2D Dilute Bose Mixture at Low Temperatures

Abstract

The thermodynamic and superfluid properties of the dilute two-dimensional binary Bose mixture at low temperatures are discussed. We also considered the problem of the emergence of the long-range order in these systems. All calculations are performed by means of the celebrated Popov path-integral approach to the Bose gas with a short-range interparticle potential.

Appendix

After transformation (4) the action of the two-component Bose mixture is

$$\begin{aligned}&S\rightarrow \int \mathrm{d}x \,\psi ^*_a(x)\left\{ \partial _{\tau }+\frac{\hbar ^2 }{2m_a}\Delta +\mu _a\right\} \psi _a(x)\nonumber \\&\quad +\int \mathrm{d}x \,\tilde{\psi }^*_a(x)\left\{ \partial _{\tau }+\frac{\hbar ^2 }{2m_a}\Delta +\mu _a\right\} \tilde{\psi }_a(x)\nonumber \\&\quad -\frac{1}{2}\int \mathrm{d}x\int \mathrm{d}x'\varPhi _{ab}(x-x')|\psi _a(x)|^2|\psi _b(x')|^2\nonumber \\&\quad -\int \mathrm{d}x\int \mathrm{d}x'\varPhi _{ab}(x-x')\left\{ \psi ^*_a(x) \psi ^*_b(x')\psi _b(x')\tilde{\psi }_a(x)+\mathrm{c.c}\right\} \nonumber \\&\quad -\int \mathrm{d}x\int \mathrm{d}x'\varPhi _{ab}(x-x')\left\{ |\psi _a(x)|^2|\tilde{\psi }_b(x')|^2+\psi ^*_a(x) \tilde{\psi }^*_b(x')\psi _b(x')\tilde{\psi }_a(x)\right\} \nonumber \\&\quad -\frac{1}{2}\int \mathrm{d}x\int \mathrm{d}x'\varPhi _{ab}(x-x')\left\{ \psi ^*_a(x)\psi ^*_b(x')\tilde{\psi }_b(x')\tilde{\psi }_a(x)+\mathrm{c.c}\right\} \nonumber \\&\quad -\int \mathrm{d}x\int \mathrm{d}x'\varPhi _{ab}(x-x')\left\{ \tilde{\psi }^*_a(x) \tilde{\psi }^*_b(x')\tilde{\psi }_b(x')\psi _a(x)+\mathrm{c.c}\right\} \nonumber \\&\quad -\frac{1}{2}\int \mathrm{d}x\int \mathrm{d}x'\varPhi _{ab}(x-x')|\tilde{\psi }_a(x)|^2|\tilde{\psi }_b(x')|^2, \end{aligned}$$

(32)

where nine interaction vertices are presented diagrammatically in Fig. 1. For a very dilute system while integrating out $\tilde{\psi }$-fields one can use simple perturbation theory considering the ideal gas action (the second term in Eq. (32)) as a zero-order approximation. Furthermore, in the low-temperature limit the only nonzero contribution to the effective action governing “slowly” varying fields is given by the graphs depicted in Fig. 2. This infinite series of diagrams is summed up to give the linear integral equation for the renormalized symmetric vertices

$$\begin{aligned} t_{ab}(P,Q|Q+K,P-K)= & {} [g_{ab}(k)+\delta _{ab}g_{aa}(p-k-q)]/2^{\delta _{ab}}\nonumber \\&-\frac{1}{2^{\delta _{ab}}\mathcal {A}\beta }\sum _{S}[g_{ab}(s)+\delta _{ab}g_{aa}(p-s-q)]\nonumber \\&\times \,G_a(P-S)G_b(Q+S)t_{ab}\nonumber \\&\quad (P-S,Q+S|Q+K,P-K), \end{aligned}$$

(33)

where $g_{ab}(k)$ is the Fourier transform of $\varPhi _{ab}(\mathbf{r})$, and we used notation for Green’s functions of the ideal gases $G_a(P)=1/[i\omega _p-\varepsilon _a(p)+\mu _a]$. In the dilute limit, where $\mu _{a}\ll \hbar ^2\varLambda ^2/m_a$ (the most natural choice in the two-component case is $\varLambda ^2\sim n_A+n_B$) and the physically relevant region of the wave-vectors integration $p,q,k\sim \sqrt{m\mu _{a}}/\hbar \ll \varLambda$, we can neglect the dependence on P and Q under the integral in Eq. (33). If it is also assumed that the interaction potentials are short-ranged, i.e., $g_{ab}(s)$ weakly depends on s, then the $t_{ab}(-S,S|0,0)$ is also independent on the transferred momentum. The latter observation allows to find the asymptotic solution of (33) (denoting $t_{ab}(0,0|0,0)\equiv t_{ab}$)

$$\begin{aligned} \frac{1}{t_{ab}}=\frac{1}{g_{ab}(0)} +\int ^{\infty }_{\varLambda }\frac{\mathrm{d}s\,s}{2\pi }\frac{g_{ab}(s)}{g_{ab}(0)} \frac{1}{\varepsilon _{a}(s)+\varepsilon _{b}(s)}. \end{aligned}$$

(34)

Actually, $t_{ab}$ enters the hydrodynamic action as a matrix of the effective coupling constants.

Fig. 1

Diagrammatic representation of the vertices appearing in Eq. (32). The solid and dashed lines stand for the $\psi _a(x)$ and $\tilde{\psi }_a(x)$, respectively

Fig. 2

Renormalization of the interaction potential in the hydrodynamic action