directed quantum chaos

VOLUME 79, NUMBER 3 P H Y S I C A L R E V I E W L E T T E R S 21 JULY 1997

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Directed Quantum Chaos

K. B. EfetovMax-Planck Institut für Physik komplexer Systeme, Heisenbergstrasse 1, 70569 Stuttgart, Germ

and L. D. Landau Institute for Theoretical Physics, Moscow, Russia(Received 10 February 1997)

Quantum disordered problems with a direction (imaginary vector potential) are discussed and maonto a supermatrixs model. It is argued that the 0D version of thes model may describe a broad classof phenomena that can be called directed quantum chaos. It is demonstrated by explicit calculathat these problems are equivalent to those of random asymmetric or non-Hermitian matrices. Aprobability of complex eigenvalues is obtained. The fraction of states with real eigenvalues provebe always finite for time reversal invariant systems. [S0031-9007(97)03575-8]

PACS numbers: 74.60.Ge, 05.45.+b, 73.20.Dx

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New very interesting phenomena may occur in systewith non-Hermitian quantum Hamiltonians. Although thHamiltonian of a closed system in equilibrium must bHermitian, non-Hermitian models can describe noneqlibrium processes [1], open systems connected to revoirs [2], dynamics of neural networks [3], and have maother interesting applications (see, e.g., [4–6]).

In a recent remarkable work [7], Hatano and Nelsconsidered a model of particles described by a randSchrödinger equation with an imaginary vector potentThis model arises as a result of mapping flux linesa sd 1 1d-dimensional superconductor to the world lineof d-dimensional bosons. Columnar defects introducexperimentally in order to pin the flux lines [8] leato the random potential in the boson system, wherthe component of the magnetic field perpendicular todefects results in the constant imaginary vector potenih [9]. The prediction made in Ref. [7] is that alreada one-dimensional chain of the bosons has to undea localization-delocalization transition as a functiondisorder. This effect is new and very unusual for physof disordered systems, which clearly shows that randmodels with the imaginary vector potentials deservfurther discussion.

The HamiltonianH of the simplest model of noninteracting particles with a disorder and the imaginary vecpotentialih has the form

H sp̂ 1 ihd2y2m 1 Usrd , (1)wherep̂ 2i= andUsrd is a random potential.

The non-Hermitian Hamiltonian, Eq. (1), differs fromHamiltonians for open dots [2]. It is real and contains tvector h that introduces a direction. Average physicquantities depend on the direction ofh, but the time-reversal symmetry is not broken. To understand bethe physical meaning of the vectorh, it is instructive towrite a lattice HamiltonianHL corresponding toH

HL 2t2

Xr

dXn1

seh?en c1r1en

cr 1 e2h?en c1r cr1en

d

1X

rUsrd c1

r cr , (2)

0031-9007y97y79(3)y491(4)$10.00

mseeui-ser-ny

onom

ial.inseddeasthetialyrgooficsomed

-tor

heal

tter

wherec1 and c are creation and annihilation operatorsandhenj are the unit lattice vectors. Considering the oneparticle Hamiltonian, we do not need to specify statisticof the particles. [Equation (2) was used in Ref. [7] fonumerical calculations.]

In Eq. (2), the hopping probability alongh is higherthan in the opposite direction. In other words, theHamiltonianHL describes adirectedhopping in a randompotential. This model can be considered as a quantucounterpart of a directed percolation model introduced bObukhov [10]. Such models may describe properties odisordered systems without a center of inversion. Thfinite h in Eq. (2) can be due to, e.g., the possibility oa virtual tunneling into a dielectric in an electric field.This would not lead to an imaginary contribution to thechemical potential as in open quantum dots, but mighmake hopping in different directions nonequivalent.

It is interesting to note that Green functions used iRef. [10] contained as the energy spectrum the combinations sp 1 iad2 with a constant vectora, whichcorresponds to the continuum version of the Hamiltonian, Eq. (1). The non-Hermitian operators describeby Eqs. (1) and (2) enter also Fokker-Planck equationderived for models of random walks in random mediwith constrained drift forces [4–6]. The so-called noisyBurgers equation studied recently in a number of work[11–13] can be reduced by the Cole-Hopf transformatioto a linear equation containing similar non-Hermitianoperators with a direction [14].

“Conventional” (nondirected) disordered systems exhibit a variety of different phenomena. Depending onthe geometry of the sample and strength of disorder, ocan have conduction, localization, or, e.g., quantum chaoOne can guess that the models described by Eqs. (1) a(2) also contain these effects although the effects maypeculiar. In analogy with the directed percolation, it isreasonable to call the corresponding phenomenadirectedquantum chaos, directed localization,etc.

The main results of Ref. [7] were derived for 1Dand 2D samples from numerical computations, althougsome qualitative conclusions were made considering

© 1997 The American Physical Society 491


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one-impurity model. At the same time, very well developed analytical methods of study of conventiondisordered systems exist and it is highly desirabto develop analogous schemes for the directed dorder problems. This would help to understand thlocalization-delocalization transitions, but also considnew phenomena like directed quantum chaos.

In this Letter it is shown that problems of directedisorder described by Eqs. (1) and (2) can be reducto calculations within a nonlinear supermatrixs model.This approach proved to be very useful for a broavariety of different problems (for a review see [15]). Ths model derived below is applicable in any dimensioand differs from previous ones by a newh-dependentterm. Problems of quantum chaos correspond to the zedimensional version and are most simple for calculationSome new results are obtained for this case classifibelow asdirected quantum chaos.

Recently a “ballistic”s model was derived by averag-ing over either rare impurities or energy [16]. Apparentlyproperh terms can be written for this case, too.

Although the imaginary vector potentialh entersEqs. (1) and (2) almost in the same way as the physivector potentialA, the presence ofi changes considerablythe derivation of thes model. A simple replacementseycd A ! ih in the s model of Refs. [15,17], would

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lead to the absence of the ground state. This is becaeigenenergies of the Hamiltonians, Eq. (1) or Eq. (2are not necessarily real, and retardedGR

e and GAe Green

functions can have now poles everywhere in the complplane of . As a result, these functions cannot be writtein the usual form of convergent Gaussian integrals, whiwas a necessary step in derivation of thes model.

First of all, one should choose a proper quantito calculate. According to a discussion of Ref. [7important information can be obtained by studying a joiprobability Pse, yd of real e

0k and imaginarye

00k parts of

eigenenergies or, in other words, the density of compleigenenergies. This function is introduced as

Pse, yd V 21

*Xk

dse 2 e0kd ds y 2 e00

k d

+, (3)

where V is the volume of the system, the sum is takeover all eigenstates, andk· · ·l stands for averaging overthe disorder. In contrast to the average density of stain conventional disordered systems, the functionPse, ydat h fi 0 distinguishes between localized and extendstates. For the localized states it is proportional tods ydbut is a nontrivial function for extended ones.

It is convenient to rewrite Eq. (3) as

Pse, yd 1

pVlimg!0

*Xk

g2

fse 2 e0kd2 1 s y 2 e

00k d2 1 g2g2

+, (4)

,

],

l

les

which allows one to express the functionPse, yd in termsof a Gaussian integral over supervectorsck. This can bedone using the fact that the ratio in the sum in Eq. (4) cbe composed of the elements of the matrixM21

k , where

Mk

µig 2 se 2 e

0kd 2is y 2 e

00k d

is y 2 e00k d ig 1 se 2 e

0kd

∂.

In order to replace the integrals over allck by integralsover supervector fieldscsrd, one should use vectorsuksrdandyksrd and their conjugatesuksrd andyksrd

uk 12

√fksrd 1 f

pksrd

fksrd 2 fpksrd

!,

yk 12

√fksrd 2 f

pksrd

fksrd 1 fpksrd

!,

wherefksrd and fksrd are right and left eigenfunctionsof the Hamiltonian H, Eq. (1) [or HL, Eq. (2)]; thesymbol p is a complex conjugation. With the vectoruksrd andyksrd one can make Fourier expansion for an2-component vector field.

Using 8-component supervectorscsrd with exactlythe same structure as those in Refs. [15,17], one crewrite Eq. (4) in terms of a functional integral with theLagrangianL

L 2iZ

csrd H csrd dr , (5)

n

y

an

where the8 3 8 matrix operatorH has the form

H H 0 2 e 1 igL 1 H 00L1 1 iyL1t3 . (6)

In Eq. (6),H 0 12 sH 1 H1d andH 00 1

2 sH 2 H1d areHermitian and anti-Hermitian parts of the Hamiltonianrespectively. In the continuum versionH 00 h=ym.The matricesL and t3 are the same as in Refs. [15,17and the matrixL1 anticommutes withL being equal to

L1

µ0 11 0

∂, (7)

where1 is the4 3 4 unit matrix. A proper preexponentiaterm is rather lengthy and is not written here.

Further steps of derivation of thes model are standard.One averages over the random potential and decoupthe effective interaction by integration over8 3 8 super-matricesQ. The integral over the eigenvalues ofQ iscalculated using the saddle-point approximation. Thes

model is finally obtained by expansion over=Q and h.As a result, the functionPse, yd takes the form

Pse, yd 2 limg!0

pn2

4V

ZAfQg exps2FfQgdDQ , (8)


d

-

ee

where the free energy functionalFfQg is written as

FfQg pn

8

ZSTrfD0s=Q 1 hfQ, L1gd2

2 4QsgL 1 yL1t3dgdr . (9)In Eq. (9),D0 is the classical diffusion coefficient,f., .g isthe commutator, STr is the supertrace, andn is the densityof states of the system without disorder ath 0. Thepreexponential functionalAfQg equals

AfQg Z

hfQ1142srd 1 Q22

42srdg fQ1124sr0d 1 Q22

24sr0dg

2 fQ2142srd 1 Q12

42 srdg fQ2124 sr0d 1 Q12

24 sr0dgj

3 drdr0. (10)Numeration of matrix elements in Eq. (10) is the saas in Refs. [15,17]; Eqs. (8)–(10) can be used in adimension. The supermatrixQ has the same symmetry aQ for the orthogonal ensemble in Refs. [15,17]. The frenergy functionalFfQg, Eq. (9), has two additional termwith respect to the functional used for “conventionadisorder problems. These terms contain the matrixL1,which leads to new effective “external fields” in the freenergy. We see that the replacement of the physvector potentialA by the imaginary quantityih changesthe symmetry ofFfQg. This reflects the fact thatA andih violate different physical symmetries.

Of course, one can include in Eqs. (1) and (2) tvector potentialA, which would result in a standarterm in Eq. (9) describing a magnetic field. If this fieis strong enough, one can use Eqs. (8)–(10) as befbut in this caseQ is a supermatrix with the structurcorresponding to the unitary ensemble of Refs. [15,1This is a system with the broken time reversal invarian

Calculation of the functionPse, yd can be carried ouusing methods presented in Ref. [15]. One can try torenormalization group methods for study of the 2D ca(corresponding to the 3D case for flux lines), transfmatrix method for the 1D case (thick films with parallline defects and magnetic field), or calculate definintegrals overQ for the 0D case (flux lines in longcylinders). Leaving 1D and 2D cases for future studlet us consider now the 0D case classified here as direquantum chaos.

In this limit one should integrate overQ, assuming thatthis variable does not depend on coordinates. Then,gradient terms inFfQg in Eq. (9) can be omitted and integration overr, r0 in Eq. (10) easily performed. The 0Dform of FfQg obtained in this way allows one to makevery interesting conclusion even without starting explicomputation. This can be done comparing Eq. (9) wresults of a recent work [18] in which the supersymmetechnique was used to study density of complex eigvalues of “almost-Hermitian” random matricesX. Thesematrices were written in the form

X A 1 iaN21y2B (11)with N 3 N statistically independent Hermitian matriceA andB, and a numbera of the order of unity. Due to the

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presence of the factorN21y2, the ensemble described byEq. (11) differs from ensembles of matrices with arbitrarycomplex elements studied previously [19]. Fyodorovet al. [18] calculated the joint probability of the realand imaginary parts of eigenvalues of the matricesXcorresponding to the functionPse, yd, Eq. (3).

Reducing in a standard way integration over the ma-trices A and B to integration over supervectors and thento supermatrices, the authors of Ref. [18] arrived in thelimit N ! ` at the 0D version of the free energyFfQg,Eq. (9), withh2 , a2. The symmetry ofQ correspondedto the supermatricesQ of Refs. [15,17] for the unitary en-semble, which is due to the hermicity of the matricesAandB. Apparently, if they used real symmetric matricesA, antisymmetric matricesB, and imaginarya they wouldobtain Eq. (9) with the supermatricesQ corresponding tothe orthogonal ensemble (below these cases are callesimply orthogonal and unitary). This demonstrates thatthe models of disorder, Eqs. (1) and (2), in a limited vol-ume are equivalent to the ensembles of weakly nonsymmetric (or non-Hermitian if the time reversal symmetry isbroken) random matrices.

For explicit calculations, Fyodorovet al. [18] usedthe parametrization of Ref. [17]. However, due to thepresence of the new terms withh and y in Eq. (9), thecalculations with this parametrization are very difficult.So, the final result was obtained for the unitary ensembleonly. As concerns the orthogonal case, computationswith this parametrization do not seem to be possibleat all. At the same time, study of the orthogonalensemble can be very important because it describes thvortices in superconductors and, as will be seen later, thfunctionPse, yd for the orthogonal ensemble at smallh isqualitatively different from that for the unitary one.

Fortunately, one can circumvent the difficulties using anew parametrization. Leaving details for another publica-tion, I want to present here only a general structure. ThesupermatrixQ is written as

Q ZQ0Z (12)

with supermatricesZ satisfying the conditionsZZ 1andfZ, L1g 0. The central partQ0 is chosen as

Q0

µcosf̂ 2t3 sinf̂

2t3 sinf̂ 2 cosf̂

∂,

f̂

√f 00 ix

!.

The most lengthy part is calculation of the Jacobian, butthen the computation is of no difficulty and the density ofcomplex eigenenergiesPse, yd for the unitary ensembletakes the form

Pse, yd n

pp

aDexp

µ2

x2

4a2

∂3

Z 1

0coshxt exps2a2t2d dt , (13)

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sum

ornt--.

.

.

.

E

,

;

tt.

.

wherex 2pyyD, a2 2pD0h2yD, andD is the meanlevel spacing.

Equation (13) is exactly the result obtained in Ref. [18For the models of weak disorder, Eqs. (1) and (2),Pse, yddepends only on the variabley becausenis constant. Forthe random matrix model, Eq. (11),n depends on andobeys the Wigner semicircle law. In the limita ¿ 1 oneobtains

Pse, yd øpn

2a2D, jxj # 2a2, (14)

which means that the functionPse, yd is practically con-stant within the intervaljxj # 2a2. Beyond this intervalthe function exponentially decays. For the random mtrix model, this behavior corresponds to the well-know“elliptic law,” Ref. [19]. For anya the functionPse, ydis smooth, which means, in particular, that the probabilitto find real eigenvalues is negligible.

The situation drastically changes if the time-reverssymmetry is not broken. Although now the computatioof Pse, yd is somewhat more lengthy, one can finallyexpress this function in a rather simple form

Pse, yd Pr 1 Pc,

Pr nds ydZ 1

0exps2a2t2d dt , (15)

Pc pn

DF

µjxj

2a

∂ Z 1

0exps2a2t2d sinhsjxjtd tdt ,

whereFsyd 2p

p

R`

y exps2u2d du.We see that the functionPse, yd contains the partPr

proportional tods yd. This means that a finite fraction ofeigenstates has real eigenenergies for any finitea. Thedistribution of complex eigenenergies is described by thsmooth functionPc. The fraction of the states with realeigenvalues vanishes only in the limita ! `. In thislimit one comes forjxj ¿ a to Eq. (14), which againresults for the random matrix model in the elliptic law.

The result about the finiteness of the fraction of eigenstates with real eigenenergies as well as Eq. (15) itselfnew, although indications of a peculiar behavior of thprobability of real eigenvalues for asymmetric real matrces can be found in Ref. [19]. In Ref. [7] a mixture ostates with real and complex eigenvalues was found nmerically near the band center for the 2D model, Eq. (2Perhaps the localization length in that region of parameteexceeded the sample size. This would correspond to t0D situation rather than 2D. If it is so, the analytical resuagrees with the numerical observation.

Study of the directed disorder problems in higher dmensions, although more difficult, is definitely of interest. Solution of thes model, Eqs. (14) and (15), inone dimension seems to be possible using the transfmatrix method. This case corresponds to a thick film ia parallel magnetic field in the problem of the vorticein a superconductor. It would be interesting to obtai

494

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a-n

y

aln

e

-is

ei-fu-).rshelt

i--

er-nsn

a localization-delocalization transition analytically. Theone-dimensionals model is also known to describe thequantum kicked rotor problem [20]. Maybe finiteh cor-responds to a dissipation in this problem and one can fia transition here, too.

In summary, quantum disorder problems with a direction (imaginary vector potential) are considered. It idemonstrated that they can be mapped onto a new supmatrix s model. The 0D version of the model describea class of phenomena that can be called directed quantchaos. Using the 0Ds model, the equivalence of thisclass of systems to ensembles of weakly asymmetricnon-Hermitian random matrices is established. The joiprobability of complex eigenvalues is computed explicitly, and it is discovered that the fraction of real eigenenergies for time reversal invariant systems is always finite

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directed quantum chaos

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