Duality and the vibrational modes of a Cooper-pair Wigner crystal

When quantum fluctuations in the phase of the superconducting order parameter destroy the off-diagonal long-range order, duality arguments predict the formation of a Cooper-pair crystal. This effect is thought to be responsible for the static checkerboard patterns observed recently in various underdoped cuprate superconductors by means of scanning tunneling spectroscopy. Breaking of the translational symmetry in such a Cooper-pair Wigner crystal may, under certain conditions, lead to the emergence of low-lying transverse vibrational modes which could then contribute to thermodynamic and transport properties at low temperatures. We investigate these vibrational modes using a continuum version of the standard vortex-boson duality, calculate the speed of sound in the Cooper-pair Wigner crystal, and deduce the associated specific heat and thermal conductivity. We then suggest that these modes could be responsible for the mysterious bosonic contribution to the thermal conductivity recently observed in strongly underdoped ultraclean single crystals of $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6+x}$ tuned across the superconductor-insulator transition.

Figure 1

(Color online) A schematic plot of the spatial variation of the order parameter $\chi$ and the field $B_d$ in the dual Abrikosov vortex lattice. ${\xi }_d$ and ${\lambda }_d$ denote the dual coherence length and dual penetration depth, respectively. The dashed line represents the magnetic field $b_d\left(\mathbf{r}\right)$ associated with a single isolated vortex; $B_d$ in a dense lattice is a superposition of such field profiles centered at individual vortices.

Figure 2

A schematic plot of the acoustic transverse mode (dashed line) and the longitudinal mode (solid line). The functions are plotted with ${\lambda }_d=10a$ where $a$ is the vortex lattice constant and $\omega \left(k\right)$ is presented in units of $v_s$.

Figure 3

(Color online) (a) A schematic illustration of the flux lines in the presence of a monopole. (b) Josephson tunneling with lateral displacement of the dual vortex. The shaded regions represent charge densities associated with a Cooper pair in the PWC.