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PHYSICAL REVIEW B, VOLUME 65, 085309
Orthogonality catastrophe in parametric random matrices
Raul O. Vallejos,1,2 Caio H. Lewenkopf,1 and Yuval Gefen31Instituto de Fı´sica, Universidade do Estado do Rio de Janeiro, R. Sa˜o Francisco Xavier 524, 20559-900 Rio de Janeiro, Brazil
2Centro Brasileiro de Pesquisas Fı´sicas, R. Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil3Department of Condensed Matter Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
~Received 29 June 2001; published 4 February 2002!
We study the orthogonality catastrophe due to a parametric change of the single-particle~mean-field! Hamil-tonian of an ergodic system. The Hamiltonian is modeled by a suitable random matrix ensemble. We show thatthe overlap between the original and the parametrically modified many-body ground states,S, taken as Slater
determinants, decreases liken2kx2, wheren is the number of electrons in the systems,k is a numerical constant
of the order of 1, andx is the deformation measured in units of the typical distance between anticrossings. Weshow that the statistical fluctuations ofS are largely due to properties of the levels near the Fermi energy.
DOI: 10.1103/PhysRevB.65.085309 PACS number~s!: 73.23.Hk, 71.10.2w, 05.45.2a
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I. INTRODUCTIONThe Anderson orthogonality catastrophe~AOC!, intro-
duced by Anderson in 1967,1 is a fundamental effect observed in many-body systems. The original work addresthe ground state of a finite system consisting ofN noninter-acting electrons. Upon the introduction of a localized pertbation, this ground state gets modified. Anderson has shthat the overlap between the original and the modifiN-electron ground states,CNuCN8 &, is proportional to anegative power ofN, and vanishes in the thermodynamlimit, hence the catastrophe. Variants of the AOC are atbasis of some central themes in solid-state physics, includthe x-ray edge singularity, zero-bias anomalies in disordesystems, and tunneling into quantum Hall systems.
The applicability of this concept and attempts to extento more generic circumstances have been the focus of ation for more than three decades. Particularly appealinthe application of AOC ideas to the field of mesoscopics. Tstudy of mesoscopic systems involves finite-size systewhere it is usually important to account for the dynamics athe thermodynamics of the electrons on a quantumechanical level. Important ingredients in characterizmesoscopic electronic systems include the strength ofambient disorder potential, the system’s size and shape,the strength of the electron-electron interaction. Evidentlymost cases such systems are too complex to be describanalyzed exactly. In such situations one needs to resovarious approximation schemes.
Finite-size ~‘‘zero-dimensional’’! conductors, i.e., quantum dots ~QD’s! have received special attention in receyears, partly due to the rich physics involved, but also duetheir experimental accessibility.2–4 The simplest scheme taccount for a QD with interacting electrons is theconstantinteractionmodel.5 The latter implies that the system Hamtonian is given by
H5H01HCI , ~1!
whereH0 is the single-particle Hamiltonian and the interation, represented by an infinite wavelength~zero mode!, isgiven by
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d
-n
d
egd
itn-ises
d-
gendn
orto
to
HCI5e2
2C~N2N0!2. ~2!
HereC is the total effective capacitance of the dot,N is theparticle number operator, andN0 represents a tunable background charge~related to the gate voltage!. For a complexQD with diffusive disorder or with chaotic dynamics, thsingle-particle energy spectrum and eigenfunctions ofH0should be the subject of a statistical description. It turnsthat for such systems spectral correlations within energy wdows up to the Thouless energyETh are well described bythe random matrix theory~RMT!; see, for instance, Refs.and 7. The individual wave functions, hence spectral colations, remain unchanged as we add/remove electrons fthe system. Other than the Coulomb gap, we do not expany signature of the AOC.
For various reasons the constant interaction model dnot provide a satisfactory framework to account for key phnomena. The latter include some features of the addispectrum, Coulomb peak-height correlations, and elecscrambling~see, e.g., Ref. 3!. To improve on that model onecan employ the best effective single-particle approach,Hartree-Fock~HF! approximation. Expressed in the basisthe exact eigenstates ofH0, denoted by$ca% with the corre-sponding eigenenergies$«a%, the HF Hamiltonian reads
HHF5(a
«ana11
2 (a,b
vababnanb . ~3!
Herevabab are the antisymmetrized matrix elements of tinteraction andna is the number operator of the stateca .Note thatHHF includes only diagonal interaction matrix elements~in the exact eigenstate basis!; for a short-ranged in-teraction it has been shown that off-diagonal matrix elemeare parametrically small.8–10 It is clear that within this ap-proximation the effective single-particle states~but not theHF energies!! are independent of the occupations of thestates, and are unchanged upon the addition/removal of etrons from the system. This is in line with the Koopmanpicture.11
Only when the single-particle states of the systemmodified upon the introduction of a ‘‘perturbation’’12 is it
©2002 The American Physical Society09-1
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RAUL O. VALLEJOS, CAIO H. LEWENKOPF, AND YUVAL GEFEN PHYSICAL REVIEW B65 085309
meaningful to address the question of the AOC. Hereperturbation can be understood as varying a localizedvoltage,13 introducing a static impurity or adding an electroto the (N11)st state. The latter motivates the study of
S5^CN11ucN11† uCN&, ~4!
whereuCN& is theN-electron system ground state andcN11†
is the corresponding electron creation operator at itssingle-particle excited state. In all these examples~includingthe HF scheme! both ground states are given by Slater detminants. Indeed the Anderson formalism deals with the ovlap of two Slater determinants. More specifically, one ofapplications we have in mind is the role of AOC in the eletronic transport through complex quantum dots. At very lotemperatures their conductance peak heights in the Coulblockade regime can be evaluated from the tunneling rateand from the corresponding many-body ground state ofelectron island.14 In the simplest approximation, these ratare given by the overlaps between a single-particle wfunction in the dot and the channel wave functions. Howevif interactions are taken into account, even in a HF mefield approximation, a many-body contribution to the tunning rates has to be considered.A priori the inclusion of anadditional electron to the island may change each individsingle-particle orbital negligibly; however, if the numberelectrons in the dot is large enough, the new many-bground state can be almost orthogonal to the old one. Saspects of the AOC in the presence of disorder have bpreviously considered, see, e.g., Ref. 15. In the present pwe try to circumvent the task of analyzing the AOC in tpresence of disorder potential, and replace this challengstudying the AOC within the framework RMT. Here the roof an added perturbation is played by a parameter whicvaried. We end up studying the AOC in the context of pametric random matrices.16 As has been argued recently,17 thisis a good model to describe electron scrambling in quandots embedding the discrete electron number in the dotcontinuous variation of a parameter.13
The paper is organized as follows. In Sec. II we rederthe expression for the ground-state overlaps, reviewingmain features of the theory and introducing the notation eployed throughout this work. Section III is devoted to tpresentation of the random matrix model used in this stuSection IV contains the main body of our analysis, preseing our analytical and numerical results for theaveragedground-state overlaps. The distribution ofuSu is the subject ofSec. V. Section VI contains some brief concluding remarWe have also included three appendixes. Appendix A qutifies the accuracy of the ‘‘unit volume approximation’’ useby Anderson in his original work.1 In Appendix B we discusssome implications of evaluating the ensemble-averavalue of the Anderson integral subject togrand canonicalconstraints. Finally in Appendix C we discuss the precisof the first-order perturbation theory used to evaluate Andson’s integral, defined in Sec. II.
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II. REMINDER ON THE ANDERSON ORTHOGONALITYCATASTROPHE: A PARAMETRIC APPROACH
Let us consider systems of electrons whose HamiltonH(X) is a function of a parameterX. In this study the genericparameterX can be either continuous or discrete. In the caof modeling the change in an external magnetic field orthe electrostatic potential,X is continuous. One can alsimagine X as modeling the change in theeffectivesingle-particle potential due to the sequential addition of electroto the system, in which caseX is discrete~in the latter case itis necessary first to subtract all systematic changes ofsingle-particle spectrum!. It has been shown recently18 thatscrambling due to the sequential addition of electrons toQD can be embedded in a parametric RM process onlinteraction matrix elements due to the accumulation of sface charge are taken into account. The subject of the prepaper is the overlapSbetween ground states of systems cresponding toH(X) and H(X1dX). In particular, this sec-tion is devoted to the presentation of different levels of aproximation for lnuSu.
Since we are dealing with noninteracting electrons,ground-state wave functions are Slater determinants.overlap S can then be expressed in terms of the occupsingle-particle wave functions overlap. Let us denoteck(X) and ck(X1dX) the ~single-particle! eigenstates ofH(X) and H(X1dX), respectively. We define the unitaroverlap matrixA as
Ai j 5^c i~X!uc j~X1dX!&. ~5!
Thus the overlap of the ground statesS can be written as
S5detAoo, ~6!
where the superscript ‘‘o’’ stands for occupied states. AcordinglyAoo corresponds to the subspace of occupied staof the matrixA defined in Eq.~5!.
The catastrophe is manifest by the fast suppression ofoverlapS as the number of fermionsn in the system is in-creased for a fixed perturbation strength. Anderson1 singledout two basic reasons that make the absolute value ofoverlap S smaller than 1: The rows ofAoo have normssmaller than unit and they do not form an orthogonal set. Iconvenient to separate these two contributions by introding a new matrixAoo with normalized rows, defined as
~Aoo! i j 5~Aoo! i j /Ni , ~7!
where the normalization factorNi reads
Ni5A(j 51
n
@~Aoo! i j #2. ~8!
uSu is then rewritten as
uSu5udetAoou3)i 51
n
Ni . ~9!
The absolute value of detAoo can be geometrically interpreted as the volume of an-dimensional parallelepiped de
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ORTHOGONALITY CATASTROPHE IN PARAMETRIC . . . PHYSICAL REVIEW B65 085309
fined by a set of nonorthogonal vectors of unit length. In R1 it is claimed thatudetAoou does not depend onn and it isclose to 1. In Appendix A we show perturbatively that indethe most important contributions touSu come from)Ni , thedeviations from the ‘‘unit volume approximation’’ being ohigher order indX. Aiming at obtaining an upper bound othe overlapS we shall neglect below the contribution oudetAoou to uSu. We thus restrict our analysis to the productnormalization factors. We can therefore write
uSu,)i 51
n
Ni[)i 51
n
~12Pi !1/2, ~10!
where we have introduced
Pi5 (j 5n11
`
u^c i~X!uc j~X1dX!&u2. ~11!
Pi measures the probability of the statei, whose energy« i issmaller than the Fermi energy, to be spread over thej com-ponents lying above the Fermi surface once the Hamiltonis changed. The product of the normalization factors,~10!, is the first level of approximation to the overlap deteminant considered here. For the sake of convenience, hafter we shall deal with lnuSu instead ofuSu itself, and define
I norm[1
2 (i 51
n
ln~12Pi !. lnuSu. ~12!
Equation~12! is still not suitable for an insightful analysisFor this purpose a higher level of approximation forS wasintroduced.1 For smalldX the ‘‘probabilities’’ Pi will also besmall. Thus by expandingI norm in a power series,
I norm51
2 (i 51
n S 2Pi2Pi
2
22 . . . D , ~13!
and retaining only its first expansion term one writes
I[21
2 (i 51
n
Pi.I norm. ~14!
By explicitly expressingI in terms of the single-particleoverlaps, one arrives at
I 521
2 (i 51
n
(j 5n11
`
u^c i~X!uc j~X1dX!&u2. ~15!
This is the well-knownAnderson integralterm. The approxi-mationsI norm andI for lnuSu will be analyzed at length in SecIV where we present numerical and analytical results forrandom matrix model introduced in the forthcoming sectio
III. RANDOM MATRIX MODEL
In this section we present a model for the statistical stuof the Anderson orthogonality catastrophe in ballistic ergosystems. The setting is the same as in the previous secWe model the single-particle HamiltonianH which dependson a parameterX by a suitable ensemble of random matrice
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In one of its simplest forms,19 this model is realized by
H~X!5H0 cosX1U sinX, ~16!
whereH0 andU are two independent matrices of dimensiN3N, both members of the same Gaussian ensemble. Instudy, we restrict our analysis to the Gaussian orthogoensemble (b51) and to the Gaussian unitary ensembleb52). Whereas the orthogonal ensemble is used to mosystems presenting time-reversal symmetry, the unitarysemble models systems where the latter symmetry is absThus the distribution ofH0 andU is6,7
P~H0 ,U !5CbN expH 22N
bl2~ tr H0H0
†1tr UU†!J ,
~17!
whereCbN is a normalization constant. As a consequencethis choice, the matrix elements ofH(X) are also Gaussiandistributed with zero mean and variance parametrized as
^Hi j* ~X!Hi 8 j 8~X!&5~d i i 8d j j 81db1d i j 8d j i 8!l2
N~18!
independent ofX. Here the symbols •••& stand for en-semble averaging.
For any arbitrary value ofX, the resulting mean levedensity is given by the well-known Wigner semicircformula6
r~«!5N
plA12S «
2l D 2
, ~19!
where « is the single-particle energy. Evidently, the melevel spacing at the center of the spectrum («50) is D5pl/N. Accordingly, the average bandwidth is«max2«min54l ~this is in line with the standard solid-state pictuwhereby the bandwidth is fixed, independent of the systesize, while the mean level spacing scales asN21).
The typical scale for the parametric variationX!, repre-senting the average distance between anticrossings, cageneral be characterized by the level velocity standdeviation20,21
D
X!5AK S d«m
dX D 2L 2 K d«m
dX L 2
, ~20!
where«m(X) is an energy level ofH(X) close to the centerof the band. In the model defined by Eqs.~16! and ~17! theaverage level velocity is zero and
X!5D
A2/~bN!l5pA b
2N. ~21!
Now it is possible to quantify the effect ofdX in the single-particle spectrum by the scaled parameterx[dX/X!.
It remains to specify the many-body part of the modThe ground-state configuration is generated by populathe single-particle levels withn5N/2 fermions~the spin de-gree of freedom is not considered here!. For a givenX, theground state is the Slater determinant made up of the low
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RAUL O. VALLEJOS, CAIO H. LEWENKOPF, AND YUVAL GEFEN PHYSICAL REVIEW B65 085309
N/2 eigenstates ofH(X). Now the single-particle Hilbertspace is finite and of sizeN, and the number of particles iN/2. The Fermi level,«F5(«N/2111«N/2)/2, lies on averageat «F50. It is important to stress that the sums in Eqs.~11!and ~15! have to be modified accordingly.
Numerical simulations of the model presented abovestraightforward to implement. We generate a pair of matriH0 andU of dimensionN whose elements are Gaussian dtributed with zero mean and variance given by Eq.~18!. Fix-ing X andx we calculate the eigenvalues and eigenfunctioof the correspondingH(X) and H(X1dX). Having com-puted the complete set of eigenfunctions, we calculateoverlap matrixAoo and its determinantSN(X,x). In order tokeep the notation simple, we shall write down the argumeof S explicitly only when necessary.
As a guide for the discussions to come let us considerepresentative member of the ensemble defined by Eq.~17!for b51 andN550. In Fig. 1 we show the parametric spetrum as a function ofX around«50 ~top panel! and thecorresponding ground-state overlapsuS(X)u for x50.4 ~bot-tom panel!, both obtained by a direct numerical proceduThe most striking feature of this plot is that large fluctuatioin uSu are closely correlated with the occurrence of narravoided crossings at the Fermi surface. In other wordseach narrow gap~avoided crossing! there correspondssmall overlapuSu. The figure also suggests that the appromation schemes presented in Sec. II fail in the vicinitynarrow gaps. These observations indicate that an accuanalytical estimate of the averageuSu and its variance re-quires a good handle of the ‘‘two-level problem’’ of narro
FIG. 1. Top panel: typical energy levels as a function ofX. Thefilled region is the gap between the last occupied and the first emsingle-particle levels. Bottom: the three curves represent the eoverlapS, exp(Inorm), and exp(I) ~see text for definitions! as a func-tion of X. They respect the orderingS,exp(Inorm),exp(I). Theparametrical distance is set tox50.4 (dX'0.13) and the dimen-sion of the single-particle Hilbert space isN550.
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avoided crossings. Had we considered integrable~or mixed!systems, this issue would become even more importantto the absence~or suppression! of level repulsion.
IV. AVERAGE GROUND STATES OVERLAP
As we shall argue in the next section, the distribution oScan be anomalously broad. As a first step though, we sconsider in this sectionensemble averagedquantities. Whenthe distribution is narrow, as is often the case with thermdynamic quantities, one may interchange
^ lnuSu&' ln^uSu&. ~22!
Evidently, when in doubt, the meaningful quantity is thright-hand side~r.h.s.! of the equation above. Technicallythough, the more accessible quantity is the left-hand s~l.h.s.! of Eq. ~22!. This section is devoted to the statisticstudy of upper bounds forlnuSu& within the approximationlevels presented in Sec. II and the random matrix modescribed in Sec. III. Our analytical results are compmented with numerical simulations.
In Eq. ~15! the ground states overlap can be easily evaated by considering the single-particle overlaps in first-orperturbation theory, namely
^c i~X!uc j~X1dX!&'dHi j
« j2« i. ~23!
Here dH'(2H0 sinX1U cosX)dX and $« j% are the eigen-values ofH(X). The matrixdH and the set of eigenvalue$« j% are statistically independent due to the invariance ofconsidered ensembles under orthogonal (b51) or unitary(b52) transformations. Recalling Eq.~15! ^ lnuSu& now reads
^ lnuSu&,^ I &[21
2 (i 51
N/2
(j 5N/211
N K udHi j u2
~« j2« i !2L . ~24!
In Appendix A it is shown that the corrections responsiblethe inequality in Eq.~24! are of fourth order in a perturbatioexpansion. Note that only off-diagonaldHi j matrix elementscontribute in Eq.~23! since thei states lie below the Fermsurface and the labelj corresponds to states above it. Thensemble average overudHi j u2, defined in Eq.~18!, yields
^ I &521
2 (i 51
N/2
(j 5N/211
Nb
2x2K D2
~« j2« i !2L , ~25!
where we have usedX! of Eq. ~21! to express lnuSu& in termsof the dimensionless variablex5dX/X!.
A satisfactory accuracy of the approximation introducby Eq. ~23! requiresdHi j /(« j2« i) to be small. Such a condition is translated to I & by examining separatelyx2 and^D2/(« j2« i)
2&. Viewed from the perspective of the parameric framework, the approximation is under control since tphysical situations in mind, see Sec. I, call forx!1, or atleast forx,12. By contrast, the average overD2/(« j2« i)
2
requires a careful discussion.The occurrence of small gaps at the Fermi surface,« j
2« i!D for j 2 i 51, causes the breakdown of the perturb
tyct
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ORTHOGONALITY CATASTROPHE IN PARAMETRIC . . . PHYSICAL REVIEW B65 085309
tion approach employed in Eq.~23!, to calculate^c i(X)uc i 11(X1dX)&. In that case even a small variationx!1 may give rise to a significant mixing of an originaloccupied level~the last one! and an empty one~the lowestoriginal vacant level!. Such a situation calls for a nonpertubative solution. Indeed it is tempting to justify the approxmation used in Eq.~23! invoking the presence of level repusion: the large errors introduced in the calculation ofuSu atsmall gaps are minimized by the rareness of such eveUnfortunately, such a scheme is doomed to fail due tofollowing reason: For neighboring levels
K D2
~« i 112« i !2L 5E
0
`
dss22Pb~s!, ~26!
where Pb(s) is the nearest-neighbor spacing distributio6
Since Pb(s);sb for s!1 the average on the l.h.s. of E~26! diverges for the orthogonal symmetry~although not forthe unitary case!. This motivates the employment of a noperturbative approach to account for the effect of the ‘‘crlevels’’ ~near the Fermi energy! on the ensemble-averageAnderson integral. With this proviso in mind it should balso realized thatj 2 i 51 corresponds to a single term in thdouble sum of Eq.~25!. As concluded from Fig. 1 this termhas amajor contribution to the large fluctuationsin uSu andwill be the subject of analysis in Sec. V. By now it is onnecessary to anticipate that the occurrence of narrow gcontributes to lnuSu& with an additional factor which dependon x and on the size of the narrow gap. Note that the produre presented in Eq.~26! represents averaging underca-nonical conditions, namely averaging over systems withgiven particle numberN,22 or averaging over both impurityrealizations andN. Such procedures are referred to as stroand weak canonical averaging, cf. Ref. 23. There are expmental setups where it is the chemical potentialm, which isthe controlled parameter. In such circumstances it is mappropriate to employ a grand canonical averaging produre. This is briefly discussed in Appendix B.
As the value ofj 2 i becomes larger the perturbative aproach works increasingly better. Moreover, at the same tthe fluctuations of« j2« i relative to (j 2 i )D decrease as aconsequence of the spectral rigidity. These matters arecussed in Appendix C. In this regime we do not expectintroduce a large error~independent ofN) replacing thespectrum ofH(X) by its average spectrum, that is
K 1
~« j2« i !2L '
1
~^« j&2^« i&!2. ~27!
The latter approximation allows us write
^ I &52b
4x2E
22l
2D/2
d« iED/2
2l
d« j
r~« i !r~« j !
~« j2« i !2
, ~28!
with the mean level densityr(«) given by the Wigner semi-circle law, Eq. ~19!. Changing variables and defininga[p/(4N) we arrive at
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^ I &52b
4x2E
a
1
duEa
1
dvA12u2A12v2
~u1v !2. ~29!
We can now isolate the singularity ata50 (N large! andrewrite the integral on the r.h.s. of Eq.~29!,
Ea
1
duEa
1
dvF 1
~u1v !21
A12u2A12v221
~u1v !2 G . ~30!
While it is the first term under the integral in Eq.~30! whichis ultimately responsible for the catastrophe, the secondgives only a constant asa→0. The final result is therefore
^ I &52b
4x2~ ln N1C!, ~31!
where C52 ln(p/2)2p2/8. The latter constant has to btaken with caution due to the approximations made.
Summarizing our results, we have several levels ofproximation to the exact value of^ lnuSu&. The product of thenormalization factors, which neglects corrections to the vume of the parallelepiped, assumed to be unity, yields
^ lnuSu&'^I norm&[1
2 (i 51
N/2
^ ln~12Pi !&. ~32!
By keeping only the first term of the expansion of the logrithm in I norm we have
^ lnuSu&'^I &[21
2 (i 51
N/2
^Pi&. ~33!
By calculatingI using first-order perturbation theory andsmoothed spectrum, we obtain the analytical estimate^ I &,given by Eq.~31!.
For small values ofx these quantities are close to eaother. They are ordered as follows:
^ lnuSu&,^I norm&,^I &,^ I &. ~34!
We turn now to our numerical analysis. For different vaues ofx and N we have evaluatedlnuSu&, ^Inorm&, ^I &, and
^ I & ensemble averaging overM realizations ofH(X) whichis defined in Sec. III. Each simulation is performed at tcost of the order ofM3(b3N)3 operations, imposing acomputational constraint on the procedure.
In Fig. 2 we show a comparison between the differeapproximations for lnuSu& as a function ofN for b51. Wechose four representative values ofx and fixedM5104. Weobserve that in all approximation schemes^ lnuSu& displays theslope predicted by Eq.~31!. In view of the large samplesizes, the fluctuations in the mean values indicate the cosponding distributions are characterized by large standdeviations, as we discuss in the following section. As alyzed in Appendix C the first-order perturbation theory esmate breaks down when used for levels in the vicinity of tFermi surface and/or for not sufficiently small values ofx. Asthe latter is increased we even expect deviations frompredicted slopes forI & versus lnN. In Fig. 2 such discrep-
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RAUL O. VALLEJOS, CAIO H. LEWENKOPF, AND YUVAL GEFEN PHYSICAL REVIEW B65 085309
ancies are only noticeable forx51 at N<50. We concludethat the power-law suppression ofuSu as a function ofN isvery robust. The discussion presented in Appendix C sgests why our theoretical estimate does not always servan upper bound for all our approximation schemes. Thisdone by noting two facts.~i! For small values ofx the per-turbation approach underestimates^uAN/2,N/21 j u2&, henceoverestimates the overlap. In such cases we are guaranan upper bound.~ii ! For larger values ofx the situation isquite the opposite. When this happens the constantC doesnot provide any longer an upper bound. Results for theb52 case behave in the very same way as forb51, withslopes following Eq.~31!. ~We have thus decided to omitcorresponding figure for the unitary case.!
Figure 2 also confirms Anderson’s claim that the ‘‘unvolume’’ corrections to lnuSu& are small, at least forx!1, asindeed is seen in panels~a! and~b!. While within our statis-tical precision we do not observe anyN dependence for theunit volume corrections, the latter cannot be ruled out.
V. DISTRIBUTION OF OVERLAPS AND THEIR LARGEFLUCTUATIONS
This section is devoted to the analysis of the distributof the overlaps between ground states,P(uSu). We show thatthe large fluctuations ofS(X) occurring in the vicinity of
FIG. 2. The average lnuS(x)u as a function ofN. The number ofrealizations for eachN is M5104 and ~a! x50.1, ~b! x50.2, ~c!x50.5, and~d! x51.0. Open dots stand for the exact^ lnuSu&, filleddots for ^I norm&, squares for I &, whereas the solid lines represe
^ I &.
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small gaps at the Fermi surface are well described by32 model, otherwise exp(^I&) provides a satisfactory estimate foruSu.
Let us start quantifying the influence of the narroavoided crossings on the overlap determinant. In the proxity of a narrow gap the single-particle overla^cN/2(X)ucN/2(X1dX)& will be small and rapidly varyingwith X. Consequently, as it was qualitatively establishedSec. II, it will dominate the fluctuations of the single-particoverlap matrix determinant. In this situation we expect ththe behavior ofS will be captured by the approximation
uSN~X,dX!u'KN~X,dX!u^cN/2~X!ucN/2~X1dX!&u.~35!
This approximation is just the first term of the expansionthe determinant ofAoo along itsN/2th row. In other words,we calculateuSu as if all fluctuations are due to the interation of the highest occupied single-particle level with tlowest empty one. The factorK accounts for the contributionof all remaining states and will be approximately constan
KN~X,dX!'^uSN21~X,dX!u&. ~36!
The results presented in Fig. 3 confirm that this approximtion works impressively wellin the vicinity of narrow gapsand for small values ofx. The system is the same as thatFig. 1, but smaller deformation is considered,x50.2, as wellas a smaller parameter range forX. The latter contains, in thepresent example, two narrow gaps~cf. Fig. 1!. The smallshift between the two curves can be attributed to theK term.The proposed approximation gives rise to spurious pewhenever an avoided crossing~narrow gap! between the oc-cupied levelsN/2 andN/221 is encountered. Such cases abeyond the scope of our approximation: the factorizationforward by Eq.~35! is no longer valid, and the gap at thFermi energy is certainly not small. The spurious peak aX'1.45 in Fig. 3 represents such a situation, as can be verfrom Fig. 1.
We have thus demonstrated heuristically that small Andson overlaps arise due to avoided crossings at the Felevel,
FIG. 3. Overlaps of Slater determinants forx50.2 as a functionof X for N550. The exact determinant is shown as a black line. Tgray line corresponds to the approximationuSu'u^cN/2(X)ucN/2(X1dX)&u.
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ORTHOGONALITY CATASTROPHE IN PARAMETRIC . . . PHYSICAL REVIEW B65 085309
involving the highest occupied and the lowest unoccuplevels. This observation leads us to conjecture that
P~ uSu!'P~K•uSu! for uSu!1, ~37!
whereS5^cN/2(X)ucN/2(X1dX)&. The derivation of an ex-pression forP(uSu), which calls for a nonperturbative approach, is discussed now. Due to level repulsion two levrarely come very close to each other. When this happens,possible to effectively model a narrow avoided crossing b232 matrix, since it is quite unlikely to have yet anothlevel very close by. This model also removes the undesoccurrence of spurious peaks as the one shown by Figthey cancel between the factorK andS. While this is quite asimple model, the calculation of the distributionP(uSu) in-volves some cumbersome multidimensional integrals.24
For the orthogonal case,b51, the integrations can bsimplified through a geometrical construct, which makespossible to obtainP(uSu) analytically within a 232 paramet-ric random matrix model. Let us write our model Hamtonian, Eq.~16!, H(X)5H0 cosX1U sinX as
H~X!5cosX~H01U tanX!. ~38!
Let us also parametrize its eigenvector asc5(cosu, sinu).Then, by defining the vectorh(X)[(2H12,H222H11), it isstraightforward to show that the eigenvector equation canwritten as
~2H12,H222H11!•~cos 2u, sin 2u!50, ~39!
so that the eigenvectors are determined solely by the veh(X). Moreover, the angle betweenc(X1dX) andc(X) ishalf the anglea betweenh(X1dX) and h(X), that is uSu5cosa/2. Now the problem is reduced to finding the distbution of a. Let us setX50 and introduce
h~X50
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RAUL O. VALLEJOS, CAIO H. LEWENKOPF, AND YUVAL GEFEN PHYSICAL REVIEW B65 085309
due toKN , a rescaling of the horizontal axis, is negligiblethis case~from Fig. 3 one finds thatK'0.98! and was notincluded.
The agreement between the output of our 232 model andthe exact diagonalization becomes even more evidentanalyzing the cumulative overlap distribution in a log-loplot, as is seen from Fig. 5.
VI. CONCLUDING REMARKS
We have studied the orthogonality catastrophe due tparametric change of the single-particle of an ergodic stem. The Hamiltonian was modelled by a suitable randmatrix ensemble. We show that the average overlap betwthe original and the parametrically modified many-boground states, taken as Slater determinants, decreasen2bx2/4, wheren is the number of electrons in the system ax is the deformation measured in units of the typical distabetween anticrossings. We have also shown that the fluctions of lnuSu are enormous. To account for the latter in torthogonal case (b51), we have put forward a simple32 matrix model and employed it to obtain a prediction fP(uSu) for uSu!1, in good agreement with our numericanalysis. In the unitary case (b52) the fluctuations aresmaller, but still quite significant. Here also it was shown24
that the 232 model works well, but no simple analyticaexpression is avaliable.
This model study constitutes a first step towards undstanding the relevance of the orthogonality catastropheballistic ergodic systems. One improvement to the thepresented in Sec. IV should arise from the fact that fogeneric dynamical system the ‘‘perturbation’’ matrix,dHi j ,will in general have a finite bandwidthb. How are our resultschanged in this case? There is extensive literature dea
FIG. 5. Log-log plot of the cumulative distribution of overlapN(uSu)5*0
uSuduSuP(uSu). We compare~i! the analytical predictionbased on the caseN52 and ~ii ! numerical simulations over 105
pairs (H0 ,U) ~dots!, for N550. In both cases the relative deformtion is x50.2.
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with the matrix elements properties of low-dimensional dnamical systems. In particular, for the case of a twdimensional chaotic ballistic system, the bandwidth for‘‘generic’’ perturbation was estimated27,28 to be
b5\2
tD, ~48!
wheret is a time needed for a particle to traverse the systei.e.,
t'L
vF. ~49!
Here L is the linear dimension of the system andvF is theFermi velocity. Relating the two-dimensional~2D! meanlevel spacing to the system’s size,D52p\2/(mL2), and us-ing «F'nD, we obtain a simple relation between the banwidth and the number of electrons, namely,b'An. In linewith the present study, we assume that within the bandu j2 i u,b, the fluctuations ofdHi j are statistically well de-scribed by RMT. Accordingly, the average of lnuSu, obtainedin Eqs.~32! and ~33!, becomes
^ lnuSu&'2b
4x2 ln An. ~50!
We stress that in order to write Eq.~50!, two important as-sumptions have been made.~i! The HamiltonianH(X) mustbe fully chaotic.27 ~ii ! The perturbation must be genericthe sense defined in Ref. 28. If such conditions are not mthe statistical distribution ofdHi j could strongly depend onu j 2 i u, for u j 2 i u,b.
The study of the overlap between of many-body stasubjected to a perturbation has also been a traditional subof investigation in nuclear physics problems.29 More recentlythere has been renewed interest in examining overamong excited states.30 This study is much more involvedthan ours, since the states considered are superpositionSlater determinants, rather than a single one as studiedA full theory for such a case is still lacking; available nmerical evidence~evidently for small systems! indicates thatthe scaling of the overlaps withn follows a Gaussian, a facwhich is yet to be explained.
ACKNOWLEDGMENTS
We thank D. Kusnezov, I. V. Lerner, E. Mucciolo, MSaraceno, and E. Vergini for helpful discussions. This wowas supported by FAPERJ, CNPq, and PRONEX~Brazil!,and by the U.S.-Israel Binational Science Foundation,DIP Foundation, the Minerva Foundation, and The IsrScience Foundation founded by the Israel AcademySciences-Center of Excellence Program~Israel!.
APPENDIX A: A PERTURBATION SERIES FOR ln S
In Sec. II we showed that the overlap of the ground staScan be written asS5detAoo, with the matrix elements ofAdefined by Eq.~5!. This is done for the case where timereversal symmetry is present~absent!, namely the orthogona~unitary! symmetry. The notation ‘‘oo’’ stands for ‘‘occupiedoccupied’’ states~in this Appendix we shall also use ‘‘oe’’ for
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ORTHOGONALITY CATASTROPHE IN PARAMETRIC . . . PHYSICAL REVIEW B65 085309
‘‘occupied empty’’!. Within the Rayleigh-Schro¨dinger pertur-bation theory the overlap matrix is expanded as
A511eB1e2C1e3D1 . . . . ~A1!
SinceAA†51, i.e., using the fact that this series is normized, it is implied that
B1B†50,
C1C†1BB†50,
D1D†1CB†1BC†50, ~A2!
etc. In particular, one knows from perturbation theory ththe first-order correctionB is an antihermitian matrix satisfying Bii 50 and
eBi j 5dHi j
Ej2Ei, for j . i . ~A3!
To obtain a perturbative series for lnuSu we use Eq.~A1! andthe identity ln det5tr ln to write
ln detAoo5tr ln~11eBoo1e2Coo1e3Doo1••• !. ~A4!
By expanding the logarithm and regrouping the terms insum order by order ine, we obtain
ln detAoo5trFeBoo1e2S Coo21
2Boo2D1•••G . ~A5!
The linear term ine vanishes sinceBii 50. Using the firsttwo equations in Eq.~A2! one arrives at
ln detAoo521
2e2tr~BoeBoe†2Coo1Coo†!1O~e3!.
~A6!
The contribution fromCoo2Coo† being purely imaginarycorresponds to a phase in detAoo, so that
lnudetAoou521
2e2trBoeBoe†1O~e3!. ~A7!
If we average over the parametric random matrix ensemEq. ~18!, the third-order terms disappear:
^ lnudetAoou&521
2e2tr^BoeBoe†&1O~e4!, ~A8!
or, equivalently,
^ lnudetAoou&521
2 (i j
K udHi j u2
~Ej2Ei !2L 1O@~dH !4#,
~A9!
wherej andi run over the occupied and empty states, resptively.
This calculation also shows that
exp lnudetAoou&2^udetAoou&5O~e4!. ~A10!
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APPENDIX B: GRAND CANONICAL AVERAGES
In this appendix we study a slightly different questiocomplementary to the ones we have addressed so far.related to the averaging procedure. Suppose we have ansemble of systems, each of which having a given numbeelectronsN, which is kept fixed as the parameterX is varied.Let us assume, though, that thepreparationof this ensembleof systems is performedgrand canonically, i.e., by attachingeach sample to a weakly coupled particle reservoir at a gichemical potentialm, equilibrating, and then removing thicoupling. The number of electrons in each system may vbut is kept fixed during the ‘‘measurement’’~i.e., varyingX).This procedure has been defined in Refs. 26 and 23 asgrandcanonical–canonical. Under these conditions we employgrand canonical averaging scheme to calculate^ lnuSu&. Herewe expect that large fluctuations due to avoided crossiwill be suppressed, since the statistical weight of an avoicrossing needs to be modified fromP(s) ~canonical! tosP(s) ~grand canonical!.22 This reflects the fact that theprobability to place the chemical potential in a given g~between the last occupied and the first vacant level! is pro-portional to the size of the gap.
The results of our simulations using the grand canonensemble are summarized by Fig. 6. There, as in Fig. 2,show ^ lnuSu& and its different approximation schemes~ex-plained in Sec. IV! as a function ofN.
By comparing the results of Fig. 2 with the ones showabove we observe that the values of^ lnuSu&, as well as I norm&and^I &, obtained by the canonical simulation are systemcally reduced with respect to the ones obtained usinggrand canonical averaging procedure. This is in line withreasoning that the grand canonical averaging supprelarge fluctuations corresponding to small gaps. Thus it elinates the long smalluSu tails of P(uSu) characteristic of thecanonical ensemble forx!1. We also observe that both ensemble averaging procedures tend to render similar resulthe value ofx is increased.~For this reason we refrain fromshowing the grand canonical averages correspondingx
FIG. 6. The average lnuS(x)u as a function ofN. The number ofrealizations for eachN is M5104 and ~a! x50.1 and~b! x50.2.Open dots stand for the exact^ lnuSu&, filled dots for^I norm&, squares
for ^I &, whereas the solid lines represent^ I &. See Sec. IV for thedefinitions.
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RAUL O. VALLEJOS, CAIO H. LEWENKOPF, AND YUVAL GEFEN PHYSICAL REVIEW B65 085309
50.5 andx51.0 as depicted in Fig. 2.!Figure 7 quantifies the discussion presented in the fore
ing paragraph. The object of study is the standard deviad lnuSu5@^(lnuSu)2&2^lnuSu&2#1/2. For a fixed matrix sizeN550andM5104 realizations we compute the ratioR between thecanonical and the grand canonicald lnuSu as a function ofx.As expectedR(x).1 for all investigated values ofx. Fur-thermore, as we increasex the occurrence of small gaps bcomes less important and the canonical fluctuationscloser to the grand canonical ones.
APPENDIX C: ACCURACY OF THE APPROXIMATIONSFOR THE AVERAGE OF SINGLE-PARTICLE
OVERLAPS
Here we exhibit a test of the combination of first-ordperturbation theory and an average spectrum to accounaverages of single-particle overlaps. We want to compareaverage
^uAN/2,N/21 j u2&5^u^cN/2~X!ucN/21 j~X1dX!&u2& ~C1!
FIG. 7. The ratio between canonical and grand canonical sdard deviationd lnuS(x)u as a function ofx for a fixed N550 andM5104.
R
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with the approximation discussed in Sec. IV,
^uAN/2,N/21 j u2&5bx2D2
2^«N/21 j2«N/2&2
. ~C2!
The quantity plotted in Fig. 8 is the ratioR5^uAN/2,N/21 j u2&/^uAN/2,N/21 j u2&. For j small as compared toN/2 the average semicircle spectrum can be approximatea ‘‘picket fence’’ with spacingD, meaning that^«N/21 j2«N/2&
25 j 2D2. The ratioR is then
R~ j ,x!52 j 2
bx2^u^cN/2~X!ucN/21 j~X1dX!&u2&. ~C3!
The figure clearly displays the following features. Forxfixed, the ratioR→1 for large values ofj. Whenx,1 theapproximation works well except for the few states with, sj <5. In the case ofj 51, even ifx!1, perturbation theorybreaks down due to very narrow avoided crossings.
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FIG. 8. Test of first-order perturbation theory plus average sptrum. We plot the ratioR, the quotient ‘‘exact/approximate’’~seetext!, as a function of the energy differencej for different values ofthe parametrical distancex5dX/X! (x50.2,0.3,0.4,0.5,0.75,1.0the curves with increasingx correspond to the ones with decreasivalues for the first abscissa pointj 51). Each curve was calculateby averaging over 10 000 pairs (H0 ,U) with N5100.
.
v.
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