Quantum theory of interacting bosons

ANNALS

OF

PHYSICS:

9,

292-324

Quantum

(1960)

Theory EUGENE

Brandeis

University,

of Interacting

Bosons*,

P. GROSS Waltham,

Massachusetts

Some qualitative features of the ground state of a system of interacting bosons are discussed using wave functions suggested by the semiclassical theory of boson wave fields. For the case where one deals with weak repulsions, one is lead to a variational extension of Bogolyubov’s work. A finite fraction of the N particles occupies the zero momentum single particle state, and the dynamic correlations are described by pair excitations. When attractive forces play a decisive role, two cases are found. In one case a finite fraction of the particles occupies a single particle state, which is now periodic in space. The dynamic correlations are described as a generalization of pair excitations which is different in character for excitation momenta of the order of the inverse of the range of the attractive forces. The single particle state and dynamic correlations are codetermined in a systematic way. The approximate ground state shows long range order which is destroyed at finite temperatures. A second case where attractions are important is the solid state of the boson system. The ground state has the property that of the order of N orthogonal single particle states are occupied, each with an average of approximately one particle. 1. INTRODUCTION

In the present work we try to clarify somepoints concerning the ground state of a system of interacting bosons. We are particularly interested in the case when attractive interactions play an important role. We will put in more explicit and detailed form some considerations reported earlier (1, S) . This work proceededfrom an analysis of the semiclassicalequations associatedwith the Heisenberg equations of motion for the quantized field operators characterizing the boson system. The approach was related to Bogolyubov’s fundamental paper (3) on the low-lying states of a system of weakly repelling bosons, and to the author’s small oscillation theory of the interaction of a particle and a field (4). In the next paragraphs we briefly summarize this approach and describe qualitatively the wave functions to which it leads. We emphasize particularly the differences we are led to expect between the case of purely repulsive interactions and the case where attractive interactions play an important role. In Section 2 the method is applied to the caseof purely repulsive interactions to obtain a variational generalization of Bogolyubov’s theory. In Section 3 we study a casewhere * Supported

in part by Office of Scientific

Research, U. S. Air Force. 292

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BOSONS

attractive interactions are important, but the system is still expected to exhibit the properties of a superfluid. In Section 4 the same theory is presented in a different formal guise. The common feature of the two cases is essentially that a finite fraction of the N particles is in a single particle state. For repulsive interactions this state is l/Q’/’ where Q is the volume of the normalization box. For strong enough attractive forces the state is a spatially periodic function &,x). In both cases the ground-state wave function is more complicated, since it also describes the dynamical correlations of the particles (essentially two-body collisions). The work of Sections 2, 3, and 4 will be chiefly concerned with these correlations. The appearance of the periodic state is, on the other hand, a longrange order effect and already arises in the Hartree approximation. These two situations are to be contrasted with the case of solid helium (Section 5). To describe the ground state of solid helium in the crudest manner, we may set up N localized one-particle functions, each centered at a different lattice site. The functions may be taken to be orthogonal to each other. Then the ground state of the solid has one particle in each of the N orthogonal one-particle states, and is thus qualitatively completely different from the wave functions of Sections 2, 3, and 4. In the language of quantum chemistry, the solid is roughly described by almost a pure Heitler-London wave function, while the superfluid is almost a pure molecular orbital wave function (where the exclusion principle does not act). This distinction is discussed in Section 5. To discuss the distinctions between the different types of ground-state wave function.s, it is necessary to introduce notation and a formal statement of the problem. Our object formally is to investigate the ground-state wave function and energy of the Bose field governed by the Hamiltonian

H = & / w+w d32 + f j-1~+(x>$+(y)v(x - y)+(x)+(y)

d32 d3y,

(1.1)

where n/r is the mass of the boson, and V(x - y) is the two-body potential between bosons. The exact states are eigenfunctions of the number operator N, = J#‘# d3r. #(x) and #+( x > are annihilation and creation operators obeying the Bose-Einstein commutat’ion rules M(x), Associated

$+(Y)l

= 6(x - Y),

with the Hamiltonian

;f$ = M,Hl

= - &

W(x),

is the Heisenberg v% + 1 V(x

a)1

= 0.

(1.2)

equation of motion,

- YM+(YMY)

d3Ydx),

(1.3)

and the complex conjugate equation. Our approach to this problem consists of two steps. First we examine the equations of motion, treating #(x,t) as a function, i.e., neglecting commutation

294

GROSS

or operator properties of $ and $+. Thus we study in detail the properties of the underlying classical wave field associated with the second quantization procedure. This step does not correspond to studying the usual classical limit of the system of N particles. In the usual classical limit Planck’s constant tends to zero but M is kept fixed. The limit contemplated here is a zero mass case where both M and fi -+ 0 in such a way that fi2/2M is fixed. But actually we are not mainly interested in this limit in which the “semiclassical” theory becomes exact. Rather, we are interested in learning some of the qualitative features of the semiclassical equations of motion, so that we can be guided in choosing appropriate quantum wave functions to describe the low-lying states of the system. The study of the properties of wave functions suggested by this procedure is the object of this paper. We may note that physically the semiclassical theory is most closely allied to the description of the ground and excited states of quantum system in the Hartree self-consistent field approximation. It is precisely the self-consistent field aspect which yields the periodic, long-range order solutions of Sections 3 and 4. The second step in our approach is to establish a formal connection between the semiclassical and quantum theories. One way to do this is to note that much of the classical analysis can be phrased as the introduction of new and more appropriate canonical dynamical variables. Furthermore, the suggested canonical transformations are sufficiently elementary so that one can immediately write down the corresponding canonical transformations of the quantum field operators and the unitary operators generating the transformations. The method thus leads to approximate wave functions of the form @ = U&, , where U is a unitary transformation and a0 is a state of the noninteracting boson system. In Section 2 we note for repulsive interactions that the semiclassical theory leads us to take U = es’es2 where X1 = ( fo 1 ( eiR#af - e-ZR$o) with R any real number. 82 is + ck “’ (42) ( eziR&+#?k - c.c.). For a given R the wave function refers to an indefinite number of particles. The parameters 1f0 1 and uk are determined by minimizing the energy subject to the constraint that the mean number of particles in N. The presence of the parameter R is a reflection of the infinite degeneracy of the solutions of the semiclassical theory. The wave function for each value of R leads to the same ground-state energy and by superposing the functions with equal weights we can construct a wave function which is an exact eigenfunction of the number operator. Thus far the theory is simply essentially a variational generalization of the work of Bogolyubov. A finite fraction of the N particles is in the zero momentum plane wave state 1/Q1’2. The factor es2 describes the spread over other momentum states arising from the presence of excited pairs of particles of opposite momenta. It arises physically from the zero point In rather

this paper, vectors occurring than boldface type.

as subscripts

and superscripts

are printed

in lightface

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295

motions of the sound wave excitations of the system, which in turn are essentially due to the binary collisions of the particles. In the semiclassical theory we are led t,o the form of es2 by the study of the excited states of the classical wave field. The study of large amplitude steady waves actually leads to a more general form for ea2 in which harmonics 2k, 3k, etc. are excited along with k and -k. But this does not alter the qualitative nature of the ground-state wave function for repelling particles, which is our main concern. In Section 3 we treat a case where attractive interactions are important. If one neglects the effects of the zero point motions of the sound waves and other excitations, the unitary transformation has the form es’ with &

=

ck

(.fklL.k+

-

.fk*tik),

i.e., linear in the field operators. The requirement that the wave function es+& represents a mean number of partides N yields c 1fk 1’ = N. If the fk are freely varied, a set which makes the energy stationary, satisfies the self-consistent field equations of the semiclassical theory. The uniform solution f~ = fi satisfies the equations, but there is also a range of spatially periodic solutions, all of which have a lower energy per particle. The special solution of lowest energy is the one realized at absolute zero, but the others play an important role at finite temperatures. Indeed, there should in general be solutions corresponding to crystal structures different from the one at absolute zero, thus leading to the possibility of a kind of order disorder transition at finite temperatures. It is interesting that nothing corresponding to these exact solutions of the semiclassical theory which differ from each other in energy per particle, has been found for the repulsive case. Section 3 treats further the relation between the different periodic solutions. We show that in general they are orthogonal to each other and that the Hamiltonian has no nonaero matrix elements for two different states in the limit N -+ CQ. It is therefore not possible to improve the wave function by forming a linear combination of the quasi degenerate wave functions. Each solution fk is, however, exactIy degenerate with another fkeikareiR where CCand R are real numbers. This fact is utilized to construct exact eigenfunctions of the momentum and number operators. The remainder of Section 3 is devoted to the improvement of the wave functions when one makes a normal mode transformation to describe close collisions. Each periodic solution has its own excited state spectrum and there is a shift in zero point energy which lowers t,he ground-state energy. In addition, the close collisions between particles modify the effective average potential in which the particles move, i.e., changes the equations which fk satisfy. Section 4 treats the samephysical content in a somewhat different formal way. We use a,n expansion of the field operators in a general basis, rather than always working .in a plane wave basis. The physical meaning of our starting point is that

296

GROSS

a finite fraction of the particles are in a single state cpo(x). In this form we obtain an algorithm for approximating the ground state of a superfluid boson system subject to any boundary conditions. For the present case of an infinite system with attractive interactions, the formal development of Section 4 suggests new approximations. One of these is the use of a tight binding approximation. In Section 5 we briefly discuss ground-state wave functions which differ drastically from those of the previous sections. The fundamental assumption, common to repulsive and attractive cases has been that there is a finite fraction of the particles in some one particle state. As already remarked, this is certainly not the case for the solid state of the boson system, where essentially N different one-particle states have occupation numbers of the order of unity. Thus for sufficiently high density and strong attraction a ground state with a finite fraction of particles in a single state is unstable. The instability is also revealed by showing that it is energetically advantageous to populate higher one-particle states. Naturally the variational theorm insures that the energy as computed in Sections 3 and 4 is an upper bound. But there may be situations with attractive interactions which are intermediate to the case of a finite fraction of N particles in a single level on the one hand and the case of the solid. Such a case might be N” levels occupied with of the order of N1-OL particles each. (CXis a fraction between 0 and 1.) A ground-state wave function could be constructed to cover such a case, if it exists, again relying on the variational principle. But it would be difficult to construct the type of systematic approach involved in Sections 24, which lends itself easily to the study of the excited states and thermodynamic properties of a system of bosons. 2. REPULSIVE

INTERACTIONS-UNIFORM

SOLUTIONS

In the case of repulsive interactions it is most opportune the field $(x,t) in a cube of volume Q. We write $i+(x,t)

= -&

C

to Fourier

analyze

$,k+emikz,

(2.1)

The Hamiltonian and the equations of motion are

(2.3) We would like to find solutions for which the number of particles, N = is definite.

xk

#k+#k

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BOSONS

The classical solution of lowest energy was #(x) = &i@e-iENt’h or & = & = 0 for k # 0. Here E = V&/22/5. This solution is independent of space and was called the uniform solution. We can go to field variables characterizing this solution, which are time independent, by performing the timedependent canonical transformation dge-iENtlfi)

$(x) g+(x) This transformation

4 #(x)eiENt’*

,

+ #+(x)emiENtlh.

induces the change in the Hamiltonian H-H-EN.

It is expedient to use a quantum way of speaking. The above canonical transformation is generated by the time-dependent unitary operator exp (iENt &h+ih) . W e will also adopt the point of view in which the new dynamical variables have the same denotation as the old, and in the Hamiltonian are replaced by their expressions in terms of the old variables. For a time-independent’ unitary transformation U, the transformed Hamiltonian UHU-’ is the same function of the #r~ , $k+ that H is of the new operators U&U-‘, U@U-‘. In the quantum theory the Schrodinger equation for the state vector 4%~) in

a%ld

=

at

-old

becomes

(2.5) with @new = U&,,d , @,,ld = u-b,,,

.

Given. an exact solution, the next step in the classical theory was to introduce dynamical variables which are the deviations of the field amplitude from the values characterizing the solution. In the present case only the Fourier component $0 is excited in the exact solution. The unitary transformation u1

=

e(fo+o+-fo*hl)

(2.6)

induces a linear shift

U4~ouF = fo+ $0, ul*o,+u~’ = j-o*+ *o+, ul$ku?

The transformed

=

#k

Hamiltonian

,

&$k+u;-’

Ul(H,,

=

- EN,,)

#k+

,

k # 0.

UT’ will be written

(2.7) as the sum

298

GROSS

of five terms x0=

xf3,, Hi - ENi

= xxi,

~&fi21fo12-El~o12,

~1 = 5

(fo* Ifo I2+o + cc.1 - E(fo*+o + fob’+),

x2 = &)k+J/k

k

-_

+F&

(2.8) (.fO*fO*J’k$-k

+

C.C.),

where we group terms according to the number of times f. and fo* occur in the potential energy. We leave f. unspecified for the time being. In the classical theory we had f. equal to &i? but it will not have precisely this value in the quantum theory. We may note that there are actually infinitely many exact uniaform solutions of the classical equations ofequal energy. In fact,

with R any real constant is an exact solution with energy E = (V0/2fi)N2, independent of R. This degeneracy is aIso present in the quantum theory even when / fo 1 # dq. All of these solutions are needed if one is to construct wave functions which correspond to a definite number of particles. We shall therefore allow fo to be complex. We will see that this step of superposing the degenerate solutions leads to a state vector a olrl = (9o+>“/N!

%ew ,

which is just the ground-state vector for a system of N noninteracting bosons when + new is the vacuum for the free-particle states. The expectation value of the Hamiltonian with this function yields the energy (Vo/2& N(N - 1) and so differs only by a term of order l/N times the leading term, i.e., a contribution to the energy for which the energy per particle vanishes. The next step is to analyze the equations of motion for the +k governed by the Hamiltonian H,, - EN,, . In the classical theory there are no restrictions on the smallness of the amplitudes so that there is always a set of motions where x3 and x4 can be neglected, provided the motions governed by X2 are stable. The condition that there be no linear increase of $0 in time then requires that X1

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BOSONS

vanishes, or E Z=-

fo

2’

(2.9)

12.

This guarantees that we are studying oscillations about a time constant mean value. For infinitesimal amplitudes the number condition N = c $k”& becomes 1Jo 1’ = N. The analysis of ~2 leads one to introduce the Bogolyubov normal modes #k’

=

U2$k@

=

#ii’

=

~,~k+~,’

ffk$k =

+

@k,&

ak*#k+

+

,

(2.10)

pk*J’-k

,

where (Yk and @k satisfy ] ak

The generator

1’ -

1 bk

1’

=

akbk

1,

of this transformation

=

a-k&-k

(2.11)

.

is (2.12)

where coshk+

u-k 1 2

(gk

= ff!s = CQ.z,

+

C-k>

, uk + u-k ,

sinhI uk ’ u-k I = 2

,& = P-k.

(2.13)

This transformation can also be defined for k = 0, but it is unimportant whether we do this or not, since the anomalous character of the k = 0 state has been treated by the transform U1 . UT induces a change of the Hamiltonian = ~Lo3P,

u2xuz1

where x UN _- ~~~Ijo121fo12

- Elfo12,

x (1)= i ,s IfoI2- E) jo*d + c.c., fi%” vo I fo I2 x (

+

(2)

=

F


CM

-

i?

+

-

ti

vk -

i f0 fi

I2

(2.14)

300 &3)

GROSS

=

z

3Jo*(~m+hJs-k~+

+

w&

(2 14) cont.

+ F 5

(fo*l+bk+$k + foQk+k+)& + cc.

In the preceding expression we have used the notation

i.e., (#z+icf-+l)+is the ordered form of @$?I (with creation operators to the left of destruction operators. In like manner (#~+#~+#z-&~+k:) + is the ordered form of @!bY&k&+k . We have also defined

(2.16)

alois a vacuum state of the number operators $k+$k . Thus $$$k’ = ($k+&)+ + #k+#k , etc. It has been left in terms of the primed operators for brevity. It is understood that the primed operators are to be replaced by their equivalent combinations of unprimed operators. If we disregard 3~‘~’and XC4)we have A’ = ( VO/~$ 1fo 12.The ak and @kare determined by requiring that 3~‘~’ be in normal form. This means that the coefficients of $k+& and #k&k must vanish for each k, i.e., that Lykand Pk satisfy

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BOSONS

fi2k2

%iF+-

Vk I jo I2 -& jo*2/?k/3-k + -& jo%k*& = 0 (2.17) ffk*@k + 4fQ > together with the complex conjugate equation. When ak , @k , and jo are real, the solution of these equations taking into account the conditions (2.11), is CYk= [l Sk

22 Yk=&+-

=

g/4]-‘, -Yk/sk

ok

+

VkIjO12 24n ’

=

&k2/6k2

Skak/%

-

4

(2.18)

6k = vkj:/&i.

To complete the theory we must choose Sew . The obvious first approximation to the ground state is to take %,,, to be the vacuum state for the operators #k+#k . Then in the approximation that we neglect SC@)and XC4), the ground-state energy 1,) *Cm

(2.19) +

sG

( j;*flk=k

+

c.c.>.

The number operator condition is N = 1Jo 1’ + c I Bk I’, and enforces a partial depletion of the state k = 0. This condition together with E = (Vohh) 1fo 1’fi xes all the parameters. We note that the expressions (2.11) and (2.17) are invariant to the substitution j. + joeCR,@k + @keiR, ffk --) ake --iR . In the quantum theory the commutation relations thus rule out motions with infinitesimal amplitudes. The inherent zero point motions of the field imply that we are always to a certain extent in the region where anharmonic corrections occur. The contributions of x.? and xC4) to the energy may not be negligible. What happens is already clear from our study of exact large amplitude classical solutions. The amplitude 1jo I’ is decreased and the (Yk and flk which determine the frequency spectrum are altered. In addition harmonics are excited, i.e., along with #k and #+k we expect $2k , etc. to have a finite value. This latter effect will not be taken into account in the present considerations. We will adopt the procedure of Ref. 4. x’~’ is written with all creation operators to the left of destruction operators. This process leads to the isolation of terms linear in tie and tie+, already written separately in Eq. (2.14). Our earlier requirement that there be no linear increase of the operators in time, i.e., that the coeficients of the 1inea:r terms in #k and +k+ vanish, now yields

IjoI2--E+

F+12

+~G,,,12)jo+

Fsjo

= o.

(2.19)

302

GROSS

With f0 real and Vk* = Vk , we get

For given Crk, Pk , Eq. (2.20) together with the number operator constraint N

=

1 .fO I2 +

T

I bk

(2.21)

I2

determines E and 1fo 12.The conditions determining E and I f. I2 are equivalent to the variational requirement that the ground-state energy E, (see Eq. (2.25) ), be a minimum with respect to variations of fo and fo* (independent of cUkand pk) , subject to the number constraint. When XC4’ is ordered in the same way there appear constant and quadratic terms in J/k , &‘. To fix LYEand @kwe examine the complete quadratic form and require that it be in normal form. The requirement that the coefficients of #k&k and #k+icf--k vanish fixes L11k , Pk . Since the quadratic terms are

Cl (!h++l)+Yl + (rc’l+J’+&l*+ (!hb-z)+h , this step yields (2.22) and the complex conjugate equation.

Here

(2.23)

=- &212 + 2M

~ ‘+

+

T

9Q

(‘&k#l-k

-

With these choices of (~1, PI , E, the Hamiltonian terms in the ordered creation and annihilation

where

@k#k)

-

c

$Q

+k#-k.

is diagonal as far as quadratic operators. We can write

INTERACTING

BOSONS

+ c &$ r$ + ‘%&$) + F 2$Q(.fo** G + cd k

303

(2.25)

and ez = -Yz(J az I2 + 1Pz I”) + 2&crz*pz + 25z*cud32*

(2.26)

If we take as our approximation to the ground state that a,,,, is a vacuum state for the operators tiki$k , E, becomes the ground-state energy. By the variational principle the true energy lies lower. The cubic and quartic terms have zero expectation value. Nevertheless they have off -diagonal matrix elements and contribute to the energy when a better Ssw is used. We may note that when j f~ j2 and ffk , Pk are fixed the unitary transform U = es’es2 is definite. Therefore for perturbation theory if U%,w generates a set of state vectors appropriate a,,, runs through a set of states consisting of creation operators acting on the vacuum state. It should be noted that while the mean number of particles corresponding to each of these states will not be N, it will differ from N by only a finite number of the order of unity if only a finite number of creation operators are used. Thus if the perturbation method converges this effect is unimportant as N ---f cc,. It is not germane to the main point of this paper to enter into this type of improvement here. Rather, our purpose is to bring the analysis of ground state for the case where attractive forces are important to the same starting point provided by the above variational generalization of Bogolyubov’s work. For the same reason we will not discuss the spectrum of excitations indicated by el by Eq. (2.26). It shares the same difficulties (a spurious energy gap as 1 + 0) as other similar theories (5, S) in which only pairs of particles of opposite momenta are excited by U, but the k = 0 state is treated in different ways. These difficulties result for the above U from too crude an assumption for the st’ates a new . The semiclassical theory points to a number of generalizations of the preceding work which can be made the basis of variational calculations. For example, the exact nonlinear waves of the semiclassical theory have the property that not only is k = 0 state depleted i.e., / fo I2 < N, but that harmonics of k and -k are excited. Thus particles with momenta which are multiples of &k are excited along with the pairs. This effect can be partially taken into account by generalizing 82 so that while it is still a quadratic form in creation and annihila-

304

GROSS

tion operators, it now contains terms such as I&& . The state vector will no longer be an eigenfunction of the momentum operator, but we can introduce the additional constraint that the expectation value of the momentum is zero. The more general U brings down additional quadratic and constant terms from x4 , and we can require that the ordered quadratic expression be in normal form. Indeed, a still more general trial state vector (partly suggested by the analysis of Section 3) is @a=

s

G( R)e”“R’,sz(R)c$o .

Here &(R) is the expression c fk(R)Gk+ - c.c., and is linear in the creation and annihilation operators. SZ is a quadratic form in the operators with parameters depending on R. G(R) is a weight factor. These wave functions lead to an appreciable improvement in the ground state energy. It would take us too far afield to work with them here. To conclude this section we discuss the infinite degeneracy connected with the fact that the equations determining the approximation are invariant with respect to the substitution f0 -+ foeiR, ok + PkeiR, CY~+ cukemiR.In accordance with general quantum mechanical ideas, we ought to superpose, with equal weights, the degenerate wave functions of different R values. Hence we form the wave function q&L M oln %dR) s

(2.27)

dR,

where R is restricted to the range 0 - 2~ to avoid redundancy. The factor M is M2 = (/- %dR)

dR, 1 %dR’)

dR’)

(2.28)

and is required to obtain a normalized Cp.The above 9 is an eigenfunction of the number operator. In fact, if we start with @m(R) for a given value of R, say R = 0, we may obtain an eigenfunction of N,, by applying a projection operator. Consider the operator 2?r

pN

=

L

ei(Nop--N)S

d,‘$.

(2.29)

2lr s 0 In a representation

where N,, is diagonal, we have (N’ 1 PN 1N) =

to any wave function, Thus we write of N,, .

Applied

$2 =

6~r.N.

@N selects the portion which is an eigenfunction

M”

=

(@N%ld(O),

@N%do>).

(2.30)

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305

BOSONS

Cp’ has the property that for zero interaction, it goes over into [(u~+)~/N!]%, i.e., the exact wave function for N particles in this limit. 6% has the properties 2 4@N = @N , @N = C& and it commutes with H. It is clear that a’ = 9. Let us now examine the revised ground-state energy & obtained when we use a. We have

(2.31) = &

(u-‘(o)~N~(o)%,,

, u-‘(o)~~(o)%W).

Now the properties of U&KU(O) = c 3CCi) are obtained from Eqs. (2.14). We have (x”’ + X,‘i”,‘)@o = 0 where ~$2 is linear portion of x’~‘. Similarly ( x’~’ + c?.$,‘,,)% = 0 where x~L, is the quadratic portion of x’*‘. Thus we have

(E, + EN)$ + (se:"' + x:"')%,. On the other hand the factor U-'PNU%~~~is u-‘xu@,,,

=

1 5, s

2r

ciNRU-'(O)U(R)%,dR.

This is a sum of terms in which a number of pairs (k, - k) (1, - 1)) etc. are excited when k,l # 0 and a certain number of zero momentum particles are excited. The term (xY’@,, has an odd number of nonzero momentum quanta and thus gives no contribution to the inner product. The portion of X:“‘% which contributes is

We thus have

I-: = E. $ EN -I- i2& Some analysis Nt to.

shows

s,?‘ epaNR (U-'(O)U(R)~odR,X:4'~o).

that the correction 3. ATTRACTIVE

to the energy per particle

(2.32) vanishes

as

INTERACTIONS

We now take up a casewhere attractive interactions play a commanding role. There are actually at least two such cases.One type of ground state corresponds to a superfluid liquid and wil1 be treated in the present section. A second type of ground state corresponds to a solid state of the boson system and will be briefly discussedin Section 5. Our construction of a superfluid ground state when attractions are important rests on the exhibition of spatially periodic solutions in the semiclassicaltheory.

306

GROSS

It was shown in Ref. .8 that such solutions are possible and in fact can occur over a range of periodicities. Each solution has a lower energy than the uniform solution and the set of periodic solutions has a continuous spectrum. The condition that a periodic solution have a lower energy than a uniform solution in the semiclassical theory is essentially that the spatial order be capable of taking advantage of the negative Fourier components of the potential. In one dimension the attractive energy per particle NVd/&? must overcome the cost in kinetic energy fizd2/2M required to give the boson field amplitude a curvature in space. The energy of a periodic state is for weak potentials 2

fi2d2/2M + 2 ;

H

c$d N

7

wli (N/Q)

where +k = &tVk = J” eik”V(x) dx. The second term is the depression of the energy below that of the uniform solution. The Fourier components C#Qwill be positive when d w 0, so that periodicities are only possible for values of d of the order of 2x/a, where a measures the range of the attractive part of the potential. At a given density p = N/8, the value of d corresponding to the lowest energy is given by minimizing the energy per particle. This condition (a/ad) (Ho/N) = 0 yields

so that +d’ must be negative for a minimum to occur for real d. With d(p) as the solution of the above equation, we can then ask for the value of p which yields the lowest energy. The condition

$ (Ho/N)= $

(Ho/N)

$ + $ (HO/N)

yields

; @c/N)= $, (Ho/N)= 6 ; (+o+ 4d)+ &-d + $ > 0. There is no minimum in the energy per particle. The system takes up the smallest value of p compatible with periodic solutions, since the energy is less than p&,/2. This corresponds to a real positive value of p when 140 1 >> 1C#J~ (, a condition which physical potentials obey. If the system is constrained to a higher density it realizes the periodic solution d(p) . The modifications of the above theory which ensue when the binary collisions between particles are taken into account, will be discussed later in this section.

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307

BOSONS

In addition to the d value corresponding to the lowest energy for a given density, other periodicities are possible. There will be a finite range, not containing very long or short wave periodicities, where the energy is lower than the uniform state. In addition, in the three dimensional case a number of crystal structures may be possible, all with energies lower than that of the uniform solution. The periodic solutions differ not only in wavelength. They also differ in that each solution has a different finite fraction of the N particles occupying the k = 0 Fourier component. In one dimension I.fo12 = NCb#3c1 Each solution has its characteristic fraction of the particles, e.g.,

Zi2d2/2JWb4d

Fourier

l.fd I2 725Wfi2d2/2M

components

excited with

a finite

+ %dd/S~#u.

Let us now proceed to the mathematical analysis leading to a ground-state wave function for the attractive case. The classical #(x,t) corresponding to a spatially periodic exact solution is # = f(x) eeiENt’*,with f(x) periodic. The timedependent factor is again handled in the quantum theory by introducing the Lagrange multiplier E, so that one studies the Hamiltonian H,, - EN, . E is fixed with the help of the conditions that the expectation value of N,, is N, and that the operators of the states occupied by a finite fraction of particles do not increase linearly with time. Actually, the approximate wave functions that we set down initially will not be eigenfunctions of the momentum operator Pop = c fik&+#k . We ought therefore to introduce an additional Lagrange multiplier vector d and consider H,, - EN,, - d.P,, with the requirement that the expectation value of P,, is zero. For the ground state, it is not necessary to go to this trouble, since by taking the parameters to be the same for wave vectors k and -k, the expectation value of P,, is automatically zero. A given periodic solution f(x) has Fourier components fk , which are nonzero for k = 0 or k = a multiple of a reciprocal lattice vector appropriate to the type of periodic structure. Our first step is to introduce the unitary transformation WI = exp[C

(3.1)

(.fk$k+ - .fk*#k>l,

which has the property Jwkwy’ The Hamiltonian X = X’ = xt=.o SC;, where

= j-k + #k ,

wl$k+m’

=

.fk*

+

J’k+.

(3.2)

H,, - EN,, becomes X’ = W~XWT’. We write it as

308

GROSS

Let us start by choosing the set jk using the condition 8%/6jk* = 0, &%0/6jk i.e., that X0 is stationary with respect to independent variations of the fk . This leads to the set of equations

(E-Z)

fk = c

$

c

mjm*fm+lfk--2

.

= jk*

0, and

(3.4)

A set of jk , jk* which satisfies these equations makes Xi vanish. Indeed the coefficient of each& and #k+ in X1 vanishes separately so that there is no constant term in the equation of motion. Equations (3.4) for fk are the sameas Eqs. (37) of ref. d with jk = Ltk . The analysis given there shows that at least for certain types of potentials with some attractive Fourier components, there are periodic solutions of lower energy than the uniform solution. The preceding equation plus the condition xk )jk 1’ = N constitutes the classical theory. Our procedure thus far amounts to computing the energy of the ground state with the trial wave function 9 old

=

~1hm

under the condition that the expectation value of the number operator is N, with a,,, a vacuum for $k+#k . The Set jk iS chosen so as to minimize the energy. It is clear that Jo = d%, fk = 0 for k # 0 always satisfies the above equations with E = (Vo/&)N. Our point is that for suitably attractive interactions there is a set jk of lower energy. There are a number of noteworthy points at this stage of the procedure. First there is an infinite degeneracy. Along with any solution jk , there are also solutions of equal energy of the form fkeikaeiR where (I! and R are real numbers. The superposition of these solutions with equal weights

INTERACTING

309

BOSONS

leads to a state vector which is an eigenfunction of both the number N,, and the momentum operator Pop = c fik&+gk . We can write @ = & &&Wl(dM’new

,

where pp is a projection operator for the total momentum tion constant. We may take @P = &

--m exp KPo, s

operator

- PI-al

(3.5) and M is a normaliza-

da.

(3.6)

For the ground state with the fluid as a whole at rest, we take P = 0. With such a @ belonging to the periodic solution d the energy is

= EN + k2 j-1 eiNR dR da! F(a,R)

(3.7)

where

(3.8) + N + c

1fk I2 e-ika--iR

.

Thus g = EN + k2 11 eiNR dR da F(a,R) TT

(3.9)

-z fl*fm*(eit”+“R - l)(e”a+iR- 1)2%fl+kfm-k with M2 = 11 eiNR dR da F(a,R). It can be shown no change in the A second point there is a range

(3.10)

that the correction is of order l/N times EN, i.e., that there is energy per particle as N -+ to. is the consideration of the fact that in the semiclassical theory of periodicities which have lower energies than the uniform

310

GROSS

solution. One of these has the lowest energy, but there are others with energies infinitesimally close to the lowest energy. Let us study the relation between these solutions. The approximate quantum state vectors are 9 old 53 WI%,,

= exp c f&k+

- fk+h&lew

,

(3.11)

where 9,,, is a vacuum for J/k’& . Each periodic solution is defined by the nonzero set fo( d) , fd , f& , etc. The plane wave states k = 0, k = multiple of d all are occupied by a finite fraction of the N particles, namely 1.fk 12.We also have the constraint N = c 1fh 12. Th e most important property is that the states corresponding to different periodicities are in general orthogonal to each other. In fact @b(d),

%,dd’))

= (Wl(dh,

Wdd’h)

- ~h*(d)fdd’) I =e -N exp [c

(3.12)

fk*(d’)fk(d)l.

For d = d’ this inner product has the value unit, as it should. For d # d’ the value is e-Ne~O*(d)fO(d’) since fo is the only Fourier component the two solutions have in common. Now 1fo( d) I* and I fo( d’) I2 are both less than N and are in fact different. Hence the inner product tends to zero as N -+ co, so that in this limit the states are orthogonal. The fact that the different periodic solutions are orthogonal does not mean that the ones of higher energy do not play any role. In the present approximation they do not enter into the description of the ground state. But at any finite temperature there is an average finite excitation energy per particle. All the periodic solutions including those belonging to different crystal structures contribute to the thermodynamic properties, with weights given by the Boltzmann factor. An interesting point concerns the phonon spectrum. Associated with each periodic solution there is a phonon spectrum which has a band character, as discussed in the semiclassical theory. The gaps occur at different wave vectors corresponding to the different underlying periodicities. Thus in our approximation there are gaps in the excitation spectrum at absolute zero. At finite temperatures the gaps are smeared out. A more careful discussion of the excitation spectrum will be given elsewhere. A third point is raised by the suggestion that one improve the ground state

INTERACTING

311

BOSONS

wave function by forming a linear combination of the wave functions belonging to the diflerent periodic solutions. We try to resolve the approximate degeneracy of these solutions by taking 9 = ; c

(3.13)

G(d)Wl(d)%.

Since the functions are orthogonal the norm L is L2 = Cd / G(d) j2. We shall now show, however, that nothing is gained by doing this because the Hamiltonian has no nonvanishing matrix elements between two different functions, W1( d)+,,, and Wl( d’)%,, . Indeed

= F*(d,d’)

z

.(fm*(d’) F(d,d’)

== exp {%c

[fk*(d)fk(d’)

5

(fl*(d’)

-

fdd)fk*(Wl)

c

G*(d)G(d’)emN

F

(3.15) I Add’)

When fk(d) and fk(d’) have only f,(d) and fo(d’) ponents, we have F(d,d’) = eCN for d # d’, F(d,d) energy is d

(3.14)

- fn*(d))fl+k(d)fm--k(d),

exp(-?hc

b c

- fl*(d))

(-iv2

- b(d)

I’>

as common Fourier com= 1. The correction to the

$!=)

d’

and vanishes as N + 00. Thus in the limit of large systems there is no gain in superposing such solutions. The approximate ground state is given by the periodic solution of lowest energy. It is possible that this conclusion must be modified if the unit cell contains several atoms and has several different mirror images. This point requires further study. The discussion thus far has corresponded to a self-consistent fieId treatment of the system. We have simply noted that with attractive interactions the best selfconsistent field may correspond to a periodic particle density. The next step is to take account of the specific collisions between particles. This is done by making a normal mode analysis of ~2 . In the semiclassical theory this corresponds to studying the small oscillations of the boson wave field about a periodic solution. This leads us to introduce in the quantum theory a unitary transformation Wz

312

GROSS

such that

(3.16) with

a suitable set of coefficients.

7 R,ctRk*ll -

R kkl and Skk’ must satisfy

S,,S,*t, = SF+ ,

the conditions

F R,clR$z - S,,S,*q = 0,

(3.17)

which insure that the transformation is unitary. The Rkl and Ski of course depend on the particular underlying solution jk . They are thus functions of d, CY,R for periodic solutions. In fact, for each periodic solution, RH is nonzero only when 1 is k plus a vector of the reciprocal lattice of the spatially periodic structure. ski is nonzero only when 1 = -k plus such a vector. These qualitative features are revealed by the semiclassical theory which neglects xs and ~4 and is concerned only with putting ~2 in normal form. In the quantum formulation this approximation consists of transforming x by means of W2 . We write X’ = WAX%’ = ~~~ CJC.(~). The choice of jk an E according to Eq. (3.4) makes X1 = 0. We therefore may concern ourselves with putting W2x2K1 in normal form. To do this we must set to zero the coefficients of This yields a set of further #k’h and $k +$4 -t and of $k+$l , 1 # k in W2X2K1. conditions on Rkl and ski . W2X2WZ1 is then in the normal form C ek#k+#k . Finally we take as the approximation to the ground state a vacuum state for the operators $k+J/k . The energy of the ground state is then E. C XO -I- x2 , neglecting a contribution from X4. The equations for jk , the conditions on & , S,U and the constraint iv

=

(w~w2~new

, N,,W~W25L,)

(3.18) define the approximation. However once the qualitative nature of the normal mode transformation is revealed, we may proceed directly to treat the full Hamiltonian as in the small oscillation theory. Subjecting x to a unitary transformation W2 , we write

W*XK1 = 5 3P, l=O

(3.19)

where the index again denotes the number of operators in the potential energy. The grouping of terms is effected after the operators are ordered with the destruction operators to the right of the creation operators. With the notation ( q )+ and (m> of Section 2, we find

INTERACTING

313

BOSONS

(3.20)

x

C3) =

(x3)+,

d4)

=

(x4)+

The transformation has brought down linear terms in #k and rf’k+ from X8 , constant terms from xZ and X4 and quadratic terms from x4 . If we now take the wave function Sew to be the vacuum for $k+J/k , the energy is just XC’) + EN. For given Rk2 and ski we determine E and fk from the condition that SC?) be zero

together with

the requirement N

=

T

1fk 1’ +

g

1 Sk2

1’.

(3.21a)

To fix Rkl and ski , we require that SC@‘)be in the normal form 3? = c ei#li$l , after the primed operators are expressed in terms of the unprimed operators. We recall that #z&n

= F RwSmv

,

$z+& ~

= q S:z’Smz’

(3.22)

and define

AZ, =(~-E)si,+~~(f~+kf~+k+~) +c* Bzm

=

c

k sti

(fm*fl-k

(fl+kfm-k +

+

ih+khr-k)*

$m+$l-k),

(3.23)

314

GROSS

Then dz) The conditions

= g hn(+z+h)+

+ (h+hn+)+Bz,

+ (IClz$%n)+B:,n.

(3.24)

that the coefficients of tiltim are zero are

The complex conjugate equation insures that the terms in #l+tim+ vanish. condition that 3~~‘) have the form c Q#~+$Q yields the final relations

Equations (3.21)) state energy is

(3.24)) and (3.25)

define the approximation.

Eo = do) + EN.

The

The ground(3.27)

With the parameters determined in the above way X(O) + 3~~‘) + x(‘) is diagonal in the number operator representation. x.‘~’ and xC4) have zero expectation value for the ground state and for single excitations. The off-diagonal contributions may be analyzed by perturbation methods. The formidable conditions for fk , E, Rkl , and Ski admit of a simple physical interpretation. The equations for fk describe the modification of the best average potential because of two-body collisions (primarily). The equations for Rkl and Ski describe, on the other hand, the modification of the two-body interaction arising from the average background potential. We note that periodic solutions are compatible with the above equations. Whether, in fact, they exist, and how the energy compares with that of the theory of Section 2, (and also with the energy of a solid modification), depends on the potential energy Vk . This must be ascertained by special investigation. From the work of Ref. 2, we expect the periodic solutions when the potential has both negative and positive Fourier components with the negative components large enough. The periodic solutions will have a range of lattice spacings. In addition a number of different crystal structures will in general be compatible with the equations. While only the structure and lattice spacing whose wave function corresponds to the lowest energy are realized at absolute zero, the other solutions, each with its own phonon spectrum, contribute in a vital way to the thermodynamic properties at finite temperatures. Practical calculations can be made with the above theory by restricting oneself to a small number of Fourier components. Alternatively one can rely on the variation principle and assume simple flexible forms for fk , RLE, and SLE . A third

INTERACTING

315

BOSONS

method results from the different presentation of the theory developed in Section 4. An interesting and very simple approximation to the ground-state energy is obtained by making the Bogolyubov linear transformation of the operators. In this case only #& and fil+ql are nonzero. The equations determining fk , E are simply (

E - ;‘$

- c $m+t,hm‘V” ;!/-“> m

fk = F 5;

c fm*f,&k--l m

(3.28)

+f:k2+=,

The analysis is very similar to that of ref. 4. For example, in a one-dimensional casewith only fo and fd different from zero, we find

(3.29)

For the spatially periodic state to be energetically preferred, we must have Vd < 0 and fd” > 0. The number of particles left in the condensedstate now plays an essential role. The ground-state energy itself is given in the present approximation by

To determine ffk and Pk and thus #k+~b = diagonalize part of Eq. (3.24)) i.e.,

@k2,

#,&-L =

akflk

we note that we can

316

GROSS

Thus we may now put

(3.32)

(fk*fTk

+

jtk$-k).

Then CYLand ok are again determined by Eq. (2.22) together with c~k2- fik2 = 1. The modifications in yz and 61 occur mainly at wave vectors near the periodicity vector d. To conclude this section, we note that the degeneracy of each periodic solution is still present and may be utilized to form eigenfunctions of the momentum and number operators. This is equivalent to forming the wave function

where M is a normalization integral. The correction to the energy comes entirely from 3P’ + X (4), but yields a vanishing energy per particle as N + 00. A detailed numerical calculation of the ground-state energy according to Eq. (3.30), and the comparison with both the theory of Section 2 and the theory of the solid state (Section 5)) will be carried out in a separate paper. 4. ATTRACTIVE

CASE-ALTERNATIVE

TREATMENT

There is an alternative way of presenting the content of the previous section, which, while completely equivalent, is more useful for general purposes. We note that the set of Fourier components f,+ defines a function CC,%(X).We can develop the quantized field operators in a one-particle complete orthonormal basis #(x>

=

c

akpk(x),

(4.1)

where F Qk*(x’Mx)

= 6(x - x’), (4.2)

s

‘pk*b>dd

C&

=

bk.2,

and (00(x) is the first function in the set. The unitary transformation #(x) t cocpO(x)+ $(x) now is the sameas uo --, CO + uo , ak -+ ak for k # 0. The creation and annihilation operators for these one-particle states of course satisfy the Bose commutation rules. We may note the connection between these operators and those appropriate to a plane wave basis. From

INTERACTING

317

BOSONS

$4x)= c wPk(d= c 925, we deduce

This 8

=

change xk,l

of basis

sk,$‘k+$‘l(~kl

is induced =

by

a unitary

transformation

eis with

sfk).

The development of # in a general basis is a convenience in studying periodic solutions for the case of attractive interactions. But it is a necessity in treating boundary value problems such as the problem of finding the stationary states of a boson system in a cylindrical container or rotating bucket. Then, the boundary conditions may require, for example, that all wave functions vanish at the container walls. This is satisfied automatically if a single particle orthonormal set of function is used, each of whose members satisfies the condition. In the new representation the Hamiltonian has the form H = c

Tnmanfa,

+ c

Gmnjkam+an+ajak ,

(4.4)

where

Tm=& / ve*vnn d3q (4.5) Gmnjlc= $6

~m*(x>w*(y>V(x

- YMXMY)

d3x d3y.

The number operator is N,, = cn a,+~, . Our first step now involves performing the unitary transformation WI = e(wo+-~O*acd 4.6) so that WlaoK’ = CO + a0 , Wgo+WT’ = CO* +’ ao+. The transformed Hamiltonian W,( H - EN,,) K1 = zf=o Xi is xo = (Too - E) I co I2 + xc:1 =

F’(Tko-

x:z

c

=

{ (Tkl

+ t:

Gocoo

I co I2 I co 12,

E) ak%,, + C.C. + c’ -

E8k.l)

+

=

c

(Gumo

+

x.4

=

c

(.&,dZk+~~+CL~Un

1 Co

I”(Gkool

ak+ 1Co +

+ C.C.,

&+‘&+Co2&oo

X:3

2

GklOm)ak+al+amCO .

effective

+

C.C.,

12Co*2Gkooo

Gokod)ak+m

+

C.C.,

(4.7)

318

GROSS

The crudest approximation corresponds to taking %POid= IV@,,, , where P,,, is a vacuum state for the number operators ak+ak . (00(x) is obtained from Gx.&~~*(x) = 0. This variational equation leads to the self-consistent field equation

- & v2po(x>+ I co12(Po(x) 1 V(x - y> I cpo(Y)I2d3Y= -@vow. This is the same as Eq. (3) of Ref. 2, with (00(x) replacing #(x). The constant CO is obtained from N = (!&id, Nop\kold) and yields j CO 1’ = N. The physical meaning of the wave function q0rd is apparent. All the particles are placed in a one-particle state, but this one-particle state is not the plane wave state l/-&j. It is instead I~O(X) , which in the case of strong attractive forces can be a spatially periodic state. The periodic state of lowest energy then becomes the starting point for a study of binary and higher order collisions. It is a result of the longrange order which sets in when the attractions are sufficiently strong. We know that a better ground-state wave function can be obtained by introducing in addition a normal mode transformation. Let us start by introducing a simple linear transformation of the Bogolyubov type. arc’ = w2ak&1 ak +I

=

= ukak + vkatk ,

w2akw’

=

uk*ak+

+

!&*a-k

with ‘& = u-k , 2’k = 2)-k and 1‘& I2 - 1vk I2 = 1. Again and (O)+ with a,+a,

= 1urn 12,

(a,‘aL)

,

We

USC?

the symbols

(a)

ama-, = umvm

= (ama-,)+

+ a,a-,

.

While the above transformation is formally similar to that used in Section 3, it is not the same. Here it applies to the ak , ak+ operators rather than the creation and annihilation operators for plane waves. The result of this transformation is

x’=W#.zK1 =go3P, where the terms are properly

ordered. We have (with

CO= CO*)

do’ = (Too - E) I co I2 + GoowI co 1’ I co I2 +

T

(Tkk

-

E

+

c

G-kkd:~

+

c

G+~,(-k,ki!&&aka_k

+

4 +

1 CO @OkOk)ak+ak

C.C.

--

klm

+

C.C.

+

2(Gkzzk

+

Gklkz)ak+Uka,+U~

,

(4.3)

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319

BOSONS

r&l) = x u$'co[Tko - E + 1co1'X&a30 +

cm (Gkmom + Gnkom)um+um + cc., -t (Gowan + Gokm(--m))a,a+J

se(‘) = 2

-t -t

(um+uk) +[ Tkm - EC%,, + 4Gokom 1co 1’ 2(G,mm + Gk~)u~+u~} 3

(uk+Gn

d3’

= x

(Gmno +

x(4)

=

Gmnn(

E

2

+ c.c.,

+ c.c.,

Gktom)co(uk+ut+um)+

ak+u~+u,u,)

The condition

+ G~~+m~a-J

+>+(~o”Gmo

(4.8) cont.

+ .

that the coefhcient of uk’co vanishes is

Tko - E + 1 co ~22Gmo + 2 c (Gkmom + Gmkom)u,+a, (II + c (Gokm(--ml + Go++&-z,u-, m For k = 0 this is the same condition filled if the (ok(x) Satkfy --

2’;

v2’+‘k*b>

+

/-

G*(

as 6X’“‘/6co*

x$)(&*(x’)

= 0

(4.9)

= 0. These conditions are ful-

d3x’ =

E&Ok*(X),

(4.10)

where G*(x,x’)

= / V(

~-Y>tl~o121~o~Y~12+CI~~121~~~Y>121 m

.d3y 6(x - x’) + V(x + y

- x’) ( c 1vm I2 pm*(x’) m b7awcp-mb’>

(4.11)

+ (Pm(X’)(C-m(X))}.

G*(x,x’) acts as a common Hermitian potential for the functions pk*(x). (00(x) satisfies the same equation with E. = E. This is a generalization of the selfconsistent field problem of the semiclassical theory. The self-consistent potential is now nonlocal and is computed as a sum over all the functions (P,,,(X). This involves a direct contribution in which the states are weighted according to the mean number of particles in the state, i.e., 1co I2 or ) vm j2. There is also an exchange contribution to the effective potential. These equations together with for a giVen N = ( co I2 + cm 1 vm I2 d et ermine E, cpo(x) as well as Ek and (ok(x) are compatible with periodic solutions, with the genset uk , tik . The equations eral features the same as for the semiclassical theory. The average self-consistent

GROSS

320

potential is merely modified by what is essentially close collisions of the particles. With our restricted normal mode transform we cannot bring x(‘) to normal form by a suitable choice of uk and vk . But we can make the terms in aka-k and ak+& vanish (for every k) . This condition yields Ukuk[Tkk -

E + 2 1 COj2Gokom + 2 F

1Vz/‘(Gkzzk + Gkzkz)]

+ Yk2(dGk(-k)oo

+

F

(4.12)

GZ;-z,(--k)kUzVz)

+ Uk2(d2Gk(-k)oo+ F G:(-z)(-k)kUzVz) = 0. The problem of solving these equations when uk and l)k are subject to ( uk I2 12)kI2 = 1, is the same as that of Section 2. The above type of normal mode transform takes into account the effect on the ground-state energy of the phonon excitations at long wavelengths. It does not do justice to the excitations with wave vectors near the vectors for which Bragg reflections can occur. We must introduce the more general normal mode transform ak’ = W2akK’

a:’ = w2ak+K1

=

c

ckklakl + Dkkta$

= c D&4& + &‘I&

,

,

(4.13)

where

In the general casethe quantities ak& are nonzero when m is -k plus a vector of the reciprocal lattice and ,&+a, are nonzero when m = k plus a vector of the reciprocal lattice. We again write x’ = w2x~1

= c 3P,

where x (') = + c x (‘) = z(!&O +

c

-__

G

kznn [ak+az+a,a, -

-+ ak+a,aK

E + 1 co 122Gkooo+ 2x

(G:zko + G:zok>aza,>ak+co

+ ak+anaz+a& (‘%zmo

+ c.c.7

+

Gkzomhi%z

(4.15)

INTERACTING

x(2)

x(3) %(4)

321

BOSONS

= mk c (am+uk)+(Tkm- E&n + 2 1co12(Gmca + Gomod -I- (Gmzrcn + Gmznlc + &mm + Gnmzdaz+a,~ + 5 (uk+am +>+(dGkmcm + G.twnzn&d+ C.C., =

=

c

(Giczmo+

c

Garwz(uk+uz+um~J+

G~~o,)Co(Uk+U~+U,)+

The revised condition

+

(4.15) cont.

c.c.,

.

for the functions

Tko - E + 1co j22Gmo + 2 c

is

(ok(x)

(Gano + G~~omh+um + (G&o

+ G~zoduzum = 0

(4.16)

which leads to --

2’;

v2Vk*(d

+

1

K(

x,x’)(dk*(x’)

d3x’ = Ekpk*(x),

(4.17)

K(x7x’) = dx - x’> / ( I coI2 I cpo(Y) I2 + c Pz*(yh%n(y)al+a,) *V(x

- y> d3g + V(x + V(x

- x’ F

- x’> F w*bhb’~al+a, uzum(cpm(x’)Pz(x)

(4.18)

+ (Pb’)dd).

Again we have a Hermitian integral operator K(x,x’) to determine the complete set $Ok*(x). The conditions that C and D must satisfy to put X(‘) in diagonal form can be readily set down. Then X’3) and x(4) alone have off-diagonal contributions and the theory is on the same footing as that of Section 2 for the repulsive case. It is apparent that the method of this section is very general and can be used to discuss the ground state and low-lying states of the boson system for general boundary value problems. For the purposes of the present paper, the most interesting application of the formalism is to investigate the periodic solutions in a “tight binding” approximation. This allows one to contrast clearly the superfluid ground state and that of the solid. The “tight binding” approximation starts from a particular form for the one-particle functions Vk(x). Let us go to the picture that the atoms are in cages of a suitable depth, size, and shape. We define a complete set of single particle localized functions x”(x - Rj) centered about each lattice point Ri . CY labels the excited states of a particle in a cage. It should be cautioned that the cages do not necessarily have a side equal to the mean distance between particles

322

GROSS

or an average of a single particle per cage. The states xa can be chosen to be orthogonalized atomic orbitals of a type s

x*~(x

- Rj> d3x = &,,6(Ri,Rj).

- R,)x’(x

(4.19)

From these we form the Bloch orbitals (Pan

= &

C exp (Zk*Rj)x”(x i

(4.20)

- Rj),

where N’ is the number of cages and k takes on values in a basic zone. These functions are orthogonal for different k and (Y values. If one particle is placed in the state V:(X) it occupies each cage an equal fraction of the time, i.e., is the analog of an electron placed in a molecular orbital. If a ground state is formed by placing N particles in the state P:(X) the state contains many configurations in which there is a high multiple occupancy of a single cage. In addition the fluctuation in number of the state p:(x) is of order dr. The burden of describing how particles avoid coming too close to each other falls entirely on the normal mode transformation. In actual calculations of the energy one can use approximations based on including only near neighbor interactions. The high fluctuations in the number of particles per cage is another way that the superfluid ground state contrasts with the solid ground state. To conclude this section we note the precise connection between the present wave functions and those of Section 3. Our wave functions in this section are of the form *,,ld = W,W2% , where !Po is a vacuum for the number operators of the (ok(x) basis. Now a state vector Q is expressed in the plane wave basis as KPwhere 9 = exp(i C skrak+al)* The first unitary Thus

= exp(i C Sklak+ar) exp(coao+ - c0*u0)W2*0.

transformation

induces a change of operators

ak +

(4.21) c

(4.22)

9 = exp( C coUo*lal+ - c.c.) W&I. I

fl = coUo*l.

This state vector has the same form as that of Section 3 with 5. THE

SOLID

STATE

OF

THE

BOSON

Ukral .

SYSTEM

In this concluding section, we comment briefly on the nature of the ground state for the solid modification of a system of bosons. To construct such a ground state we start from localized atomic orbitals of a type introduced in Section 4. Let a,(x - Rj) have the properties f

u,*(x

- Ri)&x

- R,+) d% = 8,,,&(Ri,

Rk).

(5.1)

INTERACTING

We expand the quantized

field operators

#(xl The Ha,miltonian

323

BOSONS

as

= aa, L(Wua(x ’ 3

- W.

(5.2)

takes the form

H = C ba+(Rj>b,9(&)Ta~(Rj

7Rk) + C bol’(Rj)bB+(Rk>b,(Rl>ba(Rm) eFaaya(Rj 7 Kc 7 RZ p Rn),

(5.3)

where T,b(Rj,

Rk) = &

Fa8yaCR.i 3 Rk) RZ 7 Rx)

= $5

/

- R~)VU~(X

VU,*(X

1s

a,*(~ .V(x

- Rj)aa*(y - y)a,(x

- Rk) G?X, (5.4)

- Rk)

- R&*(x

- Rn) &r&j.

The first approximation to the ground state is to place one atom at each lattice site in the lowest localized state; i.e., one atom is in each of N states uo(x - Ri). The state vector is q = bo+(Rl)

. - . bo+(Riv)\ko ,

where 90 is a vacuum for the number operators ba+(Rj)b,(Rj). of H which has a nonzero expectation value is HO = F h+(Rj>h(Rj>~oo(Rj 1 {J’ow(R~

(5.5) The only part

7 Rj) + ETR, ~o+(R~>~~~(R~)~o(Rj)bo(Rh> 2 Rk 2 Rj :Rk)

+ Fowo(Rj > Rtz) & 7 Rj)).

(5.6)

The energy is, in fact d3z d3g (1 00(x - Rd I2 / CO(Y) 1’ + uo*(x)uo(x

- Rk)uo(y)uo*(y

- Rk))V(x

- y).

It consists of a kinetic energy of localization and of direct and exchange contributions to the potential energy. This expression is a fully quantum mechaniacl extension of London’s semiheuristic calculation (7) of the cohesive energy of solid helium. As in his work, one must compare different crystal structures to find the one which gives maximum binding. Here we have the function so(x) and the lattice structure to be chosen to minimize the energy. An important calculation, to be reported later, is the comparison of the ground-state energy of Eq. (5.7) with that of (3.30) as a function of density.

324

GROSS

It is possibleto construct a schemefor the solid state that is the analog of the work of earlier sections of this paper. A theory of the lowlying states of solid helium will be developed elsewhere. It is clear that the consideration of terms in the Hamiltonian quadratic in operators b,(Rj) gives riseto an excitation spectrum. The shift in zero point energy which occurs when the excitations are considered is the analog of the contribution of Hz to the ground state energy. For the solid it is, however, of far less importance. The above solid ground state wave function, (of the Heitler-London type) already keeps particles from coming too close to each other. Thus, the normal mode correction does not have to describe this effect entirely by itself. The solid state wave function we have written down is, of course, an extreme casein that no multiple occupancy of a cell is permitted. A check on the above viewpoint is provided by the fact that the excitation spectrum obtained for the solid has two transverse and one longitudinal branches for long waves. The excitation spectrum of the periodic solutions of Sections 3 and 4 has only one longitudinal branch. RECEIVED:

July 23, 1959 REFERENCES

I. E. P. GROSS, Phys. Rev. 106, 161 (1957)) and Stevens Institute Many-Body January, 1957. 2. E. P. GROSS, 14nnuZs of Physics 4, 57 (1958). S. N. BOGOLYUBOV, J. Phys. U.S.S.R. 11, 23 (1947). 4. E. P. GROSS, Phys. Rev. 100, 1571 (1955). 6. M. GIRARDEAU AND R. ARNOWITT, Phys.Rev. 113,755 (1959). 6. J. VALATIN AND BUTLER,NUOVO cimento [lo] 10, 37 (1958). 7. F. LONDON, “Superfluids,” Vol. 2, p. 18, Wiley, New York, 1954.

Conference,