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Variational principle for dynamic susceptibilities
Physica 98A (1979) 352-358 @ North-Holland
VARIATIONAL
PRINCIPLE
Publishing
Co.
FOR DYNAMIC
SUSCEPTIBILITIES*
G. SAUERMANN Theoretische FestkSirperphysik, Institut fiir Festktirperphysik, Technische Hochschule Damstadt, D-6100 Damstadt, Germany
Received 2 April 1979
It is pointed out that the Schwinger variational principle of scattering theory directly applies to the case of dynamic susceptibilities in statistical mechanics. The relation to other dynamic approximation schemes is discussed.
1. Introduction
Variational principles are a powerful tool for calculating physical quantities. In Quantum mechanics of pure states one can use the Ritz principle to determine bound states and among others the Schwinger principle to get approximate phase shifts. For temperature dependent problems, however, there is just a minimum principle for the free energy. A variational principle for time dependent response functions has not been established. It is the purpose of this paper to point out that dynamic susceptibilities can be obtained from a Schwinger type variational principle. To derive it, no new mathematical theory is needed. Use is made of the fact that susceptibilities can be rewritten as resolvents. Thus results from scattering theory and Pad6 approximations directly apply. The physical meaning of the corresponding quantities, however, is entirely different. To discuss the variation principle in statistical mechanics connection is made to other standard approaches. This will give some feeling, how to find a suitable ansatz for dynamic susceptibilities in complicated cases.
2. Stationary variational
principle
Consider a system with Hamiltonian 2 driven by an external disturbance -X,&r,(t). The response of the observable & is given by the dynamic susceptibility x,(w). The Kubo formula for ,yvfican be expressed in different * This work was performed Frankfurt.
within the program of the Sonderforschungsbereich
352
65 Darmstadt-
VARIATIONAL
PRINCIPLE FOR DYNAMIC
SUSCEPTIBILITIES
353
ways. For our purpose it is essential to use the Liouville space formulation’) m
x,(w)
=
I
ei(W+iO’p(iL&, ei”&J,
(1)
0
where the Liouvillian
L and the scalar product are defined by B
Ld = f [X, d],
(9, ~4) = d
I
(enx9+ e-AKd),qu.
(2)
0
Regarding L = Lt and setting w + ie = z the susceptibility
is rewritten
as
(3) Therefore
it suffices to establish a variational
principle for the resolvent
In principle one has the same mathematical task in scattering theory’). Hence we can transfer the results obtained there to statistical mechanics and write down a Schwinger type functional F”J4’,
41= (4’9 J&J + (A 4) - (4’. (2 + L)4),
which has to be varied with respect to 4’ and 4. It is a simple matter to show that F, is stationary
4st = (2 + L)-‘dp, and the stationary
EJ4;t,
for
41t = (2 + L)-‘+&
(6)
value of F, is given by
4stl= (~6, (z + L)-‘dJ
= G,(z)
For the derivation it is important formulation, which implies (.%?,.%)=O
(9
ifandonlyif
.92=0.
that we have used the vector
(7) space
(8)
Starting from the functional (5) and a suitable ansatz for 4 and 4’ one can find approximations for G,(Z) i.e. the susceptibility. This is practically possible if one has some idea in which part of the Liouville space the stationary values (6) of 4 and 4’ will be. Then this subspace can be parameterized and the parameters can be determined from SF, = 0. Common
G. SAUERMANN
354
to all variational principles the accuracy of the results will crucially depend on the chosen subspace.
3. Connections
to other approaches
3.1. Relation to perturbation
theory
A simple test for the functional decomposition of the Liouvillian
F,,[4’, 41 can be made if there
is a
L, = L’O’+ I,“’ 7
(9)
where L’” is some small perturbation to L(O). According to the stationary values of 4 and (6’ (6) it is suggestive to choose 4 = (2 + I?‘)-‘&
4’ = (2 + L’o’)-‘+&,.
(10)
This leads to F,[&‘, (b] = (a,, (2 + I?‘)-‘.&)
- (a”, (2 + L’O’)-‘L”‘(z + I?‘)-‘sQ,)
= G:;(Z) + G:;(z),
(11)
which coincides with the result of first order perturbation theory for G,(z). If I,“’ is not totally fixed, but depends on some unknown parameters L”‘(C), it follows from (5) that we can find the “best values” for C by just varying the first order perturbation expression (11) with respect to C. In the general case the result for C will depend on &,, and &,, and the frequency z. Such a procedure of finding L(O)is demonstrated in the appendix. There the simple case of an isotropic Heisenberg ferromagnet is considered where L’“‘(&r) is chosen to describe independent spins in an unknown magnetic field H,e. The general spin wave approximation is obtained. 3.2. Relation to Pade’ approximations Consider a system L = L(O)+ gL”’ with fixed L(O)which leads to an expansion of G, (4), where just (z + L(O)+ gL(‘))-’ is expanded
G,(z) =
nf.o G$kn.
It has generally been provenzS4) that PadC approximations
(12) for Taylor series (13)
VARIATIONAL
with coefficients
PRINCIPLE FOR DYNAMIC SUSCEPTIBILITIES
355
of the form (14)
a, = (@IK”M)
in a suitable space can be obtained from a Schwinger type variational principle. Therefore, we need not discuss in detail the possibility of obtaining PadC approximations from the functional F, (5). For completeness we just want to consider the simplest possibility. According to (6) one is lead to take the ansatz
where C and C’ are variational parameters. notation (11) the functional (5) reads F,
= (C’* + C)G:;
Its stationary
- C’*C(Gf;
It is easy to see that in the
- G:;).
(16)
value is found to be (17)
which is the [l, 0] PadC approximation for G,. Considering just a diagonal element and comparing it with the exact expression for G,(z) which reads in the projection operator formalism3) 1
Gv = (do 4) 2 + 0, - i&(z) one notices that the stationary
(18)
value (Fm),, (17) can be written as
1 (FW)st = l/G:: _ Glt’/(G@)i
=
2 +
0, -
(,cp,,4) i@)(z) - MZ”(2)’
where 0 Y= @O’+ Y R(l) ”7
(19)
(20)
and M”(Z) = M!? + MZ” + . . .
(21)
denote the Taylor expansions with respect to g.* This shows that the simple choice (15) for 4 and 4’ corresponds to a first order expansion of the memory function M(z). * The scalarproduct is to be kept fixed.
G. SAUERMANN
356
3.3. Relation
to the general Mori-Zwanzig-theory
Let ~4, be a set of observables such that linear combinations of them lead to approximate eigenvectors of L. Then according to the desired expressions (6) for 4 and 4’ it is reasonable to take the ansatz
(22) and to determine the matrices c and c’ from the stationary condition of FyIL[+ 4’1. From a mathematical point of view this calculation is contained in the theory of matrix PadC approximations*) and we could specialize the result to our case. There is no difficulty, however, to write it down directly. Inserting the ansatz (22) into (5) we find in standard matrix notation for Fvp F[c, c’] = c’+(Y + (YC- c’+ Vc,
(23)
where qW = (&, J&7), v,
= (&, (z + L)&).
(24)
Variation of (23) with respect to c and c’ leads to the stationary
value of F
(F),, = aV-‘a,
(25)
which can be cast into the form
where P is the projector p =
onto the subspace spanned by the set &
I: l~“)WLLLW,l,
(27)
%cL
P2=P,
P=P+.
(28)
Let us compare this result for (F,,),, with the exact expression in the projector formalism of Mori3) G,(z) =
(A z + PLP
- PLQ[z
+ QLQ]-‘QLP
&’ > ’
for G,, written
(29)
where Q=l-P.
(30)
The last term of the denominator in (29) accounts for the memory effects. Inspection of the approximation (26) and the exact expression (29) shows that an ansatz of the type (22) in the variational principle neglects memory effects,
VARIATIONAL
PRINCIPLE FOR DYNAMIC SUSCEPTIBILITIES
357
but gives an exact description of the systematic part of dynamics. It is well known that approximations of the type (26) are appropriate in many cases, even to describe damping phenomena if the set of d, becomes infinite.
4. Conclusion It has been shown that the Schwinger variational principle of scattering theory is suited for statistical mechanics, where it allows a variational calculation of dynamic susceptibilities. From the connection with standard approaches it was concluded which kind of terms should enter into a reasonable ansatz. The variational method has the advantage that different physical aspects of approximation can be considered simultaneously. Thus the method might especially be useful, when the ansatz for different limiting cases is known and one wants to treat an intermediate situation.
Appendix
Consider xe,
an isotropic
Heisenberg
ferromagnet
with exchange
%’ = - yHS + 2fex,
interaction
(A.1)
where S’ =
5 s;.
(-4.2)
j=l
Let us denote by tie’ the Hamiltonian effective magnetic field Hea
of independent
spins in an unknown
tie) = - yH,&.
(A.3)
For the calculation of the transverse
susceptibility
we start from (1 l), where now
& = dfi = Sz = 3 eiqRiSf. j=l If the Liouvillian
corresponding
(A.4) to tie’ is called L(O)then
L”‘S,+ = - yH,&
holds. Since % commutes space
(A.3
with ti”,
it follows that L(O)is hermitean
in Liouville
G. SAUERMANN
358 L’W
= L(O).
64.6)
Making use of (A.5) and (A.6) and the identity 1 (J4 LB) = hp ([.@, a11
(A.7)
one finds for (11) F&G &)
= ‘3%
Ka) + Gb’)(z, EL,)
= (S& Si) (Z - ;& +$(S’) This function is stationary Y&T
=
2
l (2 - rK-fi)2.
b4.8)
for
w>
(A.9)
P(s;v %I
and its stationary (m
- (z - ;&*)z)
value is given by
1 = (St, s3 * _ 2[(s’)/p(s;,
where (A.9) is the general expression
Si)]’
(A. 10)
for the spin wave frequency3).
References 1) 2) 3) 4)
R. Kubo, Rep. Progr. in Phys. 29 (1966) 255. G. Turchetti, Fortschritte d. Physik 26 (1978) 1. H. Mori, Progr. Theor. Phys. 33 (1%5) 423. J. Nuttall, The Pad6 Approximant in Theoretical Physics (Academic Press, New York, 1970) p. 219.
VARIATIONAL
PRINCIPLE
Publishing
Co.
FOR DYNAMIC
SUSCEPTIBILITIES*
G. SAUERMANN Theoretische FestkSirperphysik, Institut fiir Festktirperphysik, Technische Hochschule Damstadt, D-6100 Damstadt, Germany
Received 2 April 1979
It is pointed out that the Schwinger variational principle of scattering theory directly applies to the case of dynamic susceptibilities in statistical mechanics. The relation to other dynamic approximation schemes is discussed.
1. Introduction
Variational principles are a powerful tool for calculating physical quantities. In Quantum mechanics of pure states one can use the Ritz principle to determine bound states and among others the Schwinger principle to get approximate phase shifts. For temperature dependent problems, however, there is just a minimum principle for the free energy. A variational principle for time dependent response functions has not been established. It is the purpose of this paper to point out that dynamic susceptibilities can be obtained from a Schwinger type variational principle. To derive it, no new mathematical theory is needed. Use is made of the fact that susceptibilities can be rewritten as resolvents. Thus results from scattering theory and Pad6 approximations directly apply. The physical meaning of the corresponding quantities, however, is entirely different. To discuss the variation principle in statistical mechanics connection is made to other standard approaches. This will give some feeling, how to find a suitable ansatz for dynamic susceptibilities in complicated cases.
2. Stationary variational
principle
Consider a system with Hamiltonian 2 driven by an external disturbance -X,&r,(t). The response of the observable & is given by the dynamic susceptibility x,(w). The Kubo formula for ,yvfican be expressed in different * This work was performed Frankfurt.
within the program of the Sonderforschungsbereich
352
65 Darmstadt-
VARIATIONAL
PRINCIPLE FOR DYNAMIC
SUSCEPTIBILITIES
353
ways. For our purpose it is essential to use the Liouville space formulation’) m
x,(w)
=
I
ei(W+iO’p(iL&, ei”&J,
(1)
0
where the Liouvillian
L and the scalar product are defined by B
Ld = f [X, d],
(9, ~4) = d
I
(enx9+ e-AKd),qu.
(2)
0
Regarding L = Lt and setting w + ie = z the susceptibility
is rewritten
as
(3) Therefore
it suffices to establish a variational
principle for the resolvent
In principle one has the same mathematical task in scattering theory’). Hence we can transfer the results obtained there to statistical mechanics and write down a Schwinger type functional F”J4’,
41= (4’9 J&J + (A 4) - (4’. (2 + L)4),
which has to be varied with respect to 4’ and 4. It is a simple matter to show that F, is stationary
4st = (2 + L)-‘dp, and the stationary
EJ4;t,
for
41t = (2 + L)-‘+&
(6)
value of F, is given by
4stl= (~6, (z + L)-‘dJ
= G,(z)
For the derivation it is important formulation, which implies (.%?,.%)=O
(9
ifandonlyif
.92=0.
that we have used the vector
(7) space
(8)
Starting from the functional (5) and a suitable ansatz for 4 and 4’ one can find approximations for G,(Z) i.e. the susceptibility. This is practically possible if one has some idea in which part of the Liouville space the stationary values (6) of 4 and 4’ will be. Then this subspace can be parameterized and the parameters can be determined from SF, = 0. Common
G. SAUERMANN
354
to all variational principles the accuracy of the results will crucially depend on the chosen subspace.
3. Connections
to other approaches
3.1. Relation to perturbation
theory
A simple test for the functional decomposition of the Liouvillian
F,,[4’, 41 can be made if there
is a
L, = L’O’+ I,“’ 7
(9)
where L’” is some small perturbation to L(O). According to the stationary values of 4 and (6’ (6) it is suggestive to choose 4 = (2 + I?‘)-‘&
4’ = (2 + L’o’)-‘+&,.
(10)
This leads to F,[&‘, (b] = (a,, (2 + I?‘)-‘.&)
- (a”, (2 + L’O’)-‘L”‘(z + I?‘)-‘sQ,)
= G:;(Z) + G:;(z),
(11)
which coincides with the result of first order perturbation theory for G,(z). If I,“’ is not totally fixed, but depends on some unknown parameters L”‘(C), it follows from (5) that we can find the “best values” for C by just varying the first order perturbation expression (11) with respect to C. In the general case the result for C will depend on &,, and &,, and the frequency z. Such a procedure of finding L(O)is demonstrated in the appendix. There the simple case of an isotropic Heisenberg ferromagnet is considered where L’“‘(&r) is chosen to describe independent spins in an unknown magnetic field H,e. The general spin wave approximation is obtained. 3.2. Relation to Pade’ approximations Consider a system L = L(O)+ gL”’ with fixed L(O)which leads to an expansion of G, (4), where just (z + L(O)+ gL(‘))-’ is expanded
G,(z) =
nf.o G$kn.
It has generally been provenzS4) that PadC approximations
(12) for Taylor series (13)
VARIATIONAL
with coefficients
PRINCIPLE FOR DYNAMIC SUSCEPTIBILITIES
355
of the form (14)
a, = (@IK”M)
in a suitable space can be obtained from a Schwinger type variational principle. Therefore, we need not discuss in detail the possibility of obtaining PadC approximations from the functional F, (5). For completeness we just want to consider the simplest possibility. According to (6) one is lead to take the ansatz
where C and C’ are variational parameters. notation (11) the functional (5) reads F,
= (C’* + C)G:;
Its stationary
- C’*C(Gf;
It is easy to see that in the
- G:;).
(16)
value is found to be (17)
which is the [l, 0] PadC approximation for G,. Considering just a diagonal element and comparing it with the exact expression for G,(z) which reads in the projection operator formalism3) 1
Gv = (do 4) 2 + 0, - i&(z) one notices that the stationary
(18)
value (Fm),, (17) can be written as
1 (FW)st = l/G:: _ Glt’/(G@)i
=
2 +
0, -
(,cp,,4) i@)(z) - MZ”(2)’
where 0 Y= @O’+ Y R(l) ”7
(19)
(20)
and M”(Z) = M!? + MZ” + . . .
(21)
denote the Taylor expansions with respect to g.* This shows that the simple choice (15) for 4 and 4’ corresponds to a first order expansion of the memory function M(z). * The scalarproduct is to be kept fixed.
G. SAUERMANN
356
3.3. Relation
to the general Mori-Zwanzig-theory
Let ~4, be a set of observables such that linear combinations of them lead to approximate eigenvectors of L. Then according to the desired expressions (6) for 4 and 4’ it is reasonable to take the ansatz
(22) and to determine the matrices c and c’ from the stationary condition of FyIL[+ 4’1. From a mathematical point of view this calculation is contained in the theory of matrix PadC approximations*) and we could specialize the result to our case. There is no difficulty, however, to write it down directly. Inserting the ansatz (22) into (5) we find in standard matrix notation for Fvp F[c, c’] = c’+(Y + (YC- c’+ Vc,
(23)
where qW = (&, J&7), v,
= (&, (z + L)&).
(24)
Variation of (23) with respect to c and c’ leads to the stationary
value of F
(F),, = aV-‘a,
(25)
which can be cast into the form
where P is the projector p =
onto the subspace spanned by the set &
I: l~“)WLLLW,l,
(27)
%cL
P2=P,
P=P+.
(28)
Let us compare this result for (F,,),, with the exact expression in the projector formalism of Mori3) G,(z) =
(A z + PLP
- PLQ[z
+ QLQ]-‘QLP
&’ > ’
for G,, written
(29)
where Q=l-P.
(30)
The last term of the denominator in (29) accounts for the memory effects. Inspection of the approximation (26) and the exact expression (29) shows that an ansatz of the type (22) in the variational principle neglects memory effects,
VARIATIONAL
PRINCIPLE FOR DYNAMIC SUSCEPTIBILITIES
357
but gives an exact description of the systematic part of dynamics. It is well known that approximations of the type (26) are appropriate in many cases, even to describe damping phenomena if the set of d, becomes infinite.
4. Conclusion It has been shown that the Schwinger variational principle of scattering theory is suited for statistical mechanics, where it allows a variational calculation of dynamic susceptibilities. From the connection with standard approaches it was concluded which kind of terms should enter into a reasonable ansatz. The variational method has the advantage that different physical aspects of approximation can be considered simultaneously. Thus the method might especially be useful, when the ansatz for different limiting cases is known and one wants to treat an intermediate situation.
Appendix
Consider xe,
an isotropic
Heisenberg
ferromagnet
with exchange
%’ = - yHS + 2fex,
interaction
(A.1)
where S’ =
5 s;.
(-4.2)
j=l
Let us denote by tie’ the Hamiltonian effective magnetic field Hea
of independent
spins in an unknown
tie) = - yH,&.
(A.3)
For the calculation of the transverse
susceptibility
we start from (1 l), where now
& = dfi = Sz = 3 eiqRiSf. j=l If the Liouvillian
corresponding
(A.4) to tie’ is called L(O)then
L”‘S,+ = - yH,&
holds. Since % commutes space
(A.3
with ti”,
it follows that L(O)is hermitean
in Liouville
G. SAUERMANN
358 L’W
= L(O).
64.6)
Making use of (A.5) and (A.6) and the identity 1 (J4 LB) = hp ([.@, a11
(A.7)
one finds for (11) F&G &)
= ‘3%
Ka) + Gb’)(z, EL,)
= (S& Si) (Z - ;& +$(S’) This function is stationary Y&T
=
2
l (2 - rK-fi)2.
b4.8)
for
w>
(A.9)
P(s;v %I
and its stationary (m
- (z - ;&*)z)
value is given by
1 = (St, s3 * _ 2[(s’)/p(s;,
where (A.9) is the general expression
Si)]’
(A. 10)
for the spin wave frequency3).
References 1) 2) 3) 4)
R. Kubo, Rep. Progr. in Phys. 29 (1966) 255. G. Turchetti, Fortschritte d. Physik 26 (1978) 1. H. Mori, Progr. Theor. Phys. 33 (1%5) 423. J. Nuttall, The Pad6 Approximant in Theoretical Physics (Academic Press, New York, 1970) p. 219.