Condensate density of interacting bosons: A functional renormalization group approach

We calculate the temperature-dependent condensate density ${\rho }^0\left(T\right)$ of interacting bosons in three dimensions using the functional renormalization group (FRG). From the numerical solution of suitably truncated FRG flow equations for the irreducible vertices we obtain ${\rho }^0\left(T\right)$ for arbitrary temperatures. We carefully extrapolate our numerical results to the critical point and determine the order parameter exponent $\beta \approx 0.32$ in reasonable agreement with the expected value 0.345 associated with the $XY$-universality class. We also calculate the condensate density in two dimensions at zero temperature using a truncation of the FRG flow equations based on the derivative expansion including cubic and quartic terms in the expansion of the effective potential in powers of the density. As compared with the widely used quadratic approximation for the effective potential, the coupling constants associated with the cubic and quartic terms lead to small corrections of the condensate density.

Figure 1

(Color online) Typical FRG flow of the condensate density, the anomalous dimension and the dimensionless interaction ${\tilde{u}}_l$ for $D=3$ and $\tilde{\mu }={\tilde{u}}_0=1$. Solid line: critical temperature (${\tilde{T}}_c=45.1318$ for our choice of parameters); dashed line: $\tilde{T}=47>{\tilde{T}}_c$; dotted line: $\tilde{T}=43<{\tilde{T}}_c$.

Figure 2

(Color online) Temperature dependence of the order parameter ${\tilde{\varphi }}_{\ast }^0=\sqrt{{\tilde{\rho }}_{\ast }^0}$ in three dimensions for $\tilde{\mu }=1$ and three different values of the dimensionless bare interaction ${\tilde{u}}_0$ defined in Eq. (3c).

Figure 3

(Color online) Iterative procedure to extract the order parameter exponent $\beta$ from the FRG results. We fit our numerical FRG results in a series of intervals $I_z$ given by Eq. (40) to power law (39) and thus determine the critical exponent $\beta \left(z\right)$ for a given value of the zoom factor $z$. The $z$ dependence of $\beta \left(z\right)$ is then extrapolated for $z\to \infty$ as shown in Fig. 4.

Figure 4

(Color online) The dependence of $\beta \left(z\right)$ on the zoom factor $z$ can be described by the indicated fit function [see also Eq. (41)] with $\alpha =0.1945$ and $\gamma =8.545$. The data are based on the numerical solution of the FRG flow equations for ${\tilde{u}}_0=\tilde{\mu }=1$.

Figure 5

(Color online) RG flow of the dimensionless couplings ${\tilde{U}}_{\Lambda }^{\left(2\right)}$, ${\tilde{U}}_{\Lambda }^{\left(3\right)}$, and ${\tilde{U}}_{\Lambda }^{\left(4\right)}$ defined in Eq. (50) for ${\tilde{u}}_0=3$, $\tilde{\mu }=1$, $D=2$, and $T=0$.

Figure 6

(Color online) RG flows of the IR-rescaled, dimensionless couplings ${\tilde{U}}_{\Lambda }^{\left(2\right)}{\left(\Lambda /{\Lambda }_0\right)}^{D-2}$, ${\tilde{U}}_{\Lambda }^{\left(3\right)}{\left(\Lambda /{\Lambda }_0\right)}^{2D-2}$, and ${\tilde{U}}_{\Lambda }^{\left(4\right)}{\left(\Lambda /{\Lambda }_0\right)}^{3D-2}$ for ${\tilde{u}}_0=3$, $\tilde{\mu }=1$ in two dimensions.

Figure 7

(Color online) RG flow of the condensate density for ${\tilde{u}}_0=3$, $\tilde{\mu }=1$, $D=2$, and $T=0$ with and without the cubic and quartic couplings ${\tilde{U}}_{\Lambda }^{\left(3\right)}$ and ${\tilde{U}}_{\Lambda }^{\left(4\right)}$.