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Single-particle approximation to the canonical density operator of a system of N interacting fermions
Vol. 19 (1984)
REPORTS
ON MATHEMATICAL
PHYSICS
No. 2
SINGLE-PARTICLE APPROXIMATION TO THE CANONICAL DENSITY OPERATOR OF A SYSTEM OF N INTERACTING FERMIONS J. MACKOWIAK Institute of Physics, N. Copernicus
University, Toruti, Poland
(Received July 4, 1979)
A necessary condition for the minimum of free energy Tr(pHN) + pm ’ Tr(plnp) is investigated for a system of N interacting fermions on the set of all density matrices of the form p = Z- ‘exp( - /?NTyh’), where f ‘: is Kummer’s (1, N)-expansion map, and h’ is a tinitedimensional one-particle operator. It is shown that for values of /I in an open circle U(O&,) in the complex plane there exists a unique bounded analytic in fi one-particle operator X l(p) such that Z - ‘exp( - /IN r: X ‘(#I)) satisfies this condition.
1. Introduction The idea to describe a system of interacting particles by an effective single-particle hamiltonian has been frequently exploited in physics. It underlies Husimi’s theory [l] (the counterpart of Hartree-Fock theory in statistical mechanics), the theory of superconductivity [2], [3], [4] an d mean field theories of lattice systems [5]. In all such theories one seeks an effective density operator peffcorresponding to a noninteracting system which serves as a substitute for the exact density operator describing the equilibrium Gibbs state of the system. The equations for perrare derived using the well known variational principle which states that the free energy as a functional of the state of the system at fixed temperature Tand with fixed number of particles (fixed mean number of particles) reaches its minimum on the canonical (grand canonical) density operator (cf. [S], [6], [7], [S]). Th us one minimizes the free energy Y [p] of the system over a set of trial equilibrium states of systems of noninteracting particles and takes for Pea the density matrix on which S[[p] assumes minimum in this set (cf. [ 11, [3], [4], [5], [9]). Following this scheme, the present paper investigates the single-particle approximation to the canonical density operator corresponding to a finite-dimensional system of N interacting fermions. It is shown that for values of /? = (kg-l (k is Boltzmann’s constant) in an open circle C(O&,) in the complex plane there exists a unique bounded analytic in p single-particle operator X”(p) = N ry X ‘(/I) such that minimizes the free energy of this p(XN(B)) = (Tr expC-PXN(B)l)-lexp(-BXN(B))
[I551
156
J. MAkKOWIAK
system on the set of density operators ((Tr exp( - /?hN))- ‘exp( - /?hN),hN = N Tyh’ }, ry standing for Kummer’s (l&)-expansion [lo]. An equation for X “(/I) is derived and its solution discussed. 2. The free energy and necessary condition for its minimum Consider a finite-dimensional hamiltonian
system of N identical interacting
HN = NT:
fermions with the
T’ + (;)r;V”,
(2.1)
where r; (p is a non-negative integer, 0 < p < N) denotes Kummer’s mapping [lo], the domain 9(HN) of HN being the N-fold Grassmann some fixed M-dimensional (M < co) manifold of square-integrable tions. The r: expansion is defined for any self-adjoint operator BP rr;‘BP = with PAP standing for the projector system (2.1) is p,(HN) = Z-’
~^N(BP
0
pd’J4)p~N
on mAq. The canonical Z[HN]
[HN]exp(-pHN),
(p,N)-expansion product !DIANof l-particle funcon !IRAp by
density operator
= Tr exp(-/3HN),
for the
/? > 0. (2.2)
As follows from von Neumann’s theorem [6], pc(HN) is the unique solution to the following variational problem: Find a self-adjoint operator p on m”” which minimizes the free energy FD [p, H “1 F,JCP,H~]
(2.3a)
= Tr(pHN) + kTr(plnp)
with p subject to Trp=l,
(2.3b)
p>O
for any density operator p.l As discussed It follows that F-,[p,HN] B F,[p,(HN),HN] in the Introduction, the principle (2.3) may serve to select from a given class of density operators the one that resembles as close as possible the state (2.2), the measure of closeness being the value of Ffl [p, H “1 - FGs [p(H N),H “1 for given /?. It is of interest, in particular, to find the relevant density operator from the class of operators pN:=
exp(-#hVr~xl), x’ = (xl)+,
‘By a density operator
we mean any operator
TrpN = 1.
on FAN fulfilling (2.3b).
(2.4)
APPR50XIMATION TO THE CANONICAL DENSITY OPERATOR
For the trial density operator
(2.4) the functional
Tr(pNHN) - NTr(pNI’y
to be minimized
15:
is
X I) - aTrpN,
(2.5)
where a is a Lagrange multiplier. Equating to zero the variation of(2.5), one obtains the equation Tr[(HN
x exp(-/?NT~X’)r~6X’] = Tr[(exp(-BNTYX
where y is a constant
‘)(NryX1
+ yPAN)TyGX
(2.6)
and
A x expB: = A + $4B + &(fW From (2.6), by the arbitrariness
+ BA) + -&(AB2 + BAB + B2A) + + MB2
+ B2AB + B3A) + .. .
of 6 X 1 follows a necessary condition
L~(HNxexp(--/INI’~X1))=L~(exp(-/UVT~X’)(NT~X’ Tr exp(-PNryX’) where Lk denotes Kummer’s (IV,l)-contraction should be determined from (2.7b). However, irrelevant, since (A x expB)expC for any C commuting the equation
‘1,
+yPAN)),
= 1,
(2.7~~ (2.7b)
operation [lo]. The constant y in (2.7a) the presence of y in (2.7a) is, in fact,
= A x exp(B + C)
with A and B, and thus X ’ may be equivalently
Lk(HN x exp(-/?NfyX’))=
for X ‘:
(2.8) determined
L~(Nr~X’exp(-/?NTyX’))
from
(2.9)
pN is then given by pN = (Tr exp(-/?NT~X1))-lexp(-/UVr~X1). A possible approach to eq. (2.9) is offered by a self-consitent iterative procedure on X ’ (fix X r in the exponentials, solve for X ’ on the r.h.s., substitute the solution into the exponentials, etc.) or by expansion of X ’ into a power series in fi and comparison of coefficients on both sides. In any case, one way of dealing with the resulting equations is to resort to S. Pruski’s decomposition of a p-particle operator ~~r~Bp into its 0 - , 1 - , 2- , . . . . p-particle projections. A brief outline of this decomposition is given in the next section. 3. The q-particle projection of a p-particle operator The linear operators on !WN constitute Hilbert-Schmidt (H.S.) scalar product
a linear space bN. Equipped
with the
158
J. MACKOWIAK
QX, Y) = Tr(X + Y),
X, YE$jN,
(3.1)
eN is a Hilbert space. One can distinguish in $JNreal subspaces Big of self-adjoint operators (r)T: P (0 < p i N). Obviously, 5: c 9,” for 4 < p. According to the theorem on orthogonal projection in Hilbert space, there exists a unique element (i)riXq~ !# which is nearest to a given (:)rI yP~$j: (q < p), with respect to the H.S. distance: (Tr((F)r:
p - (y)rfXq)2)"2
=
hfq(Tr((;)Tf p s
zNJ2)lj2 (3.2)
X4 can be found by solving the equation (;)W;
y”) = (;)J%(f:Xq)
(3.3)
which follows from necessary conditions for the minimum in (3.2): 6Tr((F)rF
P - (i)rtXq)’
= 0,
(3.4)
where variation is with respect to X4. The solution (:)rrXq to (3.3) is unique and is the orthogonal projection of (,“)rr Yponto !#. (3.3) may be also restated in the form Tr(BqL;G((F)r:P ’
- (t)riXq))
= 0,
VBq~fii.
For q = 1 the solution to (3.3) can be found using the theorem proved in [ll]:
In particular, the l-particle projection of the hamiltonian (2.1), HN = I’r;‘[NT: T’ + (t)V’],
is XN = NT:%‘, (0)
where N-l (Tr p)P’. (M - l)(M - 2)
4. The euation Li(HN x exp(-/?NTyXr))
(3.6)
=‘L~(N~I;YXlexp(-BNT~X’))
Suppose the solution NryX ‘(8) to eq. (2.9) is sought as a power series in /?. To determine the form of the series let us resort to the exact solution (2.2) of the variational problem (2.3). Obviously, the expansion of exp( - 4jN) in fl involves only non-negative powers of fi and s-limexp( -j?HN) = PAN. Analogous requirements imposed on exp (-/INTyX’)
dis!ki&z the form of NryX’(p): NT;X’@)
= N f fl”ryX’, n=O (W
(4.1)
APPROXIMATION
where
X1, Xl,... (0) (1)
TO THE CANONICAL DENSITY OPERATOR
etc., are operators
expansion
exp( -/?NTyX
equations
for the X1’s: (“)
in YJI. Let
XN:= Nf 7X’. (n) (n)
‘) = f (n!)-‘( -BNTyX n=O
NL#‘;JX’) (n)
= L;RN, Cd
159
Insertion
of the
‘)” and then (4.1) into (2.9), yields
n=Of23 , ,-..,
(4.2)
where RN=HN, (0) RN=-(XNHN+HNXN)+ (k-1) W
(k-1)
s~~~i~o~,z,,X-~...XNHN +c I -
i
k$
m=l
C ” XNX:‘T .
(4.3a)
XN . ..XNb(l.+...+lS-k+s)+(4.3b) i
x:‘L):
+ .:I’: l,,,+l -k
+m),
k = 1,2,...
11...1,+ 1 (11) (12) (‘?n+1)
It follows from (4.2) that the XN*s are orthogonal (n) there exists a unique solution
projections
of the RN,s onto 8,‘. Thus W)
(4.1) to eq. (2.9). For n = 0 (4.2) reads Lh XN = LAH”, (0)
proving that ,:” = N TYX’ is the projection of HN on $i,’ and is thus given by (3.6). (0) It will be now proved that the solution (4.1) with XN’s satisfying (4.2) is bounded V0 and analytic in /3 in some neighbourhood of fi = 0 in the complex plane, the analyticity of X N(fi)~!+jN being understood as analyticity of (40,XN(p)$) for any cp,ll/~)IJ”~. To this end let us introduce the following inner product of operators on bN: for fixed X&r, (A,@,:=
Tr((A+
x exp X)B)(Tr
exp X)-l,
A,BE!$‘.
One easily checks that I(A,B),I < 00 for A,BEJS~ and that (A,B), = (B,A),. Linearity with respect to the second argument and additivity with respect to both are immediate. To prove that (A,A), > 0 for A ~0, it suffices to use (2.8): (A,B), = (A,B),e”e-”
=
(ApB)X+apA~T
.For CIsuch that X + aPhN > 0, all terms in the expansion
aER’. of (A,A)x+apAN are positive:
160
J. MACKOWIAK
kw,+,.N
=
(TrA+A + $TrA+(X + aP~$4
+
+ $TrA(X + aP~$4’
+ ...)(Trex+aPAN)- 1
proving that (A,A), > 0 for A #O. One can now set up an iterative procedure on X N(p),the solution to (2.9).In the first step take arbitrary
YN~$f and find YNe$,l from the condition: (0) (1) (HN - YN,HN - YN)_-RYY = inf, (1)
(1)
/I > 0,
id,
Next find YNefif satisfying (2) = inf (HN - YN, I-IN - UN) _BYN (2) (1) (2) and so on. At each stage one has to solve an equation for YN = NT: Y~E!#: (m) (m) G&-IN x exp(+&Yr,))
=
m = 1,L.
(4.4)
which is necessary and sufficient for (HN-
Obviously, the solution
= inf. YN, HN - YN) (In) $-z (m)
(4.5)
Y”(p) exists and is unique at each stage. We shall prove that
for values of /3 in some%osed circle c(O,p,) in the complex plane with radius Be, centered at the origin, it can be given as a power series YN(j3)= i /3k Y N (m.k) k=Q 0@
(4.6)
convergent in norm topology on eN. Inserting
YN(fi)and UN (p) in the form of Mn) (rn-1) expansions (4.6) into eq. (4.4) and comparing coefficients at the same powers of p, one arrives at the following equations for the YN’sf!$: (m.q) L,:p =L;HN ow
(4.7a)
APPROXIMATION
161
TO THE CANONICAL DENSITY OPERATOR
and for 4 = 1, 2,...
c n=l(n+l)!~=lk=OOGp
where UN is the 2-particle consecutive
equations
projection
Y N Y N... P&p, VW
Ptt,
+ ...p” - q + n + r)
PI
(4.7b)
of HN and
(4.7a, b) for P(#?), p@),... 1 2
P E P . A simple analysis of s (m- 1,s) shows that for given m, YN does not (0)
for p’s q = O,l,..., m - 1 and that the latter are identical to the (m.q)
enter into the equations corresponding
YN...P P k I”’PP“1
equations
(4.3) on XNVs.As a consequence, (4) P = XN, @MI) (n)
4 = O,l,..., m - 1
(4.8a)
and lim lim
i
/3’ Y N = lim
k+4)mdwqq0
in norm topology
(W
on $jN, provided
k-co
i q=O
pq XN
(4.8b)
(4)
both limits exist. Now $jN is a finite-dimensional
Banach space. Thus to prove that 1 f
[’ XNI < 0~)in some norm 1 1on BN and 5 lying
k=O
(k)
within some circle c(OJ?r), it suffices to show that
is a Cauchy sequence for any ~~~(0,/?r). We turn to the proof of lo. Let I I denote the H.S. norm. A bound on I f
(k Y NI
k=O
can be obtained by noting that the H.S. no trm of the l-particle ZN~J3,N does not exceed IZN(: lYNl d
lZNl
projection
(m,k)
PE!$:
of
(4.9)
and that (4.10)
162
J. MA&OWIAK
Using the inequalities (4.9), (4.10), one finds from (4.7b)
Denote S,(5) = i jcf”j p 1.Then for 5 satisfying /[I < S,_ l([)- ‘ln2 we may write n=O @RN
i
111”/Y Ni 6
q=1
O%qt
/UN! k $ c l[lPI*-,*+q PI...{ PI Pi=1 O<:p,...p*
\lJpE+..~+p”IYy_.I Yy p, P”
(4.11) *
Taking the limit k -+ cm in (4.1I), we get
luNl(exp(tiis,-,(r))- 11+ lXNl 2 - ew&XL- XI) (0) for
/Cl< S,_,(l)-‘In2
(4.12)
proving that 1 C C4P 14 00 for i: satisfying /[I-C S,_ I([)-1~n2. The inequality (4.12) and the arbitrariness of YNnow suffice to demonstrate validity of 1”: Take YNsuch (0) (0) that i?YN
I YNI> ‘$7, (0)
(0) = 0. ag
(4.13)
Then according to (4.12)
I ~Nltew(lCllc;Nl) - 1) ~163= g 151”l p I< n=O
(1.W
--+/XNI
2 - exp(l~ll~~Nl)
(0)
(4.14)
for ICI< 1 YNj-11n2. Furthermore, since I YNI> \XN\, there exists a circle C(O$) such (0) (0) (0) that
APPROXIMATION
TO THE CANONICAL DENSITY OPERATOR
I W~WW$Nl)
for
+ l(:Nl G I YNI (0)
ew(lCll yNI) (0)
2(4.15) is equivalent
- 1)
kW4P1).
163
(4.15)
to
lil < lt~Nlplln 1 + (
I YNI - I$“’ (0)
(4.16)
l,;Nl + I UN1- If”1 >*
Thus
PI = /(zN(-‘ln
The condition
l,fl (
- lZNl < 1YNI-rln2.
1+
I YNI+ I UN1- I(;“’ (0)
>
(4.17)
(0)
l&J< YNI-‘ln2 relating to (4.14) may be therefore dropped if values of c (0)
are confined
to C(O,p,):
I”~oIn(lY)NIG I YNl .” (0)
for
(4.18)
l”~o~“,,‘.,“ld I YNl (0)
for
(4.19)
By (4.12) (4.15) (4.18)
and similarly at each stage:
l”$ocn(~)lG I YNI
for
@C(O$,),
(4.20)
m = 1,2)...
(0)
proving that the sequence
{ c k=O
ikCTb >z= 1 is uniformly IPI,
bounded
by I YNI within the (0)
circle c(O,fi,). The value of I(:“1 may now be chosen so as to maximize #I1 = PI (l,f ThisispossiblesinceB,(I
YNl)in(4.17) is a continuous (0)
I).
positive function on the interval
164
J.MAcKOWIAK
(1XN 1,co), vanishing at both boundary 10)
points, and therefore has an absolute maximum
Do at some value y, of 1YNI. Instead of (4.20) we may thus write (0)
l”~oin(m&“l d Yo
for
m = 1,2,...
~EC(O,P~),
(4.21)
This completes the proof of lo. Now 2” is easy. Let p > q and 4’~C(O,Bo). Then by (4.8a) and uniform boundedness (4.21) lim Ifln P14-‘Q,
n=O
YN-Yfc” (PJO
YNl=
n=O
(pYN ) -
lim lfc’ p,q+w
(4.“)
n=q
.n
&n n=q
YNI=O. (q?o
sequence in BN for 5 E C(O,j?,) and the series (4.8b) C”(q;lN}ym= 1 is a Cauchy n=O 3 converge in norm topology on !?jN within C(O,p,). The matrix elements of (4.1) thus converge almost uniformly in the open circle C(O,/?,) and are thus analytic functions of p in C(O,flo). This proves validity of the following
Thus { f
THEOREM. of
N
The equation (2.9) is a necessary conditionfor the minimum offree energv fermions on the set of all density operators of the for~r~
interacting
wh ere h’ is afinite-dimensional l-particle operator on 9JI. (2.9) has a unique analytic solution for ~EC(O,P~) in the form of a power series
Z-‘exp(-PNTTh’)
NTyX1(j3)
= N f
/Y’l-7X’ (a)
n=O
with X l’s satisfying (4.2), convergent in norm topology on !?jNfor 5 E C(O,p,), where PO is (n) the absolute maximum of the function
/3,(y)
on the interval (IX”l,co). (0) w4Po)*
= ( y-‘In
y-IXNI
(0)
1+ Y+
lUNI- ENI > (0)
7he matrix elements of (4.1) are analytic functions
of p in
Direct calculation of the minimum value of free energy (2.3a) on the set of density operators of the form Z-‘exp(-PNT’;Jh’)
APPROXIMATION
TO THE CANONICAL
DENSITY
OPERATOR
165
yields Rfi[p(XN),HN]
= -j-llnTr
exp(-@VryX
l(B)).
The theorem proved guarantees existence of solutions to (2.9) only at sufficiently high temperatures. For low temperatures one can expect other solutions, in general nonanalytic ones, which could, possibly, serve as a means of description of phase transitions in the system. Acknowledgements This work was carried out at Department of Physics, Queen Mary College, London and Institute of Physics, N. Copernicus University, Toruri, with financial support from the British Council and Polish Ministry of Sciences, Higher Education and Technology. I wish to thank Prof. S. Pruski and Prof. A. Kossakowski for reading the manuscript and many valuable suggestions. REFERENCES [l] [2] [3] [4] [S] [6] [7] [S] [9] [lo] [ll]
K. Husimi: Proc. Phys. Math. Sot. Japan, 22 (1940), 264. J. Bardeen, L.N. Cooper, J.R. Schrieffer: Phys. Rev. 108 (1957), 1175. N.N. Bogoliubov, D.N. Zubarev, Yu. A. Tserkovnikov: J. Exptl. Theoret. Phys. USSR 39 (1960) 120. J.G. Valatin: Nuovo Cimento, 7, (1958) 843. S.V. Tyablikov: Merhods of Quantum Theory of Magnetism, Nauka, Moscow 1975 (in Russian). J. von Neumann: Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932. D. Ruelle: Statistical Mechanics, W.A. Benjamin, New York, Amsterdam, 1969. J.M. Blatt: Theory of Superconductivity, Academic Press, New York and London, 1964. N.N. Bogoliubov: A Methodfor Studying Model Hamiltonians, Pergamon Press, Oxford, New York, 1972. H. Kummer: Journ. Math. Phys., 8, (1967) 2063. S. Pruski: Rept. Math. Phys., 15, (1979), 99.
REPORTS
ON MATHEMATICAL
PHYSICS
No. 2
SINGLE-PARTICLE APPROXIMATION TO THE CANONICAL DENSITY OPERATOR OF A SYSTEM OF N INTERACTING FERMIONS J. MACKOWIAK Institute of Physics, N. Copernicus
University, Toruti, Poland
(Received July 4, 1979)
A necessary condition for the minimum of free energy Tr(pHN) + pm ’ Tr(plnp) is investigated for a system of N interacting fermions on the set of all density matrices of the form p = Z- ‘exp( - /?NTyh’), where f ‘: is Kummer’s (1, N)-expansion map, and h’ is a tinitedimensional one-particle operator. It is shown that for values of /I in an open circle U(O&,) in the complex plane there exists a unique bounded analytic in fi one-particle operator X l(p) such that Z - ‘exp( - /IN r: X ‘(#I)) satisfies this condition.
1. Introduction The idea to describe a system of interacting particles by an effective single-particle hamiltonian has been frequently exploited in physics. It underlies Husimi’s theory [l] (the counterpart of Hartree-Fock theory in statistical mechanics), the theory of superconductivity [2], [3], [4] an d mean field theories of lattice systems [5]. In all such theories one seeks an effective density operator peffcorresponding to a noninteracting system which serves as a substitute for the exact density operator describing the equilibrium Gibbs state of the system. The equations for perrare derived using the well known variational principle which states that the free energy as a functional of the state of the system at fixed temperature Tand with fixed number of particles (fixed mean number of particles) reaches its minimum on the canonical (grand canonical) density operator (cf. [S], [6], [7], [S]). Th us one minimizes the free energy Y [p] of the system over a set of trial equilibrium states of systems of noninteracting particles and takes for Pea the density matrix on which S[[p] assumes minimum in this set (cf. [ 11, [3], [4], [5], [9]). Following this scheme, the present paper investigates the single-particle approximation to the canonical density operator corresponding to a finite-dimensional system of N interacting fermions. It is shown that for values of /? = (kg-l (k is Boltzmann’s constant) in an open circle C(O&,) in the complex plane there exists a unique bounded analytic in p single-particle operator X”(p) = N ry X ‘(/I) such that minimizes the free energy of this p(XN(B)) = (Tr expC-PXN(B)l)-lexp(-BXN(B))
[I551
156
J. MAkKOWIAK
system on the set of density operators ((Tr exp( - /?hN))- ‘exp( - /?hN),hN = N Tyh’ }, ry standing for Kummer’s (l&)-expansion [lo]. An equation for X “(/I) is derived and its solution discussed. 2. The free energy and necessary condition for its minimum Consider a finite-dimensional hamiltonian
system of N identical interacting
HN = NT:
fermions with the
T’ + (;)r;V”,
(2.1)
where r; (p is a non-negative integer, 0 < p < N) denotes Kummer’s mapping [lo], the domain 9(HN) of HN being the N-fold Grassmann some fixed M-dimensional (M < co) manifold of square-integrable tions. The r: expansion is defined for any self-adjoint operator BP rr;‘BP = with PAP standing for the projector system (2.1) is p,(HN) = Z-’
~^N(BP
0
pd’J4)p~N
on mAq. The canonical Z[HN]
[HN]exp(-pHN),
(p,N)-expansion product !DIANof l-particle funcon !IRAp by
density operator
= Tr exp(-/3HN),
for the
/? > 0. (2.2)
As follows from von Neumann’s theorem [6], pc(HN) is the unique solution to the following variational problem: Find a self-adjoint operator p on m”” which minimizes the free energy FD [p, H “1 F,JCP,H~]
(2.3a)
= Tr(pHN) + kTr(plnp)
with p subject to Trp=l,
(2.3b)
p>O
for any density operator p.l As discussed It follows that F-,[p,HN] B F,[p,(HN),HN] in the Introduction, the principle (2.3) may serve to select from a given class of density operators the one that resembles as close as possible the state (2.2), the measure of closeness being the value of Ffl [p, H “1 - FGs [p(H N),H “1 for given /?. It is of interest, in particular, to find the relevant density operator from the class of operators pN:=
exp(-#hVr~xl), x’ = (xl)+,
‘By a density operator
we mean any operator
TrpN = 1.
on FAN fulfilling (2.3b).
(2.4)
APPR50XIMATION TO THE CANONICAL DENSITY OPERATOR
For the trial density operator
(2.4) the functional
Tr(pNHN) - NTr(pNI’y
to be minimized
15:
is
X I) - aTrpN,
(2.5)
where a is a Lagrange multiplier. Equating to zero the variation of(2.5), one obtains the equation Tr[(HN
x exp(-/?NT~X’)r~6X’] = Tr[(exp(-BNTYX
where y is a constant
‘)(NryX1
+ yPAN)TyGX
(2.6)
and
A x expB: = A + $4B + &(fW From (2.6), by the arbitrariness
+ BA) + -&(AB2 + BAB + B2A) + + MB2
+ B2AB + B3A) + .. .
of 6 X 1 follows a necessary condition
L~(HNxexp(--/INI’~X1))=L~(exp(-/UVT~X’)(NT~X’ Tr exp(-PNryX’) where Lk denotes Kummer’s (IV,l)-contraction should be determined from (2.7b). However, irrelevant, since (A x expB)expC for any C commuting the equation
‘1,
+yPAN)),
= 1,
(2.7~~ (2.7b)
operation [lo]. The constant y in (2.7a) the presence of y in (2.7a) is, in fact,
= A x exp(B + C)
with A and B, and thus X ’ may be equivalently
Lk(HN x exp(-/?NfyX’))=
for X ‘:
(2.8) determined
L~(Nr~X’exp(-/?NTyX’))
from
(2.9)
pN is then given by pN = (Tr exp(-/?NT~X1))-lexp(-/UVr~X1). A possible approach to eq. (2.9) is offered by a self-consitent iterative procedure on X ’ (fix X r in the exponentials, solve for X ’ on the r.h.s., substitute the solution into the exponentials, etc.) or by expansion of X ’ into a power series in fi and comparison of coefficients on both sides. In any case, one way of dealing with the resulting equations is to resort to S. Pruski’s decomposition of a p-particle operator ~~r~Bp into its 0 - , 1 - , 2- , . . . . p-particle projections. A brief outline of this decomposition is given in the next section. 3. The q-particle projection of a p-particle operator The linear operators on !WN constitute Hilbert-Schmidt (H.S.) scalar product
a linear space bN. Equipped
with the
158
J. MACKOWIAK
QX, Y) = Tr(X + Y),
X, YE$jN,
(3.1)
eN is a Hilbert space. One can distinguish in $JNreal subspaces Big of self-adjoint operators (r)T: P (0 < p i N). Obviously, 5: c 9,” for 4 < p. According to the theorem on orthogonal projection in Hilbert space, there exists a unique element (i)riXq~ !# which is nearest to a given (:)rI yP~$j: (q < p), with respect to the H.S. distance: (Tr((F)r:
p - (y)rfXq)2)"2
=
hfq(Tr((;)Tf p s
zNJ2)lj2 (3.2)
X4 can be found by solving the equation (;)W;
y”) = (;)J%(f:Xq)
(3.3)
which follows from necessary conditions for the minimum in (3.2): 6Tr((F)rF
P - (i)rtXq)’
= 0,
(3.4)
where variation is with respect to X4. The solution (:)rrXq to (3.3) is unique and is the orthogonal projection of (,“)rr Yponto !#. (3.3) may be also restated in the form Tr(BqL;G((F)r:P ’
- (t)riXq))
= 0,
VBq~fii.
For q = 1 the solution to (3.3) can be found using the theorem proved in [ll]:
In particular, the l-particle projection of the hamiltonian (2.1), HN = I’r;‘[NT: T’ + (t)V’],
is XN = NT:%‘, (0)
where N-l (Tr p)P’. (M - l)(M - 2)
4. The euation Li(HN x exp(-/?NTyXr))
(3.6)
=‘L~(N~I;YXlexp(-BNT~X’))
Suppose the solution NryX ‘(8) to eq. (2.9) is sought as a power series in /?. To determine the form of the series let us resort to the exact solution (2.2) of the variational problem (2.3). Obviously, the expansion of exp( - 4jN) in fl involves only non-negative powers of fi and s-limexp( -j?HN) = PAN. Analogous requirements imposed on exp (-/INTyX’)
dis!ki&z the form of NryX’(p): NT;X’@)
= N f fl”ryX’, n=O (W
(4.1)
APPROXIMATION
where
X1, Xl,... (0) (1)
TO THE CANONICAL DENSITY OPERATOR
etc., are operators
expansion
exp( -/?NTyX
equations
for the X1’s: (“)
in YJI. Let
XN:= Nf 7X’. (n) (n)
‘) = f (n!)-‘( -BNTyX n=O
NL#‘;JX’) (n)
= L;RN, Cd
159
Insertion
of the
‘)” and then (4.1) into (2.9), yields
n=Of23 , ,-..,
(4.2)
where RN=HN, (0) RN=-(XNHN+HNXN)+ (k-1) W
(k-1)
s~~~i~o~,z,,X-~...XNHN +c I -
i
k$
m=l
C ” XNX:‘T .
(4.3a)
XN . ..XNb(l.+...+lS-k+s)+(4.3b) i
x:‘L):
+ .:I’: l,,,+l -k
+m),
k = 1,2,...
11...1,+ 1 (11) (12) (‘?n+1)
It follows from (4.2) that the XN*s are orthogonal (n) there exists a unique solution
projections
of the RN,s onto 8,‘. Thus W)
(4.1) to eq. (2.9). For n = 0 (4.2) reads Lh XN = LAH”, (0)
proving that ,:” = N TYX’ is the projection of HN on $i,’ and is thus given by (3.6). (0) It will be now proved that the solution (4.1) with XN’s satisfying (4.2) is bounded V0 and analytic in /3 in some neighbourhood of fi = 0 in the complex plane, the analyticity of X N(fi)~!+jN being understood as analyticity of (40,XN(p)$) for any cp,ll/~)IJ”~. To this end let us introduce the following inner product of operators on bN: for fixed X&r, (A,@,:=
Tr((A+
x exp X)B)(Tr
exp X)-l,
A,BE!$‘.
One easily checks that I(A,B),I < 00 for A,BEJS~ and that (A,B), = (B,A),. Linearity with respect to the second argument and additivity with respect to both are immediate. To prove that (A,A), > 0 for A ~0, it suffices to use (2.8): (A,B), = (A,B),e”e-”
=
(ApB)X+apA~T
.For CIsuch that X + aPhN > 0, all terms in the expansion
aER’. of (A,A)x+apAN are positive:
160
J. MACKOWIAK
kw,+,.N
=
(TrA+A + $TrA+(X + aP~$4
+
+ $TrA(X + aP~$4’
+ ...)(Trex+aPAN)- 1
proving that (A,A), > 0 for A #O. One can now set up an iterative procedure on X N(p),the solution to (2.9).In the first step take arbitrary
YN~$f and find YNe$,l from the condition: (0) (1) (HN - YN,HN - YN)_-RYY = inf, (1)
(1)
/I > 0,
id,
Next find YNefif satisfying (2) = inf (HN - YN, I-IN - UN) _BYN (2) (1) (2) and so on. At each stage one has to solve an equation for YN = NT: Y~E!#: (m) (m) G&-IN x exp(+&Yr,))
=
m = 1,L.
(4.4)
which is necessary and sufficient for (HN-
Obviously, the solution
= inf. YN, HN - YN) (In) $-z (m)
(4.5)
Y”(p) exists and is unique at each stage. We shall prove that
for values of /3 in some%osed circle c(O,p,) in the complex plane with radius Be, centered at the origin, it can be given as a power series YN(j3)= i /3k Y N (m.k) k=Q 0@
(4.6)
convergent in norm topology on eN. Inserting
YN(fi)and UN (p) in the form of Mn) (rn-1) expansions (4.6) into eq. (4.4) and comparing coefficients at the same powers of p, one arrives at the following equations for the YN’sf!$: (m.q) L,:p =L;HN ow
(4.7a)
APPROXIMATION
161
TO THE CANONICAL DENSITY OPERATOR
and for 4 = 1, 2,...
c n=l(n+l)!~=lk=OOGp
where UN is the 2-particle consecutive
equations
projection
Y N Y N... P&p, VW
Ptt,
+ ...p” - q + n + r)
PI
(4.7b)
of HN and
(4.7a, b) for P(#?), p@),... 1 2
P E P . A simple analysis of s (m- 1,s) shows that for given m, YN does not (0)
for p’s q = O,l,..., m - 1 and that the latter are identical to the (m.q)
enter into the equations corresponding
YN...P P k I”’PP“1
equations
(4.3) on XNVs.As a consequence, (4) P = XN, @MI) (n)
4 = O,l,..., m - 1
(4.8a)
and lim lim
i
/3’ Y N = lim
k+4)mdwqq0
in norm topology
(W
on $jN, provided
k-co
i q=O
pq XN
(4.8b)
(4)
both limits exist. Now $jN is a finite-dimensional
Banach space. Thus to prove that 1 f
[’ XNI < 0~)in some norm 1 1on BN and 5 lying
k=O
(k)
within some circle c(OJ?r), it suffices to show that
is a Cauchy sequence for any ~~~(0,/?r). We turn to the proof of lo. Let I I denote the H.S. norm. A bound on I f
(k Y NI
k=O
can be obtained by noting that the H.S. no trm of the l-particle ZN~J3,N does not exceed IZN(: lYNl d
lZNl
projection
(m,k)
PE!$:
of
(4.9)
and that (4.10)
162
J. MA&OWIAK
Using the inequalities (4.9), (4.10), one finds from (4.7b)
Denote S,(5) = i jcf”j p 1.Then for 5 satisfying /[I < S,_ l([)- ‘ln2 we may write n=O @RN
i
111”/Y Ni 6
q=1
O%qt
/UN! k $ c l[lPI*-,*+q PI...{ PI Pi=1 O<:p,...p*
\lJpE+..~+p”IYy_.I Yy p, P”
(4.11) *
Taking the limit k -+ cm in (4.1I), we get
luNl(exp(tiis,-,(r))- 11+ lXNl 2 - ew&XL- XI) (0) for
/Cl< S,_,(l)-‘In2
(4.12)
proving that 1 C C4P 14 00 for i: satisfying /[I-C S,_ I([)-1~n2. The inequality (4.12) and the arbitrariness of YNnow suffice to demonstrate validity of 1”: Take YNsuch (0) (0) that i?YN
I YNI> ‘$7, (0)
(0) = 0. ag
(4.13)
Then according to (4.12)
I ~Nltew(lCllc;Nl) - 1) ~163= g 151”l p I< n=O
(1.W
--+/XNI
2 - exp(l~ll~~Nl)
(0)
(4.14)
for ICI< 1 YNj-11n2. Furthermore, since I YNI> \XN\, there exists a circle C(O$) such (0) (0) (0) that
APPROXIMATION
TO THE CANONICAL DENSITY OPERATOR
I W~WW$Nl)
for
+ l(:Nl G I YNI (0)
ew(lCll yNI) (0)
2(4.15) is equivalent
- 1)
kW4P1).
163
(4.15)
to
lil < lt~Nlplln 1 + (
I YNI - I$“’ (0)
(4.16)
l,;Nl + I UN1- If”1 >*
Thus
PI = /(zN(-‘ln
The condition
l,fl (
- lZNl < 1YNI-rln2.
1+
I YNI+ I UN1- I(;“’ (0)
>
(4.17)
(0)
l&J< YNI-‘ln2 relating to (4.14) may be therefore dropped if values of c (0)
are confined
to C(O,p,):
I”~oIn(lY)NIG I YNl .” (0)
for
(4.18)
l”~o~“,,‘.,“ld I YNl (0)
for
(4.19)
By (4.12) (4.15) (4.18)
and similarly at each stage:
l”$ocn(~)lG I YNI
for
@C(O$,),
(4.20)
m = 1,2)...
(0)
proving that the sequence
{ c k=O
ikCTb >z= 1 is uniformly IPI,
bounded
by I YNI within the (0)
circle c(O,fi,). The value of I(:“1 may now be chosen so as to maximize #I1 = PI (l,f ThisispossiblesinceB,(I
YNl)in(4.17) is a continuous (0)
I).
positive function on the interval
164
J.MAcKOWIAK
(1XN 1,co), vanishing at both boundary 10)
points, and therefore has an absolute maximum
Do at some value y, of 1YNI. Instead of (4.20) we may thus write (0)
l”~oin(m&“l d Yo
for
m = 1,2,...
~EC(O,P~),
(4.21)
This completes the proof of lo. Now 2” is easy. Let p > q and 4’~C(O,Bo). Then by (4.8a) and uniform boundedness (4.21) lim Ifln P14-‘Q,
n=O
YN-Yfc” (PJO
YNl=
n=O
(pYN ) -
lim lfc’ p,q+w
(4.“)
n=q
.n
&n n=q
YNI=O. (q?o
sequence in BN for 5 E C(O,j?,) and the series (4.8b) C”(q;lN}ym= 1 is a Cauchy n=O 3 converge in norm topology on !?jN within C(O,p,). The matrix elements of (4.1) thus converge almost uniformly in the open circle C(O,/?,) and are thus analytic functions of p in C(O,flo). This proves validity of the following
Thus { f
THEOREM. of
N
The equation (2.9) is a necessary conditionfor the minimum offree energv fermions on the set of all density operators of the for~r~
interacting
wh ere h’ is afinite-dimensional l-particle operator on 9JI. (2.9) has a unique analytic solution for ~EC(O,P~) in the form of a power series
Z-‘exp(-PNTTh’)
NTyX1(j3)
= N f
/Y’l-7X’ (a)
n=O
with X l’s satisfying (4.2), convergent in norm topology on !?jNfor 5 E C(O,p,), where PO is (n) the absolute maximum of the function
/3,(y)
on the interval (IX”l,co). (0) w4Po)*
= ( y-‘In
y-IXNI
(0)
1+ Y+
lUNI- ENI > (0)
7he matrix elements of (4.1) are analytic functions
of p in
Direct calculation of the minimum value of free energy (2.3a) on the set of density operators of the form Z-‘exp(-PNT’;Jh’)
APPROXIMATION
TO THE CANONICAL
DENSITY
OPERATOR
165
yields Rfi[p(XN),HN]
= -j-llnTr
exp(-@VryX
l(B)).
The theorem proved guarantees existence of solutions to (2.9) only at sufficiently high temperatures. For low temperatures one can expect other solutions, in general nonanalytic ones, which could, possibly, serve as a means of description of phase transitions in the system. Acknowledgements This work was carried out at Department of Physics, Queen Mary College, London and Institute of Physics, N. Copernicus University, Toruri, with financial support from the British Council and Polish Ministry of Sciences, Higher Education and Technology. I wish to thank Prof. S. Pruski and Prof. A. Kossakowski for reading the manuscript and many valuable suggestions. REFERENCES [l] [2] [3] [4] [S] [6] [7] [S] [9] [lo] [ll]
K. Husimi: Proc. Phys. Math. Sot. Japan, 22 (1940), 264. J. Bardeen, L.N. Cooper, J.R. Schrieffer: Phys. Rev. 108 (1957), 1175. N.N. Bogoliubov, D.N. Zubarev, Yu. A. Tserkovnikov: J. Exptl. Theoret. Phys. USSR 39 (1960) 120. J.G. Valatin: Nuovo Cimento, 7, (1958) 843. S.V. Tyablikov: Merhods of Quantum Theory of Magnetism, Nauka, Moscow 1975 (in Russian). J. von Neumann: Mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932. D. Ruelle: Statistical Mechanics, W.A. Benjamin, New York, Amsterdam, 1969. J.M. Blatt: Theory of Superconductivity, Academic Press, New York and London, 1964. N.N. Bogoliubov: A Methodfor Studying Model Hamiltonians, Pergamon Press, Oxford, New York, 1972. H. Kummer: Journ. Math. Phys., 8, (1967) 2063. S. Pruski: Rept. Math. Phys., 15, (1979), 99.