Phonon and Optical-roton Branches of Excitations of the Bose System

Abstract

For a system of a large number of Bose particles, a chain of coupled equations for the averages of field operators is obtained. In the approximation where only the averages of one field operator and the averages of products of two operators at zero temperature are taken into account, there is derived a closed system of dynamic equations. Taking into account the finite range of the interaction potential between particles, the spectrum of elementary excitations of a many-particle Bose system is calculated, and it is shown that it has two branches: a sound branch and an optical branch with an energy gap at zero momentum. At high density, both branches are nonmonotonic and have the roton-like minima. The dispersion of the phonon part of the spectrum is considered. The performed calculations and analysis of experiments on neutron scattering allow to make a statement about the complex structure of the Landau dispersion curve in the superfluid $^4$He.

Appendices

Appendix A: General Form of Coefficients in the Dispersion Equation (55)

$$\begin{aligned}L_k&\equiv A_k+C_k,\qquad N_k\equiv A_kC_k-B_kD_k,\nonumber\\ A_k&=\Big [\varepsilon _k^{(1)}+2xnU_0\Big ]\!\Big [\varepsilon _k^{(2)}-2nU_0(x+z)\Big ] \nonumber \\&\quad +\,2xn\big (U_0+\Delta U_{k/2}\big )\big [2nU_0(2w-x)+\alpha _k\big ] \nonumber \\&\quad +\,2xn\big [2U_0+\big (\Delta U_k+\Delta U_{k/2}\big )\big ]\!\big [2nU_0(x+z)-\alpha _k\big ], \end{aligned}$$

(A1)

$$\begin{aligned}B_k&=xn\big(U_0+\Delta U_{k/2}\big)\Big [\varepsilon _k^{(1)}+\varepsilon _k^{(3)}+2nU_0(2w-z-2x)-2xn\big(\Delta U_{k}+\Delta U_{k/2}\big)\Big ], \end{aligned}$$

(A2)

$$\begin{aligned}&C_k=\Big [\varepsilon _k^{(3)}+2nU_0(2w-z-x)\Big ]^{\!2} + 2xn\big (U_0+\Delta U_{k/2}\big )\big [2nU_0(2w-z-3x)+\gamma _k-2\delta _k\big ], \end{aligned}$$

(A3)

$$\begin{aligned}D_k&=2\Big [\varepsilon _k^{(3)}+2nU_0(2w-z-x)\Big ]\!\big [2nU_0(2w-x)+\alpha _k\big ] \nonumber \\&\quad +2\Big [\varepsilon _k^{(2)}-2nU_0(x+z)\Big ]\!\big [2nU_0(2w+2z+x)+\gamma _k\big ] \nonumber \\&\quad +\,4\big [nU_0(4x+3z)+\delta _k\big ]\!\big [2nU_0(x+z)-\alpha _k\big ]. \end{aligned}$$

(A4)

$$\begin{aligned}&\alpha _k\equiv n\Delta U_k(1-2x-z), \quad \beta _k\equiv n\Delta U_k(1+2x+z), \quad \gamma _k\equiv n\Delta U_k(1+2x+3z), \nonumber \\&\delta _k\equiv n\Delta U_k(x+z) + n\Delta U_{k/2}(3x+2z), \nonumber \\&\varepsilon _k^{(1)}\equiv \varepsilon _k+\beta _k, \quad \varepsilon _k^{(2)}\equiv \varepsilon _k+\alpha _k, \quad \varepsilon _k^{(3)}\equiv \frac{\varepsilon _k}{2}+2n\Delta U_{k/2}(2-x). \end{aligned}$$

(A5)

In formulas (A1)–(A4) the notation (A5) is used.

Appendix B: Coefficients of the Dispersion Equation (55) for the Potential (26)

$$\begin{aligned}\frac{A_k}{n^2U_0^2}\equiv {\tilde{A}}_k&=4x(z+2w)+2b\Big [-\zeta z\kappa ^2-3(x+z)(1+z)f(\kappa )+6x(z+2w)f\Big (\frac{\kappa }{2}\Big )\Big ] \nonumber \\&\quad + \,b^2\Big [\zeta ^2\kappa ^4 + 6\zeta \kappa ^2f(\kappa )+9(1+z)(1-2x-z)f^2(\kappa )\Big ], \end{aligned}$$

(B1)

$$\begin{aligned}\frac{B_k}{n^2U_0^2}\equiv {\tilde{B}}_k&=2x(2w-2x-z)+3bx\!\left[ \frac{1}{2}\zeta \kappa ^2+(1+z)f(\kappa )+2(2+2w-4x-z)f\Big (\frac{\kappa }{2}\Big )\right] \nonumber \\&\quad + \,9b^2f\Big (\frac{\kappa }{2}\Big )x\!\left[ \frac{1}{2}\zeta \kappa ^2+(1+z)f(\kappa )+4(1-x)f\Big (\frac{\kappa }{2}\Big )\right] , \end{aligned}$$

(B2)

$$\begin{aligned}&\frac{C_k}{n^2U_0^2}\equiv {\tilde{C}}_k=4\big (4w^2+2xw-4x^2-2wz-xz\big ) \nonumber \\&\quad +\,2b\!\left[ (2w-x-z)\zeta \kappa ^2+3x(1+z)f(\kappa )+6\big (8w-4x-2xw-4x^2-4z-xz\big )f\Big (\frac{\kappa }{2}\Big )\right] \nonumber \\&\quad + \,b^2\left[ \frac{\zeta ^2}{4}\kappa ^4+6(2-x)\zeta \kappa ^2f\Big (\frac{\kappa }{2}\Big )+72\big (2-2x-x^2-xz\big )f^2\Big (\frac{\kappa }{2}\Big ) + 18x(1+z)f(\kappa )f\Big (\frac{\kappa }{2}\Big )\right] , \end{aligned}$$

(B3)

$$\begin{aligned}&\frac{D_k}{n^2U_0^2}\equiv {\tilde{D}}_k=8\big (4w^2+2x^2-2xw-2wz+3xz\big ) \nonumber \\&\quad +\,4b\Big [\Big (3w+2z+\frac{x}{2}\Big )\zeta \kappa ^2+3\big (4w+6x^2+5xz-4xw-2wz-5x-3z\big )f(\kappa ) \nonumber \\&\quad +\,6\big (4w-2x+6xw+4wz+xz\big )f\Big (\frac{\kappa }{2}\Big )\Big ] \nonumber \\&\quad + \,3b^2\!\left[ (3+2x+5z)\zeta \kappa ^2f(\kappa )+6(1+z)(1-2x-z)f^2(\kappa ) + 24(1-2x-z)^2f(\kappa )f\Big (\frac{\kappa }{2}\Big )\right] , \end{aligned}$$

(B4)

$$\begin{aligned}&L_k=A_k+C_k=(nU_0)^2{\tilde{L}}_k, \qquad {\tilde{L}}_k={\tilde{A}}_k + {\tilde{C}}_k, \nonumber \\&\quad N_k=A_kC_k-B_kD_k=(nU_0)^4{\tilde{N}}_k, \qquad {\tilde{N}}_k={\tilde{A}}_k {\tilde{C}}_k-{\tilde{B}}_k{\tilde{D}}_k. \end{aligned}$$

(B5)

The notation is used:

$$\begin{aligned} \begin{array}{ccc} \displaystyle { \zeta \equiv \frac{\varepsilon _a}{\chi I}=\frac{1}{\theta \chi }, \qquad b=\frac{1}{1-J}, \qquad \kappa \equiv r_0k. } \end{array} \end{aligned}$$

(B6)

The function $f(\kappa )$ is defined by formulas (59), (60).

Appendix C: Coefficients of the Dispersion Equation (55) for the Potential (26) in the Long Wavelength Limit $\kappa \ll 1$

$$\begin{aligned}&{\tilde{A}}_k\approx {\tilde{A}}_0+{\tilde{A}}_2\kappa ^2+{\tilde{A}}_3\kappa ^3+{\tilde{A}}_4\kappa ^4, \qquad {\tilde{B}}_k\approx {\tilde{B}}_0+{\tilde{B}}_2\kappa ^2+{\tilde{B}}_3\kappa ^3+{\tilde{B}}_4\kappa ^4, \nonumber \\&{\tilde{C}}_k\approx {\tilde{C}}_0+{\tilde{C}}_2\kappa ^2+{\tilde{C}}_3\kappa ^3+{\tilde{C}}_4\kappa ^4, \qquad {\tilde{D}}_k\approx {\tilde{D}}_0+{\tilde{D}}_2\kappa ^2+{\tilde{D}}_3\kappa ^3+{\tilde{D}}_4\kappa ^4, \nonumber \\&\tilde{L}_k\approx {\tilde{L}}_0+{\tilde{L}}_2\kappa ^2+{\tilde{L}}_3\kappa ^3+{\tilde{L}}_4\kappa ^4, \nonumber \\&{\tilde{L}}_0\equiv {\tilde{A}}_0+{\tilde{C}}_0, \quad {\tilde{L}}_2\equiv {\tilde{A}}_2+{\tilde{C}}_2, \quad {\tilde{L}}_3\equiv {\tilde{A}}_3+{\tilde{C}}_3, \quad {\tilde{L}}_4\equiv {\tilde{A}}_4+{\tilde{C}}_4, \nonumber \\&{\tilde{N}}_k\approx {\tilde{N}}_2\kappa ^2+{\tilde{N}}_3\kappa ^3+{\tilde{N}}_4\kappa ^4, \nonumber \\&\tilde{N}_2\equiv {\tilde{A}}_0{\tilde{C}}_2+{\tilde{C}}_0{\tilde{A}}_2-{\tilde{B}}_0{\tilde{D}}_2-{\tilde{D}}_0{\tilde{B}}_2, \nonumber \\&\tilde{N}_3\equiv {\tilde{A}}_0{\tilde{C}}_3+{\tilde{C}}_0{\tilde{A}}_3-{\tilde{B}}_0{\tilde{D}}_3-{\tilde{D}}_0{\tilde{B}}_3, \nonumber \\&\tilde{N}_4\equiv {\tilde{A}}_0{\tilde{C}}_4+{\tilde{C}}_0{\tilde{A}}_4+{\tilde{A}}_2{\tilde{C}}_2-{\tilde{B}}_0{\tilde{D}}_4-{\tilde{D}}_0{\tilde{B}}_4-{\tilde{B}}_2{\tilde{D}}_2. \end{aligned}$$

(C1)

Here

$$\begin{aligned}&{\tilde{A}}_0=4x(2w+z), \qquad {\tilde{B}}_0=2x(2w-z-2x),\nonumber \\&{\tilde{C}}_0=4\big (\!-4x^2+4w^2+2xw-2wz-xz\big ),\nonumber \\&{\tilde{D}}_0=8\big (\!2x^2+4w^2-2xw-2wz+3xz\big ),\end{aligned}$$

(C2)

$$\begin{aligned}&{\tilde{A}}_2=-2zb\zeta + u\big (2z^2+xz-2xw+2x+2z\big ),\nonumber \\&{\tilde{B}}_2=\frac{x}{2}\big [3b\zeta -u(4+z+2w-4x)\big ],\nonumber \\&{\tilde{C}}_2=2(2w-x-z)b\zeta + u\big (2x-8w+4z+2xw-xz+4x^2\big ),\nonumber \\&{\tilde{D}}_2=4\!\left( 3w+2z+\frac{x}{2}\right) \!b\zeta + 4u\Big (6x-6w+3z+xw-6x^2-\frac{11}{2}xz\Big ), \end{aligned}$$

(C3)

$$\begin{aligned}&{\tilde{A}}_3=\frac{\pi }{32}(b-1)\big (4x+4z+3xz-2xw+4z^2\big ), \nonumber \\&{\tilde{B}}_3=-\frac{\pi }{64}(b-1)\,x(6+3z+2w-4x), \nonumber \\&{\tilde{C}}_3=-\frac{\pi }{32}(b-1)\big (8w-4z+3xz-2xw-4x^2\big ), \nonumber \\&{\tilde{D}}_3=-\frac{\pi }{16}(b-1)\big (24x^2 -10xw + 21xz -4wz - 22x +20w -12z \big ), \end{aligned}$$

(C4)

$$\begin{aligned}{\tilde{A}}_4&=-6bb_4\Big (x+z+z^2+\frac{7}{8}xz-\frac{1}{4}xw\Big ) + b^2\zeta ^2 - 2b\zeta u + u^2\big (1-2x-2xz-z^2\big ), \nonumber \\{\tilde{B}}_4&=\frac{3}{8}bb_4x(10+7z-4x+2w) -\frac{3}{8}ub\zeta x + \frac{u^2}{4}x(2+z-x), \nonumber \\{\tilde{C}}_4&= \frac{1}{4}\Big (b^2\zeta ^2-2u(2-x)b\zeta + 2u^2\big (2-x-x^2\big )+\frac{3}{4}bb_4\big (4x+8w-4z+7xz-2xw-4x^2\big ), \nonumber \\{\tilde{D}}_4&=\frac{3}{2}bb_4\big (48x^2+41xz-26xw-12wz-42x+36w-24z\big ) \nonumber \\&\quad +\,4u^2\big (1-3x-z+2x^2+xz\big )-ub\zeta (3+2x+5z). \end{aligned}$$

(C5)

The notation is used:

$$\begin{aligned} \begin{array}{ccc} \displaystyle { u\equiv \frac{(1/5-J)}{2(1-J)}=\frac{1}{2}\!\left( \!1-\frac{4}{5}\,b\right) , \qquad b_4\equiv \frac{1}{120}\!\left( \frac{1}{7}+J\right) . } \end{array} \end{aligned}$$

(C6)