*Huang and Klein, op. cit.

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SUPERFLUIDS

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Fig. 13.10 Experimental liquid structure fac-

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tor (Fourier transform of pair correlation function) of liquid He 4 .

Fig.

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13.11 Comparison between the calculated excitation spectrum (solid curve) and measurements (dashed curve).

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gives an excitation energy as function of k shown in Fig. 13.11, which reproduces the roton part of the spectrum qualitatively. *

1 3.8 DILUTE BOSE GAS

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Dilute Bose gases have become accessible to experiments, furnishing us with simpler examples of superfluids. Among these are spin-aligned atomic hydrogen gas,t and liquid He 4 adsorbed in porous Vycor glass.:j: At sufficiently low temperatures, the interparticle potential may be summarized by only one parameter, the scattering length, which is a kind of effective hard-sphere diameter. The low-lying energy levels of a dilute Bose gas then correspond to those of a hard-sphere gas, which can be treated through the methods of pseudopotentials introduced in Section 10.5.

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*An improved wave function that fits the data better is given by R. P. Feynman and M. Cohen, Phys. Rev. 102,1189 (1956). iT. J. Greytak and D. Kleppner, in New Trends in Atomic Physics, Vol. II (Les Houches 38) G. Greyberg and R. Stora, eds. (North-Holland, Amsterdam, 1984). tB. C. Crooker, B. Hebral, E. N. Smith, and J. D. Reppy, Phys. Rev. Lett. 51, 666 (1983). T. D. Lee, K. Huang, and C. N. Yang, Phys. Rev. 106,1135 (1957); see also K. Huang, op. cit.

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SPECIAL TOPICS IN STATISTICAL MECHANICS

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Effective Hamiltonian

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The Hamiltonian is

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.Yt'= - 2m

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L 'il/ + j=l

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~(ri

-rij

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arij

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where a is the hard-sphere diameter. We can re-express it in the quantized field representation as

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(13.871 where

l/J(r) =

ikor

!TT '

(13.88 1

and we have imposed periodic boundary conditions:

(13.891

when n is a vector whose components are independently 0, 1, 2, .... The appearance of the operator (ajar)r seems to complicate things, but actually the effect is quite simple. The operator has the property that

a; [rj(r)] r-O = j(O)

ar [rj(r)] r=O = 0

(if j is regular at r ( if j----+ r-->or

(13.901

The operator (ajar)r weeds out, so to speak, any singularity of the Ijr type in the function to which it is applied. * We treat the interaction as a perturbation on the ideal gas. The validity of the Hamiltonian is restricted by the conditions

ajv 1/ 3 1,

ha 1

(13.91)

where v is the specific volume, and k is any wave number of an excited particle in the unperturbed state. For low-lying excited states, we assume (13.92 )

"cf. Problem 10.7.

SUPERFLUIDS

First we decompose the interaction into a part that has only diagonal matrix elements with respect to the unperturbed free-particle states and a part that has only off-diagonal elements. The part with diagonal matrix elements has been worked out and given in (A.36) of the Appendix, and discussed in Section 10.5. For the off-diagonal elements, we isolate a part that contains terms proportional to aoao or aiab. What is left contains terms both proportional to and independent of a o or ai. The matrix elements of the first part are of order N larger than those of the remaining part, which will be neglected. Thus we have

2m.Yt' = L>2a!a

+ 87TaN

(1 - ~ ~:n~) 2N

47Ta

L' (atataoa o + akakabai) k

(13.93)

Now we replace both aoao and abai by N, and obtain an effective Hamiltonian:

2m~ff =

The sum

47TaN +

I' [(k

+ 87Ta) ata k + 47Ta (ata~k + aka- k )] (13.94)

in (13.93) and (13.94) is defined by

I' fk ==

r-->O

!...-(r I Jr

(13.95)

In practice, this operation affects only the ground state energy, and has the effect of subtracting out the term proportional to a 2 .

Energy Levels

To diagonalize (13.94) we introduce a linear transformation first used by Bogolubov* : 1 a k = ~ (b k - akb~k) 1 - a~ (13.96) 1 at = ~ (bt - akb_ k ) 1 - a~ or

bk =

btk-

~ (a k + aka~k)

1 - a~ 1 1 - a~

(13.97)

~ (at + aka_ k )

where a is assumed to be a real number than one. It is clear that b k and bt

*N. N. Bogolubov. J. Phys. USSR 11,23 (1947).

SPECIAL TOPICS IN STATISTICAL MECHANICS

satisfy the same commutation rules as

[b k , b k ,] =

at, namely

(13.98 I

[bt, bt,] =

~kk'

[b k , btl

Therefore b k and bt can be interpreted, respectively, as annihilation and creatior. If we substitute (13.96) into the effective Hamilto operators, just as k and nian (13.94), we find that ~ff is diagonalized by choosing