Brouwer PhysRevB.55

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Voltage-probe and imaginary-potential models for dephasing in a chaotic quantum dot

P. W. Brouwer and C. W. J. BeenakkerInstituut-Lorentz, University of Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

Received 26 September 1996

We compare two widely used models for dephasing in a chaotic quantum dot: the introduction of a fictitious

voltage probe into the scattering matrix and the addition of an imaginary potential to the Hamiltonian. We

identify the limit in which the two models are equivalent and compute the distribution of the conductance inthat limit. Our analysis explains why previous treatments of dephasing gave different results. The distribution

remains non-Gaussian for strong dephasing if the coupling of the quantum dot to the electron reservoirs occurs

via ballistic single-mode point contacts, but becomes Gaussian if the coupling occurs via tunneling contacts.

S0163-1829 97 00808-4

I. INTRODUCTION

Extensive theoretical work has provided a detailed de-scription of the universal features of phase-coherent transportin classically chaotic systems, such as universal conductance

fluctuations, weak localization, and a non-Gaussian conduc-tance distribution.112 The advances of submicrometer tech-nology in the past decade have made these manifestations ofquantum chaos in electronic transport accessible toexperiment.1320 Although experiments on semiconductorquantum dots confirm the qualitative predictions of thephase-coherent theory, a quantitative comparison requiresthat loss of phase coherence be included in the theory. Twomethods have been used for this purpose.

The first method, originating from Buttiker,21 is to includea fictitious voltage probe into the scattering matrix. The volt-age probe breaks phase coherence by removing electronsfrom the phase-coherent motion in the quantum dot, and sub-

sequently reinjecting them without any phase relationship.The conductance G of the voltage probe in units of2e 2/h) is set by the mean level spacing in the quantum dotand the dephasing time , according to G2/ .This method was used in Refs. 7, 8, 13, and 20. The secondmethod is to include an spatially uniform imaginary poten-tial in the Hamiltonian, equal to i/2. This method wasused in Refs. 9 and 11.

The two methods have given very different results for thedistribution of the conductance G , in particular, in the casethat the current through the quantum dot flows throughsingle-mode point contacts. While the distribution P(G) be-comes a peak at the classical conductance for very strongdephasing (

0) in the voltage-probe model, P(G) peaks

at zero conductance in the imaginary-potential model. It isthe purpose of the present paper to reconcile the two meth-ods, and to compute the conductance distribution in the limitthat the two methods are equivalent.

The origin of the differences lies with certain shortcom-ings of each model. On the one hand, the imaginary-potentialmodel does not conserve the number of electrons. We willshow how to correct for this, thereby resolving an ambiguityin the formulation of the model noted by McCann andLerner.11 On the other hand, the voltage-probe model de-scribes spatially localized instead of spatially uniform

dephasing. This is perfectly reasonable for dephasing by areal voltage probe, but it is not satisfactory if one wants afictitious voltage probe to serve as a model for dephasing byinelastic processes occurring uniformly in space. A seconddeficiency of the voltage-probe model is that inelastic scat-

tering requires a continuous tuning parameter , while thenumber of modes N in the voltage probe can take on integervalues only. Although the introduction of a tunnel barrier transparency ) in the voltage probe allows the conduc-tance GN to interpolate between integer values, thepresence of two model parameters creates an ambiguity: Theconductance distribution depends on N and separately,and not just on the product N set by the dephasing time.

In this paper we present a version of the voltage-probemodel that does not suffer from this ambiguity and that canbe applied to dephasing processes occurring uniformly inspace. This version is equivalent to a particle-conservingimaginary-potential model. We show that the absorbing term

in the Hamiltonian can be replaced by an absorbing lead thevoltage probe in the limit N , 0 at fixedGN. This is the locally weak absorption limit ofZirnbauer.2 Both shortcomings of the voltage-probe modelare cured: The limit N together with ergodicity ensuresspatial uniformity of the dephasing, while the conductanceG is the only variable left to parametrize the dephasing rate.

The outline of the paper is as follows. In Sec. II we recallthe voltage-probe model and derive the limit N ,0 at fixed N from the particle-conservingimaginary-potential model. We then calculate the effect ofdephasing on the conductance distribution in the case ofsingle-mode point contacts Sec. III . The distribution nar-rows around the classical series conductance of the two pointcontacts when the dimensionless dephasing rate2/ becomes 1, but not precisely in the waywhich was computed in Refs. 7 and 8. In Sec. IV we brieflyconsider the case of multiple-mode point contacts numberof modes 1), which is less interesting. We conclude inSec. V.

II. TWO MODELS FOR DEPHASING

The system under consideration is shown in Fig. 1. Itconsists of a chaotic cavity, coupled by two point contacts

PHYSICAL REVIEW B 15 FEBRUARY 1997-IVOLUME 55, NUMBER 7

550163-1829/97/55 7 /4695 8 /$10.00 4695 1997 The American Physical Society


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with N1 and N2 propagating modes at the Fermi energyEF) to source and drain reservoirs at voltages V1 and V2. Acurrent II1I2 flows from the source to the drain. In thevoltage-probe model,21 a fictitious third lead (N modesconnects the cavity to a reservoir at voltage V. Particle

conservation is enforced by adjusting V in such a way thatno current is drawn (I0). The third lead contains a tunnelbarrier, with a transmission probability which we assumeto be the same for each mode. The scattering matrix S hasdimension MN1N2N and can be written as

Ss 11 s12 s1

s 21 s22 s2

s1 s2 s

, 1

in terms of NiNj reflection and transmission matricess i j . Application of the relations

22

Ik2e2

h lG kl Vl , k1,2,, 2a

GklklNktr s kl s kl , 2b

yields the dimensionless conductance G(h/2e 2)I/( V1V2),

GG12G1G2

G1G2. 3

Using unitarity of S we may eliminate the conductancecoefficients G kl which involve the voltage probe,

GG12 G11G12 G22G12

G11G12G21G22. 4

The remaining conductance coefficients are constructed fromthe matrix,

Ss 11 s12

s 21 s22, 5

which formally represents the scattering matrix of an absorb-ing system. The first term in Eq. 4 would be the conduc-tance if the voltage probe would truly absorb the electronswhich enter it. The second term accounts for the electrons

that are reinjected from the phase-breaking reservoir, therebyensuring particle conservation in the voltage-probe model.

The imaginary-potential model relates Sto a Hamiltonian

H with a spatially uniform, negative imaginary potentiali/4. As used in Refs. 9 and 11, it retains only the firstterm in Eq. 4 , and therefore does not conserve particles.We correct this by including the second term. We will nowshow that this particle-conserving imaginary-potential modelis equivalent to the voltage-probe model in the limitN , 0, N.

Our equivalence proof is based on the generalrelationship,23,24

S12i W EFHiWW 1W, 6

between the NN scattering matrix S(NN1N2) and the

NN Hamiltonian H the limit N is taken later on .The Hamiltonian contains an imaginary potential,

HH i/4, with H a Hermitian matrix. For achaotic cavity, H is taken from the Gaussian ensemble of

random matrix theory.25 The NN matrix W has

elements24,26

Wn2

1nN 2n

112n

11 n . 7

Here n is the transmission probability of mode n in theleads and the energy is the mean level spacing of H. We

embed W into an NN matrix by the definition Wn0for NnN, and define

Wn2Wn

2n/4. 8

Substitution into Eq. 6 shows that Sis an NN submatrixof an NN unitary matrix,

S12iW EFHiWW 1W. 9

We have neglected the difference between W and W for1N, which is allowed in the limit N . The matrixS is the scattering matrix of a cavity with three leads: Tworeal leads with N1, N2 modes, plus a fictitious lead withNN modes. The transmission probability n of a mode inthe fictitious lead follows from Eqs. 7 and 8 ,

n42Wnn

2 N

N2Wnn2 2

Nif N , 10

where we have used that Wnn2/4 for NnN.

We conclude that the particle-conserving imaginary-poten-tial model and the voltage-probe model are equivalent inthe limit NNN , /N0, N(1N/N).

III. SINGLE-MODE POINT CONTACTS

The effect of quantum interference on the conductance ismaximal if the point contacts which couple the chaotic cavityto the source and drain reservoirs have only a single propa-gating mode at the Fermi level. Then the sample-to-samplefluctuations of the conductance are of the same size as theaverage conductance itself. One thus needs the entire con-ductance distribution to characterize an ensemble of quantum

FIG. 1. Chaotic cavity, connected to current source and drain

reservoirs 1 and 2 , and to a voltage probe (). The voltage probe

contains a tunnel barrier dotted line . The voltage V is adjusted

such that I0.

4696 55P. W. BROUWER AND C. W. J. BEENAKKER


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dots. An ensemble may be generated by small variations inshape or in Fermi energy.

In the absence of dephasing, the conductance distributionP(G) is strongly non-Gaussian.3 6 For ideal point contacts transmission probabilities 121), one finds

4,5

P G 12 G

2 /2. 11

The symmetry parameter 2 (1) in the presence absenceof a time-reversal-symmetry breaking magnetic field. Forhigh tunnel barriers (1 ,21) , P(G) is maximal forG0, and drops offG3/2 for G12.

3,6 In this section,we compute the conductance distribution in the presence ofdephasing, using the voltage-probe model in the limitN , 0 at fixed N, in which it is equivalent tothe current-conserving imaginary-potential model. We focuson the case of ideal point contacts, and discuss the effect oftunnel barriers briefly at the end of the section.

The scattering matrix S is distributed according to thePoisson kernel,2629

P S 1

V

det 1S

S

M2 /2

det 1SS M2, 12

where V is a normalization constant, MN1N2N is the

dimension of S, and S is a diagonal matrix with diagonal

elements Snn1n. Here n is the transmission prob-ability of mode n (n for N1N2nM). The mea-sure dS is the invariant measure on the manifold of unitary unitary symmetric matrices for 2 (1).

We now focus on the case of ideal single-mode pointcontacts, N1N21 and 121. We seek the distribu-

tion of the 22 submatrix S defined in Eq. 5 . We start

with the polar decomposition of S,

Su 0

0 v 1t

t i t

i t 1t t u 0

0 v , 13

where u and u (v and v) are 22 (NN) unitarymatrices, and t is a N2 matrix with all elements equal to

zero except tnnTn, n1,2. In the presence of time-reversal symmetry, uuT and vvT. In terms of the polardecomposition 13 we have

Su

1T1 0

0 1T2u. 14

The two parameters T1 and T2 govern the strength of theabsorption by the voltage probe. For T1 ,T20 the matrix

S is unitary and there is no absorption, whereas for

T1 ,T21 the matrix Svanishes and the absorption is com-

plete. Substitution of the invariant measure12

dS T1T2

T1T2 N

2 /2

d u d u

dv dv

dT1dT2 15

and the polar decomposition 13 into the Poisson kernel 12

yields the distribution of Sin the form

P T1 ,T2 ,u ,u

N N2 /2 T1T2

1

V

dv

dv

T1T2 N2 /2

det 1vv N2, 16a

1 1t t . 16b

Since Eq. 16 is independent ofu and u, the matrices u andu are uniformly distributed in the unitary group, and the

distribution of S is completely determined by the joint dis-tribution P(T1 ,T2) of the absorption probabilities T1 andT2.

We must still perform the integral over v and v in Eq. 16 . This is a nontrivial calculation, which we describe inthe Appendix. The final result in the limit N , 0 atfixed N is

P T1 ,T2 18 T1

4T24exp 12 T1

1T2

1 T1

T2 2 22ee

T1T2 66e42e2

T1T2 2424e186e623

17a

for 1 presence of time-reversal symmetry , and

P T1 ,T2 12 T16

T26exp T1

1T2

1 T1T2

2 4 12ee22e 3 T1T2 48e4e 222e

22e3e 2 T12T2

2 24e2e 244e22e3e 2T1T2 2040e20e 2

1616e4282e43e4e T1T2 T1T2 1224e12e 22424e12223

23e4e T12

T22 1224e12e 22424e242122e8343e424e 17b

55 4697VOLTAGE-PROBE AND IMAGINARY-POTENTIAL . . .


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for 2 absence of time-reversal symmetry .To relate the conductance G to T1, T2, u , and u, we

substitute the polar decomposition of S into Eq. 4 , with theresult

Gi ,j1

2

u 1iu i2 u 1j* uj2 * 1Ti 1Tj

T1T21

i,j1

2

u 1 i2 uj2

2TiTj . 18

Equations 17 and 18 , together with the uniform distribu-tion of the 22 matrices u , u over the unitary group, fullydetermine the distribution P(G) of the conductance of a cha-otic cavity with two ideal single-mode point contacts. Weparametrize u , u in Euler angles and obtain P(G) a s afour-dimensional integral, which we evaluate numerically.The distribution is plotted in Fig. 2 solid curves for severalvalues of the dimensionless dephasing rate 2/ .For 1,30 the conductance distribution becomes peakedaround the classical conductance G1/2,

P G

2 1 x 1x e

x if 1, 19

where x2(G1/2). Notice that the distribution remainsnon-Gaussian for all values of . The limiting distribution 19 is plotted in Fig. 3, for 1 and 2. The average and

variance of the conductance are

G 12

12 1

1O 2 , 20a

varG 14 121 2O 3 . 20b

The effect of dephasing was previously studied in Refs. 7and 8 for the case 1 of an ideal voltage probe withouta tunnel barrier . The corresponding results are also shown inFig. 2 dotted curves . We see that the limit N ,0 results in narrower distributions at the same value ofN. In particular, the tails G0 and G1 are

strongly suppressed even for the smallest , in contrast withthe case of the ideal voltage probe. The physical reason forthe difference is that keeping N small and setting equalto 1 corresponds to dephasing which is not fully uniform inphase space, and therefore not as effective as the limitN , 0. For large , the difference vanishes, andthe distribution 19 is recovered for an ideal voltage probeas well. The fact that the conductance fluctuations aroundG1/2 are non-Gaussian was overlooked in Refs. 7 and 8.

We have shown in the previous section that the voltage-probe model in the limit N , 0 is equivalent to theparticle-conserving imaginary-potential model. The require-ment of particle conservation is essential. This is illustratedin Fig. 4, where we compare our results with those obtainedfrom the imaginary-potential model without enforcing con-servation of particles. This model corresponds to settingGG12 in Eq. 4 and was first solved in Ref. 3. For1, the imaginary potential without particle conservationyields a distribution which is maximal at G0, instead of astrongly peaked distribution around G1/2 cf. Eq. 19 .

The first two moments of the conductance can be com-puted analytically from Eqs. 17 and 18 . The resultingexpressions which are too lengthy to report here are plottedin Fig. 5. The markers at integer values of are the results ofthe ideal voltage-probe model of Refs. 7 and 8, where1 and N0,1,2, . . . . The remarkable result

8 that

G is the same for 0 and 1 is special for dephasing

FIG. 2. Solid curves: Conductance distributions of a quantum

dot with two ideal single-mode point contacts, computed from Eqs.

17 and 18 for dephasing rates 0, 0.5, 1, 2, and 5. The top

panel is for zero magnetic field (1) , the bottom panel for broken

time-reversal symmetry (2) . The dotted curves are the results of

Refs. 7 and 8 for the model of an ideal voltage probe without a

tunnel barrier , in which dephasing is not fully uniform in phase

space. For 0 the two models coincide. The value 0.5 is not

accessible in the model of an ideal voltage probe because

N can take on only integer values if1) .

FIG. 3. The limiting conductance distribution 19 for 1

solid curves . A Gaussian distribution with the same mean and

variance is shown for comparison dotted curves .



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by a single-mode voltage probe: The present model withspatially uniform dephasing has a strictly monotonic increaseof G with for 1.

So far we have considered ideal point contacts. Nonideal

point contacts i.e., point contacts with tunnel barriers cor-respond to 1 ,21 in the distribution 12 of S. This casecan be mapped onto that of ideal point contacts by theparametrization2628

SRT 1SR 1ST, 21

where R and Ti1R 2 are diagonal matrices. The onlynonzero elements of R are R 1111 andR 2212. The distribution of S is given by the Poissonkernel 12 with 121. Physically, S is the scatteringmatrix of the quantum dot without the tunnel barriers in thepoint contacts, while R (T) is the reflection transmission

matrix of the tunnel barriers in the absence of the quantumdot.26 We may restrict the parametrization 21 to the 22

submatrix S,

SRT 1SR 1ST, 22

where the matrices S, R, and T are the upper-left 22

submatrices of S, R , and T, respectively. The matrix S hasthe distribution given by Eqs. 16 and 17 . The matrices

R and T are fixed, so the distribution of S follows directlyfrom Eq. 22 .

For strong dephasing (1 ,2), we find that the con-ductance distribution becomes a Gaussian with the mean and

variance given by

G 12

12

2122

2 4/1 2

1 23 , 23a

varG41

222 1

22

212

21

2 2

1 23 . 23b

The average conductance G is the classical series conduc-tance of the two point-contact conductances 1 and 2. Fluc-tuations around the classical conductance are of order1/2. For ideal point contacts (1 ,21) the variance 23b vanishes. The higher-order fluctuations are non-

Gaussian, described by Eq. 19 .Again our result is entirely different from that of theimaginary-potential model without a particle conser-vation,3,11 where P(G) becomes sharply peaked at G0when 1 ,2. We have verified that we recover the re-sults of Ref. 3 from our Eqs. 17 and 18 if we retain onlythe first term in Eq. 4 , i.e., if we set GG12 . The resultsof Ref. 11 are recovered if we symmetrize this term, i.e., ifwe set G(G12G21)/2. This is different from G12 if2 and 0. Once particle conservation is enforced, theimaginary-potential model leads unambiguously to Eq. 23 .

IV. MULTIPLE-MODE POINT CONTACTS

In this section we consider the case N1 ,N21 of a largenumber of modes in the two point contacts. The conductancedistribution is then a Gaussian, hence it suffices to computethe first two moments of G . We first consider ideal pointcontacts (121), and discuss the effect of tunnel bar-riers at the end.

For N1 ,N21 the integration over the scattering matrixS with the probability distribution 12 can be done using thediagrammatic technique of Ref. 31. The result for the aver-age of the conductance coefficients G i j is

G i j Nii jNiNj

NN,1A i j , 24a

A i jNiNj N2NN

2

NN3

i jNi

NN, 24b

up to terms of order N1. We recall that NN1N2 . Forthe covariances cov(G i j ,G kl ) G i jG kl G i j G kl , wefind

cov G i j ,G kl A ikAjl,1A ilAjk

2NiNjNkNlN NN 2 1

NN6 .

25

FIG. 4. Solid curves: Same as in Fig. 2, bottom panel. Dotted

curves: Results of the imaginary-potential model without particle

conservation.

FIG. 5. Variance of the conductance as a function of the dephas-

ing rate , for 1 solid curve and 2 dotted curve , com-

puted from Eqs. 17 and 18 . The crosses (1) and squares

(2) at integer result from the model of Refs. 7 and 8 with the

ideal voltage probe. The inset shows the average conductance for

1. For 2 the average is trivially equal to 1/2 for all in

both models.



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In order to find the average and variance of the conductancein the presence of dephasing, we substitute Eqs. 24 and 25 into Eq. 4 . The result is

G N1N2

N 1 1

N, 26a

varG

2N12N2

2

N2 N 2 , 26b

with N.Equation 26a was previously obtained by Aleiner and

Larkin.10 Equation 26b for varG agrees with the interpola-tion formula of Baranger and Mello7. The present derivationshows that this interpolation formula is in fact a rigorousresult of perturbation theory. However, the interpolation for-mula of Ref. 7 for G differs from Eq. 26a . In the finalexpression for G and varG only the product N ap-pears, although the moments of the conductance coefficientsG i j depend on N and separately. Apparently, in large-N perturbation theory the precise choice of N and in the

voltage-probe model is irrelevant, the conductance distribu-tion being determined by the product N only. For smalldephasing rates N, Eq. 26 agrees with Efetovs result,9

which used the imaginary-potential model without enforcingparticle conservation. However, for N, our result differsfrom that of Ref. 9, indicating the importance of particleconservation once the dephasing rate and the dimension-less escape rate N through the point contacts become com-parable.

We have carried out the same calculation for the case ofnonideal point contacts. The transmission probability ofmode n is denoted by n (n1, . . . ,N1 corresponding to thefirst point contact, nN11, . . . ,N1N2 to the secondpoint contact

. The result is

G g 1g 1

g1

g 2g 12g 1

2g2

g2 g, 27a

varG2g 1

2g 1

2

g 2 g 2

4 g 14

g 2g 14

g 3g 2g 14g3g 1

4

g 4 g

3 g 14

g 22g 2

2g1

4

g4 g 2

4g 12

g12 g 2g2

g 3 g 2, 27b

gpn1

N1

np , gp

n1N1

N1N2

np , gg 1g1 . 27c

One can check that Eq. 27 reduces to Eq. 26 for ideal

point contacts when gpN1, gpN2). As in the case of

single-mode point contacts, varG2 for 1 withouttunnel barriers, while varG1 otherwise.

V. CONCLUSION

In summary, we have demonstrated the equivalence oftwo models for dephasing, the voltage-probe model and theimaginary-potential model. In doing so we have corrected anumber of shortcomings of each model, notably the nonuni-formity of the dephasing in the voltage-probe model of Refs.7 and 8 and the lack of particle conservation in the

imaginary-potential model of Refs. 9 and 11. We have cal-culated the distribution of the conductance and shown that itpeaks at the classical conductance for strong dephasing onceparticle conservation is enforced, thereby reconciling thecontradictory results of Refs. 7 and 8, on the one hand, andRefs. 9 and 11, on the other hand. We find that for idealsingle-mode point contacts no tunnel barriers , conductancefluctuations are non-Gaussian and for strong dephasing

(0). In the case of nonideal point contacts with tunnelbarriers , fluctuations are larger () and Gaussian for0.

The effect of dephasing becomes appreciable when thedimensionless dephasing rate 2/ is of the sameorder as the dimensionless escape rate g nn through thetwo point contacts. For g , the weak-localization correc-tion G G (2) G (1) and the conductancefluctuations are given by30

Ga 1g/O g/2, 28a

varGb1g/b 2 g/2O g/ 3, 28b

where a 1, b 1, and b 2 are numerical coefficients determinedby Eqs. 20 , 23 , 26 , and 27 . For the special case of twosingle-mode point contacts, we have

a 141

222

1 24 , 29a

b141

222 1

22

212

21

2 2

1 24 . 29b

The coefficient b2 is only relevant if 1 ,21, whenb1(212)/41 and b 2(121)/16. At finitetemperatures, in addition to dephasing, the effect of thermal

smearing becomes important.9 Since thermal smearing hasno effect on the average conductance, the weak-localizationcorrection G provides an unambiguous way to find thedephasing rate .

The fact that dephasing was not entirely uniform in phasespace in the model of Refs. 7 and 8 leads to small but no-ticeable differences with the completely uniform descriptionused here, in particular, for the case of single-mode pointcontacts. The differences may result in a discrepancy1 in the estimated value of the dimensionless dephas-ing rate , if the ideal voltage-probe model of Refs. 7 and 8is used instead of the model presented here. A difference1 is relevant, as experiments on semiconductor quan-

tum dots can have dephasing rates as low as 2.32

Both the voltage-probe model and the imaginary-potentialmodel only provide an effective description of dephasing.They cannot compete with a microscopic theory of inelasticscattering in quantum dots see, e.g., Refs. 33 and 34 . Atthis time, a microscopic theory for the effect of inelasticscattering on the conductance distribution does not yet exist.For the time being, the model presented here may well be themost realistic description available.

ACKNOWLEDGMENTS

We have benefitted from discussions with I. V. Lerner, C.M. Marcus, and T. Sh. Misirpashaev. This work was sup-



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ported by the Stichting voor Fundamenteel Onderzoek derMaterie FOM and by the Nederlandse organisatie voorWetenschappelijk Onderzoek NWO .

APPENDIX: CALCULATION OF PT1 ,T2

We start the calculation of P(T1 ,T2) from the integralexpression 16 , in which we may replace the double integral

of v and v by a single integral of the matrix vv over theunitary group for 2) or over the manifold of unitarysymmetric matrices for 1). We make a substitution ofvariables vvw via

vv12 w 1w 1 12. A1

The matrix was defined in Eq. 16b . One verifies that thematrix w is unitary unitary symmetric for 1). The Jaco-bian of this transformation is2628

det

dvv

dw

V

V

det 1vv N2

det 12

N2 /2

, A2

where V and V are normalization constants. This change ofvariables is a key step in the calculation, since the Jacobian A2 cancels the denominator of the integrand of Eq. 16aalmost completely,

P T1 ,T2 1

V dw 6 T1T2

j1,2

1Tj1

N6 /2

j1,2

Tj22 det 1w 2. A3

We now consider separately the integral

I dw det 1w 2

dw det 1w det 1w1 . A4

Here we have used that is a positive diagonal matrix. We

now change variables w1w

1. If the matrix wereunitary, we could write

I dwdet 1w det 1w1 , A5in view of the invariance of the measure dwdw. However, is not unitary. A theorem due to Weyl allows us to con-tinue Eq. A5 analytically to arbitrary .35

To evaluate I, we decompose w in eigenvectors andeigenphases, wUe iU, where U is an orthogonal uni-tary matrix for 1 (2), and i ji jj , 0j2. Theinvariant measure dw reads25

dwdUij

e iie ij i

di . A6

After some algebraic manipulations, we arrive at

I d1 . . . dNij e iie ij

j1

N

1eii j1

N

1 1 eij

dU detA, A7awhere the 22 matrix A is given by

A i ji j 1l1

N Uil Ujl*eilTiTj

1 1 eil

. A7b

The determinant of A is computed by a direct expansion.Since N1, we may consider the matrix elements Ukl asindependent real complex Gaussian distributed variableswith zero mean and variance 1/N for 1 (2). We writethe result of the Gaussian integrations in terms of derivativesof a generating function F,

j1

N

1 1 eij dU detADF. A8

The generating function F depends on the variables xk ,y k , and z k , where k1 for 1 and k1,2 for 2,

Fj1

N

k1

1xky k 1f xk ,y k ,z k eij ,

A9a

f x ,y ,z 1xy 1 1 1x 12T1

y 12T2 zT1T2 . A9b

The differential operator D reads

D1 12 N1 x1 y 1 N2 z1 z1, A10a

D2N2

12 x1 x2

y1 y 2

14 x1

y 2

2

N3

32 z 2 z2

12 z1 z 1

x1 y 1

N4

z1 z2 z 23 z 22 z 1 . A10b

The derivatives in Eq. A8 should be evaluated atxky kzk0 (k1,2).

We are left with an integral over the phases j which is ofthe type



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I d1 . . . dnij

e iie ij

j1

n

1eij k1

a keij . A11

The integrand is a product of secular determinantsdet(U) of a unitary matrix U. Integrals of this form wereconsidered by Haake et al.36 For 1 we can directly applythe results in their paper, for 2 we need to extend theirmethod to include a product of four secular determinants.We find

I1 1 n a 1

n3 1 3 n a 1 a 1

n1 1

a 1 13 n 1

,

A12a

I2 a 1

n2 1 a 2

n2 1

a 1 12 a 2 1

2

a1n2a 2

n2 n 2

a 1 1 a 2 2 a 1a 2.

A12b

The desired integral I is obtained from I by the substitu-tion of Eq. A12 with nN, a kf(x k ,y k ,zk) into Eqs. A7 A10 . The substitution of I into Eq. A3 then leadsto the final result 17 .

1 R. A. Jalabert, H. U. Baranger, and A. D. Stone, Phys. Rev. Lett.

65, 2442 1990 .2 M. R. Zirnbauer, Nucl. Phys. A 560, 95 1993 .3 V. N. Prigodin, K. B. Efetov, and S. Iida, Phys. Rev. Lett. 71,

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