Permutational symmetry of reduced density matrices

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XI. 577-589 (1977)

Permutational Symmetry of Reduced Density Matrices

EUGENE S. KRYACHKO Institute for Theoretical Physics, Kiev, 252130, U.S.S.R.

Abstracts

The problems of permutational symmetry of the density matrices in reduction are studied. Some necessary and sufficient conditions for N, [A [A,]-derivability problem are given.

Dans ce travail on examine la symetrie de permutation des matrices densitd sous reduction et o n cite quelques conditions nkcessaires et suffisantes pour le probltme de la N, [A ,], [A,]-derivabilite.

Es werden die Fragen der Permutationssymmetrie der Dichtematrix bei Reduzierung untersucht [A,]-Ableitbarkeit und einige notwendigen und hinreichenden Bedingungen des Problems der N, [A

angege ben.

1

A . When considering quantum mechanical systems the reduced density mat- rices offer certain advantages in comparison with the usual language of wave functions [l, 21. In this connection and in order for eigenvalues of spinless operators to be calculated [3, 41, a theory of spin-free reduced density matrices was proposed. From the viewpoint of the permutational symmetry this theory is a generalization of the theory of boson and fermion reduced density matrices.

In this paper the permutational symmetry of the reduced density matrices is studied, and some necessary and sufficient conditions of the N, [Al], [A2]- derivability are given.

B. Let U be the infinite-dimensional Hilbert space over the field of complex numbers. The elements of U will be called ket-vectors or simply kets. Assume that l& = UO UO + . .O U is the N-fold tensor product of U space. If U' is a dual space of U, i.e., the set of all linear mappings of U into the field of complex numbers, then similarly U N = U'O U'O * * - 0 U' ( N folds).

Let l$ = V , O U N be the space of tensors of the type (N, N) (i.e., tensors transforming covariantly and contravariantly on N indices relative to the transfor- mations of U space). If { l iK)}K is the basis of U space, then {lI(N))(j(N)l}I, where II(N)) = lil)li2) - * liN) and (j(N)] = (jNl(jN-l/ . (ill is the natural basis of UE. It is obvious that N-particle density matrices are (N, N)-type tensors. What is more, the set of all N-particle density matrices which correspond to pure and mixed states forms a convex subset in UZ. Finally, the transition density matrices also are tensors of the (N, N)-type.

C. Let S , be the symmetric group of permutations of N symbols and R, = c[sN] be the group algebra of SN over the field of complex numbers C. The

577 @ 1977 by John Wiley & Sons, Inc.

578 KRYACHKO

algebra R is semisimple. There is an expansion

in the direct sum of simple matric algebras of R N and

where f r A ' is the dimension of the irreducible representation [A] of SN.

{e!]} satisfying the following equation: The metric basis of R N is formed by the primitive orthogonal idempotents

e r A I [A2] = ~ [ ~ , l . [ ~ ~ l ~ ~ ~ ~ ~ ~ l rs e tu

There is a connection between the given basis of the group algebra and its regular basis:

f r A 1

P = 1 1 [ ~ I k ' e k ] [ A ] r . s = l

where [PIE1 is the (r , s)th matrix element of the irreducible representation [A] of P E s,.

The expansion of R N in a direct sum of minimal left ideals induces the expansion of uf;;

PERMUTATIONAL SYMMETRY OF REDUCED DENSITY MATRICES 579

Finally, we introduce two more notations: uc+n - T r ( m ) ( e [ h i l u N [ A z I +

d A z 1 = N e ) [A I N [A,]+ U ~ ~ ~ , [ ~ , l r ~ T r ( ~ ) ( e r r ’ UNeZt

A . Let I U ) and 1 V ) be the arbitrary elements of UN. Then the reduced density matrix of the n th order (n < N ) corresponding to kets 1 U ) and 1 V ) is defined as

DEv=Tr(”) (I U>( Vl) (n + m = N )

We consider that the ket-vectors IU) = )ul))u2) * * - IuN) and I V) = lal)lu2) - . 1 ~ ~ ) . Then

D”,,([A l]rs; [A&) = Tr(m’(eE1l) V)( VIe$’+)

are the primitive reduced density matrices. They satisfy the following relation [3] :

D”,P1P’] = D:v[rrP/P’rr-l] (11

for any rr E S , of the group of all permutations of n + 1, n + 2, . . . , N particles. Expand S, in left cosets on this subgroup S , :

Then

x D;v[7TPclP:.rr’]

By means of the substitution rr’ = 7rr-l and taking into account Eq. (l), we obtain

580 KRYACHKO

To calculate the last sum we consider the reduction S, 2 S, x S,. Then

where [A], [A,], and [A2] are the Young schemes corresponding to the symmetric groups S,, S,, and S,, respectively, and 0 denotes an outer direct product of groups.

The possible Young schemes [Al] and [A2] into which the scheme [A] breaks upon reduciion of S, = S, X S, are usualIy found by means of the Littlewood theorem [5]. Next we denote by A a nonstandard representation based on this reduction. Let ( r / ( r )A)rAJ be the elements of the transformation matrix [6] which connects the standard and nonstandard representations for the given Young scheme [A]. Then

Taking into account rr E S , we obtain

PERMUTATIONAL SYMMETRY OF REDUCED DENSITY MATRICES 581

Thus formula (2) shows that the reduced transition density matrices exist not

To consider the reduced density matrices in some particular cases it is only among various kets but also among different Young schemes.

convenient to use the following formula:

The selection rules for the reduced transition density matrices between kets which are transformed on the irreducible representations [A 1] and [A,] follow from the factor ([A:] 0 [A']+ [A,])([A:] 0 [A2]+ [A,]) in formulas (2) and (3). For exampleD",[l"];[N])=Oforn = N - 2 , N - 3 , . . . , 1 (foranyIU)andIV)),i.e., the mixture of boson and fermion N-particle functions ( N 1 4 ) does not make a contribution to the reduced second order density matrix.

In particular, if 1 U ) = I V ) and I V ) does not contain repetitive one-particle kets, i.e., the invariant [4] of IU) is equal to {k}={l, 1 , . . . , l}, then formula (2) simplifies ro

B. We consider the following example. Let I U ) = I V ) = Iu1)Iu2)1u3)1u4) and the invariant be equal to (1, 1, 1, 1). We assume also that [A,] = [2'] and [A,] = [212], and r = s = t = u = 1 in formula (4). Then

582 KRYACHKO

In this case the transformation matrices have the form [6]:

rl

(rl[A:][A:])["" = r2

r3

Thus

*This relation proves that the illustration of theorem 1 in Ref. [7 ] , via contraction of U:221[2121,

omits the term U ~ l ~ I ~ l ~ l , which can also be seen by correctly applying the theorem.

PERMUTATIONAL SYMMETRY OF R E D U C E D DENSITY MATRICES 583

where each index of summation ik, j l take values from 1 to N, and ik f. il (k # 1) with

and two indices from i ( m ) in the sum are not equal to each other or to any index from i(n). We assume that lU)= IV)= Iul)Iuz) * . luN) and the kets lu, ) are orthonormalized. Then it is easily seen that

The first formula shows that the "averaged" first order density matrix for the function e$'Iu(N)) has the same (with an accuracy of a multiplier) form as a Slater determinant (see, for example, Ref. [2]).

1 f [ A l

It is obvious that

~"urX[AI ) ' - c (~",[AIs; [AIs)) f * I s = l

is the n th order immanant (corresponding to the Young scheme [A]) of the first order density matrix D:J[A]), namely (in notation of Ref. [5]):

~ " , [ A I) = IP :d[A 1)Isr I ( A ) i.e., this is the generalization of the formula for the nth order density matrix by means of the first order density matrix for the Fock-Dirac density matrix (see, for example, Ref. [ 2 ] ) so long as the usual multiplication of the immanant elements can be considered a tensor product.

D. Let us consider once more the structure of Uzinduced by the expansion of UE on irreducible tensor components and which they can be transformed by a certain irreducible representation of the permutational group S,.

We define the superoperation [7] on U z for each element a = C P E S N aPP E RN a * X = C apPXP-', X E U E

P € S N

Then the set {el; s = 1,. . . ,PI} is an irreducible tensor; one which is transformed on the irreducible representation [A] of the group S, if and only if for any P E S,

p 0 *I= 5' [ p ] p p I = 1

584 KRYACHKO

The superoperation induces the expansion of UEin a direct sum of irreducible tensor algebras

uN- I@ , y K l

[A I N -

where

U K I = e [ A l 0 u; Assume X E UE. Then

where

characterizes the multiplicity of the [Ao] representation entering the direct pro- duct of the representations [A,] and [A,]. So far there is no sufficiently simple method to define the expansion in irreducible components of the inner direct product of two Young schemes. Thus, we have proved the following.

Theorem 1

where 6([A,] X [A,], [Ao]) is distinct from zero if and only if [A,] enters (with the nonzeroth multiplicity) the inner direct product [A 1] x [A,]. In particular, for any X E u:

where [i] is the Young scheme conjugate to [A].

3

A . In this section we consider some necessary conditions of the N, [A,], [A,]-derivability problem which is a natural generalization of the N - representability problem [l] on more general Young schemes.

PERMUTATIONAL SYMMETRY OF REDUCED DENSITY MATRICES 585

Assume X E Ui. It is said that X is N-, [Al]rs-, [A,]tu-derivable if and only if there exists a tensor such that

X=Tr (" ' ( e~ I1Ye~~Z1) ( n + m =N)

It is obvious that X is N, [A,]rs, [A2]tu-representable if and only if

x E UC31:l:s,CA*,ru

Thus N-representability in a usual meaning is either N [lN], [ a N ] - or N, [N], "1-derivability .

B. Considering the N, [ A , ] , [A,]-derivability problem we restrict ourselves to the Young schemes of the form [2P1N-2p] which corresponds to the spin-free formulation of quantum mechanics of molecules and is the most applicable for finding the eigenvalues of spinless electronic operators [8].

The most convenient representation for considering the N, [A J, [A2]- derivability problem is the seminormal representation of the permutational group SN [9]. Let r be the standard Young tableau and P, and N, be the permutation sets in the rows and columns of this Young tableau. Then E, = PrN, is the Young idempotent of the tableau r. We define the set of seminormal idempotents [9]. Let E?' be the Young idempotent that corresponds to the standard Young tableau of n squares obtained by separating the squares using the symbols n + l , n f 2 , . . . , N from the tableau r. Then the seminormal idempotents el"' are defined stepwise

. . .

where E is the unit of algebra RN. [A2]-derivability problem

will be found stepwise also. In connection with this we shall alter these conditions according to the position of a symbol N in the first or the second column of the standard tableaus of the Young schemes [A 1] and [A,].

C. The symbol N is in the first column of the standard Young tableau ([A ] ] r ) and IIAz]s). Then

The necessary conditions for solution of the N, [A

(1) l[A11r):

9 [l-(Nal)- . * . N NU^)]

, [l - (NCl) - . * - (Nc,)]

p = p ( N - ' 1 N = "N-1)

(2) J[AzIs>: p = p(N-1) N = N(N-1)

F v) CD a

Xf

I Q

2';

I Nu

;M*

r

I 0

27

1;

X trl

*$

c 11

M3

...

2.- 7

m v

I T M

=

e

I > i - - .. 6

'5:

I

m-

y-

? 2?. ?

I

x - Y I/

t v

c

Y - - t;i ?

_I-

m'

7

Yi-

Q

72

L

L

m .-

2 YI

- 7 n

\o

Y

h

m e

rl

- N u

Vl-

tr; e.

F.

rr

v)

CD

Y E

1 P i I

r-

N

Y

w x

c-c 0

3

v) 8

m ir- 2 b- " - N

Y

t R

h

V

v,

PERMUTATIONAL SYMMETRY OF REDUCED DENSITY MATRICES 587

D. For other cases considering all the possible positions of the symbol N in the Young scheme [A ,] and [A2], doing similar calculations and using the following expressions

where

we finally derive the following theorem.

Theorem 2.

This is a necessary condition of the N, [A1], [A2]-derivability problem more

A . Now we consider sufficient conditions for the N, [A,], [A,]-derivability general than that proved in Ref. [4].

problem using the primitive minimal orthogonal Young idempotents

We are also interested in Young schemes of the form [2P1N-2p] and divide as before our proof into four steps depending on the place of the symbol N in the Young tableaus l[AI]r) and /[Az1s).

B. Assume

588 KRYACHKO

Then p;N-lj[A I 1 N , (N-lXA I 1 l i (N- l ) ) ( j ( ~ - 1 ) ( ~ ~ N - 1 ) [ h 2 1 p ~ N - 1 ) [ A 2 l

= Tr‘” {E~111i(N))(j(N)IE~2”} k

(Nuu)l i (N))( i (N)IN~N-1)rA~1~~N-1)rA21 1 + 1 ~ ~ ( 1 ) {p;N-lj[A,l (N-l)[All N ,

Nr

u = l

m

r I i ( N ) )( j ( N ) 1 (Nc,)NSN- l)[* JPp- 21 } (10) + 1 Tr(lj {p(N-lj[All (N-l)CA1l

v = l

“au 1 I i ( N ) ) - f T ~ ( ~ I { ~ ~ N - ~ ) [ A , I N ~ N - ~ ) [ A , ~

u = l v = l

S 3 x ( j (N)J ( N c , ) N y 1)CA21p(N--l)h2l

The ket-vector / i ( N ) ) contains one-particle kets lil), ( i2) , . . . , ( ik) each K~~ K,, . . . , K k times. This indicates [4] that ket l i (N)) has the invariant { k } , = { K ~ , K ~ , . . . , K ~ } (numbers K , are situated in nonincreasing order). Then E?]( i (N)) Z 0 if and only if [A] a{.}, (in the sense of usual ordering of the Young schemes [9, 8 1.41).

Let us consider the ket li(N- 1)) and l i (N)) = l i (N- 1))lh). Then { k } , and {k},,, are invariants of l i (N- 1)) and l i(N)), respectively. In this case [Al-], a { ~ } ~ - ~ . For [A1], a { k } l l , (similarly to [A,], > { K } ~ ~ ) we can choose h which is not equal to any one-particle ket from ( i ( N - 1)) and Ij(N- 1)). Then in Eq. (10) the second and third right-side terms vanish. Further expanding (Nu,) in the Young idempo- tents and taking into account the left E$N-l)cA1l= EF“’ expanding only in the Young idempotents Evo1, where [A,] E {[A +I}, we obtain the sum only for those Young schemes [Ao] where [A,]E{[~ +]} (corrollary of theorem lob, Ref. [9]). Similarly to [A2].

Using the same considerations for all the possible positions of the symbol N i n the Young schemes [A,] and [A2] and based on theorem 10b [9], we finally obtain

Theorem 3.

Consequently by induction it is easy to prove a general statement. Hence we have found the sufficient conditions €or the N, [A,], [A,]-derivability problem for any Pauli-allowed Young schemes [A 1] and [A2].

Acknowledgments

The author is very much indebted to Prof. I. G. Kaplan for initiating his interest to the problems of the permutational group representation theory and to

* [A -1, denotes the Young tableau obtained by separation of the symbol N from the Young tableau \[A],). Similarly, [A +] denotes all the standard Young schemes obtained by addition of one square in any way to the standard scheme [A].

PERMUTATIONAL SYMMETRY OF REDUCED DENSITY MATRICES 589

Prof. Yu. A. Kruglyak for his supporting this work and for his attention during its development.

Bibliography

[l] A. J. Coleman, Preprint No. 23, University of Florida, 1962; Rev. Mod. Phys. 35, 668 (1963);

[2] E. S. Kryachko and Yu. A. Kruglyak, Preprint No. 162E, Institute for Theoretical Physics, Kiev,

[3] R. D. Poshusta and F. A. Matsen, J. Math. Phys. 7, 711 (1966). [4] R. D. Poshusta, J. Math. Phys. 8, 955 (1967). [S] D. E. Littlewood. The Theory of Group Characters andMatrix Representationsof Groups (Oxford

[6] I. G. Kaplan, Symmetry of Many-Electron Systems (Academic Press, New York, 1975). [7] D. J. Klein and R. W. Kramling, Int. J. Quant. Chem. Symp. 3S, 661 (1970). [S] D. J. Klein, Int. J. Quant. Chem. Symp. 3S, 675 (1970). [9] D. E. Rutherford, Substitutional Analysis (Edinburgh University Press, Scotland, 1948).

Sanibel Lectures, 1965.

1974.

University Press, New York, 1958).

Received November 4, 1975 Revised May 3, 1976