauge symmetry”.
Eq. (9), of the free energy F[Q] should be taken in the
Effective disappearance of the variables θˆ shows that
regime h>h . This leads to the σ-model in the form of
c
thecorrespondencebetweentheproblemunderconsider-
Eq. (3). Expecting that the eigenenergies become com-
ation and random flux models [16] is rather obscure be-
plex, we determine their distribution function P(ǫ,y),
causeinthelatter,thevariablesθˆplayacrucialrole. For
Eq. (6). Following the transfer-matrix technique [13,12]
thequantitieswehavestudied,Eqs.(5,6,7),thismapping
we write the function P(ǫ,y) in the form
is inadequate.
P (ǫ,y)= πν2S Ψ(Q) (Q1+P1+ Q2+P2+)dQ (18) Asinthe0Dcase,oneshouldputeverywhereθ,θ1 =0
4 Z 42 24 − 42 24 whenderivingthetransfermatrixequations. Asaresult,
differentialequations for Ψ andP contain λ and λ only
1
In Eq. (18), Ψ(Q) is the partition function of the semi-
Ψ=F(0)Ψ, P+ =L (iλ iλ)Ψ, (21)
infinite wire,the matrixfunctionP isthe partitionfunc- c 1
H H −
tion between the points r and r′ multiplied by Q′Ψ(Q′) b = 1 ∂ (1 λ2)bJ ∂ + 1 ∂ (1+λ2)J ∂ ,
andintegratedoverr′. Asusual,properdifferentialequa- H J0 λ − 0 λ J0 λ1 1 0 λ1
tions for Ψ and P are derived from F[Q], Eq. (3). b
where J =1/(λ +iλ)2.
In order to carry out these calculations it is necessary 0 1
Analysis of Eqs. (21) for hL 1 is to some extent
to choose a parametrizationof the supermatrices Q. For c ≫
similar to the one in the limit of high frequencies for the
thenon-Hermitianprobleminvolvedtheparametrization
conventionalproblem of localization [12]. We obtain
of Ref. [3] is most natural. For simplicity we consider
the unitary ensemble, where the supermatrices Q canbe Ψ exp[ h˜(λ +iy˜/2h˜2)2 h˜(λ y˜/2h˜2)2], P+ Ψ/2h˜
1
≃ − − − ≃
parametrized in the form
which gives after substitution into Eq. (18)
cosϕˆ τ sinϕˆ
Q=T (cid:16)θˆ(cid:17)Q0T¯(cid:16)θˆ(cid:17), Q0 =(cid:18)−τ3sinϕˆ −−3cosϕˆ (cid:19), P(ǫ,y)≃ 4νh˜2 (cid:26)01,, |yy˜˜|><22hh˜˜22 (22)
| |
3
TheformofthefunctionP(ǫ,y),Eq.(22),isthesameas vortex lines should abruptly change their behavior, from
the 0D result of Refs. [2,3]. This result does not depend being pinned by columns to become oriented along the
onthe dimensionalityandcorrespondstothe elliptic law magnetic field.
for strongly non-Hermitian random matrices [15]. We acknowledge useful discussions with I.L. Aleiner,
Analogous calculations for Y, Eqs. (4, 5) yield A. Altland, C.W.J. Beenakker and Y.V. Fyodorov. This
workwassupportedbySFB237“UnordnungundGrosse
Y(r) νβexp( 2β r), β = (h2 y2/4Dh2). (23) Fluktuationen”
≃ − | | −
p
Analytical study of Eqs. (21) is hardly feasible at
hL 1. To solve them numerically, we use the stan-
c
∼
dard over-relaxation method with Chebyshev accelera-
tion. The results for the distribution P(ǫ,y) are pre-
sented in Fig. 1 for severalvalues of h˜. The lowest value
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For comparison we present the 0D result of Refs. [2,3].
8384 (1998).
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Phys. Lett. A 226, 46 (1997).
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4