The condensed state of the imperfect Bose gas

Volume 76A, number 1

PHYSICS LETTERS

3 March 1980

THE CONDENSED STATE OF THE IMPERFECT BOSE GAS M. FANNES’ and A. VERBEURE Instituur voor Theoretische Fysica, Universiteit Leuven, B.3030 Lou vain, Belgium Received 11 September 1979 Revised manuscript received 6 December 1979

For the imperfect Bose gas, a rigorous proof is given for condensation in and only in the ground state, and the limit Gibbs state is explicitly constructed.

The imperfect boson gas [1] represents one of the simplest models of an interacting boson system showing condensation. Apart from a variety of physically intuitive arguments [1] there exists a rigorous proof [2] of the existence of a singularity in the mean density as a function of the chemical potential. In this note we report on the following results: (a) We give a rigorous proof of the existence of condensation in the ,

ground state and only in the ground state; (b) we show the existence of the limiting Gibbs state and give the complete solution, Let us first define our system. Consider an infinite system of Bose particles described by Weyl operators satisfying the commutation relations: W(cb)W(iI)= exp[—’}i Im(~i,~)] W(~+

0 E ~ (RV), v> 3; ~ is the space of infinitely dif. ferentiable functions with compact support. Denote by i~(the algebra generated by ~W(O)I0ECD(R~)}. To describe the model consider a cubic box A 1 ~
E

+k24ak

—

~

(1)

+-~XN~/L~’,

kEA

whereA= {kE R”IkL/2irE Z”};O
~

=

kE A Aangesteld Navorser NFWO, Belgium.

;

ak = a(fk); fk(x) = v—’!2 e ikx, x E A; fk(x) = 0, x ~ A. a, a+ are the creation and annthilation operators in the Fock representation of ~ on ~D(A). We study the equilibrium state of the system in the grand canonical ensemble. The key technique is the equivalence of the equilibrium conditions with the following correlation inequalities [3]: ,~,~*

~,

~

/

*

~/L~~X

P’~ L”L’~J’L-’\”

*

X,LI~XX L

for all bounded operators on the local Fock space in the domain of [HL, ], supplemented with ~he condition (N 3 —

L L

—

P

where p is any positive number standing for the mean density; >L represents the thermal expectation. Note that e~HL is trace class for all values of j.t. Indeed for all E R+ —~

2

—

HL

—

TL

—

/~~ONL + (X/V)[NL 2

—

—

(~i—

~u 0)V/2X]

/4~hi —

Furthermore the state can be extended to any poiynomial in the creation and annihilation operators. Hence the correlation inequalities extend to these observables. Existence of condensation in the zero mode. The structure of the proof consists in taking the thermody. namic limit with constant given particle density p, i.e., for each finite volume V LV (v ~i 3): 31

Volume 76A, number 1

p = V’(NO>L + k~* 0

(N~>J~/V,

(4)

where Nk = akak. The strategy consists in splitting the k-summation into a part close to the origin and the rest. Therefore let 6 be such that 0
=

E


0

(Nk)L,

(5)

k

We derive the following upper bounds using inequality (2): 2~ for 0< k ~6, L [eCk~ 1] 21 +X/L~’ bk(L) 62/L), for 6 < 1k, where C~(L)/3(jikI bk(L) = /3Xpv(41T)W2f(v/2) —1) ~

3 March 1980

PHYSICS LETTERS

—

—

—

Using these bounds in (5) and taking lim~0limL~ one gets from (4): 2h’2 1)~. (6)

R~dk(e~1dI L—’°°(No)L/LV +(21r)l’f

—

Now it is clear from (6) that there is a macroscopic occupation of the ground state for /3 large enough; it proves also that there can only be condensation in the zero mode. Note also that the bound (6) is exactly the one of the free Bose gas. However, the states of the condensate will turn out to be quite different in the

2O)), a~ir~(W(Ø)) = ~ the laplacian; 62W(eit~ is the infinitesimal generator of the

gauge group of autormorphisms; n is the element of ir~(s/~)’ fl ~ )“ given by the generalized strong limit L 00 of r~(NL/L~’); 7r~(9q)’is the commutant of = w~(n). The inequality (7) is of course valid for all temperatures. Now we restrict our attention to low temperatures such that according to the above there is conden. sation in the zeroth mode. In this situation let n0 be the generalized strong limit (L ~+00) of again n 0 E ~(~)‘ fl ir~(~’1)”, and let p0 = By taking X respectively equal to a0, a0a~,ak a~ —~

4,

~in eq. Xp,(2) and taking the limit L 00 one gets (8) fixing the chemical potential, independently of the particular way the limit Gibbs state is taken. Let n 0 = f~dE~be the spectral resolution of n0, then for any measurable subset S of the spectrum of n0 denote the positive functional —~

) = fdw,,~(Et )~

~

S

We show that (7) implies

130ms(x* 6X)>~0~,s(X*X) ln[0,~,s(X*X)I0~,S(XX*)]. (9) Take now X

s-lim ir~(N

casesXOandX>0. The condensate state. The main step for the description of the condensate state consists in proving that we can take w*.limit Gibbs states, which will be denoted by w~,and in showing that they satisfy the following inequality, which will admit a unique solution: /3w~(X*6X)> ~~.~(X*X)ln[w,3(X*X)/w~(XX*)], where Xis any element of~b(6) fl 1r~(9O”;‘D (6) is the domain of 6; ir~is the GNS-representation of the state w,,~7r13(g~”is the weak closure of ir,3(~~6 = 6~ + (An /.1)62; 6i is the generator of the one-parameter automorphism group (a~)~ER:

L

1/2)= U 0/(p0V)

0

in (9), then for all 5:

f~[i.i d~(E~)

—

X dw,3(E~n)]= 0.

S

Using (8) one gets

(~ —

—

dk

)fdwa(E~) = 0,

e~k_1

~ ES.

s

Therefore f~ dw~~(E~) = 0 if ~ ~ S. This means that

—

32

~

= ~ (10) Now we derive the inequality (7) for I any monom-

Volume 76A, number 1

PHYSICS LETTERS

ial in the fields ir~(a~(O1)), ir~(a(0t)),i = 1, n where = 0 (Fourier transform at zero vanishes). Multiplying I by a correct power of U0 or U~we get gaugeinvariant elements, hence 62 vanishes and using the unitarity of U0 (which is equivalent to n0 = p01) one gets the correlation inequality of the free Bose gas without condensation: ...,

6 ix) ~ w~(X*X)ln [wp(X*X)I~~z,~(XX*~ (11) Finally we get the state: =

exp

2I 0(0) I) cos a,

j

X I2ir 0

sion to the state of a result of ref. [5] in the case of the imperfect Bose gas. Note that in this model the Bogoliubov approximation is very special because there is no breaking of the gauge symmetry. Also we mention the result of ref. [6] where the author gets the state (12) by a formal application of a variational principle over quasi.free states. He does not prove that it is the unique solution. Finally the details of this letter will be published elsewhere [7]. One of us (A.V.) thanks J.T. Lewis and J.V. Pulé for very illuminating discussions on the subject matter.

[—-} (0, RØ)]

2ir

3 March 1980

(12)

da ex~2ip~’

where (Rcb)(k) = (e~k — 1)~0(k). The main difference with the condensate state of the free Bose gas consists in equality (10). It means that the density of the condensate is sharply peaked (see also ref. [2]). In the free Bose case all densities appear [4]. Furthermore, as far as the state is concerned, eqs. (11) and (12) show the exactness of the so-called Bogoliubov approximation. This is an exten-

References [1] K. Huang, Statistical mechanics ~iley,

London, 1967).

[2] E.B. Davies, Commun. Math. Phys. 28 (1972) 69. [3] M. Fannes Verbeure, (1977) 125;and 57 A. (1977) 165. Commun. Math. Phys. 55 [4] J.T. Lewis and J.V. Pul~,Commun. Math. Phys. 36 (1974) 1.

[5] J. Ginibre, Commun. Math. Phys. 8 (1968) 26.

[61 R. Critchley, Approximate equilibrium states for two

models of an interacting boson gas, Dublin Institute of Advanced Studies and Department of Math., Cork, Ireland, preprint (1978). [7] M. Fannes and A. Verbeure, The condensate phase ofthe imperfect Bose gas, preprint-KUL-TF-79/026.

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