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UDC 536.758
PHYSICS
V. V. TOLMACHEV
TEMPERATURE GREEN’S FUNCTIONS
FOR A WEAKLY NONIDEAL BOSE–EINSTEIN SYSTEM
(Presented by Academician N. N. Bogolyubov on 10 XI 1966)
Earlier in \((^{1,2})\), for a Bose system, a theory of temperature elementary excitations was constructed by analogy with the theory of elementary excitations for a Fermi system with long-range interaction \((^3)\); recently Hohenberg \((^4)\) and Havouré \((^5)\) have shown that at zero temperature \(\theta = 0\) and for a small interaction there can be no special collective branch of the spectrum, the existence of which was discussed in \((^{1,2})\). In the present work we give results obtained by the author which confirm \((^{4,5})\) (within the approach he is developing to the theory of a Bose gas \((^6)\)), as well as, apparently, some new results not known in the literature.
First of all, let us make a small remark concerning the simple secular equation (7) of \((^1)\), which is valid when exchange effects are neglected.* In this equation, in \(Q_0(q; E)\) one may put \(\varepsilon(p) = E(p)\), \(A(p) = E(p)\), \(B(p) = 0\), and obtain equation (15) of \((^1)\) without the second term. In \((^1)\) it was shown that such an equation has a phonon root as \(v(0) \to \infty\); we shall show that the root is absent in the weak-coupling theory as \(v(0) \to 0\).
Take the function
\[ \operatorname{Re}(q,\varepsilon) = \frac{1}{(2\pi)^3} \int dp\, (n_p-n_{p+q}) \frac{E(p+q)-E(p)} {-\varepsilon^2+(E(p+q)-E(p))^2}; \tag{1} \]
where the integral should be understood in the sense of the principal value. For \(\varepsilon=0\) the function is positive and, for small \(q\),
\[ \operatorname{Re}(q,0) \simeq \frac{m^2\theta}{2\hbar^2 |q|}; \tag{2} \]
as \(\varepsilon\to\infty\) the function assumes negative values, moreover
\[ \operatorname{Re}(q,\varepsilon) \simeq -\frac{2E(q)n}{\varepsilon^2} \left(\frac{\theta}{\theta_0}\right)^{3/2}; \tag{3} \]
at \(\varepsilon=E(q)=\hbar^2 q^2/2m\) the function undergoes a discontinuity with jump, for small \(q\),
\[ \Delta R \simeq \frac{m^2\theta}{4\hbar^4}\frac{1}{|q|}. \tag{4} \]
As \(q\to0\), the value of the function on the left goes to infinity, while the value on the right tends to a certain negative limit,
\[ \frac{1}{(2\pi)^2}\, \frac{2}{\theta}\, \frac{(2m\theta)^{3/2}}{\hbar^3} \int_{0}^{+\infty} x^2 dx \left( \frac{e^{x^2}}{(e^{x^2}-1)^2} - \frac{1}{x^4} \right); \tag{5} \]
the schematic behavior of \(\operatorname{Re}(q,\varepsilon)\) for small \(q\) is presented in Fig. 1.
The corresponding secular equation can be represented in the form
\[ \operatorname{Re}(q,\varepsilon)=-1/v(q); \tag{6} \]
* Exchange effects may be neglected for a small long-range interaction \((^2)\); for a small short-range interaction the exchange effects are important \((^{4,5})\).
it is quite clear that (6) has no root as \(v(0)\to 0\), in agreement with \((^{4,5})\), where in fact \(v(q)=v(0)\) is assumed. This means that there is no root for a small short-range interaction; for a small long-range interaction a root of the equation may exist.
Let us now consider the secular equation (4) from (2) in the case \(\theta\ne 0\). We linearize it in \(\Xi, H, \Gamma\), and \(\Pi\); for \(\Xi, H\) we restrict ourselves to diagrams of second order, for \(\Gamma\)—to diagrams of first order, and for \(\Pi\)—to zeroth order. Using the expansions in small \(v\) for \(A(p), B(p)\) at fixed \(p\), and for \(\Xi_2(p,E_{2k})\), \(H_2(p,E_{2k})\), \(\Gamma_1(p,E_{2k})\), and \(\Pi_0(p,E_{2k})\) at fixed \(p, E_{2k}\), we obtain the approximate equation
\[ \begin{aligned} &E_{2k}^{2}+E^{2}(p)+2n_0v(p)E(p)+2E(p)\frac{1}{(2\pi\hbar)^3}\times\\ &\times \int dp'\,\bigl(v(p-p')-v(p')\bigr)n_{p'}+(E_{2k}^{2}+E^{2}(p))\times\\ &\times \frac{v(p)}{(2\pi\hbar)^3}\int dp'\, \frac{n_{p-p'}-n_{p'}}{E(p')-E(p-p')+iE_{2k}} +\text{terms higher than } v=0, \end{aligned} \tag{7} \]
Fig. 1. Schematic depiction of the behavior of the function \(\operatorname{Re}(q,\varepsilon)\)
which, when the fourth term is neglected, coincides with (15) of \((^1)\). Taking \(p=\rho x,\ \rho=2m^{1/2}n_0^{1/2}v^{1/2}(0),\ E_{2k}=2n_0v(0)z\), i.e., \(p, E_{2k}\) are also small for small \(v\), and using the expansions of \(\Xi_2(p,E_{2k})\), \(H_2(p,E_{2k})\), \(\Gamma_1(p,E_{2k})\), and \(\Pi_0(p,E_{2k})\) in small \(v\) at fixed \(x,z\), we obtain the equation
\[ \begin{aligned} &4n_0^2v^2(0)z^2+4n_0^2v^2(0)x^2(1+x^2) +\frac{2}{\pi\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta(x^2+1)+\\ &+\frac{2}{\pi^3\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta z^2 \int dx'\,\frac{1}{e(x')} \left(\frac{x'^2-1/2}{e(x')}+\frac{(x-x')^2-1/2}{e(x-x')}\right)\times\\ &\times \frac{1}{\bigl(e(x-x')+e(x')\bigr)^2+z^2} -\frac{1}{\pi^3\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta z^2 \int dx'\left(\frac{1}{E(x')}-\frac{1}{e(x-x')}\right)\times\\ &\times\left(\frac{x'^2+1/2}{e(x')}-\frac{(x-x')^2+1/2}{e(x-x')}\right) \frac{1}{\bigl(e(x')-e(x-x')\bigr)^2+z^2}\\ &+\frac{2}{\pi^3\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta x^2 \int dx'\,\times\\ &\times \frac{e(x')e(x-x')-\tfrac12\bigl(x'^2+(x-x')^2\bigr)-3x'^2(x-x')^2-\tfrac14} {e^2(x')e(x-x')} \frac{e(x')+e(x-x')}{\bigl(e(x')+e(x-x')\bigr)^2+z^2}\\ &-\frac{1}{\pi^3\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta x^2 \int dx'\, \frac{e(x')e(x-x')+\tfrac12\bigl(x'^2+(x-x')^2\bigr)+3x'^2(x-x')^2+\tfrac14} {e^2(x')e^2(x-x')}\times\\ &\times \frac{\bigl(e(x')-e(x-x')\bigr)^2}{\bigl(e(x')-e(x-x')\bigr)^2+z^2} -\frac{1}{\pi^3\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta x^2\times\\ &\times \int dx'\, \frac{e(x')e(x-x')+x'^2(x-x')^2+\tfrac12\bigl(x'^2+(x-x')^2\bigr)} {e^2(x')e(x-x')} \frac{e(x-x')+e(x')}{\bigl(e(x-x')+e(x')\bigr)^2+z^2}+\\ &+\frac{1}{2\pi^3\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta \int dx'\, \frac{e(x')e(x-x')-x'^2(x-x')^2-\tfrac12\bigl(x'^2+(x-x')^2\bigr)} {e(x')e(x-x')}\times\\ &\times \frac{e(x')-e(x-x')}{\bigl(e(x')-e(x-x')\bigr)^2+z^2} +\text{terms higher than } v^{5/2}=0, \end{aligned} \tag{8} \]
where \(e(x)=x(x^2+1)^{1/2}\); in deriving (8), from the very beginning (4) from (2) was divided by \(1+\dfrac{v(q)}{V}\Pi\), which is equivalent to multiplying the linearized equation by \(1-\dfrac{v(q)}{V}\Pi\), i.e., to adding to it the term \(\dfrac{v(q)}{V}\Pi(E_{2k}^{2}+A^2-B^2)\). In (8) the “gap” terms (which do not vanish at \(x=0,\ z=0\)) enter virtually; they compensate each other \((^6)\).
Putting in (8) \(z^2=-s^2x^2\) and passing to the limit \(x\to 0\), we obtain
\[ 4n_0^2v^2(0)s^2 = 4n_0^2v^2(0) - \frac{1}{3\pi\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta - \]
\[ -\frac{1}{\pi^2\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0)\theta \int_0^\infty dx\, \frac{8x^6+12x^4+\frac{13}{2}x^2+1}{(x^2+1)^2(x^2+\frac{1}{2})^2} \times \]
\[ \times \left( 2+ \frac{\sqrt{x^2+1}}{2x^2+1} \ln \frac{2x^2+1-\sqrt{x^2+1}}{2x^2+1+\sqrt{x^2+1}} \right) +\text{ terms higher than }v^{5/2}, \tag{9} \]
whence for the velocity of sound we have
\[ c=\frac{n^{1/2}v^{1/2}(0)}{m^{1/2}} + \frac{mv(0)\theta}{\hbar^3}\alpha + \text{ terms higher than }v^2, \tag{10} \]
where \(\alpha\) is a certain number, \(n\) is the total density. Using formulas (8) and (9) from (7) and the thermodynamic relation \(mc^2=n\,d\mu/dn\), one can obtain the “thermodynamic” velocity of sound
\[ c=\frac{n^{1/2}v^{1/2}(0)}{m^{1/2}} - \frac{1}{4\pi\hbar^3}mv(0)\theta + \text{ terms higher than }v^2, \tag{11} \]
which, however, does not coincide with the “microscopic” velocity of sound (10). From (8) one can also obtain the damping of elementary excitations
\[ \Gamma(\rho x) = \frac{1}{8\pi\hbar^3}m^{3/2}n_0^{1/2}v^{3/2}(0)\theta(5\pi-8)x + \text{ terms }v^{3/2}\text{ higher than }x + \text{ terms higher than }v^{3/2}, \tag{12} \]
where \(\rho=2m^{1/2}n_0^{1/2}v^{1/2}(0)\); the damping at \(\theta\ne0\) is proportional to \(x\), and not to \(x^5\), as at \(\theta=0\).
We now consider equation (4) from (2) in the case \(\theta=0\). Proceeding similarly to the case \(\theta\ne0\), one can obtain for the velocity of sound
\[ c= \frac{n^{1/2}v^{1/2}(0)}{m^{1/2}} - \frac{1}{8\pi^2\hbar^3} \frac{n^{1/2}}{m^{1/2}v^{1/2}(0)} \int_0^\infty p^2dp\,\frac{v^2(p)}{E(p)} + \frac{1}{\pi^2\hbar^3}mnv^2(0) + \text{ terms higher than }v^2; \tag{13} \]
using (6), (7) from (7) and the thermodynamic relation
\[ m^2c^2(n,\theta)=n\,\partial\mu(n,\theta)/\partial n, \]
one can obtain for \(c\) an expression exactly coinciding with (13). For the damping we obtain
\[ \Gamma(\rho x) = x^5\frac{3}{20\pi\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0) + \text{ terms }v^{5/2}\text{ higher than }x^5 + \text{ terms higher than }v^3. \tag{14} \]
Formulas (13), (14) coincide with those obtained in the literature.
For the Green’s functions at \(\theta=0\) we obtain
\[ TG^{-+}=-TG^{--} = n_0v(0) - \frac{n_0}{4\pi^2\hbar^3} \int_0^\infty p^2dp\,\frac{v^2(p)}{E(p)} + \frac{3}{\pi^2\hbar^3}m^{3/2}n_0^{3/2}v^{5/2}(0) + \]
\[ + \text{ terms }v^{5/2},\text{ vanishing as }x\to0 + \text{ terms higher than }v^{5/2}, \tag{15} \]
\[ T = -s^2x^2 \left( 4n_0^2v^2(0) + \frac{4}{\pi^2\hbar^3}m^{3/2}n^{5/2}v^{7/2}(0) \right) + \]
\[ + x^2 \left( 4n_0^2v^2(0) - \frac{1}{\pi^2\hbar^3}n_0^2v(0) \int_0^\infty p'^2dp'\,\frac{v^2(p')}{E(p')} + \frac{40}{3\pi^2\hbar^3}m^{3/2}n_0^{5/2}v^{7/2}(0) \right) + \]
\[ + \text{ terms }v^2,\ v^3\text{ and }v^{7/2}\text{ higher than }x^2 + \text{ terms higher than }v^{7/2}, \tag{16} \]
where it has been set that \(E_{2k}^2=-4n_0^2v^2(0)s^2x^2\) and \(\rho=2m^{1/2}n_0^{1/2}v^{1/2}(0)x\).
The result (15), (16) is consistent with the Gavoret relation5
\[ G^{-+} \simeq \frac{n_0}{n}\,mc^2\,\frac{1}{-E^2+c^2p^2} \]
which is valid for small \(E, p\). In the approximation adopted, a similar relation also holds for \(G^{--}\).
Moscow State University
named after M. V. Lomonosov
Received
14 IX 1966
References
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V. V. Tolmachev, DAN, 135, 41 (1960). ↩