Superconductivity through Quantum Critical Fluctuations in the ...

What is a “Warp”:A warp is an arrangement of the Divergence of a Field at a point is surrounded symmetricallyFor a general vector field oneby does expectoppositetwo kinds of“charges” whose summed value is equal to the “charge”vortices, the additional topological entity to describe theat the center.FIG. 3: A phase slip between nearest neighbors in time producesa vortex and an antivortex on neighbouring plaqiettes.In time one bond has acquired a nonzero value, representedby the red bond. If the value on the bond is -1, the vorticityat site 1 is negative and on site 2 is positive.ThereforeFIG. 4: A phase slipWarpsevent in rime resultshavein a changenoin theinteractionssurrounded at theby four annearest equalneighbor butwithopposite sites. Incharge eachgeneraldistributedthe magnitudeotherlink variables. For a change of 2π at site {i, j}, thefourlinksofthe divergence at the site will equal to the number of nearestconnected to it acquire the values shown in the figure.timeThusis a phase slip1/(τ-τ’).on a link is equivalentTheirto a localproliferationin are most one latticebelowspacing apart; theainteractioncrit. valueisoflocal in space. Although this is physically obvious fromdissipation gives the “local” quantum-critical fluctuationnext section.ρ i ps Θ (τ − τ i) [(̂x + ŷ) δ (r − r ij )spectra.Field configuration of a Warpspace and time vortex current.Local phase slips and WarpsThe periodic in time boundary condition allows forδτ− ŷδ (r − r ij − aŷ)] Θ (τ − τ i )Such a vector field distribution has no curl and hence doesnot effect the vorticity. On the other hand the divergencei.e. an arrangement ofδτmonopoles in which the “charge”is nonzero and phase slip events generate field configurationsthat are orthogonal to those created by vortices.sources to generate an arbitrary distribution. For the2+1 dimensional quantum model we have, besides thewinding number sector in time. Events that change thewinding number sector, i.e. local phase slips, acts assources for a divergence in the vector field. Just as avortex is equivalent to an electric charge in the dual language,the sources created by phase slips can be shownto be a local distribution of monopoles (ρ m ). Given the“Charge” configuration of a Warpdistribution in eqn.15, the corresponding configuration ofmonopoles, which we term the charge of the phaseslip isρ m (r, τ) = ∇ · m(r, τ) (16)= [4δ (r − r ij ) − δ (r − r ij + âx)− δ (r − r ij − âx) − δ (r − r ij + aŷ)− δ (r − r ij − aŷ)] Θ (τ − τ i )The monopole distribution equivalent to a phaseslip isshown in fig.5. The total monopole charge of the configurationis zero. Since the distribution has azimuthalFIG. 5: Warps are configurations of the field m with finitesymmetry all harmonics are zero. This is the two dimensionallattice {i, j}, the realization divergenceof has thea configuration magnitude of 4 of at athe charge site anddivergence but no curl. For a warp of unit strength at site-1over a spherical shell of radius a in three dimensions. Themagnetic neighbors field due on the to the lattice. charges is confined within onein space. They come and go in time and their interaction inunit cell around the site of the phase slip and is zero outside.Thus two phase slip events can interact only if theythis discussion, ∇×m t we (r, will τ) = demonstrate ρ v (r, τ) this explicitly in the (18)m l (r, τ) = ∑Phase slip events generateia local vortex current whichis divergenceless but has − âxδ finite (r − rcurl.ij − âx)−ŷδ (r − r ij − aŷ)]5In Fourier space we getm t (k, ω n ) = ıẑ × kk 2 ρ v (k, ω n )( ) 1 − e−ık x a ̂x +m l (k, ω n ) =ıωρ ps (k, ω n ) = ∑ iρ i pse −ık·r i−ıThe locality in space is reflfactors (1 − exp (−ık x a)) andk x a, k y a ≪ 1, to leading orderm l (k, ω n ) ≈ a (k x̂x +ω nThe two components of the fieonal to each other. Comparingnent of m l generated by a phalength limit and a warp (eqn.portional to each other. In papartition function can be recas