Supplement of the Progress of Theoretical Physics, No. 46, 1970
411
Off-Diagonal Long-Range Order in Solids*)
Hirotsugu MATSUDA and Toshihiko TsuNETO*
Research Institute for Fundamental Physics
Kyoto University, Kyoto
*IJepartment
of Physics, Faculty of Science
Kyoto Uni·uersity, Kyoto
(Received November 30, 1970)
The possibility of the coexistence of the diagonal long-range order (DLRO) and
the off-diagonal long-range order ( ODLRO), which may be called 'supersolid' state, is
discussed in a system of bosons at 0°K. Examples of wave functions are given describing the state with only DLRO and with both DLRO and ODLRO. After giving
general relationships between the quantum lattice model for a system of bosons and
the corresponding spin system, the criterion for the coexistence of DLRO and ODLRO
is obtained by using the molecular field approximation for the spin system. The supersolid state corresponds to an 'intermediate' state between the antiferromagnetic and
spin-flop state in the spin system. The analysis on the basis of this criterion indicates
that the bulk solid He 4 is unlikely to become a supersolid, whereas the adsorbed He 4
film under the influence of the wall potential may become a supersolid. A speculation
is made on the possibility of a supersolid in an extremely compressed system with
nuclei of even mass number.
§1.
Introduction
Bose condensation is one of the most remarkable phenomena which the
quantum mechanical system exhibits on a macroscopic scale. In 1924 Einstein
showed that an ideal Bose gas undergoes a phase transition at a certain temperature below which a macroscopic number of particles occupy the lowestenergy single-particle state. 1 ) Since these macroscopic number of particles in
the lowest state may be looked upon as a kind of condensed particles, this
phenomenon is called Bose condensation of ideal bosons.
'I'he analogy between liquid He4 and an ideal Bose gas was first recognized
by F. London. 2 ).a) He suggested that the lambda transition in liquid He4 could
be understood as the analogy for a liquid of the transition which occurs in
the ideal Bose gas. Tisza showed that the analogy between liquid He4 and
an ideal Bose gas is also useful in understanding the peculiar transport
properties of He II, since the presence of 'condensed' particles would naturally
lead to a special two-fluid hydrodynamical description for such a gas. 4 )
*) An outline of this study was reported at the Twelfth International Conference on Low
Temperature Physics and is to be published in its proceedings. This paper is dedicated to Professor
T. Nagamiya in celebration of his sixtieth birthday.
412
H. Matsuda and T. Tsuneto
However, since the liquid He4 consists of strongly interacting He4 atoms,
the lowest-energy single particle state, on the basis of which the Bose condensation of ideal bosons was defined, loses its unambiguous meaning. Thus,
Penrose and Onsager generalized the mathematical description of Bose condensation so as to be applicable also to a system of interacting particles. 5) Bose
condensation is said to be present whenever the largest eigenvalue of the oneparticle reduced density matrix is an extensive rather than an intensive quantity.
Indeed, the idea of Bose condensation can further be generalized if one
says that the Bose condensation is present whenever the largest eigenvalue of
n-particle reduced density matrix Pn goes to infinity in the thermodynamic
limit.
Yang pointed out that this generalization is indeed useful by first showing
that if the Bose condensation is present with respect to Pn then it is also present
with respect to Pm (m>n) .6 ) He then introduced the concept of off-diagonal
long-range order (ODLRO) which has a close relation with the Bose condensation, and showed that it is reasonable to assume that both superfluid He II
and the superconductors are phases characterized by the existence of such an
order.
Let
(1·1)
be the coordinate representation of Pn· Here a(x) and a+(x) are the annihilation and creation operators of a particle at the point x, and p is the
normalized density operator of the total system. One may say broadly
that the system has the ODLRO when there exists a limiting process
Min~:i;::·.:~ I x~ Xj I ~oo for which <x~ .. ·x~ I Pn Ix1· · ·xn)~O.
In the case of the system of interacting bosons such as liquid He4 considered as an assembly of He4 atoms, to which case we restrict ourselves in
this pa:r;er, it suffices to consider only P1 for discussing the Bose condensation
or the ODLI{O. It can be proved that if the state of the system possesses
the ODLRO then the state is in the Bose condensation. Conversely,· it is
reasonable to assume that if the Bose condensation is present so is the ODLRO,
although to the authors' knowledge no rigorous proof has ever been published.
Penrose and Onsager gave the argument that liquid He II in equilibrium
shows Bose condensation. In particular, for 0°K they based their argument
on the assumption that there is no 'long-range configurational order'. Further,
they inferred that a solid does not show Bose condensation at least for 0°K
because of the presence of 'long-range configurational order'. They assumed
that a solid at 0°K is a 'perfect crystal'*)- i.e. that the value of the ground
*l From a different point of view we may understand 'a perfect crystal' to be a crystal whose
low-lying excitations consist only of longitudinal and transverse phonons. Whether or not one can
argue for or
the presence of ODLRO in a perfect crystal in this sense is an interesting
problem.
Off-Diagonal Long-Range Order in Solids
413
state eigenfunction for N bosons 1/r( x1 .. · XN) is small unless every particle is
·near each lattice site. As a concrete example of such a perfect crystal let us
assume that f' is given in the following form
N
f'
~
[}1 ¢>a;( xi),
(1·2)
where ¢>a(x) =¢>(x- Ra) is a one-particle wave function localized near the a-th
lattice point whose coordinate is Ra, and the summation is to be taken with
respect to all the possible permutations (a1az"'aN) of (1, 2, .. ·, N). Since f'
is the ground state eigenfunction of the assembly of bosons, we may assume
that ¢> is real and
¢>(x)20,
(1·3)
~¢> 2 (x)dx= 1.
(1·4)
Putting
aa1
~1/ra(x)~rl(x)dx,
(1·5)
we shall prove in Appendix I that if aa1 =f=.O only for the nearest neighbors of
1, then the ground state has no ODLRO.
However, we must note that the solid at 0°K may not necessarily be the
'perfect crystal' that Penrose and Onsager assumed. For example, from the
non-negative symmetric wave function f' 0 (x11 ... , xN+s) describing the state of
the 'perfect crystal' containing N + s bosons localized around the N + s lattice
points, one may construct the wave function
(1·6)
describing the state of a crystal containing N + s lattice points with s vacant
lattice sites. Here, c(x) is a periodic function of x with the period of the
Bravais lattice. Vve shall show in Appendix II that the state characterized by
(1· 6) certainly possesses both the order of a crystal (DLRO) and the ODLH.O
in the thermodynamic limit in which s/ N is fixed at a small but non-zero
value.*)
Recently, Andreev and Lifshitz pointed out the possibility of superfluidity
in solids, where the Bose condensation of lattice vacancies plays an essential
role for the superfluidity.7) The form of the wave function (1· 6) just corresponds to the type of the lowest-energy eigenfunction they assumed for such
a 'supersolid'. They discussed the properties of such solids based on the
assumed type of the energy eigenfunctions which represent the quantum
mechanical migration of lattice vacancies. But they showed neither why the
*) Recently, Chester proposed another type of wave function which may have both DLRO and
ODLR0. 12 )
414
H. Matsuda and T. Tsuneto
solid He4 could have such a type of energy eigenfunctions nor what property of
the Hamiltonian may lead to the ground state eigenfunction of the form (1· 6).
Because of Heisenberg's uncertainty principle, the lowest-energy eigenfunction provides for uncertainty in the position of particles in order to reduce the
kinetic energy. Since the effect of the kinetic energy becomes more important
as the mass of a particle decreases, if one could gradually decrease the mass
of each particle of a crystal at 0°K under a fixed pressure, the crystal eventually would melt and become a quantum fluid with ODLRO. Before melting,
whether the crystal provides for the uncertainty of the atomic positions simply
increasing its lattice constant or may produce lattice vacancies will depend on
the type of the interatomic interaction.
Thus, the main purpose of this paper is to inquire the properties of the
Hamiltonian that is needed to ensure the existence of ODLRO in solids. For
this purpose we invoke the quantum lattice model (QLM) for the assembly
of bosons with hard core, which was successfully introduced by Matsubara
and Matsuda (MM) .8 )
Matsubara and Matsuda pointed out the close relationship between the
QLM and the anisotropic Heisenberg model of spin 1/2. However, their
treatment was restricted to the fluid phase, since the main purpose of their
study was to discuss the anomalous behavior of He II such as the low-lying
excited states and the pressure-dependence of the A-transition.
Therefore, in order to include the occurrence of the solid phase, we give
in §2 relationships between the QLM and the anisotropic Heisenberg model in
a more general way. The notion of the DLRO and ODLRO can then be
interpreted and defined in terms of the corresponding spin system. In §3 we
discuss the possibility for the model system of the QLM to exhibit the coexistence of ODLRO and DLRO using a molecular field approximation. Finally,
in §4 we consider the case of the system under the· influence of the microscopic
external field such as He4 film adsorbed on the substrate. There we shall
discuss the possibility for the existence of ODLRO in real crystals.
§2.
Quantum lattice model and anisotropic Heisenberg model
As well-known, the lattice model for the assembly of particles is the
model introduced in order to get the mathematical simplification by restricting
the positions of particles to the lattice points of a certain fictitious lattice in
space. We consider the system with the periodic boundary condition in order
to avoid the complication arising from the boundary effect.
Let us consider the following model Hamiltonian for n bosons with hard
core m the lattice space consisting of N lattice points:
(2·1)
Off-Diagonal Long-Range Order in Solids
415
Here, at and a; are the creation and annihilation operator of a boson at the
i-th lattice point.
As Bose particles we assume the commutation relations:
(i=l=j)
(2·2)
On the other hand, in order to exclude the multiple occupation of atoms at
each lattice point corresponding to the atomic core, we impose for
the
relation:
[at, at]+= [a;, a;] +=0,
[a;, at]+
(2·3)
1.
From (2·3), n;=ata; has a value either 0 or 1. v;/=vj;) is a real
constant representing the potential between a pair of atoms occupying the i-th
and j-th lattice points. u;/ =uj;) is a real constant representing the quantum
mechanical transfer of atoms between the i-th and j-th lattice points. A is a
constant and n is the total number of atoms confined in the lattice space with
N lattice points:
:En;.
n
(2·4)
j
A simplest way to represent the kinetic energy is to take the simple
cubic lattice and to assume the total kinetic energy as
(2·5)
where 'In is the mass of an atom, d is the lattice constant, and <ij) means
to take the summation over all the nearest neighboring pair points. This corresponds to setting
( i, j are nearest neighbors)
Ull= {
A
(2·6)
0,
(otherwise)
311} /(rnd 2 ) .
In terms of Fourier transforms of at and a; defined by
at;= N- 112 2: exp(
ikRJ aj,
j
ak
N-
112
(2·7)
2: exp(ikRJai ,
j
K can be written as
(2·8)
Here Ri is the position vector of the j-th lattice point, and the 2:~ denotes
the sum over k in the first Brillouin zone. The relation (2 · 8) shows that
H. Matsuda and T. Tsuneto
416
for kd<;t;) (2 · 5) is a good approximation to the kinetic energy of the real
continuous system which is given by (fi}/2m)~kk 2 atak. At the same time,
this shows that the better approximation would be obtained to the kinetic
energy of the real system by including the transfer of particles between lattice
points other than nearest neighbor pairs, and that the lattice need not be
simple cubic.
Indeed, if we assume that K is given by
+
ft2 ~lk2 akak,
K =--L...J
2m k
(2· 8)'
which is a better approximation to the kinetic energy than (2 · 8), we obtain
for the simple cubic lattice:
u(R)=
r
2ft 2 / 1n I R I 2 ,
lo,
(for R lying on the x, y or z axis.
The upper sign is for A sublattice
and the lower sign is forB sublattice.)
(otherwise)
(2· 6)'
. Now, the operators satisfying (2 · 2) and (2 · 3) are isomorphic to a set
of those of N spins each of which is localized at each lattice point and has
the magnitude of 1/2. The isomorphism can easily be shown to be realized
by setting*)
at=Sj-iS},
(2·9)
(2·10)
Therefore, a down-spin, Sj = - ~, corresponds to a particle and an up-spm,
Sj= ~, corresponds to a hole occupying the j-th lattice point.
Substituting (2 · 9) and (2 · 10) into (2 · 1), one obtains
!J(,n =
rN- r' M + !JlRpin '
(2·11)
where
(2·12)
(2·12)'
---*l This setting, different from that in MM, where an up-spin corresponded to a particle and
····-------~---
a down-spin to a hole, is adopted here in order to have a close relationship with Lee and Yang's
paper. 9 l
Off-Diagonal Long-Range Order in Solids
417
(2·13)
and
(2·14)
Thus, Eqs. (2 · 11)- (2 ·14) show the close relationship between the QLM
and the anisotropic Heisenberg model. Prior to the appearance of the QLM,
Lee and Yang9 ) showed the relation between the isothermal magnetization
curve versus magnetic field of the Ising model and the equation of state of
the corresponding CLM (classical lattice model). We show below that the
same relation generally holds in our case, too.
Let p and V be the pressure and volume of the QLM and let T be the
temperature of the QLM and the corresponding spin system. According to
the method of grand canonical ensemble we have the relation
exp(p V/k T) = ~exp(t-tn/kT)Trn exp( -c:9tn/k T)
n
=Tr exp((.u/2
r)N/kT(exp)- (.9/.spin
HM)/kT)
=exp( (p./2-r )N/k T)Tr exp(- (.9/.spin- HM)/k T), (2 ·15)
where Tr denotes the trace in the space spanned by the eigenvectors of the
set of operators {n;} and Tr n denotes the trace in its subspace satisfying the
equation ~t n; n; t-t is the chemical potential and
H=r'
p..
(2·16)
According to the method of canonical ensemble as applied tJ the spin system
we have
Trexp(- (.9/.spin-HM)/kT) =exp( -JN/kT),
(2·17)
where JN is the Helmholtz free energy of the corresponding spin system in
the magnetic field H.
Therefore, from ( 2 · 15) . . . ., ( 2 · 17), we obtain
pV/N=
Since r and
r'
(2·18)
are constants we have
(2·19)
where ( · · ·) denotes the canonical average, so that ( M) 1s a function of T
and 1-f
We note by (2· 4), (2·10) and (2·13) that in order to have a correspondence with the QLM having n atoms in the volume Vat temperature T,
418
H. Matsuda and T. Tsuneto
the magnetic field H of the spin system must be determined by
n
N
(2·20)
(M).
Then, from (2 · 19) we obtain
(2·21)
where we used the fact that for H~oo, (M)/N~1/2 so that p~o corresponding to the vacuum.
Thus, the study of the equilibrium properties of the QLM completely
reduces to that of the corresponding anisotropic Heisenberg model in the
field H. For instance, the isothermal compressibility Kr=- (1/V) (oV/op)r,
and the thermal expansion coefficient a= (1/V)(oV/oT)P of the QLM can
be expressed in terms of the susceptibility X= (oM/oH)r and (oM/oT)H,
where we write (M)=M(H, T).
Indeed, by virtue of ( 2 · 20), we get
x __ ( an ) __ ( an ) ( aP )
aH T ap T aH r ·
Since
an )
( op
T
and (op/oH)r=
( an )
op T,V
v (a ( Vn ) jaP)
n (
T,V
av)
op
T,n
1l
K
T
n/V by (2·21), we obtain
Kr - v2
n
(2·22)
X.
From the relations
dM=(§M~) dH+(aM) dT,
aI-I
T
aT H
ap ) dH+ ( --ap ) dT
r
dp= ( --al-I T
aT H '
we get for dp=O
dH=
so that
Since
(op/oT)H dT
(apjaH)r
'
(2·23)
Off-Diagonal Long-Range Order in Solids
419
by (2·21), and
(-~1\;j)
P= -
(
:~)
p,v =
-;-(
-~~-)
P
=na,
we obtain
(2·24)
The heat capacity of the QLM at constant volume Cv is obviously equal
to the heat capacity at constant magnetization of the corresponding spin system:
(2·25)
and the heat capacity of the QLM at constant pressure CP can be expressed as
(2·26)
Now that we have established the general relationship between the thermodynamic quantities of the QLM and the spin system, let us consider possible
tyr::es of order in these systems.
In the continuous space of v-dimensions with the periodic boundary conditions, if there is no external field acting on the system of particles, the
canonical one-particle distribution function n(x) =Trn[pa+(x)a(x)] is a constant,
because of the translational invariance of the Hamiltonian. However, when
there exists a suitable external potential ¢>=e)V(x)a+(x)a(x)dx proportional
to a constant e, n(x) may depend on x in the limit e--O if one takes first
the thermodynamic limit and then make e tend to zero. We call this limit
as a e-limit.
If and only if there exists a potential ¢> such that in the e-limit the equations
n(x+aa)
n(x)
hold for any x with a suitable set of v independent vectors aa and the length
of not all of such vectors can be chosen arbitrarily small, we say that the
system has a diagonal long-range order (DLRO).
In the crystal none of the vectors aa can be chosen arbitrarily small, so
that the system is considered to have DLRO in the crystal. The fluid is
characterized by that n(x) is always considered to be constant in the above
limit, so that the system has no DLRO in the fluid.
In the QLM, where the position of each particle is restricted to the lattice
point, we must naturally modify the definition of DLRO: If and only if there
exists a potential ¢> e~; v; at a; such that in the e-limit the equations
420
H. Matsuda and T. Tsuneto
(i
1, ... , N)
hold with a suitable set of 1.1 independent lattice vectors aa, and none of the
set of the primitive translation vectors of the lattice can be chosen as the
above set of aa, then the system is said to have DLRO. Here n(R;)
= Tr n pai a;. Therefore the lattice which has DLRO has sublattices which are
not equivalent. Thus, using (2 · 10), the definition of the DLRO of the spin
system can be rephrased in terms of <Sz(R1) )=Tr [pSn. According to this
definition the paramagnetic state and the ferromagnetic state, in which <Sz(R;))
does not depend on i, has no DLRO, whereas the antiferromagnetic state has
DLRO.
The definition of ODLRO in the continuous space can easily be taken
over to the QLM: The system is said to have ODLRO if and only if
Trn[paiaj]--1->0 as IR;~RiJ~oo. Because of (2·9)
!Tr [p(a;+ai +at a;)]
Tr [paiai]
=Tr[p(S:S}+S{S})]
=
<s: Si+ sr S}).
Possible types of order of the state
Table I.
of the system are given in Table I
together with the schematic spin
0 D L R0
arrangements of the two sublattices.
NO
YES
Therefore, we must say that the correFERRO
sponding spin system has ODLRO
~
SPIN FLOP
PARA
if and only if <s: Sj +Sf S})~O as
0
z
NORMAL FLUID
SUPER FLUID
I R; Rj I~oo. The spin-flop state
(GAS, LIQUID)
0
Cl:
produced when the external magnetic
field is applied in the z-direction to
a
~INTERANTI FERRO
r
~ lI MEDIATE
the antiferromagnet has ODLRO but
STATE
w
no DLRO according to our definition.
>NORMAL SOLID
SUPERSOLID
Matsubara and Matsuda associated
such a state of the spin system with
that of He II.
Thus, finding that the question raised in § 1 can now be formulated as
the problem as to what are the properties of the anisotropic Heisenberg model
(2 · 11) for the ground state of which to have both DLRO and ODLRO, we
proceed to the next section.
il
_J
~···----
(f)
§3.
v
!
Criterion for the coexistence of the DLRO and ODLRO
in the QLM
As enunciated in §1, we discuss in this section the possibilities of the
coexistence of DLRO and· ODLRO using the QLM. In the continuous space
Off-Diago nal Long-Range Order in Solids
421
the mathematical treatment of the strongly interacting many particle system
with hard core is very difficult. Since the energy of the system is critically
dependent on the correlation between particles, the result of calculation cannot
be trusted even as a first approximation unless a due account of the correlation
is taken. In the spin system, on the other hand, a simple molecular field
approximation has often been found to be a good first approximation on the
basis of which a more elaborate calculation can be carried out. This is because
for the Hamiltonian of the spin system which we usually study as a model for
magnetic systems the energy associated for producing the short-range correlation
is not very large compared with that for producing the long-range order, the
effect of which can be taken into account by the molecular field approximation.
4
The lattice model for liquid He has been introduced in order to take
advantage of the above situation; the effect of hard core is taken into account
kinematically rather than dynamically. Using the molecular field approximation
it was shown that the system undergoes the Bose condensation and becomes
the state with the ODLRO in the present terminology.
In order to include the possibility of DLRO we assume that vii>O at
least for nearest neighbor pairs, taking a smaller lattice constant than that of
MM. Of course, the smaller the lattice constant is, the closer the QLM is to
the real system. On the other hand, since the value of vii should be about the
value of the pair potential v(R) corresponding to the distance between the i-th
and j-th lattice points R = \ R;- Ri \, we must assume large values for some
vii as the lattice constant becomes small; this assumption, however, will invalidate the molecular field approximation.
As a compromise we assume that the parameters characterizing the QLM
is so chosen that the lattice consists of two sublattices S2l and ~ in each of
which every lattice point is equivalent. Representing the state of each spin
of such a QLM as a vector of length 1/2, we denote by 0 and ifJ the angle
between the z-axis and each spin of the sublattice ~ and ~' respectively.
Then, the energy per spin of the Hamiltonian
,j{
1s
_g{,Rpin
HM
(3·1)
given by
E
H
(cosO+cos¢1) + B cosO cos¢1
(3·2)
where
1
B=----'b
8 j;
'l);j'
jE'i8, iEm:
(3. 2)'
H. Matsuda and T. Tsuneto
422
Here we took the modulus of the summation of uij in order to have each
spin oriented so as to minimize E by the values of 0, ¢ lying between 0
and n.
In the molecular field approximation the ground state is characterized by
the set of values 0 and ¢ which minimizes the values of E. In order that
the state is antiferromagnetic at least for H = 0 we assume B>O. Without
losing the generality we also assume that H?::_O, and O<O<¢<n, O<n/2.
The ground state satisfies the equations:
8~
=sino(
Iff
~!
=sin¢(
J;f -Bcos0-Ccos¢ )-Dcos¢sin0= 0.
8
B cos¢-C coso)
(3· 3).
D coso sin¢ 0,
(3·4)
Eliminating H from (3 · 3) and (3 · 4), we obtain
(cosO-cos¢) {(B
C)sinOsin¢-D( l+cosOcos¢)}
0.
(3·5)
Therefore, we have the following two types of solutions:
Symmetric solution, in which
(I)
(3·6)
0=¢.
(II)
Asymmetric solution, in which
(B-C) sinO sin¢
(3·7)
D(1 +cosO cos¢).
We first discuss the two types of solutions separately, and then by comparing their energies obtain the stable ground state in various cases.
(I)
Symmetric solution
Putting 0=¢ in (3·3) and (3·4), one obtains
sino{
J;f- (B+C+ D) coso} =0.
(3·8)
For H>U-=2(B+C +D), the only solution of (3·8) is 0=0, for which
(3·9)
E=Ev==-H+B +C-F.
For O<H<2(B+C+ D), we also have the solution
(3·10)
0=
for which
1
E=E1 -=- { T
H
2
}
(B+C+D) +D+F .
(3·11)
Off-Diago nal Long-Range Order in Solids
423
This energy is lower than Ev in the above range of H, and at H = Hv it is
equal to Ev.
Therefore, the symmetric solution for the ground state has the energy
for
for
O<H<Hv ,
H2I--Iv .
(3·12)
Asymmetr ic solution
We note that (0=0, ¢=rc) always satisfies (3·7). Since D20, for C>B
this is the only solution of (3 · 7), for which
(II)
(3·13)
B+C-F.
E=Es=
For C<B, defining
u=cosO cos¢,
t=cosO+c os¢,
(3·14)
we find that
(sinO sin¢) 2 = (1-cos 20) (1
cos2 ¢)
(1+u) 2
(3·15)
Using (3 · 5) in (3 · 7), we obtain
tz = { 1- ( B DC ) z} (1 + u) z.
(3·16)
Equation (3·16) has always a solution t=1+u=O , for which E=Es. This
is the only solution of (3 ·16) for B- D<C<B +D.
Since the replacement of t by It I gives the same or lower energy to E
in (3·2), we may assume t20 for the ground state. Thus, for C<BEq. (3 ·16) gives a solution
t=a(1+u ),
(3·17)
where
(3·18)
Substitutin g (3 ·14) and (3 ·17) in (3 · 2), we can write
E=Ea=-
IJ
a:(1+it) +Bu
+
~
{a2 (1+u) 2 -2u}-D V1-a2 (1+u)-F .
(3·19)
For C<O, Ea has no minimum as a function of u, ( -oo<u<o o). -For
--"----'from dEal du = 0, one obtains
424
H. Matsuda and T. Tsuneto
(3·20)
where
(3. 20)'
Since 1 + u2:0, Eq. (3 · 20) cannot be satisfied for O<H<H. . Therefore,
for CsO or for C>O and OsH<H. , the ground state energy of the asymmetric solution is E •.
For O<CsB D and H2:H., we obtain from (3·19) and (3·20):
1
(H-H.) 2
B+C-F,
(3·21)
which is equal to E. for H=H. and smaller than E. for H>IL.
It is to be noted, however, that (3 · 21) does not correspond to the asymmetric solution for all values of H larger than H., since there must be an
upper bound of u due to the restriction
1<cos¢~cos0<1.
(3·22)
By (3 ·14), cosO and cos¢ are the roots Z of the quadratic equation
Z2
u
0.
In order that both roots satisfy (3 · 22), it is necessary and sufficient that the
following set of inequalities holds:
1 t
u>O,
(3·23)
1 +t+u2:0 ,
(3. 23)'
lt/21
(3·24)
u-
t
4 <o.
(3·25)
Substitutin g the relation (3 ·17) in these inequalities and noting that
O<a<1 and 1+u2:0, we find that (3·23) and (3·23)' are always satisfied.
For a=O, we have from (3·17) t=O, so that from (3·25)
u<O.
(3·26)
For a>O, from (3 · 24) one obtains
2
u<--1
,
a
(3·27)
and from (3 · 25) and (3 ·18) one obtains
1<u<(
1
(3·28)
Off-Diagonal Long-Range Order in Solids
425
Since it can be shown that (3 · 28) implies (3 · 27) for a>O, and it implies
(3 · 26) for a= 0. Therefore, the inequality (3 · 28) gives the range of u for
which Eq. (3 ·17) may give the asymmetric solution for the ground state.
At u (B-C-D)/(B-C+D), we have cosO=cos¢>, so that the asymmetric
solution merges in the symmetric one at this value of u. Substituting (3 · 20)
m ( 3 · 28), we find
Ha=-Hs(B+C+ D)/(B-C+ D),
for which the energy of the asymmetric ground state is given by (3 · 21).
Now that we have obtained both the symmetric and asymmetric solutions,
let us compare the energies of the two solutions in various cases.
Case (i)
'I'he symmetric solution has energy Ev and the asymmetric solution has
energy Es. Since
(3·30)
the stable ground state is given by
(}=0,
if>
7C
(}=¢>=0
and
Case (ii)
for
H<2B
for
H2.2B.
(3·31)
-D
The asymmetric solution has energy Es and the symmetric solution has
energy E 1 for H<l-Iv and Ev for H>Hv. From (3·11) and (3·13) we
obtain
(3·32)
where
(3·33)
Since I-L 1>Hv in this case, we have Es<E1 for H<Hv. On the other hand,
from (3·30) we have Es<Ev for H<2B and Es"?::.Ev for H>2B. Therefore, the ground state is again given by (3 · 31).
Case (iii)
Here, we have O<H1 <Hv; therefore, by virtue of (3 · 32) and (3 · 30),
the ground state energy is given by
H<H 1 ,
for H,<H<Hv ,
for H"?::.Hv.
for
(3·34)
At H = Hs 1 , we have E s= E 1 , whereas the corresponding set of values of (}
H. Matsuda and T. Tsuneto
426
and ¢ m the symmetric state is different from that of the asymmetric state.
O<C<B-D
Case (iv)
The asymmetric solution has energy Es for H<H. and Ea for H.<H<Ha.
Since we have 1L<H.1 in this case, the ground state energy is Es for H<H..
Since from (3 ·11) and (3 · 21) we have
_ 1 B C+D
z
E,-Ea- BC B+C+D (H-Ha) >O,
(3·35)
the ground state energy is given by
E
Es
Ea
Et
Ev
H<Hs,
for FL<H<Ha,
for Ha<H<Hv,
for
for
(3·36)
H~Hv.
It is to be noted that at H = Ha the corresponding set of values of (} and ¢
the same for Ea and E 1 .
Case (v)
C>B
IS
D
The asymmetric solution always has energy Es. From (3 ·11) and (3 ·13)
we find E,<Es. Therefore, in this case the stable ground state corresponds
to only the symmetric solution, so that
E
H<H.,
for H>IL.
for
(3·37)
Thus, summanzmg the above results, we obtain the phase diagram as
shown in Fig. 1.
VACUUM
SUPER FLUID
-D
-(B+D}
CASE
(i}
I
(ii)
l
0
(iii)
I
(iV)
( B >D)
Fig. l(a).
c
8-D
l
(V)
~: SUPERSOLID
Off-Diagonal Long-Range Order in Solids
427
H
VACUUM
SUPER FLUID
28
/
/
NORMAL / /
SOLID//
/
/
/
/
///"
-0
-(8+0)
CASE(i)l
(ii)
1
8-D
(iii)
c
0
(V)
1
(8~0)
Fig. l(b).
§4.
Discussion
In the ·preceding section we have found within the framework of QLM
and molecular field approximation that in order to have a supersolid in which
both DLRO and ODLRO coexist the parameters specifying the model must
be of Case (iv). In the normal solid at 0°K each lattice point of sublattice
~ is occupied by an atom and no atom is present in sublattice m. In Case
(iv) as well as in Case (iii), the normal solid can be obtained only when
one applies a pressure higher than the pressure Ps which corresponds to the
field Hs.
In Case (iv) as one reduces the pressure below Pn there apJ::ear vacancies
in sublattice ~ and atoms in sublattice m. Just below Ps where the system
is the supersolid, the number of atoms in sublattice m is much smaller than
that of atoms in sublattice ~- Their difference decreases until the pressure
reaches Pa corresponding to the field Ha. At Pa the number of atoms in both
sublattices becomes equal and the supersolid transforms to the superfluid. The
transitions both from the normal solid to the supersolid and from the
supersolid to the superfluid are of second order. On the other hand, in Case
(iii) we have no supersolid and the transition from the normal solid to
superfluid is of first order.
A real normal solid at 0°K may respond when the pressure is reduced
in four ways: (1) the increase of the lattice constant, (2) the change of the
crystal symmetry, (3) the transition to a superfluid or ( 4) the transition to
a supersolid. In the QLM in this paper we disregarded (1) and (2) among
above four types of responses. In a more elaborate treatment one may include
the effect of (1) and (2) in the QLM, by allowing the lattice constant or
lattice symmetry to change in such a way as to make a free energy minimum.
However, in our treatment we only compared the energy of the normal solid
428
H. Matsuda and T. Tsuneto
with the supersolid of the same lattice and with superfluid based on the QLM
of the same lattice. We also neglected the effect of a short-range correlation
by using the molecular field approximation.
Therefore, when we apply our criterion for the existence of the supersolid
to real systems, we must use the parameters based on the QLM the ~ sublattice of which coincides with the lattice consisting of the equilibrium points
of atoms of the real normal solid which is just about to make a transition to
the supersolid or to the superfluid. Here, we neglect the possibility that the
normal solid may make a transition to a supersolid with a different lattice
constant or structure from the normal solid under consideration. With such
a reservation in mind let us discuss the possibility of a supersolid in real
systems.
Assuming that the 0-th lattice point belongs to sublattice ~' we obtain
from (3 · 2)' and (2 · 6)':
(1/8) ~
Vort-
jE.91
(n 2 /2) (fi} /md 2 ) ,
(4·1)
Since m our approximation the normal solid has the potential energy
(n/2) ~ Voj
jc.W.
V,.
(4·2)
and the kinetic energy
(4·3)
we obtain
(2 Vns+ Kns) /n,
D=3Kns/n.
(4·4)
For solid He4 at the melting point we may put K,.::::.-:::.38 cal/mole, and
Vns:::::::::.-50 cal/mole, so that we have C:::::::::.-62 cal/mole and D::::.-:::.114 cal/mole. 10 )
In order to estimate the value of B, we note the number of atoms per
lattice point of QLM is given by (2 · 10). *) Therefore letting Psot and pnq be
the density of the solid and liquid phase at the melting pressure, we have
from (3 ·10), (3 · 25) and (3 · 33):
(J
_
P~ol- Pnq _
1/2 cos{)
= - - - - - - - ---··--·····------Psol
1/2
so that
(4·5)
*) One might as well estimate the value of B from the experimental melting pressure invoking
(2·21) and (3·20)'. However, in such a case we must assume the same set of values for B, C
and D for all the pressures below the melting pressure; this assumption is by no means reliable.
Qff-Diagonal Long-Range Order in Solids
429
The molar volume of solid and liquid He4 is about 25.5 cc/mole and 27.6
cal/mole. 11 > Therefore,
cc/mole, so that a==0.076 and B==l.01(C+D)
in view of the result of the preceding section these parameters are of Case (iii).
This is consistent with the fact that at 0°K He4 is solid only under pressure.
At the same time it indicates that the bulk solid He4 will not become a
supersolid.
When He4 film is adsorbed on a substrate a layer of the film will be
under the influence of the atomic potential of the substrate. In order to add
such an effect to the QLM in the molecular field approximation we may
replace the expression of the energy E per spin .by
E
~
E
(4·6)
(cosO-cosrjJ),
where E is given by (3 · 2) and 4h is the difference of the potential of the
substrate acting on the lattice point of sublattice 2l from that acting on sublattice ~In order to see the effect of h>O on the existence range of a supersolid
in a simply way, we consider the stability of the solution (0=0, ifJ rc) noting
that this is a solution of
aE
aE
ao
8¢J
0.
(4·7)
From ( 4 · 6) and (3 · 3) we obtain at 0 0 and ifJ
B
82E
aoa¢
C+
l1
,
azE
rc:
H +B C+ h '
( 4· 7)'
=D.
The stability of the solution (0= 0,
rjJ
rc) becomes critical when
that is, when
(4·8)
On the other hand, since h>O, Eq. ( 4 · 2) has no symmetric solution
except for 0=¢= 0. However, when h is small enough, we may treat the
effect of h as a perturbation and keep on using the term symmetric and asymmetric solution as in the case h = 0. Then, in view of ( 4 · 6) the effect of h
on the energy of the symmetric solution is negligible up to the first order in h.
Therefore, at H = Hs the energy of the symmetric and asymmetric solution
is given from (3·11), (3·13) and (4·6) by
H. Matsuda and T. Tsuneto
430
H';
}
- { 4(B+C+D) +D+F +o(h)
(4·9)
and
(4·10)
so that
2
(4·11)
B+C+D (C-C+)(C-C_),
where
(4·12)
For Case (iii), as is the case for bulk solid He4 at the melting pressure,
we have H>O for h>O. Therefore, if it happens for h>O that
<C<C+'
then the energy of the solid is lower than that of the fluid at
Hs, so that
as the pressure changes the normal solid may become a supersolid before the
superfluid becomes stable.
From (4·12), for the case B+h D'2_0, we have
c_
o(h)
and
C_=
1
D
B-D h+o(h).
(4·13)
Thus, in this case, even if C<O, the supersolid may exist so long as
C>-~D/(B-D)h. However, by our estimation for the solid He\ B-D
:::::::::-61.5 cal/mole, so that unless h>61.5 cal/mole, which is too large to
regard h as a perturbation, we are not in this case.
On the other hand, for B+h-D<O we have
1
D
C+- 2 D-B h+o(h),
D-2B
C_=B D+T D-B h+o(h).
1
(4·14)
Therefore, if D<2B, the supersolid may exist even if C<B D so long
ash is just so large that C>B-D+~(D-2B)h/(D_c..B). In our estimation,
,D-2B:::::=9 cal/mole, which does not satisfy the above condition. However,
we must note that the estimation of the parameters was made for the bulk
He4 but not for the film under appropriate conditions. Therefore, if a bit
larger value should be given for B, then the film might become a supersolid.
Anyway, in view of the crudeness of the estimation we can only state at
present that the possibility still exists for the adsorbed He4 film to become a
supersolid even if it is impossible for bulk He4 •
Off-Diagonal Long-Range Ordf!r in Solids
431
For a bulk system not under the influence of an external potential to
becom~ a supersolid, it is necessary that C>O. One possibility for a solid to
have C>O when it is about to cease to be a normal solid is the case where
the solid is under extremely high pressure. At extremely high pressure, atoms
of the solid will completely be ionized and the kinetic energy effect due to
the finite mass of nuclei will become as important as the effect of potential
energy for determining the state of the matter. As a result, the normal solid
will cease to exist and eventually it will become a superfluid. Here, we note
in such a case a pair potential energy will be repulsive to make C>O. Because
of the symmetry of the Hamiltonian of the corresponding spin system of the
QLM, our results obtained in the preceding section remain valid for H<O by
replacing H by IHI, so that the criterion for a supersolid is applicable also
for the transition from the normal solid due to the increase of pressure.
Therefore, the solid with nuclei of even mass number such as D, or He4 may
become a supersolid by applying extremely high pressure before it becomes a
superfluid. It is just a speculation, because in order to ascertain the possibility
even in a confine of our approximation we must inquire whether we have
B> D; an another necessary condition for a supersolid. But we do not come
into this problem in this paper.
Finally, as a model of magnetic substances, our result suggests that an
antiferromagnet may have an 'intermediate' spin arrangement corresponding to
the supersolid before it becomes a spin-flop state when the external magnetic
field is applied parallel to the easy axis of magnetization. Here the spin
arrangement of the spin-flop state is reached continuously from that of the
antiferromagnetic state by increasing the external field. Such a situation can
be realized only when there is a suitable interaction between spins belonging
to the same sublattices. It would be interesting to inquire experimentally
whether such an 'intermediate' state really exists in nature.
Acknowledgement
One of the authors (T.T.) would like to thank Prof. E. Lieb for helpful
discussions on the subject of Appendix I.
Appendix I
In this appendix we make an attempt to prove that there is no ODLRO
m a wave function of the form (1· 2). Since we are concerned with the
ground state of a Bose system, we may take ,P; to be real and positive definite.
Let us introduce
A==
N~v ~dxdx',p(x, x'),
(A·l)
H. Matsuda and T. Tsuneto
432
where ·u is the volume of a unit cell of the lattice and ¢(x, x') 1s the one
particle density matrix defined by
¢(x, x') ===N~P'(x, Xz, · ··, xN)P'(x', Xz, · · ·, xN)dx2· .. dxN
X
{~P'2 (xl,x2, "',XN)dxl· .. dxN}-
1
(A·2)
•
In terms of A we can express the criterion for the existence of ODLRO in
the ground state as follows:
If lim A=O(l),
then ODLRO,
N_,.=
0,
then no ODLRO.
(A·3)
The normalization constant of (A· 2) for the wave function (A· 2) is equal
to N! P(aij), where
(A·4)
1s the permanent of a;j.
Similarly
\ dxdx'dx2·· ·dxN ::S ¢a1(x)¢a 2(Xz) ···¢aN(xN)
.)
{a}
N
X ::S ¢131 (x')¢a2 (xz) · · ·¢sN (xN)
{[3}
(N -1) !c2 ::S m;; ,
ij
(A·5)
where
(A·6)
and m,j
IS
the minor of the i-j element of P.
Hence we get
A
(A·7)
First consider two extreme cases. When a,j = 81h that is, when there is
no overlap, then P= 1, mij c;,j and c= Vv, so that A= I/N When a 1 j= 1
for all i and j, then
N!, m,j= (N-1)! and c= VNv so that A I/ N.
Now we proceed to the case where there is the overlap of wave functions
only for nearest neighbours, that is
ali{ a,
0,
when i and j are the nearest neighbour to each other,
otherwise.
It is instructive to formulate the problem in terms of a graph on the lattice,
which we suppose for simplicity to be simple cubic. Consider a set of oriented
graphs constructed according to the following rules:
i) At each vertex of the lattice there are one outgoing and one incoming line.
Off-Diagonal Long-Range Order in Solids
433
ii) A line can only connect a vertex with itself or with one of its nearest
neighbours.
First we will find an expression for
the permanent P. We cover all the vertices
of the lattice with closed loops in all possible
ways. An example is given in Fig. 2.
Each such graph consists of the three types
of subgraphs to which we assign weight as
follows:
c)
a)
b)
c)
A self-loop with weight 1.
A loop connecting two nearest neighbours with weight a 2•
An oriented polygon of n edges with
weight an.
Fig. 2.
The contribution to P of a particular graph is the product of the weights of
subgraphs contained in the graph, and P is equal to the sum of the contributions from all distinct graphs. In order to obtain the minor m;h we suppose
that only an outgoing or an incoming line is attached to the i-th and the j-th
vertex, respectively, and covers all vertices as above. In other words we connect the i-th and the j-th vertex with a self-avoiding walk and cover the rest
with closed subgraphs, and then sum the product of the weights of all possible
graphs.
Instead of A we now use
,_1.Im--...:::..J-c2 ~ mu
N..,.= Nv i P
ll=
for convenience.
(A·S)
According to the above rule we get
(A·9)
where fJsCi--j) designates a self-avoiding walk from the i-th to the j-th vertex
with s steps and a• W is the sum of the weight of all possible graphs containM
ing fJ.. W is defined to be equal to 0 when s is smaller than the number of
the smallest steps from i to j. Clearly for an arbitrary fJs we have
W(fJ.)<P,
(A·lO)
so that
(A·ll)
where C.(i--j) is the number of self-avoiding walks from i to j with s steps.
Obviously it is smaller than the number c; of corresponding random walks
with no immediate return. Hence
H. Matsuda and T. Tsuneto
434
(A·12)
If the sum over s converges, then tl=O.
For large s we know that
~C;(i___,..j)___,..(d-1)\
(A·13)
j
where dis the number of nearest neighbours. Consequently, if a<1/(d-1),
we have no ODLRO. To conclude the proof one only needs the inequality,
a<1/d, which one can derive from the normalization condition for ¢; using
Schwartz inequality.
Also in a general case where the overlap extends to an arbitrary finite
range, that is,
I R;-Rjl <l,
otherwise,
with l/L=O(N- 113 ) , one would expect that there is no ODLRO, but so far
we have not succeeded in extending the proof to this case.
The above formulation in terms of a lattice graph reveals similar nature
of the present problem to that of the Ising model. In the latter problem one
of course has a different type of graphs and tanh(J/kB T) in place of the
overlap constant, a;j.
Appendix II
Proof that the wave function o/(xt, ... , xN) given by (1·6) possesses both
DLRO and ODLRO.
By assumption,
(A·14)
for and only for the lattice vector a of the Bravais lattice, and
c(x+a)
c(x).
(A·15)
Assuming a periodic boundary, for any vector b we have
o/Cx1 + b, ... , xN+ b)=~ ... ~o/o(xl + b, ... , xN+ b, yb ... , Ys) i~l {c(yJdyi}
~ .. ·~o/o(xl+b, '",XN+b,yt+b, "·,ys+b)i~{c(yi+b)dyi}.
If b
(A·16)
a, we obtain by (A·14) and (A·15)
(A·17)
and this relation in general does not hold for b which 1s not a lattice vector.
Therefore, o/(xt, ···, xN) has DLRO.
Off~Diagonal
Long-Range Orde1· in Solids
435
In order to show ODLRO it suffices to show that
(A·18)
lim ¢(x, x') =FO,
jx-xlj~""
where
(A·19)
For the wave function of the 'perfect crystal' we assume that
(A· 20)
unless there exists a permutation (j(l),j(2);··,j(N+s)) of (1, 2, ···, N+s)
such that
Max
IX; x;ul I<a,
(A·21)
l<i<N+s
where a is a constant sufficiently smaller than the lattice constant of the crystal.
Therefore,
s
X o/~ (xb · ··xN, y~, ... , y;) ~ {c(yi)c* (y~)dyjdy~}dx1···dxN
J
~1
H
2
2
::::=s! ol~ ·"~I o/o(XI, .. ·, XN, yb · ", Ys) 1 1{I c(yJ l dyj} dx1" ·dxN
(A· 22)
::::=s! (J)•< I c(y) 1 2 )•,
where (J) is a constant of the order of a3 and <icC y) 12 ) is an average of
2
2
Here,
I c(yi) 1 by the probability density function I o/o(Xt, ... , xN, y1, ... , ys) 1 •
we used the approximate statistical independence of the random variables
y1, yz, ... , Yn because s~N.
Similarly we obtain
~ ... ~o/(x, Xz, ... , xN)o/*(x', Xz, ... , XN)dxz"·dxN
=
~ ... ~o/o(x, Xz, ... , XN, yh ... , y.)o/~ (x', Xz, ... , XN, y~, ... , y;)
s
X
;ri {c(yi)c(y;)dyidy;}dXz" ·dxN
J=l
::::=s2 (s-1)! (J)s+l~ ... ~'fro(X, Xz, ... , XN, X1 , yz, ... , y.)
s
X o/~ (x', Xz, ... , XN, x, yz, ... , y.)c(x')c* (x) II {I c(yi)
::::=s2 (s-1)! (J)s+lc(x')c* (x) {p(x, x')
j (N;
l
2
dyi}dxz"·dxN
j=2
5
)}
<lc(y)
1
2
)•-I,
H. Matsuda and T. Tsuneto
436
where p(x, x') is the probability for two particles to occupy the positions x
and x'. Thus we obtain for N>s:
q;,(x, x')
c* (x )c(x') p(x, x')w
<lc(y) 12 ) -
Since p(x, x') j-1->0 as
has ODLRO.
Ix
(A·23)
x' I _,..oo, we have (A ·18) so that the state
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