Thermodynamics and renormalized quasiparticles in the vicinity of the dilute Bose gas quantum critical point in two dimensions

We use the functional renormalization group (FRG) to derive analytical expressions for thermodynamic observables (density, pressure, entropy, and compressibility) as well as for single-particle properties (wave-function renormalization and effective mass) of interacting bosons in two dimensions as a function of temperature $T$ and chemical potential $\mu$. We focus on the quantum disordered and the quantum critical regime close to the dilute Bose gas quantum critical point. Our approach is based on a truncated vertex expansion of the hierarchy of FRG flow equations and the decoupling of the two-body contact interaction in the particle-particle channel using a suitable Hubbard-Stratonovich transformation. Our analytic FRG results extend previous analytical renormalization-group calculations for thermodynamic observables at $\mu =0$ to finite values of $\mu$. To confirm the validity of our FRG approach, we have also performed quantum Monte Carlo simulations to obtain the magnetization, susceptibility, and correlation length of the two-dimensional spin-$1/2$ quantum $XY$ model with coupling $J$ in a regime where its quantum critical behavior is controlled by the dilute Bose gas quantum critical point. We find that our analytical results describe the Monte Carlo data for $\mu \le 0$ rather accurately up to relatively high temperatures $T\lesssim 0.1J$.

Figure 1

Schematic phase diagram for a two-dimensional Bose gas with repulsive contact interaction in the $T\text{−}\mu$ plane close to the QCP at $\mu =T=0$. In the quantum disordered regime, $\mu <-T$, interaction effects are weak and the particle density is exponentially small, while for sufficiently large positive $\mu$ the system is in the superfluid phase with finite density even at zero temperature. This phase is separated from the normal phase by a BKT transition at the temperature $T_{\text{BKT}}$ as given in the figure, where $\tilde{\mathrm{\Lambda }}$ is a nonuniversal energy scale [2]. In the quantum critical regime (red), a quasiparticle description with free bosons is still valid, but physical quantities exhibit logarithmic corrections.

Figure 2

Comparison of our results for the renormalized state functions at $\mu =0$ with the numerical results of Rançon and Dupuis [26] (dash-dotted red line) as well as with data from three different experiments (green symbols, taken from Fig. 4 in [26]). The remaining lines correspond to the full analytical expressions in Eqs. (3.27a) and (3.27c) on the one hand (dashed blue), and to the improved equations (B33), (3.27b), and (3.35) on the other hand (solid black), where $r$ is always taken from Eq. (3.13). From top to bottom, we show the reduced pressure $\tilde{p}$, the phase-space density $\tilde{n}$, and the entropy per particle $\tilde{s}$ as a function of the effective coupling constant $g$ as defined in Eq. (3.11).

Figure 3

Density over temperature as a function of $T$ for the $XY$ model at $\mu =0$ from QMC simulations (black dots). The blue line represents our analytical prediction in Eqs. (3.21) and (3.17) with $mJ=1$ and ${\mathrm{\Lambda }}_0=23.5$. The dashed line (red) is a fit of the QMC data to our leading-order result (3.30) for the density using a larger cutoff ${\mathrm{\Lambda }}_0=205$ and a modified mass $mJ=0.8$. On the top axis, we show the corresponding $g$ values for $mJ=1$ and ${\mathrm{\Lambda }}_0=23.5$.

Figure 4

Compressibility as a function of temperature for the $XY$ model at $\mu =0$ from QMC simulations (black dots). For comparison we show as a blue line our analytical prediction from Eqs. (3.21), (3.17), and Eq. (3.25d) with $mJ=1$ and ${\mathrm{\Lambda }}_0=31$. The top axis shows the corresponding $g$ values for these parameters.

Figure 5

QMC results for the $XY$ model at $\alpha =0.1$ and 0.4 of the density over temperature (green solid symbols) and the compressibility (red open symbols). The green solid lines correspond to our analytical prediction for $n/T$ from Eqs. (3.21) and (3.17) with $mJ=1$, ${\mathrm{\Lambda }}_0=23\pm 2$ (upper green line), and ${\mathrm{\Lambda }}_0=21\pm 2$ (lower green line), while the red solid lines correspond to $\kappa$ from Eq. (3.25d) with $mJ=1$, ${\mathrm{\Lambda }}_0=27.5\pm 2$ (upper red line), and ${\mathrm{\Lambda }}_0=20\pm 2$ (lower red line).

Figure 6

QMC results for the $XY$ model at $\alpha =1.0$ and 2.0 of the density over temperature (orange solid symbols) and the compressibility (blue open symbols). The orange solid lines correspond to our analytical prediction for $n/T$ from Eqs. (3.21) and (3.17) with $mJ=1$ and ${\mathrm{\Lambda }}_0=21\pm 2$, while the blue solid lines correspond to $\kappa$ from Eq. (3.25d) with $mJ=1$ and ${\mathrm{\Lambda }}_0=20\pm 2$.

Figure 7

Comparison of the numerical results for the rescaled finite-size correlation length $\sqrt{T}\xi =1.15\sqrt{T}{\xi }^{\prime }$ (symbols) at different values of $\alpha =-\mu /T$ to our analytical prediction $\sqrt{T}\xi =1/\sqrt{2mr}$ from Eqs. (3.24) and (3.17) (solid lines) with $mJ=1$ and using a cutoff of ${\mathrm{\Lambda }}_0=22$.

Figure 8

Region of validity (red area) where the extrapolation in Eq. (4.3) works and our analytical predictions from the FRG yield accurate results. For comparison, we also show the temperature $T_{\min }$ where $n/T$ exhibits a minimum for a given $\alpha$. The dashed red line is given by $g=0.5$ for $mJ=1$ and ${\mathrm{\Lambda }}_0=22$.

Figure 9

Double logarithmic plot of our analytical result for $1-Z$ as given in Eq. (5.3) vs the dimensionless coupling $g$ for different values of $\mu /T$. For $\mu /T=\pm {10}^{-5}$ we can see that the scaling is very close to the scaling at vanishing chemical potential for larger $g$ and only begins to differ when $gW(1/g)$ is of the order of $|\mu /T|$; the deviation around $g\approx 1$ is due to the fact that for positive chemical potential we have to use (3.17) for $r$ instead of (3.13). In the case of positive $\mu$, the wave-function renormalization then shrinks until it vanishes at the phase transition where $r=0$ [see Eq. (3.18)].

Figure 10

Comparison of our analytical result for the wave-function renormalization $Z$ of hard-core bosons in Eq. (5.3) (black solid line) with numerical computations using the self-consistent $T$-matrix approximation, which is expected to be good at high temperatures (blue dots). Note that for $T\to 0$ our analytic result for $Z$ becomes exact. The intermediate regime (sketched by the gray dashed line as a simple interpolation between the results) where $g={(\frac{1}{4}ln{\beta }_0)}^{-1}$ is below, but not much smaller than, unity is not covered by either method.

Figure 11

Double logarithmic plot of our analytical result for $1-Y$ as given in Eqs. (5.7) and (5.9) vs the dimensionless coupling $g$ for different values of $\mu /T$. We can see that for $\mu /T=-{10}^{-5}$ the scaling coincides with the $\mu =0$ curve until $gW(1/g)$ is of the order of $|\mu /T|$, where it starts to fall off more rapidly.

Figure 12

Graphical representation of the FRG flow equations (A8)–(A11) for the two- and three-point vertices, where the solid and wavy arrows denote the exact elementary and HS propagators, respectively. The interchange of labels applies to all diagrams inside the brackets, while a diagram containing a cross stands for all different diagrams of this type with one of the internal propagators replaced by the corresponding single-scale propagator.

Figure 13

Graphical representation of the FRG flow equations for the four-point vertices ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\overline{a}\overline{a}aa}$, ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\overline{a}a\overline{\psi }\psi }$, and ${\mathrm{\Gamma }}_{\mathrm{\Lambda }}^{\overline{\psi }\overline{\psi }\psi \psi }$, where the solid and wavy arrows denote the exact elementary and HS propagators, respectively. The interchange of labels applies to all diagrams inside the brackets, while a diagram containing a cross represents all different diagrams of this type with one of the internal propagators replaced by the corresponding single-scale propagator.