Superconductivity through Quantum Critical Fluctuations in the ...

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ppose we separate m into the usual transverse parms of a vortex field v k, n s of a vortex field v k, n Vortex field:ik m t k, n = v k, n ik m t k, n = v k, n d a longitudinal part through introducing the wak,Split m into its transverse and longitudinal parts:a longitudinal part through introducing the w,Warp field: n m k, n = ckˆ w k, n . n m k, n = ckˆ w k, n .uite miraculously the partition function in Eq. 5 canritten exactly asThen one finds that the singular part of the action decouples as:e miraculously the partition function in Eq. 5 caexpn Z = J v , w k, k 2 vk n 2 −en exactly asS= d⇤drdr J⇥ v (r, ⇤)⇥ v (r , t)ln |r r |+ drd⇤d⇤ ⇥ w (r, ⇤)⇥ w (r, ⇤ )ln |⇤ ⇤ |Z = exp J 2 vk n 2 −J 2 k 24 wk n 2w k n 2
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
Page 1 and 2: Superconductivity through Quantum C
Page 3 and 4: Universal Phase DiagramTAntiferroma
Page 5 and 6: Fermi-Arc Phenomena:A New paradigm
Page 7 and 8: Any theoretical ideas and calculati
Page 10 and 11: Effect of Fluctuations on Single-pa
Page 12 and 13: If there is a region of Quantum-cri
Page 14 and 15: s but simply characterized temperat
Page 16 and 17: For quantum-fluctuations in the dis
Page 18 and 19: Fluctuations of the pseudogap order
Page 20 and 21: Superconductivity through Quantum C
Page 22 and 23: angle independence of the Eliashber
Page 24 and 25: Pairing VertexScattering through π
Page 26 and 27: Deducing the Fluctuation Spectra Co
Page 28 and 29: dI/dV ⇥ SdI/dV ⇥ N( )2 F (⇥)
Page 30 and 31: High Resolution Laser ARPES data. Z
Page 32 and 33: The Eliashberg functionAnalysis on
Page 34 and 35: How could one imagine superconducti
Page 36 and 37: TQuantum-critical fluctuationsof th
Page 38 and 39: Quantum Critical Fluctuations(Vivek
Page 40 and 41: t finite temperature where the corr
Page 44 and 45: Correlation function in the Quantum
Page 46: Definitive Evidence that there is a

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