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Construction of Propagators for Quantum Crystals.

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-1nnalen der Physik. 7. Folge, Band 38, Heft 6, 1981, S. 419-433
J. A. Barth, Leipzig
Construction of Propagators for Quantum Crystals
Department of Theoretical Physics,.University of Oxford, Oxford, U.K.
Abstract. We present here a mathematical basis of the iterative procedure developed earlier
by the author for strongly anharmonic crystals. We generalize the approach to the case of any strongly
nonideal nonuniform system. The cofivergence of this procedure is defied as convergence in the
mean with respect to averages of observable-quantity operators. The definitions yield self-consistent
conditions for a zero-order iteration in a natural way. Divergences in high-order iterations are
considered and two ways to eliminate them are proposed. The method is applied to a stmnglyanisotropic strongly-anharmonic quantum crystal.
Konstruktion von Propaqatoren in der Quantentheorie der Kristalle
Inhaltsubersicht. Im folgenden wird die mathematische Grundlage fiir das vom Autor schon
friiher entwickelte iterative Verfahren zur Beschreibung stark anharmonischer Kristalle dargestellt.
Wir verallgemeinern die Methode auf den Fall beliebiger stark nichtidealer und niohteinheitlicher
Systeme. Die Konvergenz dieses Verfahrens ist definiert als eine Konvergenz im Nittel beziiglich der
Mittelwverte von Operatoren beobachtbarer GroBen. Diese Definition fiihrt auf natiirlichem Wege
zu in sich widerspruchsfreien Bedingungen bei einer Iteration Nullter Ordnung. Divergenzen bei der
Iteration in hoherer Ordnung werden ebenfalls betrachtet und 2 Wege zu ihrer Eliminierung vorgeschlagen. Die Methode wird angewandt auf die Quantentheorie stark anisotropischer nnd anharmonischer Kristalle.
1. Introduetion
A most common approach in the quantum theory of crystals i s the phonon technique
[1-51, which gives the possibility of considering weakly-interacting phonon excitations
instead of strongly-interacting basic particles. ,4s a matter of fact, we cannot completely
avoid a consideration of the latter, since we have to take into account interparticle
correlations [6], because real interaction potentials [7- 101 usually contain hard cores.
The properties of a ground state (at zero temperature) can be investigated using various
wave-function methods [ll- 161. A slightly modified phonon technique is applicable
to crystals with a small number of defects [ 17- 191 and to some crystals near structuraltransition points [20]. It is possible to extend the description of different collective excitations in crystals [21] to amorphous solids [22, 231 with the help of Green functions
However, all methods mentioned above are not useful far describing a ground state
when dealing either with strongly disordered solid materials havinf nonuniform distribution of defects or with the solid-liquid phase transition [26, 271. In the theory of
melting we are forced to examine basic particles, and not only collective excitations,
because the ground state itself changes with the phase transition [28--311. And near the
fiision point the distortion of a lattice becomes really essential, which is clearly demon-
v. I. kIJKaLov
strated by molecular-dynamics calculations for two-dimensional Lennard-Jones systems
[32, 331. I t may appear that a phonon picture is not quite adequate in other extreme
situations, like high temperature or high-pressure cases [34, 351, where the interaction
between phonons can become strong in an essential way.
Thus, we see the necessity to develop a general method, which would be able to
describe strongly anharmonic as well as strongly-disordered crystals and those near
their melting points. This method must be quantum-mechanical, so that it may be used
fdr quantum crystals [3G], such as He3, He4, H, etc., which exhibit large quantum fluctuations. There are sometimes only quantitative differences in properties of quantum
and classical crystals. This concerns such characteristics as low-temperature specific
heat [37], plasticity [38], and thermal conductivity [39, 401. The laws of thermal expansion are similar for both kinds of crystals [41, 421, although the second sound can
be excited much more easily in solid He4 [43, 441 than in other solids. But sometimes
the differences between quantum and classical crystals are qualitative. This appears
for example, in the high-temperature specific heat of solid He3 [45, 461. Also, the lowtemperature diffusion in solid helium [47--49] is unusual for classical crystals [50]. To
understand the behaviour of the magnetic susceptibility of solid He3 one needs to invoke
a four-spin exchange [Bl]. Finally, He4 is the only system for which a roughening transition in a solid-liquid interface has been reported [52, 531.
In the previous papers [54, 551 a quantum approach for strongly anharmonic
crystals has been developed. The main idea of this approach is the use of a renormalized
iterative process and a new form of selfconsistent conditions [28, 561 for propagators.
The applications of such an approach have been demonstrated for a linear chain [54]
and a weakly-anisotropic anharmonic crystal [551. We present here mathematical foundations for the method, used in refs. [54, 551, emphasizing the possibility of expanding
it for disordered solids, and we show the efficiency of the method for a strongly anisotropic anharmonic crystal.
The paper is organised as follows. In Section 2 the convergence of the iteration procedure for propagators is defined as a convergence in the mean with respect to operator
averages. Such a definition gives in a natural way the conditions for a zero-order iteration, when the convergence of the iterative process becomes the most rapid. In section 3
origins of divergences in high-order iterations for averages and propagators are clarified.
These divergences can be eliminated by means of a correct definition of products of
distributions with coinciding singularities. Section 4 contains an investigation of
general properties of lattice propagators, stressing their difference for ideal and nonideal
lattices. I n Section 5 we illustrate the technique on a strongly-anisotropic quantum
2. Equations of Motion
We consider here a nonrelativistic system of spinless particles with a two-body
interaction. As in known [57], nonadditive three-body forces are small corrections. To
take into account inberparticle correlations we use a pseudopotential method, replacing
a vacuum-interaction potential by a smoothed pseudopotential @(12); here and in what
follows the abbreviation
.n) =
t , . r,, tn)
is used. We shall not discuss here explicit expressions for effective potentials @(12),
which can be obtained in a number of ways: for instance by multiplying the initial
interaction by a correlation function, which is defined by means of a constrained variation in Jastrow method (see [58]).
Construction of Propagators for Quantum Crystals
The first step in solving propagator equations of motion is the definition of a selfenergy, for which we can write a system of equations [55] presented in an operator
foriii :
Z = @n f i@Gr,
r = 1f
n =fiG,
The iipper sign corresponds to Bose-Einstein and the lower one to Fenni-Dirac statistics.
We can we likewise the following equation for the vertex function
r = 1 + i@(r
f 1+ G W l S G ) G a r .
Introdricing a binary' (two-particle) propagator by the expression
B = GG f GTG,
which is equivalent to
r = & ( G - ~ B G -~ I),
we get for the self-energy
Z= f i @ B G - l .
The binary propagator is a solution of the equation
B = BO f G(6Z/8G) G ( B - G G ) ,
It is possible to obtain an explicit expression for the self-energy using some variational
decoupling [59] of eqs. ( 1 ) or (2), or iterating these equations [55]. The result of the
iteration is a series in powers of a pseudopotential, whose form influences the convergence
rate of such a series [60]. An actual parameter of this expansion can be found by investigating a one-parametric renormalization group
Z+ r l Z , @ + z-W, G + zG,
r +r,
B +Z ~ B ,
with respect to which self-energy equations are invariant.
If the self-energy is known, there is another trouble to solve the propagator equation
+- Zax,
G(12) = G0(12)
+ J K ( 1 3 ) G(32) d ( 3 ) ,
where Go is a trial propagator, corresponding to particles in an external field Vo(.),
and the kernal function is
K ( 1 2 ) = J G0(13) [Z(32) - Vo(2)6(23)]4 3 ) .
We can seek for an iterative solution of eq. (3) according to a achenie
Gj -+ Zi+i +- Ki+l+ G j + l .
Consequently, an exact solution of eq. ( 3 ) is a series
(KO= 1, K i y Ki K i - 1 . . K1).
v. I.YmALov
There exists concern about the convergence of the series (4),that is, about the convergence of the sequence of functions
Gj = RiG,
C Ki) .
Strictly speaking, &propagatoris not a simple function; it is a distribution (a generalized
function). Therefore, the convergence of the sequence {Gi} must be defined as a convergence of some functionals. What is more, from the physical point of view we are not
interested in a propagator itself, which is not a directly measurable quantity. The
measurable quantities are the averages of dynamical operators A (-),
& J lim lim A ( 1 ) G(12) dr,.
(2b(1) 7+-0
To investigate the convergence of the distribution sequence {Gi},it is therefore reasonable
to choose { ( A i ) } as a corresponding functional sequence. T h u s we shall say that the
sequence {Gi} converges to the distribution G with respect to an observable quantity
(A), if for any 6 > 0 there is such a number J, that
R e m a r k 1. Since the number of observable quantities is finite for any physical
system, the sequence {Gi} can be made convergent with respect t o all of them.
R e m a r k 2. For a nonequilibrium case 4n average (A) is a function of the time t.
Then we should mean by the convergence of the sequence { ( A ) i ) a convergence which is
uniform in t. I n a equilibrium situation ( A ) does not depend on time.
R e m a r k 3. The average (5) is bounded for a finite system. When taking the thermodynamic limit, eq. (5) may become infinite. Then the boundedness will remain for specific averages, that is, for eq. (5) divided by a number of particles
N = * i J lim lim G( 12) dr,.
rz-wl 7+-0
Generally speaking, it is quite unnecessary and hardly probable, that the series (4)
should be absolutely convergent with respect to averages (5). It is enough to demand
that this series be asymptotic. Then, if the series (4)is asymptotic, there exists s propagator G j , which is the best approximation for G with respect to an observable (A),
so that
All Remarks 1-3 are valid for this definition.
In the framework of the condition (6) there is a n infinite number of possibilities of
varying initial iterations Go. A modification of the zero-order iteration leads to a renormalization of the whole series (4)and changes its convergence rate. We would like,
naturally, to select Go so as to provide the most rapid convergence. The definition (6)
obviously shows that the quickest convergence occurs under the @quality
If the series (4)is asymptotically convergent then by the condition (7) the zero-order
iteration itself gives the best approximation in the sense of eq. (6). In the case of absolute
convergence one can mean by J, the index of the lowest approximation realizing the
desired accuracy for each particular problem.
Construction of Propagators for Quantum Crystals
The most common condition for defining parameters of a zero-order iteration is the
variational principle for the free energy. I n contrast to this principle our condition (7)
has the following advantages:
1. we can calculate with the best accuracy directly those averages which are most
interesting for us in every specific caae;
2. to ca,lculate averages is usually easier than to determine the free energy; and if
one chooses as a dynamical operator A ( . ) the internal energy
in which ~ r ais a particle i n w , p is a chemical potential, with the connection between
the free energy P and the internal energy E,
. F = Q $ E d / l (#?O= 1 , E -
where 0 is temperature, then the convergence of Pi to F follows from the convergence
of Zj to E;
3. the criterion (7) may be applied for nonequilibriuni systems, when the free energy
is undefined; then averages (A) depend on time a s well as parametem of a trial field
V o ( . ) ;the latter must vary with time in order to be a good approximation for a system.
We shall deal here with a n equilibrium situation. Passing t o the energy representat ion
G(r, r’, to)
1G(l1’)eior d t
G(r, r’, OJ)
Go(r,r‘, o))
(t- t - t ’ ) ,
we have
+ J K ( r , r”,
G(r”,r’, O ) dr”,
K ( r , r’, o)= J Go(r,r”, Q ) [Z(r”,r’, u)) - Vo(r‘)B(r” - r’)]dr“.
Writing down the propagator of the zeroth approximation as a wavefunction expansion
n ( o ) = (ePm
we get in the place of the series (4)
v. I. YwaLov
The j-th approximation for a n average (5) becomes
( A ) i = -j=
- lini
2n r++o
2 J eioTRkn(w)G,(w) Am,(o)d w ,
So, eq. (5) can be presented as an asymptotic series
As follows from eq. (3), a n effective parameter of this expansion is
1 = sup (d)il(E)i,
the average ( ~ i having
) ~
the same form as ( A ) i , but with Am,(w) replaced by vkn(w).
3. Elimination of Divergences
Direct calculation of propagators (4)and averages (10) leads to a n appearance of
divergences. Actually, in the first approximation we meet the integrals
Imn =
f- liin J d w ' G m ( ~G,(w)
2n T 4 f O
Jmn= & - liin J eioTGm(o)G,(U) o d u ,
2n r 4 + 0
and for the second iteration
liin JeiWTGm(w)Gt(w)GJw) d w ,
Ln r++O
Jmtn= f - lim J eimTGm(w)
G,(w) G,(w) w do.
2n r++O
Construction of Propagators for Quentnm Crystals
But if a t least two indexes are equal, then a direct integration of functions Gg end G i
gives divergences. These divergences are obviously connected with the indeterniinacy
of products of distributions having coinciding singularities. To avoid such divergences
requires a regular definition of distribution products. An implicit way of doing so is
t o define the limits [54, 561
This procedure is rather long; moreover it does not explain explicitly the meaning of
a product
G:(w) cc
- wn)-'
with pole singularities.
We must define GL(w) for it to have the following properties:
a) G$(w) for j = 1 must agree with Gn(w);
b) one has to obtain the same results using G:(w) as by limiting procedures (14), (15);
c) the time representation of a propagator (9) must maintain the condition of equilibrium
[G(r, "9
' ip)lT<o'
A power of Gn(o), satisfying these three conditions, is of the form
The validity of the condition a) is evident. Integrating a n infinitely differentiable functionf(w) together with the distribution (17), we have
Taking into considemtion that
we make certain the feasibility of the condition b). Finally, going to the time representation
G(r,r‘, t) = Tr p(r’, r ) K ( t ) ,
~ ( rr’)
, = ( ~ 3 ryn(r’)),
K ( T )= { G n ( T ) } ,
K m n ( t )=
2 J Kkn(w)GJw) e-iwr-dw
2z ’
we see that the terms with coinciding singularities lead to the expressions
in which @(t) is the step function. Eq. (20) is the Fourier transform of eq. (17). Thus,
the condition c) also holds.
4. Lattice Propagators
We demonstrate an application of this approach to a quantum crystal. An effective
aniplitude of particle fluctuations in such crystals is essentially lager than in classical
crystals. The ratio of the average deviation from a lattice site to the distance between
nearest neighbours in quantum crystals is about 113, which is not a small parameter.
Therefore, the perturbation series in powers of relative deviations are not correctly
defined. I n the next Section we illustrate how to describe a quantum crystal without
an expansion in powers of these deviations. Let us first understand the general properties
of lattice propagators, including those of nonideal lattices.
If a zero-approximation field has the periodicity of the lattice, e.g. it is periodic in
lattice vectors a, so thak
V d r ) = V,(r 4,
then the zeroth propagator can be represented as an integral expansion over Bloch
or as the expansion over Wannier functions
Go(r,r’, w ) = 2 2 Gn(a, w ) y& - a) yn*(r’- a),
where v is the volume of a unit cell, n is a zone index; here and in what follows k belongs
to the Brillouin zone. These functions (Bloch and Wannier) are connected by the equat ions
Gnk(r)= 2 yln(r- a) eiL,
y,(r - a) = -J @,,(r) e--ika dk;
they form the complete systeins:
2 J @&(r)&(r’)
b(r - r ’ ) ,
42 7
Construction of Propagators for Quantum Crystals
and they can he orthogonal, so that
-1@&(r) Gnk*(r)d r = 6,6(k
$ v%(r- a) vnn(r - 6 ,
- k‘),
= drnndab.
Comparing eqs. (22) and (23), we find
G,(a, o)e-io(k-w) = 8(k - k’)GJk, w ) .
( 2 ~ a) ~
From the latter equations, with the identities
we ascertain the property
Gn(k,(9) = G,(O, w ) = G%(Q,W ) = G,(O, w ) .
Such a degeneracy is a result of the mean-field representations (22), (23) and of the
translational symmetry of the field (21) over the lattice {a}.
The density of particles must be a periodic function
n ( r ) = f - lim J eiorG(r, r , w ) dw
2n T + + o
+ a).
= n(r,
Since for any lattice-spacing -function
g ( r ) = g(r
we have
$ @%r) g ( r ) @ n d r ) d r
== 0
+ k‘)
.f y Z ( r - a) d r ) vn(r - 6 ) dr = 0
+ 6)
the condition of the periodicity (25) is valid only if the interaction potential depends
on the coordinate remainder:
@ ( r , r’) = @ ( r - r ’ ) .
In this case, using the expansion (9), one can make certain that the exaot propagator
for an ideal crystal has the form of
G(r, r’, (9) = 7
2 $ a m n ( k , w ) @&(r) @zk(r’)dk,
(2x1 mn
or of
When the crystalline lattice {a} is disordered, Bloch and Wannier functions are
not defined. However, we can use the representation (23) for a zeroth iteration, implying
that yn(r - a) are no longer Wannier functions, but other functions producing a complete set. Then the coefficients G J a , m ) , generally speaking, are not degenerated: they
v. I. Tr-KALov
depend on a. The exact propagator is as follows
G(r, r', w ) =
2 2 Gmn(a,b, w ) ym(r - a ) y$(r' - b ) .
mi& ab
It niay he noticed that for any solid the trial field Vo(r) can be presented as a sun1
C va(r))
the manifold ( a } being either a regular or irregular lattice.
5. dnisot,ropie Crystal
Let 11s take as a trial field V,(r) in the sum (30) the potential of an anisotropic
m 3
VJr) = uo - 2 u f ( r i - ai)2,
2 i=l
m being the mass of a particle. Then in the expansion (23) G,(a, w ) has the sanie form
a s G,(o) in eq. (8); yn(r - a ) are oscillator wave functions; index n is the three-index
(ni= 0, 1, 2, ...);
the corresponding spectrum
and ,u is the chemical potential. I n quantum crystals the number of particles per cell
2 J eiw'Gnm(a, a, co) d o
- lim
may differ from unity. The equation (32) enables us t o express ,u in t e r n s of the temperature 0 and the number w. Let us examine the case of a crystal which is strongly
anisotropic along one of axes, so that the vibrational frequency for this axis
u = inf {ui} (i = 1, 2, 3)
is much less than the two other frequencies. Then substituting Gn(co) for Gnn(a. a, m )
in eq. (32) and taking into account only two lower levels, we get
+2 ui.
-It low teniperature for Bose-Einstein statist)ics and any w we have
The corresponding asymptotic form of ,u for Fernii-Dirac statistics depends on whether
w < 1 or w = 1. I n any event for a monocomponent Fermi crystal w cannot be greater
than one. Otherwise eq. (33) in a Ferini case has no sense. When the number of cells is
greater than the number of particles, then
Construction of Propagators for Quantum Crystals
and if the numbers are equal
PF = & f
For both statistics
n(Eo - p ) -+ w,n(Eo
+ u - p)+0
(0 0).
That is, all particles drop into the lowest level. At high temperatures
+ 1 u + 0 In 2 i w
hut then we can not limit ourselves to several lower levels.
The trial field (31) contains four unknown parameters: uo,ul,ug,u3.We shall define
them applying the conditions (7) in the following way:
(1>1 = <1)0*
These conditions are required in order that the iteration procedure not change the
number of particles per unit cell and not spread the density ( 2 5 ) . Choosing the selfenergy in the pseudopotential Hartree-Fock form, one can obtain for the total number
of particles
vmn =
n ( o n ) (1 - Pl1 f n(mn)l (vmn 4@ n n ) l ,
~ ( w cu)m t n - Vmn,
@mtn =
@mn =
C n(mO @mtn,
J ym(r- a ) yt(r' - b ) @ ( r - r') yE(r'- b ) yn(r - a ) dr dr',
$ ym(r - a) yC(r'- b) @ ( r - r') yn(r' - a) y E ( r- b ) dr dr',
vmn =
ym(r - a ) va(r)yn(r - a ) d r .
To simplify the calculations we later assume the correctness of the inequalities
0lu Q 1,
l@nnlvn~l Q 1.
Then the first of eqs. (34) gives for Bose-Einstein stat,istics the equation
w~(ooo,ooo)= 0;
w(m, n),
while for Fermi-Dirac statistics we must take into consideration both lower levels (the
explanation of this necessity given in refs. [54, 551)
v~(ooo,ooo)+ v,(100,100)
= 0,
where we designated u1 as u. The second part of eqs. (34) leads to the equations
w(200,000)= w(020,000) = w(002,000)= 0,
which are the same for both kinds of statistics. Expanding with the use of notations
uo = u(ooo,ooo,ooo), u;,= u(100,000,100),
u, = u(200,000,000), u, = u(020,000,000),
29 Ann. Physik. 7. Folgc, Bd. 38
U(m,d, n)E UmCn,
v. I. TwaLov
we come to
To obtain eqs. (36), we did not expand the pseudopotential in powers of particle deviations. Therefore the parameters u0 and ui iinplicitly take into account all anharmonicities. Such a method has been called in ref. [ 5 5 ] a superharmonic approximation.
If calculating U,,,Uh and Ui to niake the harnionic expansion of the pseudopotential,
we get
However, this expansion is not a good approximation for quantuni crystals: in the general case ire niust use eqs. (36).
Let us now undrrstsnd when it is possible to calculate the trial parameters considering only the lowest energy level for Fernii crystals as well as for Bose. This would of
course be permissible if the forinal difference between the two statistics in eqs. (36)
could be eliminated, that is, if
uO6 - U O p = w( UO -
u/4w 0
from which we acquire the condition
c; w u, f u/2w.
According to the definitions of the matrix eleiiients Uo and Ub the approximate equality
( 3 5 ) seems to be always reasonahle. For example, in the harmonic approxiination
- U O p = (u2 - rL’G;)/4u,
thus, the condition ( 3 i ) is true. Consequently, under the correctness of eq. (37),eqs. (36)
niag be rewritten in the same fonn for both statistics:
ui = 2 4 2
+ IF)i2= l [l - 12 (1 - dii)] uj.
Construction of Propagators for Quantum Crystals
It is important that the equations (38) reniain valid for a weakly-anisotropic crystal
also. In particular, for an isotropic approximation, when U s - U ,eqa. (38) yield
uo = w
[uo- 3
i qu ] ,
us = 2 4 2 - 1/2)
Now we find expressions for the internal energy, assuming the conditions (35).
The zero-order iteration gives
Taking the self-energy (1) in the Hartree form and using the first iteration of eq. (3),
we obtain
= E,,
+ 1 2a ~(000,000).
= Eo
Because of the equality (37), El
E, for both statistics. The subsequent iterations for
the internal energy, according t o the formula (lo), yield the series in powers of v,,,,
and not in powers of particle deviations as in the standard theory of anharmonic crystals.
Moreover, as a consequence of eqs. (34) and (37), a t temperatures Pu
1 those v,,
which occur in the expansion (10) are equal t o zero, thereby giving the equations defining
trial parameters. So, E j w E, for any j. Such a remarkable result is due to the use of
the conditions (34).
Let us notice that we intentionally preserved the sunis 2 in eqs. (3g) and (40);
this is to remind one that the method considered here can be applied to amorphous
solids, when the parameters uo and ui are functions of the nonregular-lattice vector a.
I t should be emphasized that for the self-consistency of all calculations we have t o '
utilize the same form of the self-energy for the definition of trial-propagator parameters
by eq. (7) as for the computation of averages. Por example, if we define the parameters
u, and u; by means of eqs. (34) with the Hartree-type self-energy, but a t the same
time calculate the internal energy employing the self-energy of the Hartree-Fock type,
then we should have
Go = @(000,000),
@(m, n ) ,
2G1 = ~ ( 2 0 0 , 0 0 0 ) @(000,200),
2a2= @(020,000) @(000,020),
2Q3 = @(002,000) @(000,002).
The expression E , is a good approximation for the internal energy only if the teniperature is bounded below, so that the second term of E,, is small compared with the first,
A c k n o w l e d g e m e n t s . It is a pleasure for me to express my sincere gratitude,to,
Dr. D. TER HAARfor his kind hospitality and scientific support, to Prof. R. WEINSTOCK,
for correcting the manuscript, and to the British Council for a grant.
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Bei der Redaktion eingegangen am 18. Febriiar 1981.
Anschr. d. Verf.: V. I. YUKALOV
Department of Theoretical Physics
University of Oxford
1 Keble Road
Oxford OX1 3NP, U.K.
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crystals, quantum, construction, propagators
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