Reports of the Academy of Sciences of the USSR
1958. Volume 123, No. 3

PHYSICS

B. T. GEILIKMAN

ON THE APPROXIMATE SOLUTION OF THE QUANTUM MANY-BODY PROBLEM IN THE CASE OF BOSE STATISTICS

(Presented by Academician L. A. Artsimovich on 12 VII 1958)

In \((^{1})\) it was shown that an increase in the density and in the interaction energy of the particles of a Bose gas leads to a decrease in the number of particles in the condensate \(N_0\). It is therefore of interest to investigate the case for which \(N_0\) is small. In view of this, in the Hamiltonian of the system

\[ H=T+U=\sum T_k a_k^+ a_k+\frac{1}{2}\sum V_q a_k^+ a_l^+ a_{k+q}a_{l-q} \]

we shall separate out the terms with \(a_0^+\) and \(a_0\), but shall not regard them as large. We shall first assume that \(T\sim U\) and shall try, along with \(T\), to take into account the principal terms in \(U\). Since for a rarefied Bose gas the leading role is played by the interaction of particles with oppositely directed momenta \((^{1,2})\), in our case as well we shall apply a canonical transformation to new amplitudes \(\alpha_k,\ \alpha_k^+\) of the same type as in \((^{1})\): \(a_k=u_k\alpha_k+v_k\alpha_{-k}^+;\ u_k^2-v_k^2=1\). The parameters \(u_k,\ v_k\) will be found from the requirement that the mean value of \(H\) be minimal with respect to the new occupation numbers, with the additional condition that the number of particles be constant:

\[ \widetilde{E}=H_{\mathrm{av}}-\mu N =\frac{1}{2}V_0N^2-\mu N_0 +\sum \left(T_k+N_0V_k-\mu\right) \left[n_k+(2n_k+1)v_k^2\right]+ \]

\[ +N_0\sum V_k u_kv_k(2n_k+1) +\frac{1}{2}\sum V_{|l-k|} \{u_kv_k u_l v_l(2n_k+1)(2n_l+1)+ \]

\[ +\left[n_k+(2n_k+1)v_k^2\right] \left[n_l+(2n_l+1)v_l^2\right]\}. \]

Here small terms not proportional to the volume \(\Omega\) (as \(\Omega\to\infty\)) have been discarded; in the summations here and below \(k\ne0;\ l\ne0;\ n_k=\alpha_k^+\alpha_k;\ N=N_{\mathrm{av}}=N_0+\sum [n_k+(2n_k+1)v_k^2]\); \(\mu\) is a Lagrange multiplier, which coincides with the chemical potential.

Minimizing \(\widetilde{E}\) with respect to \(u_k,\ v_k\), we find

\[ 2u_kv_k\xi_k=-(u_k^2+v_k^2)g_k, \]

where

\[ \xi_k=T_k+N_0V_k-\mu+\sum V_{|l-k|}\left[n_l+(2n_l+1)v_l^2\right]; \]

\[ g_k=N_0V_k+\sum V_{|l-k|}u_lv_l(2n_l+1). \]

Hence we obtain

\[ u_k^2=\frac{1}{2}\left(\frac{\xi_k}{\varepsilon_k}+1\right); \qquad v_k^2=\frac{1}{2}\left(\frac{\xi_k}{\varepsilon_k}-1\right), \]

where

\[ \varepsilon_k=(\xi_k^2-g_k^2)^{1/2}. \]

The excitation energy \(\delta\widetilde{E}/\delta n_k\) proves to be equal to \(\varepsilon_k\).

Minimizing the free energy of the system (see (3)), we find
\(n_k=[\exp(\varepsilon_k/\theta)-1]^{-1}\). Substituting the expressions for \(u_k, v_k\) into the expressions for \(g_k, \xi_k\), we obtain integral equations for determining \(g_k, \xi_k\):

\[ g_k=N_0V_k-\sum V_{|1-k|}(n_l+1/2)g_l(\xi_l^2-g_l^2)^{-1/2}, \]

\[ \xi_k=T_k+N_0V_k-\mu+\sum V_{|1-k|}\{n_l+(n_l+1/2)[\xi_l(\xi_l^2-g_l^2)^{-1/2}-1]\}. \]

We find \(\mu\) from the condition for \(N\). As \(\theta\to0\), for all \(\varepsilon_k>0\), \(n_k\to0\), and since \(\sum v_k^2\ll N\) (see below), in order to satisfy the condition \(N=\mathrm{const}\) it is necessary that \(n(\varepsilon_{\min})=n_0\to\infty\) \((N_0\to\infty)\) as \(\Omega\to\infty\) (i.e. \(n_0\sim\Omega\) for finite \(\Omega\)). Consequently, \(\varepsilon_0=0\), i.e. \(|\xi_0|=|g_0|\). It is not difficult to see that in this case \(\varepsilon_k\to sk\) as \(k\to0\), i.e. the excitations are sound-like.

First let us solve the equations for \(g_k\) and \(\xi_k\) for the simple case when \(N-N_0\ll N\). Let
\(g_k=g_{k0}+g'_k;\ \xi_k=\xi_{k0}+\xi'_k;\ \mu=\mu_0+\mu'\), where
\(g_{k0}=N_0V_k,\ \xi_{k0}=T_k+N_0V_k,\ \mu_0=0\). Then, obviously,

\[ \mu'=\mu=\sum V_k\left\{n_{k0}+(n_{k0}+1/2)\left[\frac{\xi_{k0}+g_{k0}}{\varepsilon_{k0}}-1\right]\right\}; \]

\[ g'_k=-N_0\sum V_lV_{|1-k|}\frac{n_{k0}+1/2}{\varepsilon_{l0}}; \]

\[ \xi'_k=\sum (V_{|1-k|}-V_l)\left\{n_{l0}+(n_{l0}+1/2)\left[\frac{\xi_{l0}}{\varepsilon_{l0}}-1\right]\right\} -N_0\sum V_l^2\frac{n_{l0}+1/2}{\varepsilon_{l0}}; \]

Here \(n_{l0}=n_l(\varepsilon_{l0});\ \varepsilon_{l0}=[(2N_0V_l+T_l)T_l]^{1/2};\ N_0\) is a function of \(\theta\);
\[ N_0=N-\sum\left[\,1/2\left(\frac{\xi_k}{\varepsilon_k}-1\right)+n_k\frac{\xi_k}{\varepsilon_k}\right]. \]

Of considerably greater interest is the case when \(N_0\lesssim N\), in particular \(N_0\ll N\). From the condition \(\xi_0=g_0\) we find:

\[ \mu=\sum V_k\{n_k+(n_k+1/2)[(\xi_k+g_k)(\xi_k^2-g_k^2)^{-1/2}-1]\}. \]

For \(g_k, \xi_k\) we obtain the equations

\[ g_k=N_0V_k-\sum V_{|1-k|}(n_l+1/2)g_l(\xi_l^2-g_l^2)^{-1/2}; \]

\[ \begin{aligned} \xi_k={}&T_k+N_0V_k+\sum(V_{|1-k|}-V_l)\{n_l+(n_l+1/2)[\xi_l(\xi_l^2-g_l^2)^{-1/2}-1]\}\\ &-\sum V_l(n_l+1/2)g_l(\xi_l^2-g_l^2)^{-1/2}. \end{aligned} \]

The solution of these equations will be considered in another paper. It is not difficult to see that in this case, for \(\theta\ne0\), a negative sound dispersion is also possible. The expression written above for \(\mu\), obviously, is valid not only for \(\theta=0\), but also for \(\theta\ne0\), if a condensate exists \((N_0\sim\Omega)\), i.e. in the superfluid phase. Using the formula \(\mu=\partial E/\partial N\), one can find an expression for \(\mu\) also in the normal phase. The case \(N_0\ll N\) makes it possible to consider the transition of the superfluid phase into the normal one. The phase-transition temperature \(\theta_k\) is determined by the condition

\[ N_0=N-\frac{\Omega}{(2\pi)^3}\int\left[\,1/2\left(\frac{\xi_k}{\varepsilon_k}-1\right)+n_k\frac{\xi_k}{\varepsilon_k}\right]\,dk=0. \]

The jump in the heat capacity is connected with the vanishing of \(N_0\) and the change in the form of \(\varepsilon_k\).

Let us now consider the condition of applicability of our theory. If \(V_q\) contains a small dimensionless parameter \(g\), then, in addition to the requirement that \(g\) be small, it is also necessary to require that the correction to the \(\Psi\)-function of the system,

associated with the terms in \(T\) that are nondiagonal in \(\alpha_k, \alpha_k^+\), be small. These terms have the form

\[ T_{\text{n.d.}}=\sum u_k v_k T_k(\alpha_k^+ \alpha_{-k}^+ + \alpha_k \alpha_{-k}). \]

The condition of applicability, obviously, has the form \(\sum v_k^2 \ll N\). For \(N-N_0 \ll N\), i.e., for \(\theta \sim 0\), this condition coincides with that found in \((1)\):

\[ \left(\frac{N}{\Omega}\right)^{1/3} \ll \frac{\hbar^2}{m\Omega V_0}. \]

But now the case \(N_0 \ll N\), i.e., \(\theta \sim \theta_k\), is also possible. For \(\theta \ne 0\) the condition for the density \(N/\Omega\) may turn out to be weaker.

The approximation can be somewhat improved if, instead of the transformation \(a_k=u_k\alpha_k+v_k\alpha_{-k}^+\), one uses the more complicated transformation:
\(a_k=u_k\alpha_k+v_k\alpha_{-k}^+ + \widetilde u_k\alpha_{-k}+\widetilde v_k\alpha_{-k}^+\);
\((u_k^2-v_k^2+\widetilde u_k^{\,2}-\widetilde v_k^{\,2}=1;\; u_k\widetilde u_k=\widetilde v_k v_k)\).
Here \(\widetilde u_k,\widetilde v_k\) should be regarded as small quantities compared with \(u_k, v_k\). The corresponding transformation can easily be written also in the case of Fermi statistics. A further approximation can be obtained by using the still more general transformation
\(a_k=\sum_l (u_{kl}\alpha_l+v_{kl}\alpha_{-l}^+)\), where \(v_{kk}\gg v_{kl}\), \(u_{kk}\gg u_{kl}\) for \(k\ne l\). Terms in \(H\) containing \(u_{kl}, v_{kl}\) with \(k\ne l\) may be treated as a perturbation.

Moscow State Pedagogical Institute
named after V. I. Lenin

Received
10 VII 1958

REFERENCES

\(^1\) N. N. Bogolyubov, Izv. AN SSSR, ser. fiz., 11, 77 (1947).
\(^2\) N. N. Bogolyubov, ZhETF, 34, 58 (1958); N. N. Bogolyubov, V. V. Tolmachev, D. V. Shirkov, A New Method in the Theory of Superconductivity, Moscow, 1958.
\(^3\) J. Bardeen, L. Cooper, J. Schrieffer, Phys. Rev., 108, 1175 (1957).