Dynamic Many-Body Theory: Dynamic Structure Factor of Two-Dimensional Liquid $^4$He

Abstract

We calculate the dynamic structure function of two-dimensional liquid $^4$He at zero temperature employing a quantitative multi-particle fluctuations approach up to infinite order. We observe a behavior that is qualitatively similar to the phonon–maxon–roton-curve in 3D, including a Pitaevskii plateau (Pitaevskii, Sov. Phys. JETP 9:830, 1959). Slightly below the liquid–solid phase transition, a second weak roton-like excitation evolves below the plateau.

Appendices

Appendix 1: Long-Wavelength Dispersion in 2D

In this appendix, we study the analytic structure of the self-energy as a function of an external energy $\hbar \omega$ in the limit $\hbar \omega - 2 \varepsilon _0(k/2)\rightarrow 0$. We assume that the solution $\varepsilon _0(k)$ of the implicit equation (16) has a negligible imaginary part.

We look for processes where a state of wave vector $\mathbf{k}$ decays into two phonons of wave vectors $\mathbf{p}_1$ and $\mathbf{p}_2$. In general one expects, for long wavelengths, a phonon dispersion relation of the form (21). In fact, it is easily shown that Eq. (16) leads to such a dispersion relation.

The calculation is best carried out in relative and center of mass momenta, i.e., we set

$$\begin{aligned} {\mathbf {p}}_1 = {\mathbf {q}}- \frac{1}{2}{\mathbf {k}}\quad {\mathbf {p}}_2 = -{\mathbf {q}}- \frac{1}{2}{\mathbf {k}}. \end{aligned}$$

Then, it is clear that

$$\begin{aligned} \varepsilon _0(|{\mathbf {k}}/2+{\mathbf {q}}|) +\varepsilon _0(|\mathbf{k}/2-{\mathbf {q}}|) \end{aligned}$$

(26)

has, for all angles $\cos \theta \equiv x \equiv \hat{\mathbf {q}}\cdot \hat{\mathbf {k}}$, a relative extremum at $q=0$, viz.

$$\begin{aligned} \varepsilon _0(p_1)+\varepsilon _0(p_2) = 2\varepsilon _0(k/2) + \left[ \frac{2\varepsilon _0'(k/2)}{k}(1-x^2) + \varepsilon _0''(k/2)x^2\right] q^2 +{\fancyscript{O}}(q^3). \end{aligned}$$

(27)

For further reference, abbreviate for $\gamma k^2\ll 1$:

$$\begin{aligned} \varepsilon _0'\equiv \varepsilon _0'(k/2) = \hbar c,\quad \varepsilon _0''\equiv \varepsilon _0''(k/2) = 6\hbar c\gamma k. \end{aligned}$$

and define with $\Delta E\equiv \hbar \omega - 2\varepsilon _0(k/2)$

$$\begin{aligned} q_-=\sqrt{\frac{k|\Delta E|}{2\varepsilon _0'}} \quad q_+=\sqrt{\frac{|\Delta E|}{|\varepsilon _0''|}}. \end{aligned}$$

(28)

The value $2\varepsilon _0(k/2)$ is, at $x=1$, a relative minimum if $\varepsilon _0'' > 0$ (anomalous dispersion) and a relative maximum for $\varepsilon _0'' < 0$ (normal dispersion). We are interested only in the non-analytic behavior as $\Delta E\rightarrow 0$ which is caused by the second-order singularity of the energy denominator as $q\rightarrow 0$.To calculate this behavior, we write

$$\begin{aligned} \left| V^{(3)}\left( \mathbf{k};\mathbf{q}-\frac{1}{2}\mathbf{k}, -\mathbf{q}-\frac{1}{2}\mathbf{k}\right) \right| ^2&= \left| V^{(3)}\left( \mathbf{k},-\frac{1}{2}\mathbf{k}.-\frac{1}{2}\mathbf{k}\right) \right| ^2 + \Delta V(\mathbf{k},\mathbf{q})\nonumber \\&\equiv \left| V^{(3)}(\mathbf{k})\right| ^2 + \Delta V(\mathbf{k},\mathbf{q}). \end{aligned}$$

(29)

$\Delta V(\mathbf{k},\mathbf{q})$ still contributes to the imaginary part, but not to the non-analytic behavior. We do the calculation first for the case of anomalous dispersion and keep only $\left| V^{(3)}(\mathbf{k})\right| ^2$. Since $\left| V^{(3)}(\mathbf{k})\right| ^2$ does not depend on the momentum transfer $q$, we must introduce a cutoff wave number $Q$ to make the integrals convergent; this does not affect the result on the non-analyticity of the self-energy. We find

$$\begin{aligned} \varSigma (k,\hbar \omega )&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{2(2\pi )^2\rho } \int \frac{\mathrm{{d}}^2q}{\hbar \omega -\varepsilon _0(p_1)-\varepsilon _0(p_2)+\mathrm{i}\eta }\nonumber \\&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{2(2\pi )^2\rho } \int \frac{\mathrm{{d}}^2q}{e_0(q)+(e_1(q)-e_0(q))\cos ^2\theta +\mathrm{i}\eta }\nonumber \\&= \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho } \int _{0}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{e_0(q)e_1(q)}}\text {sign}(e_0(q)), \end{aligned}$$

(30)

where $e_0(q)\equiv \Delta E-\frac{2\varepsilon _0'}{k}q^2$ and $e_1(q)\equiv \Delta E -\varepsilon _0''q^2$ are the values of the energy denominator at $x=0$ and $x=1$. For $\Delta E>0$ the integrand is real for $0\le q\le q_-$ and $q_+\le q$ and imaginary for $q_-\le q\le q_+$,

$$\begin{aligned} \varSigma (k,\hbar \omega )&= \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho } \int _{0}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{e_0(q)e_1(q)}}\text {sign}(e_0(q))\nonumber \\&= \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho } \sqrt{\frac{k}{2\varepsilon '_0\varepsilon _0''}} \Biggl [\int _{0}^{q_-} \frac{q \mathrm{{d}}q}{\sqrt{(q_-^2-q^2)(q_+^2-q^2)}}\nonumber \\&\quad -\,\mathrm{i}\int _{q_-}^{q_+} \frac{q \mathrm{{d}}q}{\sqrt{(q^2-q_-^2)(q_+^2-q^2)}}-\int _{q_+}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{(q^2-q_-^2)(q^2-q_+^2)}}\Biggr ] \nonumber \\&\approx \frac{\left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho } \sqrt{\frac{k}{2\varepsilon '_0\varepsilon _0''}} \left[ \ln q_+ -\ln 2Q -\mathrm{i}\frac{\pi }{2}\right] , \end{aligned}$$

(31)

For $\Delta E<0$, the integral is real and identical to the real part of Eq. (31), thus

$$\begin{aligned} \varSigma (k,\hbar \omega )\!\approx \! \frac{\left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho } \sqrt{\frac{k}{2\varepsilon '_0\varepsilon _0''}} \left[ \ln q_+\!\! -\!\mathrm{i}\frac{\pi }{2}\theta (\Delta E)\right] \!=\! \frac{\left| V^{(3)}(\mathbf{k})\right| ^2}{8\pi \rho } \sqrt{\frac{k}{2\varepsilon '_0\varepsilon _0''}} \!\ln \!\left( \!-\!\!\frac{\Delta E}{\varepsilon _0''}\right) .\nonumber \\ \end{aligned}$$

(32)

where $Q \gg q_+\gg q_-$ was used. The sign of the imaginary part was chosen to guarantee causality.

Arguing along the same lines as for anomalous dispersion, one finds for $\varepsilon _0''<0$ and $\Delta E > 0$

$$\begin{aligned} \varSigma (k,\hbar \omega )&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\int _{0}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{e_0(q)e_1(q)}}\text {sign}(e_0(q))\nonumber \\&= \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}} \left[ \int _{0}^{q_-} \frac{q \mathrm{{d}}q}{\sqrt{(q_-^2-q^2)(q^2+q_+^2)}}\right. \nonumber \\&\left. -\,\mathrm{i}\int _{q_-}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{(q^2-q_-^2)(q^2+q_+^2)}}\right] \nonumber \\&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}}\left[ \frac{q_-}{q_+}-\mathrm{i}\ln 2Q+\frac{\mathrm{i}}{2} \ln (q_-^2+q_+^2)\right] \nonumber \\&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}}\mathrm{i}\ln (q_+) \end{aligned}$$

(33)

and for $\Delta E<0$

$$\begin{aligned} \varSigma (k,\hbar \omega )&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\int _{0}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{e_0(q)e_1(q)}}\text {sign}(e_0(q))\nonumber \\&= \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}} \left[ -\int _{0}^{q_+} \frac{q \mathrm{{d}}q}{\sqrt{(q^2+q_-^2)(q_+^2-q^2)}}\right. \nonumber \\&\left. -\,\mathrm{i}\int _{q_+}^{Q} \frac{q \mathrm{{d}}q}{\sqrt{(q^2+q_-^2)(q^2-q_+^2)}}\right] \nonumber \\&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}}\left[ -\frac{\pi }{2}-\mathrm{i}\ln 2Q+\frac{\mathrm{i}}{2} \ln (q_-^2+q_+^2)\right] \nonumber \\&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}}\left[ -\frac{\pi }{2}+\mathrm{i}\ln (q_+)\right] . \end{aligned}$$

(34)

In this case the real part of the self-energy has a discontinuity whereas the imaginary part has a logarithmic singularity, and we can again combine

$$\begin{aligned} \varSigma (k,\hbar \omega )&\approx \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{4\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0|\varepsilon _0''|}}\left[ -\theta (-\Delta E)\frac{\pi }{2}+\mathrm{i}\ln (q_+)\right] \nonumber \\&= \frac{ \left| V^{(3)}(\mathbf{k})\right| ^2}{8\pi \rho }\sqrt{\frac{k}{2\varepsilon '_0\varepsilon _0''}} \ln \left( -\frac{\Delta E}{\varepsilon _0''}\right) . \end{aligned}$$

(35)

Hence Eq. (25) is valid for both $\varepsilon _0''$ and $\varepsilon _0''<0$.

Appendix 2: Three-Body Vertex

Normally, the three-body vertex (15) is calculated in convolution approximation. An improvement can be achieved by summing a set of three-body diagrams contributing to $\tilde{X}_3({\mathbf {k}},{\mathbf {p}},{\mathbf {q}})$, which corresponds topologically to the hypernetted chain (HNC) summation. The first few diagrams are shown in Fig 7.

Fig. 7

The figure shows the leading order diagrams contributing to the irreducible three-body vertex $X_3({\mathbf {r}}_1,{\mathbf {r}}_2,{\mathbf {r}}_3)$. The usual diagrammatic conventions apply: circles correspond to particle coordinates, filled circles imply a density factor and integration over the associated coordinate space. Solid lines represent correlation factors $h({\mathbf {r}}_i,{\mathbf {r}}_j)=g({\mathbf {r}}_i,{\mathbf {r}}_j)-1$ and the shaded triangle represents a three-body function $u_3({\mathbf {r}}_1,{\mathbf {r}}_2,{\mathbf {r}}_3)$

The equations to be solved are best written in momentum space and relative and center of mass momenta, i.e.,

$$\begin{aligned} \tilde{X}({\mathbf {p}}_1,{\mathbf {p}}_2,{\mathbf {p}}_3) \equiv \tilde{X}({\mathbf {q}}/2+{\mathbf {k}},{\mathbf {q}}/2-{\mathbf {k}},{\mathbf {q}})\equiv \tilde{X}_{{\mathbf {q}}}({\mathbf {k}}). \end{aligned}$$

(36)

The integral equation to be solved is

$$\begin{aligned} \tilde{X}_{{\mathbf {q}}}({\mathbf {k}})&= \int \frac{ \mathrm{d}^dp}{(2\pi )^d\rho } \tilde{h}({\mathbf {k}}-{\mathbf {p}})\tilde{N}_{{\mathbf {q}}}({\mathbf {p}})\nonumber \\ \tilde{N}_{{\mathbf {q}}}({\mathbf {k}})&= \tilde{N}^{(CA)}_{{\mathbf {q}}}({\mathbf {k}}) + \tilde{s}_{{\mathbf {q}}}({\mathbf {k}})\delta \tilde{X}_{{\mathbf {q}}}({\mathbf {k}}), \end{aligned}$$

(37)

where $\tilde{N}_{{\mathbf {q}}}({\mathbf {k}})$ is the set of nodal diagrams, and

$$\begin{aligned} \tilde{N}^{(CA)}_{{\mathbf {q}}}({\mathbf {k}}) = \tilde{h}\left( \frac{{\mathbf {q}}}{2}+{\mathbf {k}}\right) \tilde{h}\left( \frac{{\mathbf {q}}}{2}-{\mathbf {k}}\right) + \tilde{u}_3\left( \frac{{\mathbf {q}}}{2}+{\mathbf {k}},\frac{{\mathbf {q}}}{2}-{\mathbf {k}},{\mathbf {q}}\right) \end{aligned}$$

is the convolution approximation for this quantity. Also, we have abbreviated

$$\begin{aligned} \tilde{s}_{{\mathbf {q}}}({\mathbf {k}}) = \left[ S(|{\mathbf {p}}+{\mathbf {q}}/2|)S(|{\mathbf {p}}-{\mathbf {q}}/2|)-1\right] . \end{aligned}$$

(38)

The equations can be easily solved by expanding all functions in terms of $k$, $q$, and the angle between the two vectors, e.g.,

$$\begin{aligned} \tilde{h}(|{\mathbf {k}}_1-{\mathbf {k}}_2|) =\sum \limits _{n=0}^\infty \tilde{h}_n(k_1,k_2) \cos (n\phi _{12}). \end{aligned}$$

(39)

This gives us the three-body vertex in the form

$$\begin{aligned} \tilde{X}_{{\mathbf {q}}}({\mathbf {p}}) = \sum \limits _m\cos ( m\phi ) X_m(q,p). \end{aligned}$$

(40)