Ground-State Properties of a Dilute Two-Dimensional Bose Gas

Abstract

We revisit the problem of the calculation of zero-temperature properties for the dilute two-dimensional Bose gas. By using Popov’s hydrodynamic approach and perturbation theory on the two-loop level, we recover not only the known expansion for the ground-state energy but also calculate for the first time the condensate density and Tan’s contact.

Appendices

Appendices

1.1 A

The explicit analytic expressions for the diagrams determining corrections to the partition function logarithm $-\beta \Delta E$ and presented in Fig. (1) are the following

$$\begin{aligned} a= & {} -\frac{\beta }{8N}\sum _{\mathbf{k},\mathbf{q}}\frac{\varepsilon _k+\varepsilon _q}{\alpha _k\alpha _q}, \end{aligned}$$

(21)

$$\begin{aligned} b= & {} \frac{\beta }{96 N}\sum _{\mathbf{k}+\mathbf{q}+\mathbf{s}=0}\frac{1}{\alpha _k\alpha _q\alpha _s}\frac{(\varepsilon _k+ \varepsilon _q+\varepsilon _s)^2}{E_k+E_q+E_s}, \end{aligned}$$

(22)

$$\begin{aligned} c= & {} \frac{\beta }{32 N}\sum _{\mathbf{k}+\mathbf{q}+\mathbf{s}=0}\left( \frac{\hbar ^2}{m}\mathbf{kq}\right) ^2\frac{\alpha _k\alpha _q}{\alpha _s}\frac{1}{E_k+E_q+E_s}, \end{aligned}$$

(23)

$$\begin{aligned} d= & {} \frac{\beta }{16 N}\sum _{\mathbf{k}+\mathbf{q}+\mathbf{s}=0}\frac{\hbar ^2}{m}{} \mathbf{ks}\frac{\hbar ^2}{m}\mathbf{qs}\frac{\alpha _s}{E_k+E_q+E_s}, \end{aligned}$$

(24)

$$\begin{aligned} e= & {} -\frac{\beta }{16 N}\sum _{\mathbf{k}+\mathbf{q}+\mathbf{s}=0}\frac{\hbar ^2}{m}\mathbf{kq}\frac{1}{\alpha _s}\frac{\varepsilon _k+\varepsilon _q+ \varepsilon _s}{E_k+E_q+E_s}, \end{aligned}$$

(25)

(recall that $|\mathbf{k}|,|\mathbf{q}|, |\mathbf{s}|<\varLambda$ and $\alpha _k=E_k/\varepsilon _k$), where the frequency sums in the diagrams b and c

$$\begin{aligned} \frac{1}{\beta }\sum _{\omega _k,\omega _q}\frac{1}{\omega _k^2+E_k^2} \frac{1}{\omega _q^2+E_q^2}\frac{1}{(\omega _k+\omega _q)^2+E_{s}^2} \rightarrow \frac{\beta }{E_kE_qE_s}\frac{1}{E_k+E_q+E_s}, \end{aligned}$$

(26)

and in e and d

$$\begin{aligned} \frac{1}{\beta }\sum _{\omega _k,\omega _q}\frac{\omega _k}{\omega _k^2+E_k^2} \frac{\omega _q}{\omega _q^2+E_q^2}\frac{1}{(\omega _k+\omega _q)^2+E_{s}^2} \rightarrow -\frac{\beta }{E_s}\frac{1}{E_k+E_q+E_s}, \end{aligned}$$

(27)

were evaluated with the help of residue theorem in the zero-temperature limit.

1.2 B

In this section, we present some details of calculation of integrals determining constants in the ground-state energy (17), condensate depletion (18) and contact (20). We will not stop on the evaluation of variational derivatives $\left( \frac{\delta E}{\delta \varepsilon _k}\right) _{n,\mathcal {T}}$, $\left( \frac{\delta E}{\delta \mathcal {T}_k}\right) _{n}$ with the following integration in $\mathbf{k}$-space and only give the dimensionless expressions written in terms of triple integrals. After elimination of the explicit dependence on cutoff parameter $\varLambda$ for $\text {const}_E$, we obtained

$$\begin{aligned} \text {const}_E= & {} \frac{32}{\pi }\int ^{\infty }_{0}{\text {d}}k\left( 1-\frac{k}{\sqrt{k^2+1}}\right) \int ^{\infty }_{0}{\text {d}}q\left( 1-\frac{q}{\sqrt{q^2+1}} \right) \nonumber \\&\times \int ^{k+q}_{|k-q|}{\text {d}}s s\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\left( s^2-\frac{s^3}{\sqrt{s^2+1}}-\frac{1}{2}\right) \nonumber \\&-\frac{32}{3\pi }\int ^{\infty }_{0}\frac{{\text {d}}kk}{\sqrt{k^2+1}} \int ^{\infty }_{0}\frac{dqq}{\sqrt{q^2+1}} \int ^{k+q}_{|k-q|}{\text {d}}s s\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\nonumber \\&\times \frac{s}{\sqrt{s^2+1}}\frac{{\tilde{f}}^2(k,q,s)}{k\sqrt{k^2+1} +q\sqrt{q^2+1}+s\sqrt{s^2+1}}, \end{aligned}$$

(28)

here and below

$$\begin{aligned} {\tilde{f}}(k,q,s)=\frac{s^2-k^2-q^2}{2}\left( \sqrt{1+1/k^2}-1\right) \left( \sqrt{1+1/q^2}-1\right) +\text {perm.} \end{aligned}$$

The second-order correction to condensate density is determined by the following constant

$$\begin{aligned} \text {const}_{N_0}= & {} \frac{8}{\pi }\int ^{\infty }_{0}{\text {d}}k\left( 1-\frac{k}{\sqrt{k^2+1}}\right) \int ^{\infty }_{0}dq\left( 1-\frac{q}{\sqrt{q^2+1}} \right) \nonumber \\&\times \int ^{k+q}_{|k-q|}{\text {d}}s s\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\left( 1-\frac{s}{\sqrt{s^2+1}}-\frac{s}{2(s^2+1)^{3/2}} \right) \nonumber \\&-\frac{8}{\pi }\int ^{\infty }_{0}{\text {d}}k\left( 1-\frac{k}{\sqrt{k^2+1}}\right) \int ^{\infty }_{0}dq\left( q^2-\frac{q^3}{\sqrt{q^2+1}}-\frac{1}{2}\right) \nonumber \\&\times \int ^{k+q}_{|k-q|}{\text {d}}s\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\frac{1}{(s^2+1)^{3/2}}\nonumber \\&-\frac{4}{\pi }\int ^{\infty }_{0}\frac{{\text {d}}kk}{\sqrt{k^2+1}} \int ^{\infty }_{0}\frac{dqq}{\sqrt{q^2+1}} \int ^{k+q}_{|k-q|}{\text {d}}s \left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\nonumber \\&\times \frac{\partial }{\partial s}\frac{s}{\sqrt{s^2+1}} \frac{{\tilde{f}}^2(k,q,s)}{k\sqrt{k^2+1}+q\sqrt{q^2+1}+s\sqrt{s^2+1}}. \end{aligned}$$

(29)

Finally, $\text {const}_{\mathcal {C}}$ after some rearrangements is given by

$$\begin{aligned} \text {const}_{\mathcal {C}}= & {} -\frac{16}{\pi }\int ^{\infty }_{0}{\text {d}}k \left( 1-\frac{k}{\sqrt{k^2+1}}\right) \int ^{\infty }_{0}dq\left( 1-\frac{q}{\sqrt{q^2+1}}\right) \nonumber \\&\times \int ^{k+q}_{|k-q|}{\text {d}}s s\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\left( 1-\frac{s^3}{(s^2+1)^{3/2}}\right) \nonumber \\&+\frac{32}{\pi }\int ^{\infty }_{0}\frac{{\text {d}}kk}{(k^2+1)^{3/2}} \int ^{\infty }_{0}dq\left( q^2-\frac{q^3}{\sqrt{q^2+1}}-\frac{1}{2} \right) \nonumber \\&\times \int ^{k+q}_{|k-q|}{\text {d}}ss\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\left( 1-\frac{s}{\sqrt{s^2+1}}\right) \nonumber \\&-\frac{32}{\pi }\int ^{\infty }_{0}\frac{{\text {d}}kk}{\sqrt{k^2+1}} \int ^{\infty }_{0}\frac{dqq}{\sqrt{q^2+1}} \int ^{k+q}_{|k-q|}{\text {d}}ss\left\{ 1-\frac{(s^2-k^2-q^2)^2}{4k^2q^2} \right\} ^{-1/2}\nonumber \\&\times \frac{s}{\sqrt{s^2+1}}\frac{{\tilde{f}}^2(k,q,s)}{k\sqrt{k^2+1}+q\sqrt{q^2+1}+s\sqrt{s^2+1}}. \end{aligned}$$

(30)

The results of numerical calculations of these integrals are presented in main text.