Critical Capacitance and Charge-Vortex Duality Near the Superfluid-to-Insulator Transition

Using a generalized reciprocity relation between charge and vortex conductivities at complex frequencies in two space dimensions, we identify the capacitance in the insulating phase as a measure of vortex condensate stiffness. We compute the ratio of boson superfluid stiffness to vortex condensate stiffness at mirror points to be 0.21(1) for the relativistic O(2) model. The product of dynamical conductivities at mirror points is used as a quantitative measure of deviations from self-duality between charge and vortex theories. We propose the finite wave vector compressibility as an experimental measure of the vortex condensate stiffness for neutral lattice bosons.

Figure 1

Critical energy scales near the SIT computed by QMC calculations.The superfluid is characterized by the mass of the amplitude mode, $m_H$, and the superfluid stiffness, ${\rho }_s$, the insulator by the single particle gap, $\mathrm{\Delta }$, and the vortex condensate stiffness, ${\rho }_v$. The amplitude ratios $m_H(-\delta g)/\mathrm{\Delta }(\delta g)=2.1(3)$[17], ${\rho }_s(-\delta g)/\mathrm{\Delta }(\delta g)=0.44(1)$ [7], and ${\rho }_v(\delta g)/\mathrm{\Delta }(\delta g)=2.1(1)$ are universal.

Figure 2

The conductivity as a function Matsubara frequency. The curves differ by the detuning parameter $\delta g$. In the insulator, the low frequency conductivity is linear, ${\sigma }_{\text{ins}}\sim {\omega }_m$ indicating capacitive behavior. In the superfluid, the conductivity diverges as ${\sigma }_{\text{sf}}\sim 1/{\omega }_m$ indicating inductive response.

Figure 3

(a) Scaling function of the dynamical conductivity in the superfluid ($\delta g<0$) and insulator ($\delta g>0$). Data for different values of the detuning $\delta g$ collapse to two universal curves. (b) Measure of charge vortex duality of the $O(2)$ model. Universal scaling function for $\mathcal{R}({\omega }_m)$ defined in Eq. (2). Deviation of this function from unity quantifies the difference between charge and vortex matter.