Reports of the Academy of Sciences of the USSR

1962, Volume 143, No. 4

PHYSICS

Yu. A. TSERKOVNIKOV

ON THE THEORY OF A NONIDEAL BOSE GAS AT A TEMPERATURE DIFFERENT FROM ZERO

(Presented by Academician N. N. Bogolyubov, 10 XI 1961)

In the work of N. N. Bogolyubov \((^{1})\), a theory was constructed for a nonideal Bose gas at zero temperature, which was subsequently developed in a number of works (see, for example, \((^{2})\)). The method of two-time temperature functions \((^{3,4})\), used in the present work, makes it possible to carry out an investigation at temperatures different from zero.

Consider a Bose gas consisting of particles interacting pairwise with one another and described by a Hamiltonian of the form

\[ H=\sum_p\left(\frac{p^2}{2m}-\mu\right)a_p^+a_p+ \frac{1}{2V}\sum_{(p_1+p_2=p'_1+p'_2)} \mathrm{v}(p_1-p'_1)a_{p_1}^+a_{p_2}^+a_{p'_2}a_{p'_1}, \tag{1} \]

where \(a_p\) and \(a_p^+\) are the Bose creation and annihilation operators for particles with momentum \(p\); \(\mu\) is the chemical potential; \(\mathrm{v}(p)\) is the Fourier component of the potential of interaction of the particles.

We perform the canonical transformation of the operators

\[ a_p=\varphi_p+b_p,\qquad \varphi_p=\sqrt{N_0}\,\delta(p),\qquad \delta(p)=1,0\quad (p=0,\ p\ne0) \tag{2} \]

(a shift of the operators by a \(c\)-number). The quantity \(N_0\) has the meaning of the number of particles in the condensate. The quantum equation of motion will have the form

\[ i\frac{db_p}{dt}= \sqrt{N_0}\left(-\mu+\frac{N_0}{V}\mathrm{v}(0)\right)\delta(p) +\xi_p b_p+\frac{N_0}{V}\mathrm{v}(p)b_{-p}^+ +\frac{\sqrt{N_0}}{V}\sum V_{pp_1} +\frac{1}{V}\sum W_{pp_2p'_2p'_1}, \tag{3} \]

where

\[ \xi_p=\frac{p^2}{2m}-\mu+\frac{N_0}{V}\bigl(\mathrm{v}(0)+\mathrm{v}(p)\bigr), \]

\[ V_{pp_1}=\{(\mathrm{v}(p)+\mathrm{v}(p_1))b_{-p_1}^+ +\mathrm{v}(p_1)b_{p_1}\}b_{p-p_1}, \]

\[ W_{pp_2p'_2p'_1} =\mathrm{v}(p-p'_1)\delta(p+p_2-p'_1-p'_2)b_{p_2}^+b_{p'_2}b_{p'_1}. \]

In the right-hand side of (3), the summation is carried out over all indices except \(p\).

From the condition \(\langle b_p\rangle=\langle b_{-p}^+\rangle=0\) for \(p=0\) (\(\langle\ldots\rangle\) denotes averaging over the grand canonical ensemble), we obtain the expression for the chemical potential in terms of the correlation functions:

\[ \mu=\frac{N_0}{V}\mathrm{v}(0) +\frac{1}{V}\sum_{p_1}\{(\mathrm{v}(0)+\mathrm{v}(p_1))\langle b_{p_1}^+b_{p_1}\rangle +\mathrm{v}(p_1)\langle b_{p_1}b_{-p_1}\rangle\} \]

\[ +\frac{1}{\sqrt{N_0}V}\sum_{pp_1} \mathrm{v}(p_1)\langle b_{p_1+p}^+b_{p_1}b_p\rangle . \tag{4} \]

Introducing the matrix notation

\[ \mathcal{B}_p=\begin{pmatrix} b_p\\ b_{-p}^+ \end{pmatrix}, \qquad \mathcal{V}_{pp_1}= \begin{pmatrix} V_{pp_1}\\ V_{-p,-p_1}^+ \end{pmatrix}, \qquad \mathcal{W}_{pp_2p'_2p'_1}= \begin{pmatrix} W_{pp_2p'_2p'_1}\\ W_{-p,-p_2,-p'_2,-p'_1}^+ \end{pmatrix}, \]

\[ L_p= \begin{pmatrix} \xi_p & \dfrac{N_0}{V}\mathrm{v}(p)\\ \dfrac{N_0}{V}\mathrm{v}(p) & \xi_p \end{pmatrix}, \qquad \alpha= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, \qquad \beta= \begin{pmatrix} 0&1\\ 1&0 \end{pmatrix}, \]

it is convenient to represent equation (3) and the equation conjugate to it in the form

\[ \alpha i\,\frac{d\mathfrak{B}_p}{dt} = \sqrt{N_0}\left(-\mu+\frac{N_0}{V}v(0)\right)\delta(p)+L_p\mathfrak{B}_p+ \]
\[ +\frac{\sqrt{N_0}}{V}\sum \mathcal{V}_{pp_1} +\frac{1}{V}\sum \mathcal{W}_{pp_2p'_2p'_1}, \tag{5} \]

We define the matrix Green function \(G_p\) as follows:

\[ G_p(t-t')=\langle\!\langle \mathfrak{B}_p(t);\mathfrak{B}_p^+(t)\rangle\!\rangle = \left\langle\!\left\langle \begin{pmatrix} b_p(t)\\ b_{-p}^+(t) \end{pmatrix} \bigl(b_p^+(t'),\, b_{-p}(t')\bigr) \right\rangle\!\right\rangle = \]
\[ = \begin{pmatrix} \langle\!\langle b_p;b_p^+\rangle\!\rangle & \langle\!\langle b_p;b_{-p}\rangle\!\rangle\\ \langle\!\langle b_{-p}^+;b_p^+\rangle\!\rangle & \langle\!\langle b_{-p}^+;b_{-p}\rangle\!\rangle \end{pmatrix}, \tag{6} \]

where, for example, for the retarded Green function
\(\langle\!\langle b_p(t);b_p^+(t')\rangle\!\rangle = -i\theta(t-t')\langle [b_p(t);b_p^+(t')]\rangle\) \((^{3,4})\), etc. (the arguments \(t,t'\) on the right-hand side of (6) and below are omitted for brevity of notation).

Using the equation of motion (5), passing to the Fourier components of the Green functions,

\[ \langle\!\langle A_p(t);B_p(t')\rangle\!\rangle = \frac{1}{2\pi}\int_{-\infty}^{\infty} \langle\!\langle A_p\mid B_p\rangle\!\rangle_E e^{-iE(t-t')}\,dE \]

and dropping, to simplify the notation, the index \(E\), we obtain

\[ (\alpha E-L_p)G_p = 1+\frac{\sqrt{N_0}}{V}\sum \langle\!\langle \mathcal{V}_{pp_1}\mid\mathfrak{B}_p^+\rangle\!\rangle +\frac{1}{V}\sum \langle\!\langle \mathcal{W}_{pp_2p'_2p'_1}\mid\mathfrak{B}_p^+\rangle\!\rangle, \]
\[ \langle\!\langle \mathcal{V}_{pp_1}\mid\mathfrak{B}_p^+\rangle\!\rangle (\alpha E-L_p) = \frac{\sqrt{N_0}}{V}\sum \langle\!\langle \mathcal{V}_{pp_1}\mid\mathcal{V}_{pp_1}^+\rangle\!\rangle + \frac{1}{V}\sum \langle\!\langle \mathcal{V}_{pp_1}\mid\mathcal{W}_{pp_2p'_2p'_1}^+\rangle\!\rangle, \tag{7} \]
\[ \langle\!\langle \mathcal{W}_{pp_2p'_2p'_1}\mid\mathfrak{B}_p^+\rangle\!\rangle (\alpha E-L_p) = \langle [\mathcal{W}_{pp_2p'_2p'_1},\mathfrak{B}_p^+]\rangle\,\alpha + \]
\[ +\frac{\sqrt{N_0}}{V}\sum \langle\!\langle \mathcal{W}_{pp_2p'_2p'_1}\mid\mathcal{V}_{pp_1}^+\rangle\!\rangle + \frac{1}{V}\sum \langle\!\langle \mathcal{W}_{pp_2p'_2p'_1}\mid\mathcal{W}_{\bar p\bar p_2\bar p'_2\bar p'_1}^+\rangle\!\rangle . \]

Multiplying the second and third equations (7) on the right by the matrix \(G_p^{(0)}=(\alpha E-L_p)^{-1}\), which is the “zero approximation” to the function \(G_p\), substituting the expressions found into the right-hand side of the first of equations (7), and multiplying this equation on the left by \(G_p^{(0)}\), we obtain

\[ G_p=G_p^{(0)}+G_p^{(0)}K_pG_p^{(0)} = G_p^{(0)}+G_p^{(0)}M_pG_p, \tag{8} \]

where \(M_p=K_p(1+G_p^{(0)}K_p)^{-1}\) is the mass operator, and \(K_p\) is defined by relation (12) (the meaning of the parameter \(\varepsilon\) will be explained below).

Equation (8) may also be written in the form \(\Sigma_pG_p=1\), \(G_p=\Sigma_p^{-1}\), where

\[ \Sigma_p=\alpha E-L_p-M_p = \alpha E-L_p-K_p(1+G_p^{(0)}K_p)^{-1}. \tag{9} \]

The elements of the matrix \(\Sigma_p\) satisfy the following symmetry properties \((^{5})\):

\[ \Sigma_{22}(p,E)=\Sigma_{11}(p,-E),\qquad \Sigma_{21}(p,E)=\Sigma_{12}(p,E). \tag{10} \]

Thus the problem of finding the Green functions \(G_p\) has been reduced to calculating the elements of the matrix \(\Sigma_p\). To calculate them we shall use the following perturbation theory. We introduce a formal dimensionless small parameter \(\varepsilon\), which in the final results we set equal to unity, and make the replacement \(v(p)\to\varepsilon v(p)\), \(N_0\to \varepsilon^{-1}N_0\). The second replacement corresponds to the assumption of a large number of particles in the condensate. Then, taking into account that \(\mathcal{V}\) and \(\mathcal{W}\) contain \(v\), for \(K_p\) and \(\mu\) (4) we shall have

\[ \mu= \frac{N_0}{V}v(0) +\varepsilon\,\frac{1}{V}\sum \{(v(0)+v(p_1))\langle b_{p_1}^+b_{p_1}\rangle +v(p_1)\langle b_{-p_1}b_{p_1}\rangle\} + \]
\[ +\frac{\varepsilon^{1/2}}{\sqrt{N_0}V} \sum v(p_1)\langle b_{p_1+p_2}^+b_{p_1}b_{p_2}\rangle; \tag{11} \]

\[ K_p=\varepsilon\,\frac{1}{V}\sum\{(\nu(0)+\nu(p-p_1))\langle b_{p_1}^{+}b_{p_1}\rangle+\nu(p-p_1)\langle b_{-p_1}b_{p_1}\rangle\beta\}+ \]
\[ +\varepsilon\,\frac{N_0}{V^2}\sum\langle\langle\mathcal V_{pp_1}\mid\mathcal V^{+}_{pp_1}\rangle\rangle +\varepsilon^{3/2}\frac{\sqrt{N_0}}{V^2}\left\{\sum\langle\langle\mathcal V_{pp_1}\mid\mathcal W^{+}_{pp_1p_2'p_2'p_1'}\rangle\rangle+ \]
\[ +\sum\{\mathcal W_{pp_2p_2'p_1'}\mid\mathcal V^{+}_{pp_1}\rangle\rangle\right\} +\frac{\varepsilon^2}{V^2}\sum\langle\langle\mathcal W_{pp_2p_2'p_1'}\mid\mathcal W^{+}_{pp_2p_2'p_1'}\rangle\rangle . \tag{12} \]

In the zeroth approximation \(\mu^{(0)}=\dfrac{N_0}{V}\nu(0)\), the function \(\hat G_p\) is equal to \(G_p^{(0)}\),

where
\[ \xi_p\to \xi_p^{(0)}=\frac{p^2}{2m}+\frac{N_0}{V}\nu(p) \quad\text{and}\quad E_p=\sqrt{\frac{p^2}{2m}\left(\frac{p^2}{2m}+2\frac{N_0}{V}\nu(p)\right)} . \]

In the first approximation in \(\varepsilon\), the matrix \(\Sigma_p\) (9) has the form
\[ \Sigma_p=\alpha E-L_p-K_p, \tag{13} \]
where in \(L_p\), in the expression for
\[ \xi_p=\frac{p^2}{2m}-\mu+\frac{N_0}{V}(\nu(0)+\nu(p)), \]
one must substitute
\[ \mu=\mu^{(1)}=\frac{N_0}{V}\nu(0)+\frac{1}{V}\sum\{(\nu(0)+\nu(p_1))\langle b_{p_1}^{+}b_{p_1}\rangle^{(0)}+ \]
\[ +\nu(p_1)\langle b_{-p_1}b_{p_1}\rangle^{(0)}\}, \tag{14} \]
\[ M_p=K_p=\frac{1}{V}\sum\{(\nu(0)+\nu(p-p_1))\langle b_{p_1}^{+}b_{p_1}\rangle^{(0)}+ \]
\[ +\nu(p-p_1)\langle b_{-p_1}b_{p_1}\rangle^{(0)}\beta\} +\frac{N_0}{V^2}\sum\langle\langle\mathcal V_{pp_1}\mid\mathcal V^{+}_{pp_1}\rangle\rangle^{(0)} . \tag{15} \]

The distribution functions \(\langle b_p^{+}b_p\rangle^{(0)}\) and \(\langle b_{-p}b_p\rangle^{(0)}\) entering (14) and (15) are calculated by means of spectral representations (see \({}^{(3,4)}\)) on the basis of the Green function of the zeroth approximation \(G_p^{0}\) (with \(\mu=\mu^{(0)}=\dfrac{N_0}{V}\nu(0)\)) and are equal to
\[ \langle b_p^{+}b_p\rangle^{(0)}= \frac{1}{2}\left(\frac{\xi_p^{0}}{E_p}\operatorname{cth}\frac{E_p}{2\theta}-1\right), \quad \langle b_{-p}b_p\rangle^{(0)}= -\frac{N_0\nu(p)}{V\,2E_p}\operatorname{cth}\frac{E_p}{2\theta}, \tag{16} \]
where \(\theta\) is the temperature of the Bose-particle system in energy units. The matrix elements
\(\langle\langle\mathcal V_{pp_1}\mid\mathcal V^{+}_{pp_1}\rangle\rangle\) are linear combinations of Green functions of the form
\(\langle\langle b_{-p_1}^{+},\,b_{p-p_1}\mid b_{p-p_1}^{+}-b_{-p_1}\rangle\rangle\),
\(\langle\langle b_p,\,b_{p-p_1}\mid b_{p-p_1}^{+}-b_{-p_1}\rangle\rangle\), etc., which in the zeroth approximation are expressed in terms of one-particle Green functions. One can also decouple the Green functions
\(\langle\langle\mathcal V_{pp_1}\mid\mathcal V^{+}_{pp_1}\rangle\rangle\)
by means of Bogolyubov’s canonical \(uv\)-transformation \({}^{(1)}\)
\[ b_p(t)=u_p\beta_p e^{-iE_pt}+v_p\beta_{-p}^{+}e^{iE_pt}, \quad u_p^2-v_p^2=1, \quad u_p^2=\frac{1}{2}\left(1+\frac{\xi_p^{(0)}}{E_p}\right), \tag{17} \]
performing, in the expressions obtained after substituting (17), a pairing of the operators \(\beta_p\):
\[ \langle\beta_p^{+}\beta_p\rangle^{(0)}= \frac{1}{2}\left(\operatorname{cth}\frac{E_p}{2\theta}-1\right), \quad \langle\beta_{-p}\beta_p\rangle^{(0)}=0. \]
Then, for the elements of the matrix \(\Sigma_p\) (13), one obtains the expressions
\[ \frac{1}{2}\bigl(\Sigma_{11}(p,E)-\Sigma_{11}(p,-E)\bigr)= \]
\[ =E\left\{1-\frac{N_0}{2V^2}\sum \operatorname{cth}\frac{E_{p_1}}{2\theta} \left(\frac{A_{pp_1}B_{pp_1}}{E^2-(E_{p_1}+E_{p-p_1})^2} +\frac{C_{pp_1}D_{pp_1}}{E^2-(E_{p_1}-E_{p-p_1})^2}\right)\right\}, \]
\[ -\frac{1}{2}\bigl(\Sigma_{11}(p,E)+\Sigma_{11}(p,-E)\bigr)+\Sigma_{12}(p,E)= \]
\[ =\frac{p^2}{2m}+\frac{1}{V}\sum(\nu(p-p_1)-\nu(p_1))\langle b_{p_1}^{+}b_{p_1}\rangle^{(0)}- \]
\[ -\frac{1}{V}\sum(\nu(p-p_1)+\nu(p_1))\langle b_{-p_1}b_{p_1}\rangle^{(0)}+ \]

\[ +\,\frac{N_0}{2V^2}\sum \operatorname{cth}\frac{E_{p_1}}{2\theta} \left( \frac{E_{p_1}+E_{p-p_1}}{E^2-(E_{p_1}+E_{p-p_1})^2}B_{pp_1}^{\,2} - \frac{E_{p_1}-E_{p-p_1}}{E^2-(E_{p_1}-E_{p-p_1})^2}D_{pp_1}^{\,2} \right), \]

\[ -\frac12\bigl(\Sigma_{11}(p,E)+\Sigma_{11}(p,E)\bigr)-\Sigma_{12}(p,E)= \]

\[ = \frac{p^2}{2m} +\frac1V\sum\bigl(v(p-p_1)-v(p_1)\bigr) \left(\langle b_{p_1}^{+}b_{p_1}\rangle^{(0)} +\langle b_{-p_1}b_{p_1}\rangle^{(0)}\right) +2\,\frac{N_0}{V}v(p)+ \]

\[ +\frac{N_0}{2V^2}\sum \operatorname{cth}\frac{E_{p_1}}{2\theta} \left( \frac{E_{p_1}+E_{p-p_1}}{E^2-(E_{p_1}+E_{p-p_1})^2}A_{pp_1}^{\,2} - \frac{E_{p_1}-E_{p-p_1}}{E^2-(E_{p_1}-E_{p-p_1})^2}C_{pp_1}^{\,2} \right), \tag{18} \]

where

\[ \begin{aligned} A_{pp_1}={}&(2v(p)+v(p_1)+v(p-p_1))(u_{p-p_1}v_{p_1}+u_{p_1}v_{p-p_1})+\\ &+(v(p_1)+v(p-p_1))(u_{p-p_1}u_{p_1}+v_{p-p_1}v_{p_1}),\\[4pt] B_{pp_1}={}&(v(p_1)-v(p-p_1))(u_{p-p_1}v_{p_1}-u_{p_1}v_{p-p_1})+\\ &+(v(p_1)+v(p-p_1))(u_{p-p_1}u_{p_1}-v_{p-p_1}v_{p_1}),\\[4pt] C_{pp_1}={}&(2v(p)+v(p_1)+v(p-p_1))(u_{p-p_1}u_{p_1}+v_{p-p_1}v_{p_1})+\\ &+(v(p_1)+v(p-p_1))(u_{p-p_1}v_{p_1}+u_{p_1}v_{p-p_1}),\\[4pt] D_{pp_1}={}&(v(p_1)-v(p-p_1))(u_{p-p_1}u_{p_1}-v_{p-p_1}v_{p_1})+\\ &+(v(p_1)+v(p-p_1))(u_{p-p_1}v_{p_1}-u_p v_{p-p_1}). \end{aligned} \tag{19} \]

Setting in (18) \(p=0\), \(E=0\), and taking (16) into account, we find that

\[ \Sigma_{11}(0,0)-\Sigma_{12}(0,0)= \]

\[ = \frac{2}{V}\sum v(p_1)\langle b_{-p_1}b_{p_1}\rangle^{(0)} -\frac{N_0}{2V^2}\sum \operatorname{cth}\frac{E_{p_1}}{2\theta}\, \frac{4v^2(p_1)(u_{p_1}^{2}-v_{p_1}^{2})^2}{-2E_{p_1}} =0. \]

Thus, the components found for the matrix \(\Sigma_p\) satisfy the necessary condition for the absence of a gap in the spectrum of elementary excitations (see \((^{5,6})\)). The energy denominators in the right-hand sides of (18) lead to the appearance of damping in the Green function \(G_p\). At temperature \(\theta\to0\), \(\operatorname{cth}\dfrac{E_{p_1}}{2\theta}\to1\), and in (18), in the last terms, the second summands under the summation signs, owing to their oddness with respect to the replacement \(p_1\to p-p_1\), drop out. For the phonon part of the spectrum \(E\sim p\), and therefore for \(\theta=0\) and \(p\to0\) the quantity \(E\) in the denominators of expressions (18) may be neglected. Consequently, at zero temperature the damping of the phonon part of the spectrum in the first approximation in \(p\) is absent (see \((^2)\)). Assuming that \(E\sim p\) as \(p\to0\) and \(\theta\to0\), we obtain

\[ C_{11}(p,E)\simeq -C_{12}(p,E) = \frac{N_0}{V}\, \frac{v^*(0)}{E^2-p^2\,\dfrac{N_0v^*(0)}{Vm^*}}, \]

where

\[ v^*(0)=v(0)\left[ 1+\frac{v(0)}{2V}\sum \left(\frac{v_{p_1}}{v(0)}\right)^2 E_{p_1}^{-3} \left(\frac{p_1^2}{2m}\right)^2 \right]^{-1}, \]

\[ m^*=m\left[ 1+\frac{N_0}{3V^2}\sum \frac{v^2(p_1)}{E_{p_1}^{3}}\, \frac{p_1^2}{2m} \left( 1-\frac{p_1}{2v(p_1)}\frac{\partial v(p_1)}{\partial p_1} \right) \right]^{-1}, \]

and the damping tends to zero as \(\theta\to0\).

In conclusion I express my gratitude to Acad. N. N. Bogolyubov, D. N. Zubarev, and S. V. Tyablikov for discussion of the results of the work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
6 XI 1961

CITED LITERATURE

1. N. N. Bogolyubov, Izv. AN SSSR, ser. fiz., 11, 67 (1947).
2. S. T. Belyaev, ZhETF, 34, 417, 433 (1958).
3. N. N. Bogolyubov, S. V. Tyablikov, DAN, 126, 53 (1959).
4. D. N. Zubarev, UFN, 71, No. 1 (1960).
5. N. N. Bogolyubov, Quasi-averages in Problems of Statistical Mechanics, preprint, Joint Institute for Nuclear Research, 1961.
6. N. Hugenholtz, D. Pines, Phys. Rev., 116, 489 (1959).