Microscopic theory of novel pseudogap phenomena and Bose-liquid superconductivity and superfluidity in high-$T_c$ cuprates and other systems

Abstract

A unified and empirically adequate microscopic theory of novel pseudogap phenomena and Bose-liquid superconductivity and superfluidity in high-$T_c$ cuprates and other systems is developed based on the original ideas of the pseudogap state and unusual superconducting/superfluid states of matter. This theory establishes the following laws: (i) the high-$T_c$ cuprates and other systems with low Fermi energies $\varepsilon _F\sim \varepsilon _A$ (where $\varepsilon _A$ is the energy of the attractive interaction between fermionic quasiparticles) are bosonic superconductors and superfluids exhibiting pseudogap behavior above the superconducting/superfluid transition temperature $T_c$ and a $\lambda$-like phase transition at $T_c$, (ii) the pseudogap state and bosonic Cooper pairs in such systems (with $\varepsilon _F\lesssim 2\varepsilon _A$ and Bardeen–Cooper–Schrieffer (BCS)-like gap $\Delta _F > rsim 0.17\varepsilon _F$) are formed above $T_c$, (iii) only a minority of preformed bosons condenses into a Bose superfluid at $T_c$ and (iv) only the systems with $\varepsilon _F>>\varepsilon _A>>\Delta _F$ become BCS-type conventional or topological fermionic superconductors and superfluids. A modified BCS-like model describes the precursor Cooper pairing of fermionic quasiparticles and the formation of bosonic Cooper pairs above $T_c$. The criteria for the bosonization of Cooper pairs and fermion–boson transitions are formulated. The mean-field theory describing new laws of condensation of attracting bosons into Bose superfluids below $T_c$ is presented. The proposed microscopic theory explains all the emerging pseudogap behaviors and unusual superconducting/superfluid states and properties of high-$T_c$ materials and other systems. In high-$T_c$ cuprates, the unconventional electron–phonon interactions and polaronic effects give rise to in-gap states, Fermi-surface reconstruction, two distinct pseudogaps and unusual normal-state properties, a quantum critical point and crossover from BCS superconductivity to Bose-liquid superconductivity. The theory of three-dimensional (3D) and two-dimensional (2D) Bose superfluids describes fairly well the novel superconducting states (i.e., the so-called A and B phases below $T_c$ and an extended A phase and related vortex-like state above $T_c$) and properties of high-$T_c$ cuprates (e.g., $\lambda$-like transition at $T_c$, first-order phase transition at lower temperatures and other unusual features) in accordance with the experimental data. The reasons for suppression and enhancement of superconductivity by disorders in high-$T_c$ cuprates are discussed. Strongly enhanced 2D Bose-liquid superconductivity emerging within a 3D cuprate superconductor (with the highest bulk $T_c$) persists up to room temperature in multi-lamellar blocks and at grain boundaries and interfaces. Most enhanced 3D Bose-liquid superconductivity can emerge at room temperature in high-$T_c$ hydrides under high pressures. Superconducting/superfluid states and properties of heavy-fermion and organic compounds, ruthenate $(\textrm{Sr}_{2}\textrm{RuO}_{4})$ and possibly high-$T_c$ hydrides, quantum liquids ($^3$He and $^4\hbox {He}$) and atomic Fermi gases are also well explained by the proposed theory of Bose superfluids. Finally, new criteria and principles of unconventional superconductivity and superfluidity are formulated.

Appendices

Appendices

Appendix A: Boltzmann transport equations for Fermi components of Cooper pairs and bosonic Cooper pairs

The Boltzmann transport equation for the excited Fermi components of Cooper pairs in the relaxation time approximation can be written as

$$\begin{aligned} f^0_C(k)-f_C(k)=\frac{\tau _\textrm{BCS}(k)}{\hbar }\vec {F} \frac{\partial f_C}{\partial {k}}, \end{aligned}$$

(A.1)

where $f^0_C(k)$ is the equilibrium Fermi distribution function, $\tau _\textrm{BCS}(k)$ is the relaxation time of the Fermi components of Cooper pairs in the BCS-like pseudogap regime, $\vec {F}$ is a force acting on a charge carrier in the crystal.

We consider the conductivity of hole carriers in the presence of the electric field applied in the x-direction. Then we can write eq. (A1) as

$$\begin{aligned} f^0_C(k)-f_C(k)= & {} \frac{\tau _\textrm{BCS}(k)}{\hbar }\vec {F_x} \frac{\partial f_C(k)}{\partial k_x}\nonumber \\= & {} \frac{\tau _\textrm{BCS}(k)}{\hbar }\vec {F_x} \frac{\partial f_C(k)}{\partial E}\frac{\partial E}{\partial k_x}\nonumber \\= & {} \frac{\tau _\textrm{BCS}(k)}{\hbar }\vec {F_x}\hbar V_x \frac{\partial f_C(k)}{\partial E}, \end{aligned}$$

(A.2)

where $E(k)=\sqrt{\xi ^2(k)+\Delta ^2_F}$, $\xi (k)=\varepsilon (k)-\varepsilon _F$, $\varepsilon (k)=\hbar ^2(k^2_x+k^2_y+k^2_z)/2m_p$, $V_x=\frac{1}{\hbar }\frac{\partial E(k)}{\partial k_x}=v_x\frac{\xi }{E}$, $v_x=\hbar k_x/m_p$.

The number of the Fermi components of Cooper pairs is determined from the relation

$$\begin{aligned} n^*_F=2\sum _k u_kf_C(k)=2\sum _k\frac{1}{2}\left( 1+\frac{\xi }{E}\right) f_C(k). \end{aligned}$$

(A.3)

Then the density of these Fermi quasiparticles is defined as

$$\begin{aligned} n^*_F=\frac{1}{(2\pi )^3}\int \left( 1+\frac{\xi }{E}\right) f_C(k)d^3k \end{aligned}$$

(A.4)

Using eqs (A.2) and (A.4) the current density in the x-direction can be defined as

$$\begin{aligned} J^*_x= & {} \frac{e}{(2\pi )^3}\int v_x\left( 1+\frac{\xi }{E}\right) f_C(k)d^3k \nonumber \\= & {} \frac{e}{(2\pi )^3}\int v_x\left( 1+\frac{\xi }{E}\right) f^0_C(k)d^3k-\frac{e}{(2\pi )^3}\nonumber \\{} & {} \times \int v^2_x\tau _\textrm{BCS}(k)F_x\frac{\xi }{E}\left( 1+\frac{\xi }{E}\right) \frac{\partial f_C(k)}{\partial E}d^3k, \nonumber \\ \end{aligned}$$

(A.5)

where $\xi$ and E are even functions of k, while $v_xf^0_C(k)$ is an odd function of $v_x$. Since integration with respect to $dk_x$ ranges from $-\infty$ to $+\infty$, the first term in eq. (A.5) becomes zero, and only the second term remains, resulting in (for $F_x=+eE_x$)

$$\begin{aligned} J^*_x=-\frac{e^2E_x}{8\pi ^3}\int v^2_x\tau _\textrm{BCS}(k)\frac{\xi }{E}\left( 1+\frac{\xi }{E}\right) \frac{\partial f_C(k)}{\partial E}d^3k. \nonumber \\ \end{aligned}$$

(A.6)

Similarly, the current density of bosonic Cooper pairs in the x-direction is given by (for $F_x=+2eE_x$)

$$\begin{aligned} J^B_x= & {} \frac{2e}{(2\pi )^3}\int v_x\left[ f^0_B(k)-\tau _B(k)v_xF_x\frac{\partial f_B}{\partial \varepsilon }\right] d^3K\nonumber \\= & {} -\frac{e^2E_x}{2\pi ^3}\int v^2_x\tau _B(k)\frac{\partial f_B}{\partial \varepsilon }d^3K, \end{aligned}$$

(A.7)

Appendix B: Possible analytical solutions of the integral equations in the theory of a 3D Bose superfluid for $T=0$

For the model potential (122), $\Delta _B(\vec {k})$ will be approximated as

$$\begin{aligned} \Delta _B(\vec {k})= \left\{ \begin{array}{lll} \Delta _{B1} &{} \text {for} \ \ |\varepsilon (k)|,|\varepsilon (k')|\le \xi _{BA},\\ \Delta _{B2} &{} \text {for} \ \ \xi _{BA}<|\varepsilon (k)|,|\varepsilon (k')|\le \xi _{BR},\\ 0 &{} \text {for} \ \ \varepsilon (k) \text { or } \varepsilon (k')>\xi _{BR}. \end{array} \right. \end{aligned}$$

(B.1)

Then eqs (116) and (118) at $T=0$ are reduced to the following equations:

$$\begin{aligned}{} & {} \Delta _{B1}=-D_B(V_{BR}-V_{BA})\Delta _{B1}I_A-V_{B1}\Delta _{B2}I_R,\nonumber \\{} & {} \Delta _{B2}=-V_{BR}\Delta _{B1}I_A-V_{BR}\Delta _{B2}I_R, \end{aligned}$$

(B.2)

and

$$\begin{aligned} \chi _{B1}=(V_{BR}-V_{BA})\rho _{B1}+V_{BR}\rho _{B2}, \end{aligned}$$

(B.3)

where

$$\begin{aligned}{} & {} I_A=D_B\int ^{\xi _{BA}}_{0}\frac{\sqrt{\varepsilon }d\varepsilon }{2\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta ^2_{B1}}}, \end{aligned}$$

(B.4)

$$\begin{aligned}{} & {} I_R=D_B\int ^{\xi _{BR}}_{\xi _{BA}}\frac{\sqrt{\varepsilon }d\varepsilon }{2\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta ^2_{B2}}}, \end{aligned}$$

(B.5)

and $\rho _{B1}=\frac{1}{\Omega }\sum ^{k_A}_{k=0}n_B(\vec {k})$, $\rho _{B2}=\frac{1}{\Omega }\sum ^{k_R}_{k=k_A}n_B(\vec {k})$, $n_B(\vec {k})=[\exp (E_B(\vec {k})/k_BT)-1]^{-1}$, $\xi _{BA}=\varepsilon (k_A)$, $\xi _{BR}=\varepsilon (k_R)$.

For 3D Bose systems, $D_B={m_B}^{3/2}/\sqrt{2}\pi ^2\hbar ^3$. From eq. (B.2), we obtain

$$\begin{aligned} \tilde{V}_BI_A=[V_{BA}-V_{BR}(1+V_{BR}I_R)^{-1}]I_A=1. \end{aligned}$$

(B.6)

At $\xi _{BA}>>\tilde{\mu }_B$, $\Delta _{B2}$, we obtain from eq. (B.5),

$$\begin{aligned} I_R\simeq D_B\int \limits _{\xi _{BA}}^{\xi _{BR}}\sqrt{\varepsilon }\frac{d\varepsilon }{2\varepsilon }=D_B\left[ \sqrt{\xi _{BR}}-\sqrt{\xi _{BA}}\right] \end{aligned}$$

(B.7)

and

$$\begin{aligned} I_R\simeq \frac{1}{2}\int ^{\xi _{BR}}_{\xi _{BA}} \frac{d\varepsilon }{\varepsilon }=\frac{D_B}{2}\ln \frac{\xi _{BR}}{\xi _{BA}} \end{aligned}$$

(B.8)

for 3D and 2D Bose systems, respectively.

We can assume that almost all Bose particles have energies smaller than $\xi _{BA}$ and $\rho _B\simeq \rho _{B1}$. Using eq. (117) the expression for $\rho _B$ can be written as

$$\begin{aligned} 2\rho _B\simeq D_B\int \limits _0^{\xi _{BA}}\sqrt{\varepsilon } \left[ \frac{\varepsilon +\tilde{\mu }_B}{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _{B1}^2}} -1\right] d\varepsilon \nonumber \\ \end{aligned}$$

(B.9)

At $\rho _{B2}<<\rho _{B1}$, the result of eq. (B.3) allows us to determine the renormalized chemical potential as $\tilde{\mu }_B=-\mu _B+2\rho _B(V_{BR}-V_{BA})$. While eq. (B.4) determines the coherence parameter $\Delta _B=\Delta _{B1}$. From eq. (B.6), it follows that the model interboson interaction potential defined by eq. (122) reduces to the following simple BCS-like potential:

$$\begin{aligned} V_B(\vec {k}-\vec {k'})=\left\{ \begin{array}{ll} -\tilde{V}_{B},\\ \ \ 0, \end{array} \ \ \begin{array}{ll} \quad |\xi (k)|, |\xi (k')|\leqslant \xi _{BA},\\ \quad \text {otherwise}\\ \end{array} \right. \nonumber \\ \end{aligned}$$

(B.10)

For a 3D Bose system, we obtain from eqs (B.4) and (B.9),

$$\begin{aligned} \frac{1}{D_B\tilde{V}_B}=\sqrt{\xi _{BA}+2\tilde{\mu }_B}-\sqrt{2\tilde{\mu }_B} \end{aligned}$$

(B.11)

and

$$\begin{aligned} \frac{3\rho _B}{D_B}= & {} \lim _{\xi _{BA}\rightarrow \infty } \sqrt{\xi _{BA}+2\tilde{\mu }_B}(\xi _{BA}-\tilde{\mu }_B)\nonumber \\{} & {} +\tilde{\mu }_B\sqrt{2\tilde{\mu }_B}-\xi _{BA}^{3/2} \simeq \tilde{\mu }_B\sqrt{2\tilde{\mu }_B}. \end{aligned}$$

(B.12)

Equation (B.11) reduces to the relation (123) and then the substitution of eq. (123) into eq. (B.12) gives the relation (124). Further, $2\rho _B/D_B=2.612\sqrt{\pi }(k_BT_\textrm{BEC})^{3/2}$ [228] and eq. (125) follows from eq. (B.12). For $E_B(0)=0$, eqs (119)–(121) can now be written as

$$\begin{aligned}{} & {} \Delta _B(\vec {k})=-V_B(\vec {k})\rho _{B0}\frac{\Delta _B(0)}{|\Delta _B(0)|} -\frac{1}{\Omega }\sum _{k'\ne 0}V_B (\vec {k}-\vec {k}')\nonumber \\{} & {} \qquad \qquad \times \frac{\Delta _B(\vec {k}')}{2E_B(\vec {k}')}(1+2n_B(\vec {k}')), \end{aligned}$$

(B.13)

$$\begin{aligned}{} & {} \rho _B=\rho _{B0}+\frac{1}{\Omega }\sum _{k\ne 0}n_B(\vec {k}), \end{aligned}$$

(B.14)

$$\begin{aligned}{} & {} \chi _B(\vec {k})=V_B(\vec {k})\rho _{B0}+\frac{1}{\Omega }\sum _{k'\ne 0}V_B(\vec {k}-\vec {k}')n_B(\vec {k}').\nonumber \\ \end{aligned}$$

(B.15)

Replacing the summation in eqs (B.14) and (B.15) by an integration and taking into account the approximation (B.10), we obtain

$$\begin{aligned}{} & {} 2(\rho _B-\rho _{B0})=D_B\int \limits _0^{\xi _{BA}}\sqrt{\varepsilon }\left[ \frac{\varepsilon +\tilde{\mu }_B}{\sqrt{\varepsilon ^2+2\tilde{\mu }_B}}-1\right] d\varepsilon , \nonumber \\ \end{aligned}$$

(B.16)

$$\begin{aligned}{} & {} \tilde{V}_B\rho _{B0}=\tilde{\mu }_B\left[ 1-\tilde{V}_BD_B\int \limits _0^{\xi _{BA}}\frac{\sqrt{\varepsilon }d\varepsilon }{2\sqrt{\varepsilon ^2+2\tilde{\mu }_B\varepsilon }}\right] .\nonumber \\ \end{aligned}$$

(B.17)

Equations (127) and (128) follow accordingly from eqs (B.16) and (B.17). In the case of 2D Bose systems, $D_B=m_B/2\pi \hbar ^2$ and the multiplier $\sqrt{\varepsilon }$ under the integrals in eqs (B.16) and (B.17) will be absent.

At $\gamma _B<\gamma ^*_B$, evaluating the integrals in eq. (131), we have

$$\begin{aligned}{} & {} W_0=D_B\Omega \Big \{\frac{2}{5}[(\xi _A+2\Delta _B)^{5/2})-\xi ^{5/2}_A]\nonumber \\{} & {} \quad +\frac{4}{15}(2\Delta _B)^{5/2} -\frac{2\Delta _B}{3}\xi ^{3/2}_A-\frac{4\Delta _B}{3}(\xi _A+2\Delta _B)^{3/2}\nonumber \\{} & {} \quad -\Delta ^2_B[(\xi _A+2\Delta _B)^{1/2}-(2\Delta _B)^{1/2}]\Big \}. \end{aligned}$$

(B.18)

In order to simplify eq. (B.18) further, we can expand the brackets $(\xi _{BA}+2\Delta _B)^{5/2}$, $(\xi _{BA}+2\Delta _B)^{3/2}$ and $(\xi _{BA}+2\Delta _B)^{1/2}$ in this equation in powers of $2\Delta _B/\xi _{BA}$ as

$$\begin{aligned}{} & {} (\xi _{BA}+2\Delta _B)^{5/2}=\xi ^{5/2}_{BA}\left( 1+\frac{2\Delta _B}{\xi _{BA}}\right) ^{5/2}\nonumber \\{} & {} \quad \simeq \xi ^{5/2}_{BA}\left\{ 1+\frac{5\Delta _B}{\xi _{BA}}+\frac{15}{2}\left( \frac{\Delta _B}{\xi _{BA}}\right) ^2\right. \nonumber \\{} & {} \quad \left. +\frac{5}{2}\left( \frac{\Delta _B}{\xi _{BA}}\right) ^3\cdots \right\} , \end{aligned}$$

(B.19)

$$\begin{aligned}{} & {} (\xi _{BA}+2\Delta _B)^{3/2}=\xi ^{3/2}_{BA}\left( 1+\frac{2\Delta _B}{\xi _{BA}}\right) ^{3/2}\nonumber \\{} & {} \quad \simeq \xi ^{3/2}_{BA}\left\{ 1+\frac{3\Delta _B}{\xi _{BA}}+\frac{3}{2}\left( \frac{\Delta _B}{\xi _{BA}}\right) ^2-\cdots \right\} , \end{aligned}$$

(B.20)

$$\begin{aligned}{} & {} (\xi _{BA}+2\Delta _B)^{1/2}=\xi ^{1/2}_{BA}\left( 1+\frac{2\Delta _B}{\xi _{BA}}\right) ^{1/2}\nonumber \\{} & {} \quad \simeq \xi ^{1/2}_{BA}\left\{ 1+\frac{\Delta _B}{\xi _{BA}}-\cdots \right\} . \end{aligned}$$

(B.21)

Substituting eqs (B.19), (B.20) and (B.21) into eq. (B.18), we find

$$\begin{aligned} W_0= & {} D_B\Omega \Bigg \{\frac{2}{5}\Bigg [\xi ^{5/2}_{BA}+5\Delta _B\xi ^{3/2}_{BA} +\frac{15}{2}\Delta ^2_B\xi ^{1/2}_{BA}\nonumber \\{} & {} +\frac{5}{2}\Delta ^3_B/\xi ^{1/2}_{BA}-\xi ^{5/2}_{BA}\Bigg ] -\frac{4}{15}(2\Delta _B)^{5/2}-\frac{2\Delta _B}{3}\xi ^{3/2}_{BA}\nonumber \\{} & {} -\frac{4\Delta _B}{3}\Bigg [\xi ^{3/2}_{BA}+3\Delta _B\xi ^{1/2}_{BA}+\frac{3}{2} \Delta ^2_B/\xi ^{1/2}_{BA}\Bigg ]\nonumber \\{} & {} -\Delta ^2_B\xi ^{1/2}_{BA}-\Delta ^3_B/\xi ^{1/2}_{BA}+\Delta ^2_B(2\Delta _B)^{1/2}\Bigg \}\nonumber \\= & {} D_B\Omega \Bigg \{2\Delta _B\xi ^{3/2}_{BA}-\frac{2\Delta _B}{3}\xi ^{3/2}_{BA} -\frac{4\Delta _B}{3}\xi ^{3/2}_{BA}\nonumber \\ {}{} & {} +3\Delta ^2_B\xi ^{1/2}_{BA}- 4\Delta ^2_B\xi ^{1/2}_{BA}+\Delta ^3_B/\xi ^{1/2}_{BA}-2\Delta ^3_B/\xi ^{1/2}_{BA}\nonumber \\{} & {} -\Delta ^2_B\xi ^{1/2}_{BA}-\Delta ^3_B/\xi ^{1/2}_{BA}- \frac{4}{15}(2\Delta _B)^{5/2}\nonumber \\{} & {} +\Delta ^2_B(2\Delta _B)^{1/2}\Bigg \}\nonumber \\= & {} D_B\Omega \Bigg \{3\Delta ^2_B\xi ^{1/2}_{BA}-5\Delta ^2_B\xi ^{1/2}_{BA} \nonumber \\{} & {} -2\Delta ^3_B/\xi ^{1/2}_{BA}-\frac{\sqrt{2}}{15}\Delta ^{5/2}_B\Bigg \}=D_B\Omega \Delta ^2_B\xi ^{1/2}_{BA}\nonumber \\{} & {} \times \Bigg \{-2-2\frac{\Delta _B}{\xi _{BA}}-\frac{\sqrt{2}}{15}\Big (\frac{\Delta _B}{\xi _{BA}}\Big )^{1/2}\Bigg \}. \end{aligned}$$

(B.22)

At $\Delta _B/\xi _{BA}<<1$, we have

$$\begin{aligned} W_0\simeq -2D_B\Delta ^2_B\xi ^{1/2}_{BA}\Omega . \end{aligned}$$

(B.23)

Appendix C: Possible analytical solutions of the integral equations in the theory of a 3D Bose superfluid for the temperature range $0<T\le T_c$

Using the model potential (122), we can write eqs (116) and (117) as

$$\begin{aligned}{} & {} \frac{2\rho _B}{D_B}=\int \limits _0^{\infty }\sqrt{\varepsilon }\left[ \frac{(\varepsilon +\tilde{\mu }_B)}{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2}}-1\right] d\varepsilon \nonumber \\{} & {} +2\int \limits _0^{\infty }\frac{\sqrt{\varepsilon }(\varepsilon +\tilde{\mu }_B)d\varepsilon }{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2}\left[ \exp (\frac{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2}}{k_BT})-1\right] }, \nonumber \\ \end{aligned}$$

(C.1)

$$\begin{aligned}{} & {} \frac{2}{D_B\tilde{V}_B}=\int \limits _0^{\xi _{BA}}\frac{\sqrt{\varepsilon }d\varepsilon }{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2}}\nonumber \\{} & {} +2\int \limits _0^{\xi _{BA}}\frac{\sqrt{\varepsilon }d\varepsilon }{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2} \left[ \exp (\frac{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2}}{k_BT})-1\right] }. \nonumber \\ \end{aligned}$$

(C.2)

According to eq. (B.12) the first integral in eq. (C.1) at $\tilde{\mu }_B=\Delta _B$ is equal to $2\tilde{\mu }_B\sqrt{2\tilde{\mu }_B}/3$. From eq. (B.11), it follows that the first integral in eq. (C.2) at $\tilde{\mu }_B=\Delta _B$ is equal to $2[\sqrt{\xi _{BA}+2\tilde{\mu }_B}-\sqrt{2\tilde{\mu }_B}]$. The main contributions to the latter integrals in eqs (C.1) and (C.2) come from small values of $\varepsilon$, so that for $T<<T_c$ and $\tilde{\mu }_B=\Delta _B$ the latter integrals in eqs (C.1) and (C.2) can be evaluated approximately as

$$\begin{aligned}{} & {} 2\int \limits _0^{\infty } \frac{\sqrt{\varepsilon }(\varepsilon +\tilde{\mu }_B)d\varepsilon }{\sqrt{\varepsilon ^2+2\varepsilon \tilde{\mu }_B} \Big [\exp \Big [\sqrt{\frac{\varepsilon ^2+2\varepsilon \tilde{\mu }_B}{k_BT}}\Big ]-1\Big ]}\nonumber \\{} & {} \quad \approx \sqrt{2\tilde{\mu }_B}\int ^{\infty }_{0}\frac{d\varepsilon }{\exp \Big [\frac{\sqrt{2\varepsilon \tilde{\mu }_B}}{k_BT}\Big ]-1}\nonumber \\{} & {} \quad =\frac{(\pi k_BT)^2}{3\sqrt{2\tilde{\mu }_B}}, \end{aligned}$$

(C.3)

$$\begin{aligned}{} & {} 2\int \limits _0^{\infty }\frac{\sqrt{\varepsilon }d\varepsilon }{\sqrt{\varepsilon ^2+2\varepsilon \tilde{\mu }_B}\Big [\exp \Big [\frac{\sqrt{\varepsilon ^2+2\varepsilon \tilde{\mu }_B}}{k_BT}\Big ]-1\Big ]}\nonumber \\{} & {} \quad \approx \frac{2}{\sqrt{2\tilde{\mu }_B}}\int ^{\infty }_{0}\frac{d\varepsilon }{\exp \Big [\frac{\sqrt{2\varepsilon \tilde{\mu }_B}}{k_BT}\Big ]-1}\nonumber \\{} & {} \quad =\frac{(\pi k_BT)^2}{3\sqrt{2}\tilde{\mu }_B^{3/2}}.\nonumber \\ \end{aligned}$$

(C.4)

At $\tilde{\mu }_B=\Delta _B$, according to eqs (B.13) and (B.14), the term $2\rho _{B0}/D_B$ should be present in eq. (C.1), while the term $2\rho _{B0}/\tilde{\mu }_BD_B$ would be present in eq. (C.2). Therefore, at $T<T_c^*<<T_c$, eqs (C.1) and (C.2) can be written as

$$\begin{aligned}{} & {} \frac{2\rho _B}{D_B}\simeq \frac{2\rho _{B0}(T)}{D_B}+\frac{(2\tilde{\mu }_B)^{3/2}}{3}+\frac{(\pi k_BT)^2}{3\sqrt{2\tilde{\mu }_B}}, \end{aligned}$$

(C.5)

$$\begin{aligned}{} & {} \frac{2\tilde{\mu }_B}{D_B\tilde{V}_B}\simeq \frac{2\rho _{B0}(T)}{D_B}+2\tilde{\mu }_B\nonumber \\{} & {} \qquad \qquad \quad \times \Big [\sqrt{\xi _{BA}+2\tilde{\mu }_B}-\sqrt{2\tilde{\mu }_B}\Big ]+\frac{(\pi k_BT)^2}{3\sqrt{2\tilde{\mu }_B}},\nonumber \\ \end{aligned}$$

(C.6)

from which we obtain eqs (134) and (135).

The first integrals in eqs (C.1) and (C.2) at $\Delta _g\ne 0$ (or $\rho _{B0}=0$) and $\tilde{\mu }_B>>\Delta _B$ can be evaluated approximately using the Taylor expansion

$$\begin{aligned} \frac{1}{\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B}}\simeq \frac{1}{\varepsilon +\tilde{\mu }_B}\left[ 1+\frac{\Delta ^2_B}{2(\varepsilon +\tilde{\mu }_B)^2}\right] . \nonumber \\ \end{aligned}$$

(C.7)

Performing the integration and using also the expansion

$$\begin{aligned} \arctan \sqrt{\frac{\xi _{BA}}{\tilde{\mu }_B}}\simeq \frac{\pi }{2}-\sqrt{\frac{\tilde{\mu }_B}{{\xi _{BA}}}} +\frac{1}{3}\left( \frac{\tilde{\mu }_B}{\xi _{BA}}\right) ^{3/2}-\cdots , \nonumber \\ \end{aligned}$$

(C.8)

we obtain the following results for the first integrals in equations (C.1) and (C.2):

$$\begin{aligned} \frac{\pi \Delta _B^2}{4\sqrt{\tilde{\mu }_B}}\ \ \ \,\text {and}\, \ \ \ \ 2\sqrt{\xi _{BA}}\left[ 1+\frac{3\pi }{32}\left( \frac{\Delta _B}{\tilde{\mu }_B}\right) ^2 \sqrt{\frac{\tilde{\mu }_B}{\xi _{AB}}}\right] . \nonumber \\ \end{aligned}$$

(C.9)

The latter integrals in eqs (C.1) and (C.2) can be evaluated near $T_c$ making the substitution $t=\sqrt{(\varepsilon /\tilde{\mu }_B)^2+2\varepsilon /\tilde{\mu }_B}$, $a^2_1t^2+a^2_2=[(\varepsilon +\tilde{\mu }_B)^2-\Delta ^2_B]/(k_BT)^2$ [228], where $a_1=\tilde{\mu }_B/k_BT$, $a_2=\sqrt{\tilde{\mu }_B^2-\Delta ^2_B}/k_BT$. The second integral in eq. (C.1) can be written in the form

$$\begin{aligned} I_2= & {} \tilde{\mu }_B^{3/2}\nonumber \\{} & {} \times \int ^\infty _0\frac{\sqrt{\sqrt{t^2+1}-1}t dt}{\sqrt{t^2+(\frac{a_2}{a_1})^2}\Big [\exp (a_1\sqrt{t^2+(\frac{a_2}{a_1})^2})-1\Big ]}, \nonumber \\ \end{aligned}$$

(C.10)

where $a_1<<1$, $a_2{<<}1$, and $\Delta _B{<<}\tilde{\mu }_B$ near $T_c$.

This integral can be evaluated using the method presented in [212] and the final result has the following form [228]

$$\begin{aligned} I_2\simeq & {} \frac{\sqrt{\pi }}{2}(k_BT)^{3/2} \Bigg [2.612-\sqrt{2\pi }\sqrt{\frac{\tilde{\mu }_B}{k_BT}+\frac{\Delta _g}{k_BT}}\nonumber \\{} & {} +1.46\frac{\tilde{\mu }_B}{k_BT}\cdots \Bigg ]\nonumber \\{} & {} =\frac{\sqrt{\pi }}{2}(k_BT)^{3/2}\nonumber \\{} & {} \Bigg [2.612-\sqrt{2\pi }\sqrt{\frac{\tilde{\mu }_B}{k_BT}+\frac{\tilde{\mu }_B}{k_BT}\Big (1-\frac{\Delta ^2_B}{2\tilde{\mu }^2_B}\Big )}\nonumber \\{} & {} +1.46\frac{\tilde{\mu }_B}{k_BT}\Bigg ]. \end{aligned}$$

(C.11)

The second integral $I'_2$ in eq. (C.2) is also evaluated in the same manner as follows:

$$\begin{aligned}{} & {} I_2^{'}\simeq \frac{\pi k_BT}{\sqrt{2\tilde{\mu }_B}}\Bigg [\sqrt{\frac{\tilde{\mu }_B}{\tilde{\mu }_B+\Delta _g}}-1.46\sqrt{\frac{2\tilde{\mu }_B}{\pi k_BT}}+\cdots \Bigg ]\nonumber \\{} & {} \quad =\frac{\pi k_BT}{2\sqrt{\tilde{\mu }_B}}\Bigg [\sqrt{\frac{1}{1-\Delta ^2_B/4\tilde{\mu }_B^2}} -1.46\sqrt{2}\sqrt{\frac{2\tilde{\mu }_B}{\pi k_BT}}\Bigg ].\nonumber \\ \end{aligned}$$

(C.12)

By expanding the expressions $\sqrt{1-\Delta ^2_B/4\tilde{\mu }_B^2}$ and $\sqrt{1/(1-\Delta ^2_B/4\tilde{\mu }_B^2)}$ in powers of $\Delta _B/4\tilde{\mu }_B$ and replacing $1.46\sqrt{2}$ by 2, we obtain from eqs (C.1), (C.2), (C.9), (C.11) and (C.12) (with an accuracy to $\sim \tilde{\mu }_B(T)$)

$$\begin{aligned}{} & {} \frac{1}{D_B\tilde{V}_B}\simeq \sqrt{\xi _{BA}}\left[ 1+\frac{3\pi }{32} \Big (\frac{\Delta _B}{\tilde{\mu }_B}\Big )^2\sqrt{\frac{\tilde{\mu }_B}{\xi _{BA}}}\right] \nonumber \\{} & {} \qquad \qquad +\frac{\pi k_BT}{2\sqrt{\tilde{\mu }_B}}\left[ \Big (1+\frac{\Delta ^2_B}{8\tilde{\mu }_B^2}\Big )-2\sqrt{\frac{2\tilde{\mu }_B}{\pi k_BT}}\right] , \end{aligned}$$

(C.13)

$$\begin{aligned}{} & {} \frac{2\rho _B}{D_B}=2.612\sqrt{\pi }(k_BT_{BEC})^{3/2}\nonumber \\{} & {} \qquad \quad \simeq \frac{\pi \Delta ^2_B}{4\sqrt{\tilde{\mu }_B}} +\sqrt{\pi }(k_BT)^{3/2}\nonumber \\{} & {} \qquad \qquad \times \left[ 2.612-2\sqrt{\frac{\pi \tilde{\mu }_B}{k_BT}}\left( 1-\frac{\Delta ^2_B}{8\tilde{\mu }^2_B}\right) \right] . \end{aligned}$$

(C.14)

For $k_BT/\xi _{BA}\sim 1/2\pi$, the relation (138) follows from (C.13). After some transformations in eq. (C.14), we have

$$\begin{aligned}{} & {} \sqrt{\pi }(k_BT_\textrm{BEC})^{3/2}\nonumber \\{} & {} =\frac{\sqrt{\pi }(k_BT)^{3/2}}{2.612}\Bigg [\frac{\sqrt{\pi }}{4}\Bigg (\frac{\Delta _B}{\tilde{\mu }_B}\Bigg )^2\Bigg (\frac{\tilde{\mu }_B}{k_BT}\Bigg )^{3/2}\nonumber \\{} & {} \qquad +2.612-2\sqrt{\frac{\pi \tilde{\mu }_B}{k_BT}}\Bigg (1-\frac{\Delta ^2_B}{8\tilde{\mu }^2_B}\Bigg )\Bigg ]. \end{aligned}$$

(C.15)

Equation (137) follows from eq. (C.15). In order to determine the temperature dependencies of $\tilde{\mu }_B$ and $\Delta _B$ near $T_c$, eqs (137) and (138) can be written as

$$\begin{aligned}{} & {} \sqrt{\pi }(k_BT_c)^{3/2}\left[ 2.612-2\sqrt{\frac{\pi \tilde{\mu }_B(T_c)}{k_BT_c}}\right] \nonumber \\{} & {} \quad \simeq \sqrt{\pi }(k_BT)^{3/2}\nonumber \\{} & {} \quad \times \left[ 2.612 -2\sqrt{\frac{\pi \tilde{\mu }_B(T)}{k_BT}}\Big (1-\frac{\Delta ^2_B(T)}{8\tilde{\mu }_B^2(T)}\Big )\right] , \end{aligned}$$

(C.16)

$$\begin{aligned}{} & {} \frac{\pi k_BT_c}{2}\sqrt{\frac{\xi _{BA}}{\tilde{\mu }_B(T_c)}} \nonumber \\{} & {} \quad \simeq \frac{\pi k_BT}{2}\sqrt{\frac{\xi _{BA}}{\tilde{\mu }_B(T)}}\left( 1+\frac{\Delta ^2_B(T)}{8\tilde{\mu }_B^2(T)}\right) . \end{aligned}$$

(C.17)

The quantities $\tilde{\mu }_B(T)$ and $\Delta _B(T)$ near $T_c$ can be determined by eliminating $\Delta ^2_B/8\tilde{\mu }^2_B$ from these equations. Making after that some algebraic transformations, we obtain

$$\begin{aligned}{} & {} \frac{-1.306\sqrt{k_B}}{\sqrt{\pi \tilde{\mu }_B(T)}}\left( \frac{T_c^{3/2}-T^{3/2}}{T}\right) +\sqrt{\frac{\tilde{\mu }_B(T_c)}{\tilde{\mu }_B(T)}}\frac{T_c}{T} \\{} & {} \quad =1-\frac{\Delta ^2_B(T)}{8\tilde{\mu }^2_B(T)}, \sqrt{\frac{\tilde{\mu }_B(T)}{\tilde{\mu }_B(T_c)}}\frac{T_c}{T}=1+\frac{\Delta ^2_B(T)}{8\tilde{\mu }^2_B(T)}. \end{aligned}$$

Combining now these equations, we have

$$\begin{aligned}{} & {} \sqrt{\frac{\tilde{\mu }_B(T)}{\tilde{\mu }_B(T_c)}}+\sqrt{\frac{\tilde{\mu }_B(T_c)}{\tilde{\mu }_B(T)}}\nonumber \\{} & {} \quad \times \Bigg [1-\frac{1.306\sqrt{k_B}}{\sqrt{\pi \tilde{\mu }_B(T_c)}}\left( \frac{T_c^{3/2}-T^{3/2}}{T_c}\right) \Bigg ] \nonumber \\{} & {} \quad \quad -2\frac{T}{T_c}=0. \end{aligned}$$

(C.18)

Now this equation has the solution

$$\begin{aligned}{} & {} \sqrt{\frac{\tilde{\mu }_B(T)}{\tilde{\mu }_B(T_c)}}=\frac{T}{T_c} \\{} & {} \quad +\sqrt{\left( \frac{T}{T_c}\right) ^2-1+\frac{1.306\sqrt{k_B}}{\sqrt{\pi \tilde{\mu }_B(T_c)}}\left( \frac{T_c^{3/2}-T^{3/2}}{T_c}\right) }. \end{aligned}$$

Next, taking into account that near $T_c$,

$$\begin{aligned}{} & {} \frac{T_c^{3/2}-T^{3/2}}{T_c} \simeq \frac{T^3_c-T^3}{2T_c^{5/2}}\\{} & {} \quad =\frac{(T_c-T)(T^2_c+T_cT+T^2)}{2T^{5/2}_c} =\frac{3T^2_c(T_c-T)}{2T_c^{3/2}}, \end{aligned}$$

we obtain

$$\begin{aligned} \sqrt{\frac{\tilde{\mu }_B(T)}{\tilde{\mu }_B(T_c)}}=\frac{T}{T_c}+\sqrt{\left[ \frac{3.918}{2\sqrt{\pi }}\sqrt{\frac{k_BT_c}{\tilde{\mu }_B}}-2\right] \frac{(T_c-T)}{T_c}}. \end{aligned}$$

After combining this equation with eq. (138) follows approximately eq. (141) at $k_BT_c/\tilde{\mu }_B(T_c)>>1$. From eqs (C.17) and (141), we obtain eq. (142). Assuming that $\tilde{\mu _B}(T)/k_BT^*_c<<1$, we examine the behaviors of $\tilde{\mu _B}(T)$ (or $\Delta _B(T)$) and $n_{B0}(T)$ near the characteristic temperature $T^*_c<T_c$. Replacing the summation in eqs (B.13)–(B.15) by an integration and using eqs (C.11) and (C.12) at $\Delta _g=0$, we obtain following equations for determining $\tilde{\mu _B}(T)$ and $\rho _{B0}(T)$ near $T^*_c$:

$$\begin{aligned}{} & {} \frac{2(\rho _B-\rho _{B0}) }{D_B}\simeq \frac{2\tilde{\mu }_B^{3/2}}{3}+\sqrt{\pi }(k_BT)^{3/2}\nonumber \\{} & {} \qquad \times \left[ 2.612-\sqrt{\frac{2\pi \tilde{\mu }_B}{k_BT}}+1.46\frac{\tilde{\mu }_B}{k_BT}\right] , \end{aligned}$$

(C.19)

$$\begin{aligned}{} & {} \frac{1}{\gamma _B} \simeq \frac{\rho _{B0}}{D_B\tilde{\mu }_B(\xi _{BA})^{1/2}}+\sqrt{1+\frac{2\tilde{\mu }_B}{\xi _{BA}}}-\sqrt{\frac{2\tilde{\mu }_B}{\xi _{BA}}} \nonumber \\{} & {} \qquad \quad +\frac{\pi k_BT}{\sqrt{2\tilde{\mu }_B\xi _{BA}}}\left[ 1-1.46\sqrt{\frac{2\tilde{\mu }_B}{\pi k_BT}}\right] . \end{aligned}$$

(C.20)

At $T=T_c^*$ and $\rho _{B0}=0$ (which corresponds to a complete depletion of the single particle condensate), eqs (C.19) and (C.20) at $\tilde{\mu }_B/k_BT_c^*<<1$ can be written as

$$\begin{aligned} \frac{2\rho _B}{D_B}\simeq & {} \sqrt{\pi }(k_BT^*_c)^{3/2}\nonumber \\{} & {} \times \left[ 2.612-\sqrt{\frac{2\pi \tilde{\mu }_B}{k_BT^*_c}}+1.46\frac{\tilde{\mu }_B}{k_BT^*_c}\right] \end{aligned}$$

(C.21)

and

$$\begin{aligned} \frac{1}{\gamma _B}\simeq \frac{\pi k_BT^*_c}{\sqrt{2\tilde{\mu }_B\xi _{BA}}}. \end{aligned}$$

(C.22)

At $T \lesssim T_c^*$, eqs (C.19) and (C.20) become

$$\begin{aligned}{} & {} 2.612\sqrt{\pi }(k_BT^*_c)^{3/2}-\pi k_BT^*_c\sqrt{2\tilde{\mu }_B(T^*_c)}\nonumber \\{} & {} \quad =\frac{2\rho _{B0}(T)}{D_B} +2.612\sqrt{\pi }(k_BT)^{3/2}\nonumber \\{} & {} \qquad - \pi k_BT\sqrt{2\tilde{\mu }_B(T)} \end{aligned}$$

(C.23)

and

$$\begin{aligned}{} & {} \frac{\pi k_BT^*_c}{\sqrt{2\tilde{\mu }_B(T^*_c)\xi _{BA}}}\nonumber \\{} & {} \quad =\frac{\rho _{B0}(T)}{D_B\tilde{\mu }_B(T)\sqrt{\xi _{BA}}} +\frac{\pi k_BT}{\sqrt{2\tilde{\mu }_B(T)\xi _{BA}}}. \end{aligned}$$

(C.24)

Eliminating $\rho _{B0}(T)$ from these equations (i.e., after substituting $\rho _{B0}(T)$ from eq. (C.24) into eq. (C.23)) and making some algebraic transformations, we obtain the equation for $\tilde{\mu }_B(T)$, which is similar to eq. (C.18). The solution of this equation near $T^*_c$ leads to a simple expression (143). Then, substituting $\tilde{\mu }_B(T)/\tilde{\mu }_B(T^*_c)$ from eq. (143) into eq. (C.23), we obtain the relation (144).

Appendix D: Calculation of the superfluid parameters of a 2D Bose-liquid for the temperature range $0< T \le T_c$

We now calculate the superfluid parameters of a 2D Bose-liquid. For a 2D interacting Bose system, eq. (116) after replacing the sum by the integral and making the substitution $y=\sqrt{(\varepsilon +\tilde{\mu }_B)^2-\Delta _B^2}/2k_BT$ takes the following form:

$$\begin{aligned} \frac{2}{\gamma _B}=\int \limits _{y_1}^{y_2}\frac{\coth y~dy}{\sqrt{y^2+(\Delta _B^*)^2}}, \end{aligned}$$

(D.1)

where $y_1=\Delta _{g}/2k_BT$, $y_2=\sqrt{(\xi _{BA}+\tilde{\mu }_B)^2-\Delta _B^2}/2k_BT$, $\Delta _B^*=\Delta _B/2k_BT$. In order to evaluate the integral in eq. (D.1) in the intervals $y_1<y<1$ and $1<y<y_2$, one can take accordingly $\coth y\approx 1/y$ and $\approx 1$. Then, after performing the integration in eq. (D.1), we obtain

$$\begin{aligned} \frac{2}{\gamma _B}\simeq & {} \ln \Bigg \{\Bigg [\frac{y_1(\Delta _B^*+\sqrt{1+(\Delta ^*_B)^2})}{\Delta _B^*+\sqrt{y^2_1+(\Delta _B^*)^2}}\Bigg ]^{-1/\Delta _B^*}\nonumber \\{} & {} \times \Bigg [\frac{y_2+\sqrt{y^2_2+(\Delta _B^*)^2}}{1+\sqrt{1+(\Delta _B^*)^2}}\Bigg ]\Bigg \}. \end{aligned}$$

(D.2)

It is clear that at $T{<<}T_c$, $\Delta ^*_B>>1$, $y_2>>\Delta _B^*$ and $\Delta _B^*>>y_1$. Taking into account that $\ln y_1$ is negligible, we can write eq. (D.2) in the form

$$\begin{aligned} \frac{2}{\gamma _B}\simeq \left( \frac{2y_2}{1+\Delta _B^*}\right) . \end{aligned}$$

(D.3)

This expression after some algebraic transformations reduces to eq. (156). At temperatures close to $T_c$, $\Delta _B^*{<<}1$. We then obtain from eq. (D.2) (with an accuracy to $\sim (\Delta _B^*)^2$)

$$\begin{aligned} \frac{2}{\gamma _B}\simeq -\frac{1}{\Delta _B^*}\ln \left| \frac{y_1(1+\Delta _B^*)}{y_1+\Delta _B^*}\right| +\ln y_2. \end{aligned}$$

(D.4)

Assuming that $y_2^{\Delta _B^*}\simeq 1$ and $\Delta _B^*/\gamma _B<<1$, we obtain from eq. (D.4)

$$\begin{aligned} \frac{1+\Delta _B^*/y_1}{1+\Delta _B^*}\simeq \exp \left( \frac{2\Delta _B^*}{\gamma _B}\right) \simeq 1 +\frac{2\Delta _B^*}{\gamma _B}+\cdots , \end{aligned}$$

which reduces to

$$\begin{aligned} \Delta _B(T)=\gamma _Bk_B T\left[ \frac{2k_BT}{\Delta _g(T)}-\frac{\gamma _B+2}{\gamma _B}\right] . \end{aligned}$$

(D.5)

One can assume that $\Delta _g(T)$ varies near $T_c$ as $\sim \alpha _0(2k_BT)^q$, where $\alpha _0$ is determined at $T=T_c$ from the condition $\Delta _B(T_c)=0$), q is variable parameter. Thus, in a 2D Bose superfluid $\Delta _B(T)$ and $\tilde{\mu }_B(T)$ can be determined accordingly from eqs (157) and (158).

For 2D Bose systems, the multiplier $\sqrt{\varepsilon }$ under the integral in eq. (171) will be absent. In order to estimate such an integral at $\Delta _g<2k_BT$, we make the substitution $x=E_B(\varepsilon )/2k_BT$. Then, taking into account that the function $\sinh x$ is close to x for $x<1$ and this function close to $(1/2\exp (x))$ for $x>1$, we obtain the following expression for the specific heat of a 2D Bose superfluid:

$$\begin{aligned} C_v(T)\simeq & {} 4\Omega D_Bk_B^2T\Bigg \{\int \limits _{y_1}^1\frac{x dx}{\sqrt{x^2+(\Delta ^*_B)^2}}\nonumber \\{} & {} + 4\int \limits _1^\infty \frac{x^3\exp (-2x)dx}{\sqrt{x^2+(\Delta ^*_B)^2}}\Bigg \}. \end{aligned}$$

(D.6)

At low temperatures, $\Delta ^*_B>>1$, so that the expression $\sqrt{x^2+(\Delta ^*_B)^2}$ in the integrands can be replaced by $\Delta ^*_B$. Then, calculating the integrals in eq. (D.6) with this approximation, we obtain eq. (175).

Appendix E: Calculation of the characteristic temperature $T_0^*$ in a 2D Bose-liquid

In the case of a 2D Bose-liquid, the expressions for $\tilde{\varepsilon }_B(k)$ and $\rho _B$ presented in §7.1 and Appendix B after replacing the summation by an integration can be written as

$$\begin{aligned} \tilde{\varepsilon }_B(k)= & {} \varepsilon (k)-\mu _B+V_B(0)\rho _B\nonumber \\{} & {} +\int \limits _0^\infty dk'{k}'V_B(\vec {k}-\vec {k}')\frac{1}{\exp [\tilde{\varepsilon }_B(k')/k_BT]-1}\nonumber \\ \end{aligned}$$

(E.1)

and

$$\begin{aligned} \rho _B=\frac{1}{2\pi }\int \limits _0^\infty dk'{k}'\frac{1}{\exp [\tilde{\varepsilon }_B(k')/k_BT]-1}. \end{aligned}$$

(E.2)

Here

$$\begin{aligned} V_B(\vec {k}-\vec {k}')&=\frac{1}{{2(2\pi )}^2}\int _0^{2\pi }d\psi V_B[(k^2+(k')^2\\&\quad -2kk'\cos \psi )^{1/2}]. \end{aligned}$$

In eq. (168) the function $J_0(kRx)$ can be expanded in a Taylor series around $kR=0$ as

$$\begin{aligned} J_0(kRx)=1-\left( \frac{kRx}{2}\right) ^2+\cdots \ . \end{aligned}$$

(E.3)

From eq. (168), we then obtain

$$\begin{aligned} V_B(k)\simeq V_B(0)\left[ 1-\frac{k^2}{k^2_R}\right] , \quad 0\le k\le k_R. \end{aligned}$$

(E.4)

Here $V_B(0)=2\pi WR^2I_1$, $k_R=4I_1/I_3R^2$ and $I_n=\int _0^\infty dxx^n\Phi (x)$.

The subsequent analytical calculations are similar to the case of a 3D Bose gas [212]. Therefore, we present the following final results for $\varepsilon _B(k)$, $m_p^*$ and $\rho _B$:

$$\begin{aligned}{} & {} \tilde{\varepsilon }_B(k)=\tilde{\varepsilon }_B(0)+\frac{\hbar ^2k^2}{2m^*_B}, \end{aligned}$$

(E.5)

$$\begin{aligned}{} & {} \frac{1}{m_B^*}=\frac{1}{m_B}-\frac{V_B(0)}{\pi \hbar ^2k^2_R}\nonumber \\{} & {} \qquad \quad \times \int \limits _0^{k_A}\frac{dk'k'}{\exp [(\tilde{\varepsilon }(0)+\hbar ^2k^2/2m_B^*)/k_BT]-1}, \nonumber \\\end{aligned}$$

(E.6)

$$\begin{aligned}{} & {} \rho _B=\frac{1}{2\pi }\int \limits _0^{k_A}\frac{dk'k'}{\exp [(\tilde{\varepsilon }(0)+\hbar ^2k^2/2m_B^*)/k_BT]-1}.\nonumber \\ \end{aligned}$$

(E.7)

Using these equations, we obtain the same relation as (165). Thus, the characteristic temperature $T_0$ is now replaced by $T_0^*=2\pi \hbar ^2\rho _B/m_B^*$.