Geometry of random interactions

5 Pages·0.21 MB·

by  P. Chau Huu-Tai

Geometry of random interactions
P. Chau Huu-Tai1, A. Frank2,3, N. A. Smirnova4, and P. Van Isacker1
1Grand Acc´el´erateur National d’Ions Lourds,
Boˆıte Postale 55027, F-14076 Caen Cedex 5, France
2Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico,
Apartado Postal 70-543, 04510 M´exico, D.F., M´exico
3Centro de Ciencias F´ısicas, Universidad Nacional Aut´onoma de M´exico,
Apartado Postal 139-B, 62251 Cuernavaca, Morelos, M´exico and
4University of Leuven, Instituut voor Kern- en Stralingsfysica,
Celestijnenlaan 200D, B-3001 Leuven, Belgium
It is argued that spectral features of quantal systems with random interactions can be given a
geometric interpretation. This conjecture is investigated in the context of two simple models: a
systemofrandomlyinteractingdbosonsandoneofrandomlyinteractingfermionsinaj = 7 shell.
2
3
Inbothexamplestheprobabilityforagivenstatetobecomethegroundstateisshowntoberelated
0
toageometricpropertyofapolygonorpolyhedronwhichisentirelydeterminedbyparticlenumber,
0
shellsize,andsymmetrycharacterofthestates. Extensionstomoregeneralsituationsarediscussed.
2

n
a Recent studies in the nuclear shell model [1, 2, 3, 4] and particle states.
J
the interacting boson model [5, 6] with random interac- We begin by considering the special case when a ba-

1 tions have unveiled a high degree of order. In particu- sis |jnαJMi can be found in which the expansion (1) is
2
lar,a markedstatisticalpreference wasfound forground diagonal. In this case the Hamiltonian matrix elements

states with J = 0+. In this Letter it is argued that reduce to the energy eigenvalues
1
v spectral features of quantal systems with random inter-
1 actions can be given a geometric interpretation which n(n−1)
E = bL G , (2)
6 allows the computation of the probability for the quan- nαJ 2 nαJ L
0 L
tum mechanical ground state to have a specific angular X
1
0 momentum, based on purely geometric considerations. withbL ≡ cL 2. Thisisobviouslyasimplificationof
3 Although these results are obtained in the context of a nαJ nαJ
themoregeneralproblem(1)butneverthelessawideva-
0 varietyofsimplemodelswhichdonotcoverthefullcom- (cid:0) (cid:1)
rietyofsimplemodelsituationscanbeaccommodatedby
/ plexity of random interactions, we believe them to be
h it. Forexample,thispropertyisvalidforanyinteraction
-t sufficiently generalto conjecture the possibility ofanen- between identical fermions if j ≤ 7 and remains so ap-
l tirely geometric analysis of the problem. 2
c proximatelyforlargerj values;itisalsoexactlyvalidfor
Consider a system consisting of n interacting particles
u p,d,orf bosons. We shallrefertothisclassofproblems
n (bosonsorfermions)carryingangularmomentumj,inte- as diagonal. For a constant interaction, G = 1, all n-
: gerorhalf-integer. Eigenstatesofarotationallyinvariant L
v particleeigenstatesaredegeneratewithenergy 1n(n−1)
Hamiltonian are characterizedas |jnαJMi where J and 2
i and consequently the coefficients bL satisfy the prop-
X M are the total angular momentum and its projection, nαJ
erties bL =1 and 0≤bL ≤1. Equation (2) can
r and α is any other index needed for a complete labeling L nαJ nαJ
a thus be rewritten in terms of scaled energies as
of the state. Although spectral properties of a Hamil- P
tonian with both one- and two-body interactions can be 2E
analyzedinthe wayexplainedbelow,weassumeforsim- enαJ ≡ n(nn−αJ1) =GL′ + bLnαJ(GL−GL′), (3)
plicity that the one-body contribution is constant for all L
X
eigenenergies and that the energy spectrum is generated
for arbitrary L′. This shows that, in the case of N in-
bytwo-bodyinteractionsonly. Underthisassumptionits
teractionmatrixelementsG ,the energyofanarbitrary
matrix elements can be written as L
eigenstatecan,uptoaconstantscaleandshift,berepre-
hjnαJ|Hˆ|jnα′Ji= n(n2−1) cLnαJcLnα′JGL, (1) dseiffnteerdenacsesapofoimntaitnriaxveelcetmoernstpsa.ceNsoptaennthedatbythme p≡osNit−ion1
XL of these states is fixed by bL and hence interaction-
nαJ
where M is omitted since energies do not depend on independent. Furthermore,since0≤bL ≤1,allstates
nαJ
it. The quantities G are two-particle matrix elements, are confined to a compact region of this space with the
L
G ≡ hj2L|Hˆ |j2Li, and completely specify the two- size of one unit in each direction. For independent vari-
L 2
body interaction while cLnαJ are interaction-independent ables GL with covariancematrix hGLGL′i=δLL′, states
coefficients. They can be expressed in terms of coeffi- are represented in an orthogonal basis. The differences
cients offractionalparentage(CFP)[7]and,assuch,are in (3) are not independent but have a covariance matrix
entirely determined by the symmetry characterof the n- of the form 1+δLL′; this leads to a representation in a
2
G
m-simplex basis (i.e., an equilateral triangle in m = 2 2
dimensions, a regular tetrahedron for m=3,...).
The following procedure can now be proposed to de-
termine the probability P for a specific state nαJ to
nαJ
become the groundstate. First, construct all points cor-
responding to the energies e . Next, build from them n=2 n=3
nαJ
G G
the largest possible convex polytope (i.e., convex poly- 4 0
hedron in m dimensions). All points (i.e. states) inside
this polytope canneverbe the groundstate forwhatever
choice of matrix elements G and thus have P = 0.
L nαJ
Finally, the probabilityPnαJ of any other state at a ver- n=5 n=6
texofthe polytopeisafunctionofsomegeometricprop-
erty at that vertex.
This general procedure can be illustrated with some
examples. Consider first a system of j = 2 d bosons.
In this case there are three interaction matrix elements,
G , G , and G with G ≡ hd2L|Hˆ |d2Li; thus, N = 3 n=8 n=50
0 2 4 L 2
and the problem can be represented in a plane. Because
of the U(5) ⊃ SO(5) ⊃ SO(3) algebraic structure, an FIG.1: Polygonscorrespondingtoseveralsystemsofinteract-
analyticsolutionisknownforninteractingdbosonswith ingdbosons. Allstatesarerepresentedbyadot. Thesmaller
eigenenergies [8, 9, 10] dots are inside or on the edge of a convex polygon (with the
larger dots as vertices and indicated in grey) and are never
2n(n−2)+2ν(ν+3)−J(J +1) the ground state. The probability of the vertex states to be
enτJ = 7n(n−1) (G2−G4) theground state is related to theangle at thevertex.
n(n+3)−ν(ν+3)
+ (G −G )+G , (4)
5n(n−1) 0 4 4
TABLEI:Theangleortheanglesum θ andthever-
where ν is the seniority quantum number which counts f∋v vf
tex probability Pv of regular polygons (m = 2) and polyhe-
the number of d bosons not in pairs coupled to angular drons(m=3). P
momentum zero. Any energy e can be represented
nνJ
asa pointinside anequilateraltrianglewithverticesG0, m θvf Pv m θvf Pv
G , and G of which the position is determined by the f∋v f∋v
2 4 X X
appropriatevaluesontheedgesG0−G4andG2−G4. Ex- triangle 2 31π 31 tetrahedron 3 π 41
amples for severalboson numbers n are shown in Fig. 1. square 2 1π 1 octahedron 3 4π 1
2 4 3 6
Forn=2therearethreestateswithL=0,2,4andener- pentagon 2 3π 1 icosahedron 3 5π 1
5 5 3 12
gies e = G0,G2,G4; clearly, they have equal probability hexagon 2 23π 61 cube 3 32π 18
ofbeingthegroundstate. Asnincreases,morestatesap- p-gon 2 p−2π 1 dodecahedron 3 9π 1
p p 5 20
pearinthetriangle. Themajorityofstates,shownasthe
smaller dots, are inside the convex polygon indicated in
grey and can never be the groundstate for whatever the
vertex to be the ground state. Table II compares prob-
choice of G . If we translate or rotate the convex poly-
L abilities calculated with the analytic relation (5) with
gon inside the triangle, its properties should not change
those obtained from numerical tests for several systems
since the points G , G , andG areequivalentand since
0 2 4 of randomly interacting d bosons with boson numbers
the distribution depends only on G2. Thus the prob-
i i n=5,6,10,18. Thenumericalprobabilitiesareobtained
ability for a point to be the ground state can only be
from 20000 runs with random interaction parameters.
P
related to the angle subtended at that vertex. The rela-
They agree with the analytic result (5).
tioncanbe formallyderivedbutalsoinferredfroma few
A second example concerns a system of four j =
simpleexamples. Ifthepolygonisanequilateraltriangle, 7 fermions which was discussed recently by Zhao and
square,regularpentagon,...eachvertexisequallyproba- 2
Arima [3]. In this case there are four interaction ma-
ble with probability 13, 41, 15,...One deduces the relation trix elements, G0, G2, G4, G6 with GL ≡hj2L|Hˆ2|j2Li,
(see the m=2 polygons in Table I)
and this leads to a problem that can be represented in
three-dimensional space. Any state in the j = 7 shell
Pv(2) = 21 − 2θπv (5) can be labeled with particle number n, seniority2ν, and
total angular momentum J with analytically known ex-
between the angle θv at the vertex v of the polygon and pansion coefficients bLnνJ [7]. For n = 4 there are eight
the probability P(2) for the state associated with that different states and the corresponding coefficients bL
v nνJ
3
TABLE II: Probabilities PnνJ (in %) of some d-boson states
obtained analytically with formula (5) and numerically with
Gaussian parameters.
Probability PnνJ Probability PnνJ
n J(ν) Analytic Gaussian n J(ν) Analytic Gaussian
5 0(3) 4.08 3.96 10 0(0) 20.87 20.77
2(1) 20.11 19.96 0(6) 0.50 0.41
2(5) 36.19 36.59 0(10) 37.56 37.96
10(5) 39.63 39.49 10(10) 41.08 40.85
6 0(0) 22.32 22.13 18 0(0) 19.89 19.87
0(6) 38.05 38.37 0(18) 37.56 38.37
12(6) 39.63 39.49 36(18) 42.06 41.75
are given in Ref. [3]. These eight states can be repre-
sented in three-dimensional space spanned by the three
differences G −G , G −G , and G −G , and six of
0 6 2 6 4 6
them define a convex polyhedron (see Fig. 2). The two
remainingpoints[correspondingtothe stateJ(ν)=4(2)
and 5(4)] are inside this polyhedron and are never the
ground state. The relation between the geometry of the
polyhedronand the probabilityP(3) of eachvertex state
v
to be the ground state can again be inferred from a few
simple examples. The relevant quantity in this case is
θ where the sum is over all faces that contain
f∋v vf
Pthe vertex v and θvf is the angle at vertex v in face f. FIG.2: Thepolyhedronforasystemofn=4j = 27 fermions.
Afewexampleswithregularpolyhedrons(seethem=3
Theupperpartindicatesthepositionofthepolyhedroninthe
polyhedrons in Table I) demonstrate that the relation is tetrahedral coordinate system. The left part shows thesame
(enlarged) polyhedron with vertices that correspond to the
P(3) = 1 − 1 θ . (6) states J(ν) that can become the ground state with a proba-
v 2 4π vf bilitygivenbytheexteriorangle. Twoadditionalstateswith
Xf∋v J(ν)=4(2)and5(4)lieinsideoronthefaceofthepolyhedron
and are never the ground state. For representation purposes
Table III compares the probabilities for the different
the polyhedral face 0(0)–2(4)–8(2) has been removed in the
states to become the ground state as calculated in var-
left part.
ious approaches. The second column gives the analytic
resultsobtainedfrom(6)whilethethirdcolumnlistsnu-
merical results obtained from 20000 runs with random
matrix elements with a Gaussian distribution. As our TABLE III: Probabilities PnνJ (in %) of n= 4 j = 27 states
code does not distinguish between states with the same obtainedwiththeanalyticformula(6),fromanumericalcal-
culation, and from the integral representation of Zhao and
angular momentum J but different seniority ν, only the
Arima [3].
summedprobabilityforeachJ isgiven. The lastcolumn
shows the results of Zhao and Arima [3] who calculate Probability PnνJ
the probability as a multiple integral.
J(ν) Analytic Numerical Integral
These notions canbe generalizedin severalways. The
0(0) 18.33 18.38 18.19
first is towards energies that depend on a set of con-
2(2) 1.06 — 0.89
tinuous variables {t ,...,t } as follows [compare with
1 q 2(4) 33.22 — 33.25
Eq. (2)]:
2(2&4) 34.28 34.88 34.14
4(2) 0 — 0.00
E(t ,...,t )= b (t ,...,t )G . (7)
1 q L 1 q L
4(4) 23.17 — 22.96
L
X
4(2&4) 23.17 22.83 22.96
The analogous problem now consists in the determina-
5(4) 0 0.00 0.00
tion of the probability density dP(t ,...,t ) for obtain-
1 q 6(2) 0.05 0.07 0.02
ing the lowest energy at {t ,...,t } with random inter-
1 q 8(2) 24.16 23.83 24.15
actions G . We assume by way of example that the
L
4
number of variables q is one less than the number N tests can be understood in this context as sampling ex-
of interactions G , q = N −1. In that case Eq. (7) rep- perimentsonthisgeometry. InthisLetterwehaveshown
L
resents a q-dimensional hypersurface Σq embedded in a thatgeometricaspectsofn-bodysystemsdeterminesome
(q+1)-dimensionalEuclideanspaceEq+1 (themetricfol- oftheir essentialcharacteristics. Inparticular,for diago-
lows from the covariance matrix hGLGL′i = δLL′). Let nalHamiltonianssurfacecurvaturedefinesprobabilityto
us suppose that Σq is the closed orientable manifold. It be the groundstate. Other correlationscould alsobe re-
can then be shown that the probability density is given lated to geometric features. These results generalizeand
by Gauss’ spherical map [11] Σq → Sq where Sq is a q- putontoafirmbasisthepreviousworkwhichhintedata
dimensional hypersphere of a unit radius. If the degree purely geometric interpretation of randomly interacting
of the spherical map is one (as it is for closed convex boson systems [5, 6], as well as provide an explanation
surfaces), the probability density reads for the method of Zhao and collaborators [3]. In fact,
in the latter reference, the authors have advanced some
1 qualitative reasons for certain states to dominate and
dP(t ,...,t )= K (t ,...,t )dv, (8)
1 q Sq q 1 q later provided an approximate procedure (which is not
always accurate [14]) to estimate the ground state prob-
where K is the Gaussian curvature of Σq, S is the vol-
q q abilities, although no reason was offered for its success.
ume of the unit hypersphere Sq and dv denotes an in-
Our study, at least for the case where the Hamiltonian
finitesimal element of Σq. In the simplest example of
is diagonal, clearly shows that the procedure of Zhao et
one parameter t ≡ t and two interactions G and G ,
1 1 2 al is equivalent to a projection of the considered above
the energy is parametrized as a curve in a plane. It can
polyhedra on the axes defined by the two-body matrix
then be shown that the probability density is given by
elements,whichtendtocorrelatewellwiththeangleswe
dP(t)=(2π)−1K (t)ds, where ds is the infinitesimal arc
1 introduce. This connection will be elaborated in detail
length. In fact, this formula is also valid for piecewise
elsewhere [15]. Further work is required to fully explore
curvesandpreciselyleadstotheresult(5)forapolygon.
thegeometryanditsconsequencesforourunderstanding
The validity of the result can be checked by comparing of n-body dynamics.
theprobabilityobtainedbyintegrationof(8)overapart
of Σq to the numerically calculated one. As an exam- Acknowledgments: AF is supported by CONACyT,
plewediscussatwo-dimensionalellipsoidinE3 withone Mexico,underproject32397-E.NASthanksL.Vanhecke
semi-axis c different from the others a. The probability for a helpful discussion.
associatedwiththe partofthe surfacewithsphericalco-
ordinates (θ,φ) satisfying 0 ≤ θ ≤ θ and 0 ≤ φ ≤ φ is
0 0
given by
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θ acosφ
P(θ ,φ )= 0 1− 0 . (9) Lett. 80, 2749 (1998);
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[2] R. Bijker, A. Frank, and S. Pittel, Phys. Rev. C 60,
p
Thisexpressionhasbeencomparedwiththenumerically 021302 (1999);
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072502 (2002).
have analyzed likewise the case of a three-dimensional
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(2001).
generalizedtohigherdimensions. Theseresultsalsoopen
[4] Y.M. Zhao, A. Arima, and N. Yoshinaga, Phys. Rev. C
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with an appropriate characteristic for each vertex of the Lett. 85, 4016 (2000).
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v 11, 20 (2001).
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(1976).
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5
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