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Energy dependence of determinantal measures
Nuclear Physics A363 ( 1981) 189-204 @ North-Holland Publishing Company
ENERGY DEPENDENCE
OF DETERMINANTAL
MEASURES
I. KELSON Department of Physics and Astronomy, Tel Aviv University, Tel Aviv, krael Received 30 July 1979 (Revised 18 December 1980) Abstract:
The concept of determinantal measures, in the context of a generalized Hartree-Fock approximation, is reviewed. The energy dependence of determinantal measures in small model spaces is quantitatively investigated. It is shown that the few-determinant approximation is much better at tow excitation energies than for arbitrary states. A pamme~~tion of the dete~~na~tal measure versus energy graph is constructed, which demonstrates thts enhancement effect, and provides indications for possible extrapolation to larger spaces.
1. Introduction The use of independent (dete~inantal) many-particle states is very widespread in all branches of many-body physics. This is so for two reasons, which are of a fundamentally different nature. The first reason is a physical one - some systems can indeed by approximated by such simple wave functions. This is so either because of the properties of the many-body hamiltonian, or due to a preculiarity in the system. The second reason is mathematical-technical - the independent particle basis is eminently suited for expanding and describing complex systems. The use of a single independent-particle wave function as an approximation to the ground state is seldom physically justified. Rather, it serves as a starting point from which better approximations can be obtained through a variety of manipulations or generalizations. In molecular physics, for example, one constructs trial wave functions which are composed of many-body independent-particle wave functions - different con~gurations in one single-particle representation ‘). In nuclear physics, for example, one obtains eigenstates by projecting components of angular momentum out of the Hartree-Fock state ‘). One can, in fact, view the Hartree-Fock approximation as the first (and simplest) of a hierarchy of approx~ations. In this approach one describes the many-body state as a linear combination of some determinantal wave functions, each one defined in its own single-particle basis. The number of such components is variable, and determines the order of the approximation in the hierarchy. Some of the problems encountered in using this framework, have been dealt with in previous publications 3- 5,. The purpose of this paper is twofold. First, to review some of the relevant concepts which have been introduced. Second, to in189
190
I. Kelson 1 Determinantal
measures
vestigate in particular the dependence of these approximations on excitation energy. Sect. 2 provides a general review of definitions and fundamental considerations. Although most of the material contained in this section is not new, we believe such a review to be both useful and essential. In sect. 3 we present numerical results for small model spaces, where complete calculations can be carried out. We also present the methods employed in obtaining those results. It is interesting to try and parametrize the various results, both because a convenient mode of presentation is thus obtained, and because extrapolations to higher dimensional spaces may be indicated. This is done in sect. 4, where a number of speculative statements are also made and discussed. Finally, a concise summary is given in sect. 5, along with indications of planned future research.
2. Basic concepts We consider a system of k fermions, which are confined to move in a set of N single-particle states. It is important to note that the identity of the fermions is quite immaterial to the presentation which follows. Clearly, however, we deal with a finite system. The fact that we use discrete single-particle states, implies that we deal with a finite size system (although one could envisage an approximation to an infinite system - this is beyond the scope of this work). The number, N, of the singleparticle states, and their properties, represents a specific way of truncating the many-body space, and is model dependent. Again, there are many interesting questions that can be raised regarding this truncation procedure, but most of them are outside the framework of this discussion. Our starting point is the truncated space, along with an (effective) hamiltonian defined in it. Thus, our analysis transcends the specific properties and peculiarities of any special finite many-body system. Rather, its purpose is to investigate the mathematical milieu common to them all. The set of N orthonormal single-particle wave functions, ‘pi = Ii) - atlo),
i = 1,. . ., N,
(1)
forms the basis of a complex Hilbert space, which we designate &‘(N, 1). The multiparticle space, is an antisymmetrized direct product of k identical single-particle spaces, d{X(N, 1) x X(N, 1). . . x X(N, l)>, and is a complex Hilbert space, X(N, k). As its orthonormal basis, we may take the set of (z) wave functions ‘il,....ik
E
d{~pi,(l).
. . Cpik(kl}.
(2)
We now define S(N, k) as the unit sphere in #(N, k); all the spaces with which we deal from now on are subspaces of S(N, k) and thus contain normalized wave functions only. Any wave function + in S (the symbols N and k will henceforth be omitted
where they are self-understood)
can be expanded in this basis
with
We now define a series, or a hierarchy, of subspaces Sg”’of S. For the sake of clarity, we begin by defining S6”; this subspace contains all states which can be expressed as a pure single-particle determinant in some representation. Namely, Sg) contains all those (normalized) states for which a representation exists in which only one of the coefficients &, . ., ir is God-v~ishing. Clearly, the property of being a pure determinant cannot always be directly inferred from an observation of the specific representation (3) of the state; we shall not discuss here the procedure involved in this determination 3). The subspace S$) comprises all states in S which can be expressed as a linear combination of II, or less, states belonging to Sg’. It is impo~ant to emphasize that these n states are determinantal wave functions in different, unrelated single-particle representations [cf., e.g. ref. 6)]. It is also worth noting that these are not linear spaces. We have, obviously, constructed a series of spaces, which, for some YE@, becomes identical to S: sg” c S(,z)c . . * c St) 55 s
(5)
To find what no is for given N and K, we count the number of inde~nde~t real parameters which are associated with each of the spaces mentioned above. We shall designate this number as dim(space). Clearly, dim(~(N,
k)) = 2
,
N k
dim(S) = 2
- 1.
0 To evaluate dim (S’,“‘),we have to count the number of independent real parameters needed to specify uniquely a given single-particle determinant. We begin with the Nk complex (or 2Nk real) expansion coefficients of the k states in the N wave function basis. From this we have to subtract the number of restrictions imposed by the orthonormality requirements. There are k conditions of no~alization (the lengths being perforce real numbers) and 2(z) conditions of orthogonality. Since the dete~inant is invariant under an arbitrary unitary transformation of the k single-partible states, we have to further subtract k’, which is the number of real parameters defining such a transfo~ation. We note that the multiplication of a given state by an arbitrary phase is included as such a unitary tr~sfo~ation. Thus, dim(S#
= 2Nk-k-2
k -k2 = 2k(N-k). 02
192
I. Kelson / Dererminantal measures
For Sg) we simply take n unrelated determinants (the condition that no two of them are equal does not reduce the dimensionality), associate with each one complex (i.e., two real) coefficient, and subtract one for over-all normalization. Hence dim($))
= ndim(Sg))+2n-1
= Znk(N-k)+2n-1.
(8)
The number n, is simply the smallest integer satisfying dim (Sg)) 1 dim (S) which is
(9)
where the brackets stand for the largest integer smaller than the enclosed expression. We would like to add the following two remarks. First, the dimensionality of SF’ can be identified as the number of independent infinitesimal variations which conserve the determinantal nature of a state. This is just the number of different particlehole excitations. Second, we may repeat the entire discussion and analysis for real Hilbert spaces, and corresponding real subspaces. We would then have dim (real S) = (f)- 1; dim (real SE’) = nk(~ -k) + n - 1; but n, is still given by eq. (9). We now turn to the central concept of dete~inantal measures, which is related to the extent to which an arbitrary state in S can be approximated by a state in Sl;’ (1 g n 5 no). We de&e a sequence of functionals D’“’ (n = 1,. . ., no) on S by @%V = rn;;) l(@l+>l.
(10)
II will be referred to as the “‘determinantal measure of order n”, or “n-determinantal measure”, and the state @, belonging to SK), for which the maximum is obtained will be called the “n-determinant projection of the state +“. A number of simple properties of the determinantal measures can be trivially inferred. For example, for any Y D@+l)(iy) 2 D’“‘(y3,
(11)
D(“O)( Y) = 11
(12)
l(“)(Y)
- -L Jno’ 2
(13)
Procedures of a rather elaborate nature are required, when space averages of determinantal measures are to be calculated 4). We have, thus far, considered only general properties of the multiparticle space (and its hierarchy of subspaces), which were related to structural, morphological and combinatorial aspects. We now turn our attention to the role played in this connection, by various operators defined over the space S, and in particular by the hamiltonian H. The primary effect of the hamiltonian is that each (no~alized)
I. Kelson / Determinantal measures
193
state in S has a numerical function associated with it, namely, the expectation value of the hamiltonian E(Y) = (YMYu>
(14)
Barring pathological singularities in the hamiltonian, this function, E(Y), is bounded both from below and from above by Emin and E,,,, respectively. That it is bounded from below is a natural property of a physical system, such as we purport to describe by our model. The fact that it is also bounded from above is a consequence of the truncation which we have, in effect, carried out when going from the infinitely dimensional Hilbert space to our finite dimensional model space. Presumably, by properly selecting a set of ever-increasing model spaces, the true situation can be successively better approximated. For any particular model space, however, such as we confine ourselves to, it is thus only natural that we concentrate our attention on the lower part of the energy spectrum. Defining a median (which we assume to be identical to, and use interchangeably with the mean) energy E, we shall in any case discuss only the region [Emi,, ET ; the region [E, E,,,] is essentially symmetric to it, but this symmetry is obviously spurious. For any range of energies [E,, E,], such that Emin s E, 2 E, 5 E,,,, we define a subspace S(E,, E,) which contains all states Y, YE S(E,, E2)
if
E, 2 E(Y) 5 E,.
(15)
This is a multiply contiguous subspace, of the same dimensionality as the entire space itself, which, using this nomenclature is simply S(Emi,, E,,). It is possible in principle, though not generally in practice, to calculate the “area” a(E,, E2) inscribed in this hypersurface. By allowing E, to approach E,, we obtain the density of states at a given energy p(E) = C,l;yo &
o(E, E+dE).
(16)
Since only averages are ever calculated, the constant C is arbitrary, and can be adjusted at will. One possible choice, for example, requires that upon integration E
max
p(E)dE = dim (S).
(17)
JEmin
This density is a measure of the locus of all points in S with a given E($), which is a subspace we designate S(E). Since the expectation value of H is always a real number, restricting it to have a specific value implies that S(E) is one dimension less than S, dim (S(E)) = dim (S)- 1 = 2(T)- 2. It is worthwhile mentioning that p(E), as defined above, does not coincide with the density of eigenstates of the hamiltonian H at that energy. A quantity of particular interest is the average value of the determinantal measures at a given energy, which can be calculated once a well defined weight is assigned in the space. In fact, it is this quantity which is the subject of this paper.
Before we continue to the next section, we would like to stress and clarify our notation: D(“){Y) stands for the n-determinantal measure of a particular state $+ D@‘(E)stands for the average of the ~-det~rmina~tal measure over all states (puints in 5) with given E. This definition extends to the dependence on any variable other than the energy, suitably defined on S. F stands for the n-determinantal average ever the entire space. In all cases the dependence on iV and k is implicit.
In this section we present results of the energy dependence sfdeterminantal measure averages in small model spaces, The spaces are small enough so that such results can be directly evaluated with a high degree of statistical confidence, and for a variety of hamilt~nians~ We begin by introducing a simplifying feature into our numerical procedure. Namely, we consider the spaces defined over the real numbers (rather than complex ones). There is no manifest reasun why this limitation should affect the values of averages such as we compute; but, on the other hand, we have no formal proof that it does not, and we must leave it as a working hypothesis. All normalized states in our vector space, thus form a unit sphere in a (:I dimensional Cartesian space, with the function E(M) defined over this hypersurface. The density of states in this vector space is taken ta be directly proportional to the surface area of this hypersphere, In the absence of direct anaIyticaf methods for evaluating averages of complicated f~~ti~na~s~ we need a procedure fctr selecting random points on this unit sphere. This is done with the following sampling procedure, First, a set of v f zz {f) - 1) real numbers yl, yz, . . ., y, are selected, the first V- I randomly picked in the range [O, n] and the last in the range [O, 2123,A new set of values ci, t2,. . ., (, is obtained by the fallowing relationships
where F$
is the inverse of the function sin”‘“+ tdt .
L,,1@) = sin”-‘-’ tdt
Go
I. Kelson / Determinantal measures
195
Now the Cartesian coordinates X1, X,, . . ., X,, 1 are determined through XL
= cos 51,
X*
= sin {I cos c2,
X”
=sin51sin52...sin5,_1cos5,,
X v+1
=
sin
Obviously, this produces a randomly distributed set of points for any orthonormal basis that one works with. Thus, each selection may be extended to a multiple set of points in a given basis by performing any arbitrary permutation of the {Xi} among themselves. Once a wave function has been selected, its determinantal measure (of any order) is evaluated, as well as E(Y), the expectation value of the hamiltonian. Space averages can be computed with a relatively small number of points. This is so because the values of the determinantal measures are strongly peaked around their mean values. However, since the density of states is also strongly peaked around the median energy, the simple procedure described above is highly inefficient in selecting states with energies far from the mean and close to the ground state. To overcome this difficulty we have to introduce a strong bias for selecting states of low energy. Such a bias can be introduced in a number of ways, which do not have any special technical significance. The basic approach is to use as an orthonormal basis the eigenstates of the hamiltonian defined in our space
HYY,= E,Y,,
u =
l,...,
(22)
with E, arranged in ascending order. Each X, is multiplied by a different factor, with the factors decreasing monotonically with u, and the set (X,} subsequently normalized. The statistical behaviour near the point E,i,, is determined utilizing the dominance of the (non-degenerate) ground-state component Yr. A set of states are chosen of the form Y = Y,cos8+Y,,sin0,
(23)
where cos 0 is varied in small, fixed increments and Yy,, is chosen randomly from the Cartesian [(F) - l] dimensional unit hypersphere, conscribed by the states Y,, . . ., Y (3. This choice follows again the procedure described above. The average determinantal measure of order n at a given energy E was obtained by averaging over the numerical values obtained within an energy range LIE around E; typical values used where LIE/(&,,, - I&) = 0.03. The only apparent exception was the average at the ground-state energy itself, which, in the absence of degeneracy, was directly obtained.
196
I. Kelson / Determinantal measures
In all spaces, characterized by N and k, the hamiltonians with which we performed the calculations were a combination of a diagonal one-body force of evenly spaced single-particle energies and a randomly constructed, uncorrelated two-body interaction. Thus,
with t, = Lx&,
(25)
and the %t)[(T)+ l] independent two-body matrix elements of V were randomly selected real numbers from the range [ - 1; + 11. The variable parameter E determines, apart from an irrelevant energy scaling, the relative importance of the onebody and two-body parts of the hamiltonian. For E = 0 we have complete zeroorder degeneracy of the particles, which is only removed by their interaction; for E B 1 we have the case of completely independent single-particle motion. The even spacing of the single-particle energies is arbitrary. We shall not treat here the case where partial degeneracies do occur. The complete independence of the twobody matrix elements equally results from the fact that no internal symmetries among the particles are assumed.
. 0
I 2 3 4 5 6 7 8 9 IO11 12 E
Fig. 1. Complete determinantal measure versus energy graph for the space N = 8, k = 4, with degenerate to single-particle energies (i.e., E = 0). Since for this space no = 5, there are curves corresponding n = 1,2, 3,4. For n = 1 and n = 2 the determinantal measures of the eigenstates of the hamiltonian are also plotted as a series of points. The lowermost scale gives the excitation energy above I?,,( = I?,,,,,) of the eigenstate, and the positions of the eigenenergies are marked on the scale above it.
In fig. 1 we present one complete graph for all determinantal measures with in a particular space as a function of energy. The space has N (= 8) states and k (= 4) particles, and there is no single-particle splitting (E = 0). The energy extends from Emin to E, and the eigenvalues of the hamiltonian are marked on the energy scale. The points correspond to the determinantal measures of these eigenn < no,
I. Kelson / Determinantal
197
measures
states; here, of course, the values are exact. The smooth curves correspond to the appropriate space averages and do involve a certain degree of uncertainty, stemming inherently from the special procedure which we have described above. These curves (as well as the general trend of the points) exhibit what is the major feature in which we are interested. Namely, the fact that at low excitations the determinantal measures are considerably larger than for average states (or for states with average energy). This is all the more significant in this case, because of the complete absence of a one-body force, which is a natural conducive agent to independent particle motion. Moreover, this enhancement is in fact more significant when we recall that the determinantal measures give amplitudes (or overlaps), while it is probabilities (or amplitude squared) which are quantitatively important in deciding the quality of the relevant approximations. To exhibit the effect of a one-body ‘force, we append to the same interaction in the same space, a single-particle spectrum with E = 1. The corresponding determinantal measure (of order n = 1 only) is shown in fig. 2.
.& ’ ’ “I ’ “““’ “’ “‘I ?I
I’ll II
IIW
P F
,
23456769D11121314 E
Fig. 2. The n = 1 determinantaf measure for the space N = 8, k = 4, where a single-particle splitting with E = 1 is added to the hamiltonian used in fig. 1. The same format is used for this figure in presenting the results.
Comparing this curve with the n = 1 curve of fig. 1, demonstrates that while Df’) at E is practically unchanged, the enhancement of Do’ near Emin (SE E,) is much stronger when a single-particle force is included. To show that this is indeed a consequence of the presence of the one-body force, rather than a peculiarity of the two-body interaction, we did the following. A sequence of ten different two-body interactions were selected (all according to the same procedure), and the corresponding average determinantal measures were computed, all for E = 0. The particular one for which D”‘(E,) was largest is shown in fig. 3. The value D”‘(E,) = 0.90 is considerably smaller than the one shown in fig. 2 for E = 1, while the behaviour at E is, again, practically identical. These numerical analyses were performed for spaces with (k = 2; N = 6,8,10, 12,14,16,18,20), (k = 3; N = 6,8,10,12,14) and (k = 4; N = 8,10,12). In the
198
I. Kefson / De~erm~an~a~ measures
0
I234567891olll2 E
Fig. 3. The n = I dete~inanta~ measure in the space N = 8, k = 4, with no single-particle splitting (E = 0), again using the same format as tigs. 1 and 2. Here the two-body interaction was varied over a set of ten interactions, maximizing the value of DC’)(&).
larger spaces, typically with (i) > 100, a complete statistical analysis was not performed for practical reasons, so that values of energy dependent averages were not obtained with the desirable small errors. It is, partially, for this reason that we find it advantageous to construct a parametric representation to describe the behaviour of graphs such as fig. 1. The determination of a small number of parameters can be done much more reliably than the computation of a complete energy dependent set of curves. Moreover, by using appropriate parameters, we may gain useful insight into the behaviour of the quantities involved, and perhaps be able to extrapolate them to higher spaces. This is done in the next section, where the rest of the numerical results are presented in a parametric format.
4. Parametric representation of determinantal measures Before proceeding with a step-by-step description of the parametrization of the determinantal measure versus energy graph, one general remark is in order. The subject matter of the present discussion is a particular approximation and its energy dependence. Being over demanding when dealing with an approximation may be counterproductive in the sense that the physical, intuitive significance of the approximation may be obscured by exacting, mathematical details which are not inherent to the problem. Thus, we follow the principle that simplifying procedures are permissible when they reproduce the essential features of the approach, at the expense of strict formal rigour. (i) We begin by employing in ill cases the dimensionless variable E-E. Cl=;* E - Emin
(26)
This variable goes from 0 to 1 over the region of interest, and is highly suitable for
199
1. Keison / Determinantal measures
a treatment which seeks to unify all relevant systems, irrespective of their specific energy scale or structural complexity. (ii) As is apparent from fig. 1, the determinantal measure versus energy graph consists of a series of curves, D”‘(or),JY2)(cc),. . ., D(“$x), the last of which being identically unity. Apparently, there is a progressive relationship within these series of curves, which stems qualitatively from the basic properties of determinantal measures. Although this is by no means a universal relationship, we shall assume that the ED; n 2 2) are determined uniquely by the function D”‘(a). Namely, that all higher order determinantal measure averages as a function of energy are known, once the simple determinantal measure average is known. Thus, it is sufficient to evaluate and parametrize the lowest of all the curves in the hierarchy, and to get the rest of them by an appropriate recursion relation. To construct explicitly an approximate recursion relation, we proceed as follows. Consider a state I/J,whose n-determinantal measure is D”“($). Thus, we may write (27) where #,, (E S’,“‘)is the so-called n-determinantal remainder state satisfies
projection of il/, and the normalized
Similarly, of course, we could write the corresponding, in terms of its (n+ 1) determinantal projection lj
=
D’““‘(l&$“+
I+
1 1 -lP+
analogous expansion of I,& 2
W)
Xn+
1’
(29)
Eqs. (27) and (29) involve the solution of distinct variational problems, and cannot be easily related to each other rigorously. We shall, however, approximate #n+l through a sequential expansion, starting from the expansion (27). Namely, let us write xn in terms of its one-determinantal projection 4p1(and a remainder state x’) (30) and substitute it in eq. (27). $ = D’“‘(li/)#,+jl
-zP’($)zD”‘(x,)$J1
+J’.
(31)
The determinant 41 is linearly independent of the n determinants into which 4, may be decomposed, or else (p,, would not be - by definition - the n-determinantal projection of +, satisfying condition (10). Hence the first two terms in the progressive expansion of +, are proportional to a state in Sg+ l). Our procedure is based on the assumption that this state is approximately equal to (6, + 1_All the above discussion refers to one particular state rl/. If we were to consider an “average” state $, or alternativaly, average over the entire space S, then we should, obviously, replace ~(“)(~) by p and Dt”+ i)($4 by r. Furthermore, we shall also assume that we may approximately replace D(‘)(x,) by t)m; namely, that the space average of the
200
I. Kelson / Determinantal measures
determinantal measure of an “average” remnant state is the same as that of an average arbitrary state. This substitution will clearly underestimate the average of D(‘)(x,), particularly for larger n. For example, if rz = ~~ - 1, then by definition D(‘)(x,,_ J = 1 ! Nevertheless, adopting these assumptions results in a very simple recursion relation, which yields a good approximation for n < n,, and always gives a lower bound on D@).It reads, J_ Or, defining the remainder p
= J_&D”‘z.
(32)
through @$‘=
l_FZ,
(33)
we can write R(“+l) = pp.
(34)
R(.+m)=pp.
(35)
This can be easily generalized to
Clearly, this is only an approximation, since we have treated successive approximations as independent. Having found a particular higher order determinantal projection satisfying (31), implies that in reality we shall have R(n+d
__~ 5
R(n) R(m).
We now further assume that this relationship is a valid approximation energy E, or equivalently a, R(“+“‘)(a) = R(“)(~1)R(~)(tl),
(36) for any given (37)
where we have followed the same notation for R’“’ as for DC”). Thus, through this ansatz, the behaviour of all curves is determined by the n = 1 curve. A seeming shortcoming of this approximation is that R”“(cc)becomes identically zero only for n = co, while in fact, it should do so at n = n,. But, in any event we are more interested in the structure of the curves for n < n,. (iii) We assume that at the mean (which we take as identical with median) energy E, the average value of D(r) is approximately the entire space average of D(l). This is generally justified in view of the high concentration of states around the mean. Thus D”‘(Cr = 1) Z Err.
(38)
The particular value p, which is clearly independent of the hamiltonian, could, in principle, be taken as one of the parameters required for the quantitative representation of the graph. Relying, however, on numerical results, we may attempt and obtain it in closed form. To do so we plot in fig. 4 the values of a large number of space averages of n-determinantal measures, as a function of the variable n/n,,
I. Kelson / Determinantal measures
201
a 6 5 4 3 .2 _I I ‘0
11
I
I
I1
I
II
I
I .2 .3 .4 .5 .6 7 .8 .9 ID
Fig. 4. Values of space averaged determinantal measures of order n, plotted against the parameter n/n,,. The continuous curve is the functional relation (n/no)0~s9+(l - Dc”~)o~59= 1.
where n, is given by eq. (9). The points correspond to these averages, and there is clearly a strong correlation between the two variables. The smooth curve is an attempt to lit the points with a function of the form
0; C+(l-Fy=
1,
with the numerical value p = 0.59( kO.02).
(40)
The fact that TD” is expressed as an explicit function of N, k is, of course, very useful. However, one must remember when considering this for extrapolation to higher spaces, that on fig. 4 the entire extrapolation region is compressed into a small area in the vicinity of the origin. One point, if only marginal, in favor of an expression such as eq. (39), is that it can be shown that for /3 > 0.5, and for sufficiently large spaces, the inequality of eq. (36) is indeed rigorously satisfied. (iv) We further notice that D(‘)(U) is a monotonically decreasing function of CY, with the enhancement over D”‘(1) fastly decreasing from its maximum at a = 0 (namely, low excitation energy). We parametrize this typical behaviour in terms of the two parameters D(‘)(O) and the exponent of decay y, so that D(‘)(a) -I);m #“(O)__p
-a/r =
e
.
(41)
It is easy to notice that this can potentially provide a better fit to points in the neighbourhood of the ground state (a = 0), which is indeed the region of interest. The entire space average F, when recalculated from the curve D”‘(a), will become, as a consequence of our formal expression, somewhat larger than its actual value. Only for large N and k, where y becomes smaller and the density of states more concentrated at E, will @n be self-consistently calculable from D”‘(a). This, again, is in accordance with the remark with which we opened this section.
I. Kelson / Determinantal measures
202
To sum up, we have constructed a parametric representation of the determinantal measure versus energy graph, which utilizes two parameters, D”‘(O) and y, and which is, explicitly, the following. The value of D(“)(a) for the space S(N, k), is given by D’“‘(a) = ,/l - {1 - [D’D(a)]2)n,
(42)
where D(“(cc) = p+
{D(l)(0)-_je-a’Y,
iP = l- {I-
(43)
n,(;,k)8jl’a.
(44)
We recall for clarity, the relevant equalities and definitions which have appeared above, a _
[es.W91,
E_-Emin
lZ-
Emin
N
0 ~-----+l k(N-k)+l k
n, =
/3 = 0.59kO.02
1 [es. (9)1,
[eq.
WN.
Using this representation, the results for spaces S(N, k), for hamiltonians with E = 0, 1,2, are summarized and presented in fig. 5. Perhaps the most striking feature
0
-
(6*3k-x+
E=Q
.4 -
Y .3 (12,410
U2,3,0~d
9’ Ol4,3) -
Fig. 5. A representation of a variety of graphs by means of the two parameters y, D”‘(O). All the results utilize the same two-body, randomly chosen interaction. Three possible values for the single-particle splittings are used for each space N, k-8 = 0, I, 2. The curves connecting points belonging to the same space are for clarity only. Note the very weak dependence of y on E within each space.
203
I. Kelson / Determinantal measures
of this figure is that the introduction of a strong single-particle force increases the value of D(‘)(O), but leaves almost unchanged the value of y, namely the fractional energy range over which the enhancement of the determinantal measure persists. From this we may be led to the conjecture that y depends (approximately) only on the space (N and k) and not on the hamiltonian. What could this dependence be, and how reliable is this conjecture for extrapolating to higher spaces? In fig. 6, we
01 0
I
1
I
I
2
3
I
4 I”($)
t
I
1
5
6
7
Fig. 6. Values of I/y as a function of In(:), indicative of a possible approximate linear relationship between the two. The dashed curve, drawn to guide the eye, satisfies 1/y = 0.4+0.6 In(i),
have plotted, for all available results, l/y as a function of In(T), where l/r is an average value of the results obtained in the corresponding space for the different hamiltonians. It seems that l/y is roughly linear in In(f) ; we are in no position to determine the reliability of this statement at this stage. The parameter D”‘(O), which is obviously strongly hamiltonian dependent is even less amenable to extrapolation to higher spaces. We shall only make at this stage the following, very general conjecture: It is possible, by studying the behaviour of D”‘(O) in small spaces, to extrapolate its behaviour to larger spaces, for hamiltonians with the same characteristics +.
5. Summary and conclusions
In finite many-fermion spaces (k particles in N single-particle states) we have considered a hierarchy of subspaces, characterized by n, the minimal number of determinantal states (in unrelated representations) needed to describe a state in them. This gives rise, also, to a hierarchy of approximations. One is a direct generalization of the Hartree-Fock approach, attempting to minimize the expectation value of the hamiltonian not in the lowest, but in progressively higher spaces up the hierarchy. The other approach centers around the concept of determinantal measures, which ’ The major characteristic, m this context, is the relative strengths of the one- and two-body parts in the hamiltonian.
I. Kelson / Determinantal measures
204
involves maximizing overlaps with a given state. Since, in general. the state which is to be approximated is not known, what we mainly investigate is the general statistical behaviour of this quantity and its dependence on various characteristics and parameters. In particular, by concentrating on states with a given expectation value of the energy, we constructed a graph which gave the average determinantal measure of any order n, as a function of excitation energy. This was cast in a universal, dimensionless format. The central feature of this graph, which stood quantitatively out in numerical model calculations, was the enhancement of the determinantal measures at low energies. There are a number of applications for a graph like this in given systems, once it is even approximately known. For example, if one demands a particular minimum overlap of an approximate state with the ground state, then the required number of determinants (namely, the value of n) can be read off the graph. Alternatively, if the number n is given, the expected quality of the fit can be easily determined, along with its dependence on excitation energy. We have introduced a special representation, using two parameters only, to describe this graph for a given hamiltonian in a given space. There is a large degree of arbitrariness in our representation, but the two parameters, - apart from being few in number, and hence convenient to use - are related to the two central, meaningful features of the graph. One is D”‘(O), the single determinantal measure at the ground state, and the other is y, related to the fraction of energy over which the “decays” to its mean value. enhancement of D (I) We have attempted to indicate a possible extrapolation of y to higher spaces. However, a reliable quantitative determination of D"'(O) and y for higher spaces calls for further theoretical study.
References 1) See, e.g., D. N. Nanda and P. T. Narasimham, 2) 3) 4) 5) 6)
R. E. Peierls and J. Yoccoz, I. Kelson and G. Shadmon, I. Kelson and G. Shadmon, G. Shadmon and I. Kelson, A. Faessler, A. Plastino and
Proc. Phys. Sot. Ann. of Phys. 63 J. of Math. Phys. Nucl. Phys. A241 K. W. Schmidt,
Int. J. Quantum Chem. 6 (1972) 819 A70 (1957) 381 (1971) 497 16 (1975) 2224 (1975) 407 Phys. Rev. Lett. 834 (1971) 31
ENERGY DEPENDENCE
OF DETERMINANTAL
MEASURES
I. KELSON Department of Physics and Astronomy, Tel Aviv University, Tel Aviv, krael Received 30 July 1979 (Revised 18 December 1980) Abstract:
The concept of determinantal measures, in the context of a generalized Hartree-Fock approximation, is reviewed. The energy dependence of determinantal measures in small model spaces is quantitatively investigated. It is shown that the few-determinant approximation is much better at tow excitation energies than for arbitrary states. A pamme~~tion of the dete~~na~tal measure versus energy graph is constructed, which demonstrates thts enhancement effect, and provides indications for possible extrapolation to larger spaces.
1. Introduction The use of independent (dete~inantal) many-particle states is very widespread in all branches of many-body physics. This is so for two reasons, which are of a fundamentally different nature. The first reason is a physical one - some systems can indeed by approximated by such simple wave functions. This is so either because of the properties of the many-body hamiltonian, or due to a preculiarity in the system. The second reason is mathematical-technical - the independent particle basis is eminently suited for expanding and describing complex systems. The use of a single independent-particle wave function as an approximation to the ground state is seldom physically justified. Rather, it serves as a starting point from which better approximations can be obtained through a variety of manipulations or generalizations. In molecular physics, for example, one constructs trial wave functions which are composed of many-body independent-particle wave functions - different con~gurations in one single-particle representation ‘). In nuclear physics, for example, one obtains eigenstates by projecting components of angular momentum out of the Hartree-Fock state ‘). One can, in fact, view the Hartree-Fock approximation as the first (and simplest) of a hierarchy of approx~ations. In this approach one describes the many-body state as a linear combination of some determinantal wave functions, each one defined in its own single-particle basis. The number of such components is variable, and determines the order of the approximation in the hierarchy. Some of the problems encountered in using this framework, have been dealt with in previous publications 3- 5,. The purpose of this paper is twofold. First, to review some of the relevant concepts which have been introduced. Second, to in189
190
I. Kelson 1 Determinantal
measures
vestigate in particular the dependence of these approximations on excitation energy. Sect. 2 provides a general review of definitions and fundamental considerations. Although most of the material contained in this section is not new, we believe such a review to be both useful and essential. In sect. 3 we present numerical results for small model spaces, where complete calculations can be carried out. We also present the methods employed in obtaining those results. It is interesting to try and parametrize the various results, both because a convenient mode of presentation is thus obtained, and because extrapolations to higher dimensional spaces may be indicated. This is done in sect. 4, where a number of speculative statements are also made and discussed. Finally, a concise summary is given in sect. 5, along with indications of planned future research.
2. Basic concepts We consider a system of k fermions, which are confined to move in a set of N single-particle states. It is important to note that the identity of the fermions is quite immaterial to the presentation which follows. Clearly, however, we deal with a finite system. The fact that we use discrete single-particle states, implies that we deal with a finite size system (although one could envisage an approximation to an infinite system - this is beyond the scope of this work). The number, N, of the singleparticle states, and their properties, represents a specific way of truncating the many-body space, and is model dependent. Again, there are many interesting questions that can be raised regarding this truncation procedure, but most of them are outside the framework of this discussion. Our starting point is the truncated space, along with an (effective) hamiltonian defined in it. Thus, our analysis transcends the specific properties and peculiarities of any special finite many-body system. Rather, its purpose is to investigate the mathematical milieu common to them all. The set of N orthonormal single-particle wave functions, ‘pi = Ii) - atlo),
i = 1,. . ., N,
(1)
forms the basis of a complex Hilbert space, which we designate &‘(N, 1). The multiparticle space, is an antisymmetrized direct product of k identical single-particle spaces, d{X(N, 1) x X(N, 1). . . x X(N, l)>, and is a complex Hilbert space, X(N, k). As its orthonormal basis, we may take the set of (z) wave functions ‘il,....ik
E
d{~pi,(l).
. . Cpik(kl}.
(2)
We now define S(N, k) as the unit sphere in #(N, k); all the spaces with which we deal from now on are subspaces of S(N, k) and thus contain normalized wave functions only. Any wave function + in S (the symbols N and k will henceforth be omitted
where they are self-understood)
can be expanded in this basis
with
We now define a series, or a hierarchy, of subspaces Sg”’of S. For the sake of clarity, we begin by defining S6”; this subspace contains all states which can be expressed as a pure single-particle determinant in some representation. Namely, Sg) contains all those (normalized) states for which a representation exists in which only one of the coefficients &, . ., ir is God-v~ishing. Clearly, the property of being a pure determinant cannot always be directly inferred from an observation of the specific representation (3) of the state; we shall not discuss here the procedure involved in this determination 3). The subspace S$) comprises all states in S which can be expressed as a linear combination of II, or less, states belonging to Sg’. It is impo~ant to emphasize that these n states are determinantal wave functions in different, unrelated single-particle representations [cf., e.g. ref. 6)]. It is also worth noting that these are not linear spaces. We have, obviously, constructed a series of spaces, which, for some YE@, becomes identical to S: sg” c S(,z)c . . * c St) 55 s
(5)
To find what no is for given N and K, we count the number of inde~nde~t real parameters which are associated with each of the spaces mentioned above. We shall designate this number as dim(space). Clearly, dim(~(N,
k)) = 2
,
N k
dim(S) = 2
- 1.
0 To evaluate dim (S’,“‘),we have to count the number of independent real parameters needed to specify uniquely a given single-particle determinant. We begin with the Nk complex (or 2Nk real) expansion coefficients of the k states in the N wave function basis. From this we have to subtract the number of restrictions imposed by the orthonormality requirements. There are k conditions of no~alization (the lengths being perforce real numbers) and 2(z) conditions of orthogonality. Since the dete~inant is invariant under an arbitrary unitary transformation of the k single-partible states, we have to further subtract k’, which is the number of real parameters defining such a transfo~ation. We note that the multiplication of a given state by an arbitrary phase is included as such a unitary tr~sfo~ation. Thus, dim(S#
= 2Nk-k-2
k -k2 = 2k(N-k). 02
192
I. Kelson / Dererminantal measures
For Sg) we simply take n unrelated determinants (the condition that no two of them are equal does not reduce the dimensionality), associate with each one complex (i.e., two real) coefficient, and subtract one for over-all normalization. Hence dim($))
= ndim(Sg))+2n-1
= Znk(N-k)+2n-1.
(8)
The number n, is simply the smallest integer satisfying dim (Sg)) 1 dim (S) which is
(9)
where the brackets stand for the largest integer smaller than the enclosed expression. We would like to add the following two remarks. First, the dimensionality of SF’ can be identified as the number of independent infinitesimal variations which conserve the determinantal nature of a state. This is just the number of different particlehole excitations. Second, we may repeat the entire discussion and analysis for real Hilbert spaces, and corresponding real subspaces. We would then have dim (real S) = (f)- 1; dim (real SE’) = nk(~ -k) + n - 1; but n, is still given by eq. (9). We now turn to the central concept of dete~inantal measures, which is related to the extent to which an arbitrary state in S can be approximated by a state in Sl;’ (1 g n 5 no). We de&e a sequence of functionals D’“’ (n = 1,. . ., no) on S by @%V = rn;;) l(@l+>l.
(10)
II will be referred to as the “‘determinantal measure of order n”, or “n-determinantal measure”, and the state @, belonging to SK), for which the maximum is obtained will be called the “n-determinant projection of the state +“. A number of simple properties of the determinantal measures can be trivially inferred. For example, for any Y D@+l)(iy) 2 D’“‘(y3,
(11)
D(“O)( Y) = 11
(12)
l(“)(Y)
- -L Jno’ 2
(13)
Procedures of a rather elaborate nature are required, when space averages of determinantal measures are to be calculated 4). We have, thus far, considered only general properties of the multiparticle space (and its hierarchy of subspaces), which were related to structural, morphological and combinatorial aspects. We now turn our attention to the role played in this connection, by various operators defined over the space S, and in particular by the hamiltonian H. The primary effect of the hamiltonian is that each (no~alized)
I. Kelson / Determinantal measures
193
state in S has a numerical function associated with it, namely, the expectation value of the hamiltonian E(Y) = (YMYu>
(14)
Barring pathological singularities in the hamiltonian, this function, E(Y), is bounded both from below and from above by Emin and E,,,, respectively. That it is bounded from below is a natural property of a physical system, such as we purport to describe by our model. The fact that it is also bounded from above is a consequence of the truncation which we have, in effect, carried out when going from the infinitely dimensional Hilbert space to our finite dimensional model space. Presumably, by properly selecting a set of ever-increasing model spaces, the true situation can be successively better approximated. For any particular model space, however, such as we confine ourselves to, it is thus only natural that we concentrate our attention on the lower part of the energy spectrum. Defining a median (which we assume to be identical to, and use interchangeably with the mean) energy E, we shall in any case discuss only the region [Emi,, ET ; the region [E, E,,,] is essentially symmetric to it, but this symmetry is obviously spurious. For any range of energies [E,, E,], such that Emin s E, 2 E, 5 E,,,, we define a subspace S(E,, E,) which contains all states Y, YE S(E,, E2)
if
E, 2 E(Y) 5 E,.
(15)
This is a multiply contiguous subspace, of the same dimensionality as the entire space itself, which, using this nomenclature is simply S(Emi,, E,,). It is possible in principle, though not generally in practice, to calculate the “area” a(E,, E2) inscribed in this hypersurface. By allowing E, to approach E,, we obtain the density of states at a given energy p(E) = C,l;yo &
o(E, E+dE).
(16)
Since only averages are ever calculated, the constant C is arbitrary, and can be adjusted at will. One possible choice, for example, requires that upon integration E
max
p(E)dE = dim (S).
(17)
JEmin
This density is a measure of the locus of all points in S with a given E($), which is a subspace we designate S(E). Since the expectation value of H is always a real number, restricting it to have a specific value implies that S(E) is one dimension less than S, dim (S(E)) = dim (S)- 1 = 2(T)- 2. It is worthwhile mentioning that p(E), as defined above, does not coincide with the density of eigenstates of the hamiltonian H at that energy. A quantity of particular interest is the average value of the determinantal measures at a given energy, which can be calculated once a well defined weight is assigned in the space. In fact, it is this quantity which is the subject of this paper.
Before we continue to the next section, we would like to stress and clarify our notation: D(“){Y) stands for the n-determinantal measure of a particular state $+ D@‘(E)stands for the average of the ~-det~rmina~tal measure over all states (puints in 5) with given E. This definition extends to the dependence on any variable other than the energy, suitably defined on S. F stands for the n-determinantal average ever the entire space. In all cases the dependence on iV and k is implicit.
In this section we present results of the energy dependence sfdeterminantal measure averages in small model spaces, The spaces are small enough so that such results can be directly evaluated with a high degree of statistical confidence, and for a variety of hamilt~nians~ We begin by introducing a simplifying feature into our numerical procedure. Namely, we consider the spaces defined over the real numbers (rather than complex ones). There is no manifest reasun why this limitation should affect the values of averages such as we compute; but, on the other hand, we have no formal proof that it does not, and we must leave it as a working hypothesis. All normalized states in our vector space, thus form a unit sphere in a (:I dimensional Cartesian space, with the function E(M) defined over this hypersurface. The density of states in this vector space is taken ta be directly proportional to the surface area of this hypersphere, In the absence of direct anaIyticaf methods for evaluating averages of complicated f~~ti~na~s~ we need a procedure fctr selecting random points on this unit sphere. This is done with the following sampling procedure, First, a set of v f zz {f) - 1) real numbers yl, yz, . . ., y, are selected, the first V- I randomly picked in the range [O, n] and the last in the range [O, 2123,A new set of values ci, t2,. . ., (, is obtained by the fallowing relationships
where F$
is the inverse of the function sin”‘“+ tdt .
L,,1@) = sin”-‘-’ tdt
Go
I. Kelson / Determinantal measures
195
Now the Cartesian coordinates X1, X,, . . ., X,, 1 are determined through XL
= cos 51,
X*
= sin {I cos c2,
X”
=sin51sin52...sin5,_1cos5,,
X v+1
=
sin
Obviously, this produces a randomly distributed set of points for any orthonormal basis that one works with. Thus, each selection may be extended to a multiple set of points in a given basis by performing any arbitrary permutation of the {Xi} among themselves. Once a wave function has been selected, its determinantal measure (of any order) is evaluated, as well as E(Y), the expectation value of the hamiltonian. Space averages can be computed with a relatively small number of points. This is so because the values of the determinantal measures are strongly peaked around their mean values. However, since the density of states is also strongly peaked around the median energy, the simple procedure described above is highly inefficient in selecting states with energies far from the mean and close to the ground state. To overcome this difficulty we have to introduce a strong bias for selecting states of low energy. Such a bias can be introduced in a number of ways, which do not have any special technical significance. The basic approach is to use as an orthonormal basis the eigenstates of the hamiltonian defined in our space
HYY,= E,Y,,
u =
l,...,
(22)
with E, arranged in ascending order. Each X, is multiplied by a different factor, with the factors decreasing monotonically with u, and the set (X,} subsequently normalized. The statistical behaviour near the point E,i,, is determined utilizing the dominance of the (non-degenerate) ground-state component Yr. A set of states are chosen of the form Y = Y,cos8+Y,,sin0,
(23)
where cos 0 is varied in small, fixed increments and Yy,, is chosen randomly from the Cartesian [(F) - l] dimensional unit hypersphere, conscribed by the states Y,, . . ., Y (3. This choice follows again the procedure described above. The average determinantal measure of order n at a given energy E was obtained by averaging over the numerical values obtained within an energy range LIE around E; typical values used where LIE/(&,,, - I&) = 0.03. The only apparent exception was the average at the ground-state energy itself, which, in the absence of degeneracy, was directly obtained.
196
I. Kelson / Determinantal measures
In all spaces, characterized by N and k, the hamiltonians with which we performed the calculations were a combination of a diagonal one-body force of evenly spaced single-particle energies and a randomly constructed, uncorrelated two-body interaction. Thus,
with t, = Lx&,
(25)
and the %t)[(T)+ l] independent two-body matrix elements of V were randomly selected real numbers from the range [ - 1; + 11. The variable parameter E determines, apart from an irrelevant energy scaling, the relative importance of the onebody and two-body parts of the hamiltonian. For E = 0 we have complete zeroorder degeneracy of the particles, which is only removed by their interaction; for E B 1 we have the case of completely independent single-particle motion. The even spacing of the single-particle energies is arbitrary. We shall not treat here the case where partial degeneracies do occur. The complete independence of the twobody matrix elements equally results from the fact that no internal symmetries among the particles are assumed.
. 0
I 2 3 4 5 6 7 8 9 IO11 12 E
Fig. 1. Complete determinantal measure versus energy graph for the space N = 8, k = 4, with degenerate to single-particle energies (i.e., E = 0). Since for this space no = 5, there are curves corresponding n = 1,2, 3,4. For n = 1 and n = 2 the determinantal measures of the eigenstates of the hamiltonian are also plotted as a series of points. The lowermost scale gives the excitation energy above I?,,( = I?,,,,,) of the eigenstate, and the positions of the eigenenergies are marked on the scale above it.
In fig. 1 we present one complete graph for all determinantal measures with in a particular space as a function of energy. The space has N (= 8) states and k (= 4) particles, and there is no single-particle splitting (E = 0). The energy extends from Emin to E, and the eigenvalues of the hamiltonian are marked on the energy scale. The points correspond to the determinantal measures of these eigenn < no,
I. Kelson / Determinantal
197
measures
states; here, of course, the values are exact. The smooth curves correspond to the appropriate space averages and do involve a certain degree of uncertainty, stemming inherently from the special procedure which we have described above. These curves (as well as the general trend of the points) exhibit what is the major feature in which we are interested. Namely, the fact that at low excitations the determinantal measures are considerably larger than for average states (or for states with average energy). This is all the more significant in this case, because of the complete absence of a one-body force, which is a natural conducive agent to independent particle motion. Moreover, this enhancement is in fact more significant when we recall that the determinantal measures give amplitudes (or overlaps), while it is probabilities (or amplitude squared) which are quantitatively important in deciding the quality of the relevant approximations. To exhibit the effect of a one-body ‘force, we append to the same interaction in the same space, a single-particle spectrum with E = 1. The corresponding determinantal measure (of order n = 1 only) is shown in fig. 2.
.& ’ ’ “I ’ “““’ “’ “‘I ?I
I’ll II
IIW
P F
,
23456769D11121314 E
Fig. 2. The n = 1 determinantaf measure for the space N = 8, k = 4, where a single-particle splitting with E = 1 is added to the hamiltonian used in fig. 1. The same format is used for this figure in presenting the results.
Comparing this curve with the n = 1 curve of fig. 1, demonstrates that while Df’) at E is practically unchanged, the enhancement of Do’ near Emin (SE E,) is much stronger when a single-particle force is included. To show that this is indeed a consequence of the presence of the one-body force, rather than a peculiarity of the two-body interaction, we did the following. A sequence of ten different two-body interactions were selected (all according to the same procedure), and the corresponding average determinantal measures were computed, all for E = 0. The particular one for which D”‘(E,) was largest is shown in fig. 3. The value D”‘(E,) = 0.90 is considerably smaller than the one shown in fig. 2 for E = 1, while the behaviour at E is, again, practically identical. These numerical analyses were performed for spaces with (k = 2; N = 6,8,10, 12,14,16,18,20), (k = 3; N = 6,8,10,12,14) and (k = 4; N = 8,10,12). In the
198
I. Kefson / De~erm~an~a~ measures
0
I234567891olll2 E
Fig. 3. The n = I dete~inanta~ measure in the space N = 8, k = 4, with no single-particle splitting (E = 0), again using the same format as tigs. 1 and 2. Here the two-body interaction was varied over a set of ten interactions, maximizing the value of DC’)(&).
larger spaces, typically with (i) > 100, a complete statistical analysis was not performed for practical reasons, so that values of energy dependent averages were not obtained with the desirable small errors. It is, partially, for this reason that we find it advantageous to construct a parametric representation to describe the behaviour of graphs such as fig. 1. The determination of a small number of parameters can be done much more reliably than the computation of a complete energy dependent set of curves. Moreover, by using appropriate parameters, we may gain useful insight into the behaviour of the quantities involved, and perhaps be able to extrapolate them to higher spaces. This is done in the next section, where the rest of the numerical results are presented in a parametric format.
4. Parametric representation of determinantal measures Before proceeding with a step-by-step description of the parametrization of the determinantal measure versus energy graph, one general remark is in order. The subject matter of the present discussion is a particular approximation and its energy dependence. Being over demanding when dealing with an approximation may be counterproductive in the sense that the physical, intuitive significance of the approximation may be obscured by exacting, mathematical details which are not inherent to the problem. Thus, we follow the principle that simplifying procedures are permissible when they reproduce the essential features of the approach, at the expense of strict formal rigour. (i) We begin by employing in ill cases the dimensionless variable E-E. Cl=;* E - Emin
(26)
This variable goes from 0 to 1 over the region of interest, and is highly suitable for
199
1. Keison / Determinantal measures
a treatment which seeks to unify all relevant systems, irrespective of their specific energy scale or structural complexity. (ii) As is apparent from fig. 1, the determinantal measure versus energy graph consists of a series of curves, D”‘(or),JY2)(cc),. . ., D(“$x), the last of which being identically unity. Apparently, there is a progressive relationship within these series of curves, which stems qualitatively from the basic properties of determinantal measures. Although this is by no means a universal relationship, we shall assume that the ED; n 2 2) are determined uniquely by the function D”‘(a). Namely, that all higher order determinantal measure averages as a function of energy are known, once the simple determinantal measure average is known. Thus, it is sufficient to evaluate and parametrize the lowest of all the curves in the hierarchy, and to get the rest of them by an appropriate recursion relation. To construct explicitly an approximate recursion relation, we proceed as follows. Consider a state I/J,whose n-determinantal measure is D”“($). Thus, we may write (27) where #,, (E S’,“‘)is the so-called n-determinantal remainder state satisfies
projection of il/, and the normalized
Similarly, of course, we could write the corresponding, in terms of its (n+ 1) determinantal projection lj
=
D’““‘(l&$“+
I+
1 1 -lP+
analogous expansion of I,& 2
W)
Xn+
1’
(29)
Eqs. (27) and (29) involve the solution of distinct variational problems, and cannot be easily related to each other rigorously. We shall, however, approximate #n+l through a sequential expansion, starting from the expansion (27). Namely, let us write xn in terms of its one-determinantal projection 4p1(and a remainder state x’) (30) and substitute it in eq. (27). $ = D’“‘(li/)#,+jl
-zP’($)zD”‘(x,)$J1
+J’.
(31)
The determinant 41 is linearly independent of the n determinants into which 4, may be decomposed, or else (p,, would not be - by definition - the n-determinantal projection of +, satisfying condition (10). Hence the first two terms in the progressive expansion of +, are proportional to a state in Sg+ l). Our procedure is based on the assumption that this state is approximately equal to (6, + 1_All the above discussion refers to one particular state rl/. If we were to consider an “average” state $, or alternativaly, average over the entire space S, then we should, obviously, replace ~(“)(~) by p and Dt”+ i)($4 by r. Furthermore, we shall also assume that we may approximately replace D(‘)(x,) by t)m; namely, that the space average of the
200
I. Kelson / Determinantal measures
determinantal measure of an “average” remnant state is the same as that of an average arbitrary state. This substitution will clearly underestimate the average of D(‘)(x,), particularly for larger n. For example, if rz = ~~ - 1, then by definition D(‘)(x,,_ J = 1 ! Nevertheless, adopting these assumptions results in a very simple recursion relation, which yields a good approximation for n < n,, and always gives a lower bound on D@).It reads, J_ Or, defining the remainder p
= J_&D”‘z.
(32)
through @$‘=
l_FZ,
(33)
we can write R(“+l) = pp.
(34)
R(.+m)=pp.
(35)
This can be easily generalized to
Clearly, this is only an approximation, since we have treated successive approximations as independent. Having found a particular higher order determinantal projection satisfying (31), implies that in reality we shall have R(n+d
__~ 5
R(n) R(m).
We now further assume that this relationship is a valid approximation energy E, or equivalently a, R(“+“‘)(a) = R(“)(~1)R(~)(tl),
(36) for any given (37)
where we have followed the same notation for R’“’ as for DC”). Thus, through this ansatz, the behaviour of all curves is determined by the n = 1 curve. A seeming shortcoming of this approximation is that R”“(cc)becomes identically zero only for n = co, while in fact, it should do so at n = n,. But, in any event we are more interested in the structure of the curves for n < n,. (iii) We assume that at the mean (which we take as identical with median) energy E, the average value of D(r) is approximately the entire space average of D(l). This is generally justified in view of the high concentration of states around the mean. Thus D”‘(Cr = 1) Z Err.
(38)
The particular value p, which is clearly independent of the hamiltonian, could, in principle, be taken as one of the parameters required for the quantitative representation of the graph. Relying, however, on numerical results, we may attempt and obtain it in closed form. To do so we plot in fig. 4 the values of a large number of space averages of n-determinantal measures, as a function of the variable n/n,,
I. Kelson / Determinantal measures
201
a 6 5 4 3 .2 _I I ‘0
11
I
I
I1
I
II
I
I .2 .3 .4 .5 .6 7 .8 .9 ID
Fig. 4. Values of space averaged determinantal measures of order n, plotted against the parameter n/n,,. The continuous curve is the functional relation (n/no)0~s9+(l - Dc”~)o~59= 1.
where n, is given by eq. (9). The points correspond to these averages, and there is clearly a strong correlation between the two variables. The smooth curve is an attempt to lit the points with a function of the form
0; C+(l-Fy=
1,
with the numerical value p = 0.59( kO.02).
(40)
The fact that TD” is expressed as an explicit function of N, k is, of course, very useful. However, one must remember when considering this for extrapolation to higher spaces, that on fig. 4 the entire extrapolation region is compressed into a small area in the vicinity of the origin. One point, if only marginal, in favor of an expression such as eq. (39), is that it can be shown that for /3 > 0.5, and for sufficiently large spaces, the inequality of eq. (36) is indeed rigorously satisfied. (iv) We further notice that D(‘)(U) is a monotonically decreasing function of CY, with the enhancement over D”‘(1) fastly decreasing from its maximum at a = 0 (namely, low excitation energy). We parametrize this typical behaviour in terms of the two parameters D(‘)(O) and the exponent of decay y, so that D(‘)(a) -I);m #“(O)__p
-a/r =
e
.
(41)
It is easy to notice that this can potentially provide a better fit to points in the neighbourhood of the ground state (a = 0), which is indeed the region of interest. The entire space average F, when recalculated from the curve D”‘(a), will become, as a consequence of our formal expression, somewhat larger than its actual value. Only for large N and k, where y becomes smaller and the density of states more concentrated at E, will @n be self-consistently calculable from D”‘(a). This, again, is in accordance with the remark with which we opened this section.
I. Kelson / Determinantal measures
202
To sum up, we have constructed a parametric representation of the determinantal measure versus energy graph, which utilizes two parameters, D”‘(O) and y, and which is, explicitly, the following. The value of D(“)(a) for the space S(N, k), is given by D’“‘(a) = ,/l - {1 - [D’D(a)]2)n,
(42)
where D(“(cc) = p+
{D(l)(0)-_je-a’Y,
iP = l- {I-
(43)
n,(;,k)8jl’a.
(44)
We recall for clarity, the relevant equalities and definitions which have appeared above, a _
[es.W91,
E_-Emin
lZ-
Emin
N
0 ~-----+l k(N-k)+l k
n, =
/3 = 0.59kO.02
1 [es. (9)1,
[eq.
WN.
Using this representation, the results for spaces S(N, k), for hamiltonians with E = 0, 1,2, are summarized and presented in fig. 5. Perhaps the most striking feature
0
-
(6*3k-x+
E=Q
.4 -
Y .3 (12,410
U2,3,0~d
9’ Ol4,3) -
Fig. 5. A representation of a variety of graphs by means of the two parameters y, D”‘(O). All the results utilize the same two-body, randomly chosen interaction. Three possible values for the single-particle splittings are used for each space N, k-8 = 0, I, 2. The curves connecting points belonging to the same space are for clarity only. Note the very weak dependence of y on E within each space.
203
I. Kelson / Determinantal measures
of this figure is that the introduction of a strong single-particle force increases the value of D(‘)(O), but leaves almost unchanged the value of y, namely the fractional energy range over which the enhancement of the determinantal measure persists. From this we may be led to the conjecture that y depends (approximately) only on the space (N and k) and not on the hamiltonian. What could this dependence be, and how reliable is this conjecture for extrapolating to higher spaces? In fig. 6, we
01 0
I
1
I
I
2
3
I
4 I”($)
t
I
1
5
6
7
Fig. 6. Values of I/y as a function of In(:), indicative of a possible approximate linear relationship between the two. The dashed curve, drawn to guide the eye, satisfies 1/y = 0.4+0.6 In(i),
have plotted, for all available results, l/y as a function of In(T), where l/r is an average value of the results obtained in the corresponding space for the different hamiltonians. It seems that l/y is roughly linear in In(f) ; we are in no position to determine the reliability of this statement at this stage. The parameter D”‘(O), which is obviously strongly hamiltonian dependent is even less amenable to extrapolation to higher spaces. We shall only make at this stage the following, very general conjecture: It is possible, by studying the behaviour of D”‘(O) in small spaces, to extrapolate its behaviour to larger spaces, for hamiltonians with the same characteristics +.
5. Summary and conclusions
In finite many-fermion spaces (k particles in N single-particle states) we have considered a hierarchy of subspaces, characterized by n, the minimal number of determinantal states (in unrelated representations) needed to describe a state in them. This gives rise, also, to a hierarchy of approximations. One is a direct generalization of the Hartree-Fock approach, attempting to minimize the expectation value of the hamiltonian not in the lowest, but in progressively higher spaces up the hierarchy. The other approach centers around the concept of determinantal measures, which ’ The major characteristic, m this context, is the relative strengths of the one- and two-body parts in the hamiltonian.
I. Kelson / Determinantal measures
204
involves maximizing overlaps with a given state. Since, in general. the state which is to be approximated is not known, what we mainly investigate is the general statistical behaviour of this quantity and its dependence on various characteristics and parameters. In particular, by concentrating on states with a given expectation value of the energy, we constructed a graph which gave the average determinantal measure of any order n, as a function of excitation energy. This was cast in a universal, dimensionless format. The central feature of this graph, which stood quantitatively out in numerical model calculations, was the enhancement of the determinantal measures at low energies. There are a number of applications for a graph like this in given systems, once it is even approximately known. For example, if one demands a particular minimum overlap of an approximate state with the ground state, then the required number of determinants (namely, the value of n) can be read off the graph. Alternatively, if the number n is given, the expected quality of the fit can be easily determined, along with its dependence on excitation energy. We have introduced a special representation, using two parameters only, to describe this graph for a given hamiltonian in a given space. There is a large degree of arbitrariness in our representation, but the two parameters, - apart from being few in number, and hence convenient to use - are related to the two central, meaningful features of the graph. One is D”‘(O), the single determinantal measure at the ground state, and the other is y, related to the fraction of energy over which the “decays” to its mean value. enhancement of D (I) We have attempted to indicate a possible extrapolation of y to higher spaces. However, a reliable quantitative determination of D"'(O) and y for higher spaces calls for further theoretical study.
References 1) See, e.g., D. N. Nanda and P. T. Narasimham, 2) 3) 4) 5) 6)
R. E. Peierls and J. Yoccoz, I. Kelson and G. Shadmon, I. Kelson and G. Shadmon, G. Shadmon and I. Kelson, A. Faessler, A. Plastino and
Proc. Phys. Sot. Ann. of Phys. 63 J. of Math. Phys. Nucl. Phys. A241 K. W. Schmidt,
Int. J. Quantum Chem. 6 (1972) 819 A70 (1957) 381 (1971) 497 16 (1975) 2224 (1975) 407 Phys. Rev. Lett. 834 (1971) 31