Vortex - boson duality in four space - time dimensions

A continuum version of the vortex-boson duality in (3+1) dimensions is formulated and its implications studied in the context of a pair Wigner crystal in underdoped cuprate super-conductors. The dual theory to a phase fluctuating superconductor (or superfluid) is shown to be a theory of bosonic strings interacting through a Kalb-Ramond rank-2 tensorial gauge field. String condensation produces Higgs mass for the gauge field and the expected Wigner crystal emerges as an interesting space-time analog of the Abrikosov lattice. When particles in quantum many body systems interact very strongly standard perturbation techniques break down and, in dimensions greater than 1, dualities often provide the only insights into the physics of such systems. Duality transformations, in general, map the strongly coupled sector of one theory onto the weakly coupled sector of another. The original Kramers-Wanier duality [1] for the Ising ferromagnet represents a prime example of such a mapping. Dualities permeate modern statistical, condensed matter and particle physics, and have emerged recently as a key tool in string theory. In condensed matter physics perhaps the most useful and influential duality is the one connecting vortices and bosons in two spatial dimensions [2–4]. This duality, hereafter referred to as Lee-Fisher duality, maps the system of interacting bosons in (2+1)D onto a fictitious super-conductor in an external magnetic field whose flux in the temporal direction is proportional to the density of the original bosons. It shows that Mott insulator, proximate to the phase fluctuating boson condensate, can be viewed as the Abrikosov vortex lattice of the dual superconductor. This deep connection has been exploited in modeling systems ranging from quantum spins to fractional quantum Hall effect, and most recently cuprate superconductors. In cuprates such considerations are motivated by the experimental findings of static checkerboard patterns in the charge density of very underdoped samples [5–7] which have been interpreted as evidence for a Cooper pair Wigner crystal (PWC) [8]. The latter can be most naturally understood by appealing to the Lee-Fisher duality [9–11]. However, recent analysis of the vibrational modes of such a PWC [12, 13] indicates that it is 3-dimensional (in the sense that vibrations propagate in all 3 space dimensions) and it is thus unclear how the inherently two-dimensional Lee-Fisher duality applies to this situation. The problem can be stated as follows. A key role in the formulation of the Lee-Fisher duality is played by vor-tices which appear (in pairs of opposite …