Level spacings at the metal-insulator transition in the Anderson Hamiltonians and multifractal random matrix ensembles

12 Pages·0.27 MB·English

by  Shinsuke M. Nishigaki

Level spacings at the metal-insulator transition in the Anderson Hamiltonians and
multifractal random matrix ensembles
Shinsuke M. Nishigaki∗
Institute for Theoretical Physics, University of California at Santa Barbara, Santa Barbara, California 93106
(September9, 1998)
9 Weconsiderorthogonal,unitary,andsymplecticensemblesofrandommatriceswith(1/a)(lnx)2
9
potentials, which obey spectral statistics different from the Wigner-Dyson and are argued to have
9
multifractaleigenstates. Ifthecoefficientaissmall,spectralcorrelationsinthebulkareuniversally
1
governedbyatranslationallyinvariant,one-parametergeneralization ofthesinekernel. Weprovide
n
analytic expressions for the level spacing distribution functions of this kernel, which are hybrids of
a
the Wigner-Dyson and Poisson distributions. By tuning the single parameter, our results can be
J
excellentlyfittedtothenumericaldataforthreesymmetryclassesofthethree-dimensionalAnderson
4
Hamiltonians at the metal-insulator transition, previously measured by several groups using exact
1
diagonalization.

]
n PACS number(s): 05.45.-a, 05.40.-a, 71.30.+h, 72.15.Rn
n
-
s
I. INTRODUCTION independent stochastic variables derived from a uniform
i
d distribution on an interval [ W/2,W/2], and the sum
t. Quantummechanicsdescribedbystochasticensembles r,r′ is over all pairs of nea−rest-neighboring sites. The
a h i
m ofHamiltonians[1],andbyHamiltonianswithclassically antiunitary symmetry of the AH [Eq. (1)] is orthogonal
chaotic trajectories [2], have been a subject of intense (β = 1), as it respects the time-reversal symmetry and
d- study for years. In contrast to Hamiltonians of classi- has no spin dependence. For smallvalues of disorderW,
n cally integrable systems whose energy levels are mutu- alleigenstatesareextended. Anoverlapofsuchextended
o allyuncorrelated,chaoticHamiltoniansgenerallyexhibit statesΨi(r)andΨj(r)isexpectedtoinducerepulsionbe-
c strong correlation among levels. To simulate this level tweenassociatedeigenvalues,ofthe form xi xj . Then
[ repulsion in chaotic or disordered Hamiltonians, Wigner thestatisticalfluctuationoftheseeigenvalu|es−isde|scribed
3 introduced the random matrix ensembles (RME) [3]. In wellbytheGaussianorthogonalensembleofrandomma-
v the RME defined as an integral over N N matrices, trices whose joint probability distribution consists of the
147 ospnelycttehde,aanndtiuitnsistparaytiasylmstmruecttruyreofistchoemHpalem×teilltyodniisacnarisderde-. apnroddGucatusosfiaanVfaacntdoerrsmonide−exd2i.eteTrhmeinqaunatdrQati<icjp|xoite−ntxiaj|l
Despite this extreme idealization, analytic predictions inthelatterismerelyforthesakeoftechnicalsimplicity,
9
0 from RME’s beautifully explain the spectral statistics of and the deformation oQf the potential by generic polyno-
8 chaotic or disordered Hamiltonians [4]. This success is mialsuniversallyleadstothesameWigner-Dysonstatis-
9 accountedforby the factthat the RME isnotmerely an tics [8]. As weincreasethe disorderW,states associated
t/ idealization of disordered Hamiltonians but it is indeed with eigenvalues close to the edges of the spectral band
a equivalenttothelatterunderasituationwherethemean are believed to start localizing. When the disorder is as
m
energylevelspacing∆ismuchsmallerthantheThouless large as to induce the metal-insulator transition (MIT)
d- energy Ec (inverse classical diffusion time) that the di- Ec/∆ ≃ 1, eigenstates are observed to be multifractal
mensionality does become unimportant [5]. This on the [9], characterized by an anomalous scaling behavior of
n
o otherhand implies thatin aregionwhere the meanlevel the moments of inverse participation ratio [10–12]
c spacing is equal to or larger than the Thouless energy,
v: and the associated states tends to localize due to diffu- Ψi(r)2p L−Dp(p−1). (2)
sion, standard RME’s cannot provide good quantitative h| | i∝
i r
X descriptions. X
r Asaconcreteexampleofsystemsofdisorderedconduc- This property implies a slowly decreasing overlap be-
a tors,letustake the Andersontight-binding Hamiltonian tween states [13] (for xi xj ∆),
| − |≫
(AH) [6,7],
Ψ (r)2 Ψ (r)2 x x −(1−D2/d). (3)
i j i j
H = εra†rar+ a†rar′, (1) Xr h| | | | i∝| − |
Xr hXr,r′i For large enough disorder, the off-diagonal (hopping)
describing free electrons in a random potential. Here term of the AH becomes negligible, leading to mutually
a† and a are creation and annihilation operators of an uncorrelatedeigenvalues. Eacheigenstateisalmostlocal-
r r
electron at a site r on a three-dimensional (3D) toroidal izedtoasinglesite. ThusthespectrumoftheAHshowsa
lattice of size L3, ε ’s are site energies that are mutually gradualcrossoverfromthe Wigner-Dysonto the Poisson
r
1
as one increases the disorder W, while keeping the size the Wigner-Dyson statistics for the Gaussian ensembles.
L fixed finite. A characteristic observable in these stud- The validity of the RM description for such critical sys-
ies of spectralstatistics is the probability E(s) ofhaving tems is of course far from clear, because the existence of
no eigenvalue in an interval of width s, or equivalently theMITcruciallyreliesuponthedimensionality,whereas
theprobabilitydistributionofspacingsoftwoconsecutive the RME has no spatial structure. Nevertheless, under
eigenvaluesP(s)=E′′(s). Theseobservablescapturethe an assumption that the spectra of the AH be described
behavioroflocalcorrelationsofalargenumberofenergy by RME’s, an attempt [32] was made to reconstruct a
eigenvalues, as the former consists of an infinite sum of random matrix potential out of the macroscopic spectra
integrals of regulated spectral correlators (the subscript ofthe AH. There,itwasobservedthata potentialofthe
reg denotes its regular part, i.e., with δ-functional peaks form
at coincident x ’s subtracted),
i 1
V(x)= [ln(1+bx)]2 (7)
∞ ( 1)n s/2 2a | |
E(s)= − dx dx ρ(x ) ρ(x ) ,(4)
n! 1··· nh 1 ··· n ireg explainswellthenumericaldata. Theabovepotentialin-
n=0 Z−s/2
X deedviolatestheuniversalityoftheWigner-Dysonstatis-
and are more conveniently measured by the exact di- tics that is guaranteed only for polynomial potentials
agonalization of random Hamiltonians than the spec- [33]. This observation leads to a speculation that the
tral correlators. The Poisson distribution is character- critical level spacing distribution might be derived from
ized by P (s) = exp( s), and the Wigner-Dyson dis- a RME with a potential (7). Later the LSDF of orthog-
P
−
tribution is well approximated by the Wigner surmise onal RME’s of type (7) has been measured by using the
P (s)=(πs/2)exp( πs2/4). Monte Carlo simulation [34] in order to compare it with
W
−
Recent technical developments [14–26] on the exact that from exact diagonalization of the AH [22]. Excel-
diagonalization of the AH on a large size of lattices lentagreementbetweenthe twowasfoundbytuning the
prompted analytical studies on its spectrum [27–31]. It parameter a to 2.5. Motivated by this success, we shall
was noticed in Ref. [14] that at the MIT point with dis- derive analytic forms of the LSDF’s of RME’s with or-
order W 16.5, the level spacing distribution function thogonal, unitary and symplectic symmetries. (The uni-
∼
(LSDF) P(s) is independent of the size L. From this tary case has already been reported [35].) In doing so,
finding these authors have argued that in the thermo- we shall retain all perturbative (polynomial in a) parts
dynamic limit there exist only three universality classes: of the spectral kernel, and discard unphysical nontrans-
Wigner-Dyson,Poisson,and the third, critical statistics. lationally invariant parts of order O(e−π2/a).
The presence of critical LSDF’s was also observed for Thispaperisorganizedasfollows. InSec.IIwereview
the unitary (β =2) and symplectic (β =4) cousins, i.e., nonstandard features of RME’s with log-squared poten-
AH’s under a magnetic field [18,20,24], tials. In Sec. III we follow the method of Tracy and
Widom [47] to derive the LSDF’s from an approximated
H = εra†rar+ Vrr′a†rar′, (5) translationallyinvariantkernel. InSec.IVwe shallcom-
Xr hXr,r′i pareourresultswiththenumericaldataoftheAH’swith
V =e∓2πiαry, V =V =1, orthogonal, unitary, and symplectic symmetries. In Ap-
r,r±xˆ r,r±yˆ r,r±ˆz
pendix A we collect standardresults on Fredholm deter-
and with spin-orbit coupling [16,25,21], minants in random matrix theories that are relevant for
our purpose.
H = εra†rσarσ+ Vrσ,r′σ′a†rσar′σ′, (6)
r,Xσ=± hr,rX′i,σ,σ′
II. RME WITH LOG-SQUARED POTENTIAL
V = e∓iθσi (ˆi=xˆ,yˆ,zˆ).
rσ,r±ˆiσ′ σσ′
The critical LS(cid:0)DF’s a(cid:1)re found to be independent of the In this section we review properties of RME’s with
log-squared potentials. For small enough a, we derive a
strength of the magnetic field α or the spin-orbit cou-
translationally invariant kernel, whose level spacing dis-
pling θ. For all values of β, the Wigner-Dyson-like be-
tribution will be our subject in this paper.
haviors P (s) sβ for small s have been confirmed.
(cr)
∝ We consider N N random real symmetric (β = 1),
There were disputes over the large s asymptotic behav- ×
complex hermitian (β = 2), and quaternion selfdual
ior of the LSDF’s, but accurate measurements on large
(β =4) matrix ensembles, whose joint probability densi-
lattices [23] strongly support the Poisson-like behavior
ties of eigenvalues are given by
P (s) exp( const s) for large s, excluding a non-
(cr)
∼ − ×
trivialexponentP (s) exp( const s1+γ)predicted
(cr) N
∼ − ×
inRef.[29]. Inthelightofthesuccessoftherandomma- (λ ,...,λ ) e−V(λi) λ λ β. (8)
β 1 N i j
trix(RM)descriptionofextendedstates,anaturalresort P ∝ | − |
i=1 1≤i<j≤N
Y Y
todescribethiscriticalstatisticsistoconsideradeformed
RME that violates the above mentioned universality of with a potential growing as
2
V(λ) 1 (lnλ)2 (λ 1). (9) λ Nπ
∼ 2a ≫ ψN(λ) cos π ρ(λ)dλ+ , (16)
∼ 2 !
Z
The potential is assumed to be regularizedat the origin, λ λ′
sin[π( ρ ρ)]
as in Eq. (7). For a particular form of the potential K(λ,λ′) − . (17)
∼ λ λ′
∞ R − R
V(λ)= ln[1+2qncosh(2arcsinhλ)+q2n], (10) Hereρ(λ)standsfortheexactunnormalizedspectralden-
n=1 sity, K(λ,λ). In the spectral bulk of the RME with
X
polynomially increasing potentials, the spectral density
(0<q <1),whichbehavesasEq.(9)witha=ln(1/q)/2,
dividedbyN isbounded,andislocallyapproximatedby
corresponding orthogonal polynomials are known as the
a constant when measuring λ in unit of the mean level
q-Hermitepolynomials[36]. Usingtheirasymptoticform,
spacing. Thisslowly-varyingfunctioniscalledthemean-
Muttalibetal.[37]haveobtainedtheexactkernelforthe
field spectral density ρ(λ), given by [40]
unitary ensemble with the potential (10) in the large N
limit,
N 1 R dµ R2 λ2V′(µ)
ρ(λ) + − .
K(exact)(x,y)=const √cosh2axcosh2ay (11) ≡ π√R2−λ2 π2 −Z−R λ−µsR2−µ2 2
× cosha(x+y) ×
(18)
ϑ (x+y,e−π2/a) ϑ (x y,e−π2/a)
4 1
− .
ϑ (2x,e−π2/a)ϑ (2y,e−π2/a) sinha(x y) Here Rarethe endpointsofthe spectrum,determined
4 4 − R±
by ρ(λ)dλ = N. After replacing the exact ρ(λ) by
Hepre ϑ are the elliptic theta functions, and the variable −R
ν the mean-field ρ(λ), the unfolding map
x λ/(2a)issorescaledthattheaveragelevelspacingis R
≡
unity. In orderto eliminate nonuniversaleffects involved
λ
by the regularization of the potential in the vicinity of λ x= ρ(λ)dλ (19)
7→
the origin from the bulk correlation, we need to take
Z
x,y 1, with x y =bounded. (12) becomes merely a linear transformation, leading univer-
≫ | − | sally to the sine kernel. On the other hand, in our case
Then we obtain an asymptotic form of the kernel of the potential (9), the mean-field spectral density (18)
behaves as
K(asympt)(x,y)=const
× 1
ϑ (x+y,e−π2/a) ϑ (x y,e−π2/a) ρ(λ) , (20)
4 1 − . (13) ∼ 2aλ
ϑ (2x,e−π2/a)ϑ (2y,e−π2/a) sinha(x y) | |
4 4 −
implying an unusual unfolding map
Tpheabovekernelisstillnottranslationallyinvariantdue
to the reason explained below. Now we make a further 1
λ x= sgn(λ)ln λ, (21)
simplification of the kernel by using an approximation.
7→ 2a | |
For e−π2/a 1, we can discard subleading orders from
≪
theqexpansionofthethetafunctionsintermsoftrigono- while the formula (17) stays valid [41]. Then the kernel
metricfunctions. Thenweobtainatranslationallyinvari- (17) universally reduces to Eq. (14) after this unfolding.
ant kernel, up to O(e−π2/a), It is clear from the form of the kernel (14) that a set
of eigenvalues with x x 1/a obeys the Poisson
i j
| − | ≫
a sinπ(x y) statistics, i.e., is uncorrelated. On the other hand, a set
K(x,y)= − . (14)
πsinha(x y) of eigenvalues with xi xj 1/a obeys the Wigner-
− | − | ≪
Dyson statistics, because Eq. (14) is then approximated
The universality of this deformed kernel within RME’s bythesinekernel,uptoO(a2). Tobeprecise,thekernel
is observed for the q-Laguerre unitary ensemble [38], (14) signifies the multifractality of the eigenstates [42].
the finite-temperature Fermi gas model [39], and subse- Toseethis,wefirstnotethattheproperty(3)ofthefrac-
quently for unitary ensembles whose potentials have the tal states leads to a compressible gas of eigenvalues, i.e.,
asymptotics(9)[34]. Thisuniversalitycanbeconsidered alinearasymptoticsofthenumbervarianceΣ2 withinan
as an extension of the universality of the sine kernel [8], energywindowofwidthS [43][Y (x) 1 ρ(x)ρ(0) ],
2 reg
≡ −h i
sinπ(x y)
S
K(x,y)= − , (15)
π(x y) Σ2(S) S 2 dx(S x)Y2(x) (22)
− ≡ − Z0 −
for polynomially increasing potentials, proven via the 1 D2
1 S χS (S 1). (23)
asymptotic WKB form of the wave functions ∼ 2 − d ≡ ≫
(cid:18) (cid:19)
3
The RME’s with the scalar kernel (14) indeed enjoy levelrepulsionpropertyof the Gaussianensembles is de-
this asymptotic behavior with the level compressibility formed slightly but not to the extent that the crystal-
χ given by lization of eigenvalues becomes prominent. In the case
of the q-Hermite ensembles, we can estimate this scale
χ= a +O(e−2π2/a) (24) to be characterized by the value of a where nontransla-
π2 tional invariance of the exact kernel becomes manifest,
i.e., e−π2/a 1. We assume this estimate to be valid
for unitary [38,42] and orthogonal ensembles, and for ≃
genericallyforRME’swithlog-squaredpotentials(9). In
the symplectic ensemble the exponentialcorrectionisre-
placed by O(e−π2/a). Moreover, the multifractality of view ofthis, for our purposeofreproducing the LSDF of
the AH, we shall concentrate on the approximated uni-
the RME’s with the deformed kernel (14) has been con-
versalkernel(14). Thiswillbejustifiedaposteriori,after
cluded [42] through the equivalence between the finite-
confirming that the best-fit value of the parameter a for
temperature Fermi gas model (having the universal de- the MIT point of the AH’s are such that e−π2/a 1.
formed kernel) and a Gaussian banded RME that is ≪
proven to have multifractal eigenstates [44]. This brings
forth the possibility of describing the spectral statistics
III. LEVEL SPACING DISTRIBUTIONS OF THE
of the AH’s at the MIT by RME’s with the deformed
DEFORMED KERNEL
kernel.
We should emphasize an important fact that the
In this section we analytically compute the LSDF
RME’s with log-squared potentials cannot describe dis-
P (s) from the deformed kernel (14) for all values of the
orderedsystemswithlargedisorder. WhiletheAHinthe β
Dysonindex β. Our result completes earlierattempts to
W limit leads to the Poisson statistics, the RME
→ ∞ compute P (s) numerically [37] or asymptotically [46],
with Eq. (9) does not obey the Poisson statistics in the 2
and is consistent with those.
limit a [34,45]. It is because the joint probability
→ ∞ We notice that the kernel (14) is equivalent to that of
distribution of RME’s after the unfolding (21) leads to
Dyson’s circular unitary ensemble at finite N [3],
(x , ,x )=const
β 1 N sin(N/2)(x y)
P ··· ×
K(x,y)= − , (27)
e2a|xi| e2a|xj| β N e−2ax2ie2a|xi|. (25) Nsin(1/2)(x−y)
± ∓
by the following analytic continuation
1≤iY<j≤N(cid:12) (cid:12) iY=1
(cid:12) (cid:12)
(cid:12) (cid:12) πi 2a
In the limit a , each factor of e2a|xi| e2a|xj| β N , x x. (28)
→ ∞ ∓ → a → i
is dominated by an exponential with a larger modulus.
Thus the Vandermonde determinant is(cid:12)(cid:12)approximated b(cid:12)(cid:12)y Tracy and Widom [47] have proven that the diagonal
resolvent kernel of Eq. (27) is determined by a second-
N order differential equation that is reduced to a Painlev´e
e2aβ(i−1)|xi| (for x < < x ). (26) VI equation [48]. We shall follow their method below.
1 N
| | ··· | |
i=1 The kernel (14) is written as
Y
Consequently (x ,...,x ) tends to a product of very φ(x)ψ(y) ψ(x)φ(y)
narrow [of varPiaβnce1σ2 = 1N/(4aβ) 1] Gaussian distri- K(x,y)= e2ax−e2ay , (29)
≪ −
butions obeyed by x whose center is at [β(i 1)+1]/2.
i 2a 2a
This “crystallization” of eigenvalues invalida−tes naively φ(x)= eaxsinπx, ψ(x)= eaxcosπx.
π π
expected mutual independence of distributions of eigen- r r
values, and drives the spectrum toward an exotic statis- These component functions satisfy
ticsdifferentfromPoissonian[45]. Thisphenomenoncan
be rephrased in the context of using the WKB formula φ′ =aφ+πψ, ψ′ = πφ+aψ. (30)
−
(17)toderivethekernel(14). AlthoughEq.(17)remains
We use the bra-ket notation φ(x) = xφ and so forth
valideveninthecasea ,useofthemean-fieldspec- h | i
→∞ [50]. Due to our choice of the component functions to
tral density ρ(λ) = 1/(2aλ) in place of the exact ρ(λ)
be realvalued(unlike[47],Sec.VD),wehave xOφ =
is not justifiable, because the crystallization of eigenval- h | | i
φOx andasimilarsituationforψwithanyself-adjoint
ues leads to a rapidly oscillating ρ(λ). In the case of the h | | i
operator O and real x. Then Eq. (29) is equivalent to
q-Hermite ensemble (10), the potential itself has a oscil-
lation of the same type, leading again to crystallization [e2aX,K]= φ ψ ψ φ, (31)
[45]. Therefore, the RME with log-squared potentials, | ih |−| ih |
despite the fact that it is constructed from the macro- where X and K are the multiplication operator of the
scopicspectraofthe AH,shouldbeconsideredasagood independent variable and the integral operator with the
model of the latter only for small values of a where the kernel K(x,y)θ(y t )θ(t y), respectively. Below we
1 2
− −
4
1
willnotexplicitlywritethedependenceontheendpoints p˜p+q˜q = (R˜sinh2at)·, (40)
oftheunderlyinginterval[t ,t ]. ItfollowsfromEq.(31) π
1 2
that R˙ =2R˜2, R¨ =4R˜R˜˙. (41)
K 1 1
e2aX, = (φ ψ ψ φ) , (32) The left-hand sides of Eqs. (39) and (40) satisfy an ad-
1 K 1 K | ih |−| ih | 1 K
(cid:20) − (cid:21) − − ditional constraint,
that is, (p˜p+q˜q)2+(p˜q q˜p)2 =(p˜2+q˜2)(p2+q2). (42)
−
(e2ax e2ay)R(x,y)=Q(x)P(y) P(x)Q(y), (33)
− − By eliminating p˜,p,q˜,q, R˜, and R˜˙ from Eqs. (39)–(42),
Q(x) x(1 K)−1 φ ,
≡h | − | i we obtain for R(s) (s 2t, ′ =d/ds)
P(x) x(1 K)−1 ψ . ≡
≡h | − | i 2
sinhas
At a coincident point x=y we have acoshasR′(s)+ R′′(s) +[πsinhasR′(s)]2 =
2
(cid:20) (cid:21)
2ae2axR(x,x)=Q′(x)P(x) P′(x)Q(x). (34) R′(s) [aR(s)]2+asinh2asR(s)R′(s)+[sinhasR′(s)]2 .
−
(43)
Now, by using the identity (cid:0) (cid:1)
∂K It is equivalent to Eq. (5.70) of Ref. [47] after the an-
=( 1)iK ti ti (i=1,2), (35) alytic continuation (28), accompanied by a redefinition
∂t − | ih |
i
R(s) (i/2a)R(s). This is slightly nontrivial because
→
we obtain Ref. [47] has used p˜ = p∗ and q˜ = q∗, which follow
from the analytic properties of its component functions,
∂Q(x)
=( 1)iR(x,t )Q(t ), (36a) φ( x) = φ(x)∗ and ψ( x) = ψ(x)∗. In the limit a 0.
i i
∂t − − − →
i it clearly reduces to the σ form of a Painlev´eV equation
∂P(x)
=( 1)iR(x,t )P(t ). (36b)
i i s 2
∂ti − R′(s)+ R′′(s) +[πsR′(s)]2 =R′(s)[R(s)+sR′(s)]2,
2
Ontheotherhand,byusingtheidentity(Disthederiva- h i (44)
tion operator)
derived for the sine kernel (15) of the Gaussian ensem-
[D,K]=K(t t t t ), (37)
1 1 2 2 bles [49]. Note that for β =1, a replacement a a/2 is
| ih |−| ih |
→
necessary because of our convention (A1).
whichfollowsfromthetranslationalinvarianceoftheker-
Finally, the LSDF’s P (s) are expressed in terms of
nel (∂ +∂ )K(x y)=0, we also have β
x y − the diagonal resolvent R(s) via Eqs. (A12), (A14), and
∂Q(x) the first of Eq. (41):
= xD(1 K)−1 φ
∂x h | − | i ′′
= x(1 K)−1 φ′ P1(s)= e−21 0sds[R(s)+√R′(s)] , (45a)
h | − | i
+ x(1 K)−1[D,K](I K)−1 φ (cid:20) R (cid:21)
h | − − | i s ′′
− dsR(s)
=aQ(x)+πP(x)+R(x,t1)Q(t1) R(x,t2)Q(t2), P2(s)= e 0 , (45b)
−
(38a) (cid:20) R (cid:21)
∂P∂(xx)=−πQ(x)+aP(x)+R(x,t1)P(t1)−R(x,t2)P(t2). P4(s)=(cid:20)e−21R02sdsR(s)cosh(cid:18)12Z02sds R′(s)(cid:19)(cid:21)′′. (45c)
p
(38b)
The boundary condition follows from the expansion (4)
of E (s) in terms of the correlationfunctions:
Nowwesett = t,t =t,x,y = tort, andintroduce 2
1 2
− −
notations q˜=Q( t),q =Q(t),p˜=P( t),p=P(t), and
s/2
− −
R˜ = R( t,t) = R(t, t),R = R(t,t) = R( t, t). The E (s)=1 dxK(x,x)
2
− − − − −
lasttwoequalitiesfollowfromtheevennessofthekernel. Z−s/2
Then Eqs. (33) and (34) read, after using Eqs. (38), 1 s/2
+ dx dx det K(x ,x )
1 2 i j
p˜q q˜p=2R˜sinh2at, (39a) 2Z−s/2 i,j=1,2 −···
− 2 =1 s+O(s4), (46)
p˜2+q˜2 = π(R˜2sinh2at+Rae−2at), (39b) R(s)= −[lnE2(s)]′ =1+s+ . (47)
− ···
2
p2+q2 = (R˜2sinh2at+Rae2at). (39c) Our main result consists of Eqs. (43), (45), and (47).
π
For s 1/a, we can Taylor-expand hyperbolic func-
≪
The total t derivatives of Eqs. (39) lead to ( =d/dt) tions, to obtain a perturbative solution to Eq. (43),
·
5
π2+a2 5(π2+a2) (π2+a2)(75 4π2 6a2)
R(s)=1+s+s2+ 1 s3+ 1 s4+ 1 − − s5+ , (48)
− 9 − 36 − 450 ···
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
4π2+a2 4π2+a2 12π2+7a2 π2+a2 4π2+a2
P (s)= s s3+ s4
1
24 − 2880 1080
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
4π2+a2 48π4+72π2a2+31a4 s5 π2+a2 4π2+a2 12π2+13a2 s6
+ + , (49a)
322560 − 226800 ···
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
π2+a2 (π2+a2)(2π2+3a2) (π2+a2)(π2+2a2)(3π2+5a2)
P (s)= s2 s4+ s6+ , (49b)
2
3 − 45 945 ···
16 π2+a2 π2+4a2 s4 128 π2+a2 π2+4a2 3π2+13a2 s6
P (s)= + . (49c)
4
135 − 14175 ···
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
The above perturbative expansions are correct for the a = 2.95,3.55,3.90. Preference could alternately be put
kernel(13) of the q-Hermite ensemble in any polynomial onbestmatchingsinthesmallervaluesofs(<2),which
orders of a, while they lack nonperturbative terms of or- would lead to a = 3.2 for β = 1 [although th∼e difference
dere−π2/a,whichdependonthe referencepoint. Onthe in P1(s) between a = 2.95 and a = 3.2 is tiny]. In Figs.
other hand, for s 1/a, it can be proven from Eq. (43) 5–7 we exhibit linear and logarithmic plots of LSDF’s of
≫
that R(s) approachs a constant. Then Eqs. (45) imply the RME’s and the AH’s. The numerical data fit excel-
lently with ouranalytic resultfromthe kernel(14), for a
1 large energy range 0 s < 6 where the LSDF’s vary by
lnP1(s)∼−2Ra2(∞)s, (50a) four to five ordersof m≤agn∼itude. The use ofthis approx-
lnP2,4(s) Ra( )s, (50b) imated kernel is justifiable because e−π2/a 1 holds for
∼− ∞ ≪
these values ofa. Small systematic deviations canbe at-
for s . In Fig. 1 we exhibit the decay rate κ(a) tributed to the errors involvedin determining the values
→ ∞ ≡
Ra(s = ) for 0 < a < 4 computed numerically from ofafromthedecayratesofnumericalLSDF’s,andpossi-
∞
Eq. (43). At present we could not find an analytic form bly to anessentialdifference of orderO(e−π2/a) between
of κ(a). For small a(< 0.5), it is well approximated by
the RME’s and the AH’s.
1/κ(a) 0.202a, which agrees extremely well with the
Furthermore,weexhibitinFig.8thenumbervariance
value 2/≈π2 =0.2028 expected from Eq. (24) and the an- Σ2(S) of the orthogonal AH obtained in Ref. [16], to-
alytic formula [27] that holds in generality,
gether with the RME result (22) with (β =1)
1 ∞
χ= . (51) Y (x)=K(x)2+K′(x) K(y)dy (52)
2κ 2
Zx
Eqs. (49) and (50) tells that our LSDF’s are indeed hy- ata=3.2. Wecanconfirmthatnotonlythe asymptotic
brids of the Wigner-Dyson-like [Pβ(s) sβ for s small] slopes χ of Σ2(S) (first pointed out by Canali [34], who
and the Poisson-like distributions [Pβ(∼s) e−κs for s computed χ by the Monte Carlo simulation of RME’s),
∼
large]. In Figs.2–4 we exhibit plots ofthe LSDF’s Pβ(s) but their full functional forms are in a good agreement
for β = 1,2,4 and for various a such that e−π2/a 1, for L<10. To recapitulate, we have the following three
≪
obtained by numerically solving Eq. (43). distinc∼t functional observables (consisting of the corre-
lation functions in the second column) that agree well
between the critical AH’s and the deformed RME’s:
IV. ANDERSON HAMILTONIANS AT MIT
Quantity Correlation function
Potential V(λ) 1-level
In this section we make comparison between the
Number variance Σ2(S) unfolded 2-level
LSDF’s and the level number variance in the exact di-
Level spacing P(s) unfolded n-level (n 2)
agonalization of the AH’s and our analytic results from ≥
multifractal RME’s. In addition, both the critical AH’s [9] and the deformed
Asnumericaldatatocomparewith,weadoptRef.[23] RME’s [42] are shown to have multifractal eigenstates,
for the AH [Eq. (1)] (β = 1), Ref. [24] for the AH un- although the sequences of the multifractal dimensions
der a magnetic field (5) with α = 1/5 (β = 2), and are yet to be compared. Agreements in the unfolded
Ref. [25] for the AH with spin-orbit coupling (6) with quantities shouldnot be considereda tautologicalconse-
θ = π/6 (β = 4), at their MIT points. We choose the quenceofthefirstline;oneshouldrecallthatanidentical
best fit values of a from the exponential decay rates κ of semicirclespectrumcouldaswellbeobtainedeitherfrom
the numericaldata using Fig. 1. Basedon the numerical invariant RME’s with Gaussian potentials (obeying the
resultsκ=1.9,1.8,1.7forβ =1,2,4,respectively,wees- Wigner-Dyson statistics), or from diagonal random ma-
timate the parameter a in the potentials of RME’s to be trices whose entries are independently derived from the
6
semicircledistribution(obeyingthePoissonstatistics),or detailed proofs. Subsequently we shall concentrate on
from any intermediate ensembles. From these grounds, the case where the spectral correlation is translationally
we conclude that the interaction between unfolded en- invariant after unfolding.
ergy levels of the 3D AH on equilateral (L =L =L ) ThejointprobabilitydensitiesofeigenvaluesofN N
x y z
×
toroidallattices atthe MIT is verywelldescribedby the random real symmetric (β = 1), complex Hermitian
form e2ax e2ax′ β,whichiscommontotheRME’swith (β = 2), and quaternion self-dual (β = 4) matrices are
| − |
log-squared potentials (9), in contrast to the standard given by
form x x′ β of the Gaussian RME’s and the AH’s in
the m|eta−llic|regime. We surmise that the dimensionality N
(x , ,x )=const w (x ) x x β,
and the fractal dimensionality enter the critical spectral β 1 N β i i j
P ··· × | − |
statisticsprimarilythroughasingleparametera,aslong iY=1 1≤iY<j≤N
asthemultifractalityofthe criticalwavefunctionsisnot w (x)=e−V(x)/2, w (x)=e−V(x). (A1)
1 2,4
too strong. Further work is needed to explain this form
of the level repulsion from the multifractality (3) of the We use the above convention between the weight func-
wave functions, and its origin from microscopic models. tions and the potentials so as to simplify the relations
Finally, remarks related to novel numerical results on between kernels [Eq. (A5) below]. We introduce the
critical AH’s are in order. Recently it was observedthat “wavefunctions” ψi(x) i=0,1,... byorthonormalizingthe
{ }
the critical LSDF of the 3D AH is sensitive to the ge- sequence xie−V(x)/2 , and the projection operator K
{ }
ometryofthe lattice,i.e.,the topology(boundarycondi- to the subspace spanned by the first N wave functions.
tion) [51,52] and the aspect ratio [53], due to the coher- As an integration operator acting on the Hilbert space
ence of the critical wave functions maintained over the spanned by the wave functions, K is associatedwith the
wholelattice. Since RME’streatrandomnessonallcites kernel (we shall use the same letter for an operator and
and bonds on equal footing, it is likely that our RME’s the kernel associated with it),
describe best the critical AH’s on maximally symmetric
lattices, i.e., equilateral tori, but not those on less sym- N−1
K(x,y)= ψ (x)ψ (y). (A2)
metric lattices, such as unequilateral toroidal lattices or i i
latticeswithboundaries. Besides,sincethevalidityofour i=0
X
expressionsfor the LSDF’s is limited to the case ofweak
Then the joint probability densities are expressed in
multifractality (relatively small a), it will not properly
terms of determinants of the kernels [3]:
describe the critical orthogonal AH in four dimensions
[54], where the level compressibility was observed to be (x ,...,x )= det K (x ,x ), (A3)
β 1 N β i j
largerthan the value in three dimensions (χ 0.27 [22]) P 1≤i,j≤N
≈
and close to its upper bound χ=0.5.
K (x,y)=K(x,y), (A4a)
2
S (x,y) S D(x,y)
ACKNOWLEDGMENTS K (x,y)= 1 1 , (A4b)
1 ǫS (x,y) ǫ(x,y) S (y,x)
1 1
(cid:18) − (cid:19)
S (2x,2y) S D(2x,2y)
I thank I.K. Zharekeshev, L. Schweitzer, T. Kawara- K (x,y)= 4 4 . (A4c)
4 ǫS (2x,2y) S (2y,2x)
bayashi,S.N. Evangelou,E.Hofstetter, andC.M. Canali 4 4
(cid:18) (cid:19)
for kindly providing me with their numerical data, and
Heredetistobeinterpretedasaquaterniondeterminant
H. Widom, J.T. Chalker, A. Zee, and E. Kanzieper for
in the case of β =1 and 4 [3]. D stands for the differen-
discussions and comments. This work was supported in
tiation operator, and ǫ, S and S stand for integration
part by JSPS ResearchFellowships for Young Scientists, 1 4
operators with kernels [56]:
byGrant-in-AidNo.411044fromtheMinistryofEduca-
tion, Science and Culture, Japan,by the Nishina Memo- 1
rial Foundation, and by NSF Grant No. PHY94-07194, ǫ(x,y)= sgn(x y),
2 −
and these supports are gratefully acknowledged.
S (x,y)=[1 (1 K)ǫKD]−1K(y,x), (A5a)
1
− −
S (x,y)=[1 (1 K)DKǫ]−1K(x,y). (A5b)
4
− −
APPENDIX A: FREDHOLM DETERMINANT IN
A composite operator such as ǫS is defined to have a
RANDOM MATRIX THEORY ∞1
convolutedkernel,ǫS (x,y)= dzǫ(x,z)S (z,y)and
1 −∞ 1
so forth, and [ ]−1 stands for an inverse operator.
In the Appendix we collect known results on random ··· R
TheprobabilityE [J]offindingnoeigenvaluesinaset
β
matrixtheoriesthatarerelevanttoourpurposeofevalu-
of intervals J is defined as
atingtheLSDFofthreesymmetryclassesofRME’s. We
follow Mehta’s classicalbook [3]and the worksby Tracy
E [J]= dx dx (x ,...,x ). (A6)
and Widom [47,55,56]. Readers are referred to them for β 1··· NPβ 1 N
Zxi6∈J
7
By virtue of Eq. (A3) and the identity detM d 1 dK
{ni} i,j ninj ∝ dtlnE2[−t,t]=tr 1 K dt
detM , it is expressedin terms of the FPredholm deter- (cid:18) − (cid:19)
n,m nm 1
= tr K( t t + t t)
minant of the scalar or matrix kernel [55]: − 1 K |− ih− | | ih |
(cid:18) − (cid:19)
K
E2[J]=det(1− K2|J), (A7a) =−2 t 1 K t . (A13)
(cid:28) (cid:12) − (cid:12) (cid:29)
E1,4[J]= det(1− K1,4|J), (A7b) Eqs. (A12b,c) can be p(cid:12)(cid:12)roven (cid:12)(cid:12)analogously from Eqs.
q (cid:12) (cid:12)
(A4b), (A4c), and (A7b). These relations are equiv-
where represents restriction of the kernel to the inter-
J
| alent to Eqs. (6.5.19) and (10.7.5) of Ref. [3] because
val J.
t
exp dt[R(t,t) R( t,t)] is a Fredholm determi-
In the following we assume that the scalar (unitary) {− 0 ± − }
nant of the kernel K(x,y) K( x,y) (although Ref. [3]
kernel K is translationally invariant and symmetric (the
R ± −
concerns primarily the sine kernel, the proofs of these
latter property is respected for invariant RME’s by con-
equationsareequallyvalidforanytranslationallyinvari-
struction (A2), but it is violated for RME’s with partly
ant and symmetric kernel).
deterministic matrix elements [50]),
Now we set 2t=s and denote E (s)=E [ s/2,s/2].
β β
−
K(x,y)=K(y,x)=K(x y). (A8) The probability Pβ(s) for a pair of consecutive eigenval-
− ues to have a spacing s is clearly equal to the probabil-
If K has these two properties, Eqs. (A5) immediately ity of finding an eigenvalue in an infinitesimal interval
reduce to [ s/2 ǫ,s/2],another in [s/2,s/2+ǫ′] and none in be-
− −
tween [ s/2,s/2],divided by ǫǫ′. Thus we have
−
S (x,y)=S (x,y)=K(x y), (A9)
1 4 − P (s)=E′′(s). (A14)
β β
because of the relations DK = KD and ǫD = 1,
and the orthogonality of the two projection operators,
(1 K)K = 0. We assume that lim and the alge-
N→∞
−
braic manipulation that lead to the relation(A5) can be
interchanged. InthefollowingweletthenotationK rep-
resent its restriction to J, formerly denoted as K . If J
J
|
consists of a single interval [ t,t], we can simplify E [J]
− β ∗ OnleavefromDepartmentofPhysics,TokyoInstituteof
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9
0.5
0.4
)
(a 0.3
κ
1/
0.2
0.1
1 2 3 4
a
FIG.1. The decay rate κ(a)=Ra(∞) of theLSDF.
0.8
a=0(GOE)
a=1
0.6 a=2
a=3
a=4
)
(s 0.4
P1
0.2
0.5 1 1.5 2 2.5 3
s
FIG.2. TheLSDFP1(s) of theorthogonal ensemble with thekernel(14).
1
a=0(GUE)
0.8 a=1
a=2
a=3
0.6 a=4
)
(s
P2 0.4
0.2
0.5 1 1.5 2 2.5 3
s
FIG.3. The LSDF P2(s) of theunitary ensemblewith the kernel(14).
1.2 a=0(GSE)
a=1
1 a=2
a=3
0.8 a=4
)
(s
P4 0.6
0.4
0.2
0.5 1 1.5 2 2.5 3
s
FIG.4. TheLSDFP4(s) of thesymplectic ensemble with thekernel(14).
10