Functional renormalization-group approach to interacting bosons at zero temperature

We investigate the single-particle spectral density of interacting bosons within the nonperturbative functional renormalization group technique. The flow equations for a Bose gas are derived in a scheme which treats the two-particle density-density correlations exactly but neglects irreducible correlations among three and more particles. These flow equations are solved within a truncation which allows to extract the complete frequency and momentum structure of the normal and anomalous self-energies. Both the asymptotic small momentum regime, where the perturbation regime fails, as well as the perturbative regime at larger momenta are well described within a single unified approach. The self-energies do not exhibit any infrared divergences, satisfy the U(1) symmetry constraints, and are in accordance with the Nepomnyashchy relation, which states that the anomalous self-energy vanishes at zero momentum and zero frequency. From the self-energies we extract the single-particle spectral density of the two-dimensional Bose gas. The dispersion is found to be of the Bogoliubov form and shows the crossover from linear Goldstone modes to the quadratic behavior of quasifree bosons. The damping of the quasiparticles is found to be in accordance with the standard Beliaev damping. We furthermore recover the exact asymptotic limit of the propagators derived by Gavoret and Nozières and discuss the nature of the nonanalyticities of the self-energies in the very small momentum regime.

Figure 1

Diagrammatic representation of the exact FRG flow equation for the condensate density. The circles with two external legs on the left-hand side denote the normal and anomalous self-energy. The circles with three external legs on the right-hand side denote irreducible three-point vertices. Crossed circles with a dot denote the derivative of the order parameter with respect to the RG cutoff $\Lambda$, and crossed solid arrows represent normal or anomalous single-scale propagators as defined in Eq. (3.21). Arrows pointing out of a vertex represent the fields $\mathrm{\psi ̄}$, while incoming arrows represent fields $\psi$.

Figure 2

Diagrammatic representation of the exact FRG flow equation for the normal self-energy ${\Sigma }_{\Lambda }^N(K)$. The circle with four external legs denotes the irreducible four-point vertex, while arrows represent cutoff-dependent normal or anomalous propagators. The dot above the two-point vertex on the left-hand side represents a derivative with respect to $\Lambda$. The other symbols are defined as in Fig. 1.

Figure 3

Diagrammatic representation of the exact FRG flow equation for the anomalous self-energy ${\Sigma }_{\Lambda }^A(K)$. All symbols are defined as in Figs. 1 and 2.

Figure 4

Typical RG flows of the interaction parameter $u_{\Lambda }$ and the condensate density ${\rho }_{\Lambda }^0$ for $D=2$. The initial values are ${\mathrm{\mu ̃}}_0=2m\mu {\Lambda }_0^{-2}=0.4$ and ${\mathrm{ũ}}_0=2m{u}_{{\Lambda }_0}{\Lambda }_0^{D-2}=4$. The Ginzburg scale $k_G$ and the crossover scale $k_c=2\mathit{mc}$ are indicated by vertical lines.

Figure 5

Typical RG flows of the coupling parameters $Y_{\Lambda }$, $Z_{\Lambda }$, $V_{\Lambda }$, and the dynamical exponent $z_{\Lambda }$ and both anomalous dimensions ${\eta }_{\Lambda }^z$ and ${\eta }_{\Lambda }^y$ in $D=2$. The initial values are the same as in Fig. 4.

Figure 6

Real and imaginary parts of the normal Matsubara Green’s function as a function of $\omega$, calculated within the non-self-consistent FRG approach. The parameters are ${\mathrm{ũ}}_0=0.8$ and ${\mathrm{\mu ̃}}_0=0.08$, which yields $k_c/k_G\approx 45$. The Green’s function is calculated for $k=0.01k_c\approx 0.45k_G$. The dashed vertical line shows the position of the minimum of the imaginary part, which coincides with the predicted resonance at $\omega =ck$.

Figure 7

FRG results for the single-particle spectral density $A(k,\omega )$ as a function of the real frequency $\omega$ for different values of $k$ (left figure) and quasiparticle damping ${\gamma }_k$ (right figure). The shown results are for ${\mathrm{\mu ̃}}_0=0.15$ and ${\mathrm{ũ}}_0=15$. In this case $k_G/k_c\approx 0.2$ which is indicated by the labels in the figures. The inset in the left figure shows the quasiparticle dispersion $E_k$ which deviates at large $k$ from linearity but is well described by a Bogoliubov-type expression $E_k=\sqrt{{\epsilon }_k^2+c^2{k}^2}$ with renormalized velocity $c=(2mVZ){}^{-1/2}$ (black dots). The peaks of the spectral function correspond to (from left to right) $k/k_c=0.2,$ $0.3,0.4,0.5$, and $0.6$, where $k_c=2\mathit{mc}$. In the right plot, black dots are extracted from the spectral density, while the solid line fits them as ${\gamma }_k\approx 0.194k^3/2{\mathit{mk}}_c$.

Figure 8

Diagrams giving rise to the IR-divergent contribution (A2) to the anomalous self-energy. Black dots denote the bare vertices while the shaded triangles correspond to fully renormalized vertices.

Figure 9

Comparison of the full numerical solution of the flow equation for the interaction $u_{\Lambda }$ as obtained from Eqs. (3.35a), (3.35b), and (3.39a) through (3.39c) (solid lines) in $D=2$ with the approximate solution Eq. (C5) (dashed lines) for different values of the bare parameters. Solutions 1 are calculated with ${\mathrm{ũ}}_0=0.1$, ${\mathrm{\mu ̃}}_0=0.001$ which implies $k_c/k_G\approx 340$. Solutions 2 were calculated with ${\mathrm{ũ}}_0=0.8$ and ${\mathrm{\mu ̃}}_0=0.08$ with $k_c/k_G\approx 45$. Solutions 3 are obtained with ${\mathrm{ũ}}_0=15$ and ${\mathrm{\mu ̃}}_0=0.15$ with $k_c/k_G\approx 5$. The vertical dotted lines show positions of the Ginzburg scale for each solution.

Figure 10

${\mathrm{\alpha ̃}}_{\Lambda }={\Lambda }_0^D{\alpha }_{\Lambda }$ and ${\mathrm{\beta ̃}}_{\Lambda }=2m{\Lambda }_0^{D-2}{\beta }_{\Lambda }$, calculated in $D=2$ for the initial values of the dimensionless chemical potential ${\mathrm{\mu ̃}}_0=2m\mu {\Lambda }_0^{-2}=0.4$ and dimensionless interaction ${\mathrm{ũ}}_0=2m{u}_{{\Lambda }_0}{\Lambda }_0^{D-2}=4$.