Superconductivity through Quantum Critical Fluctuations in the ...

t finite temperature where the correlation funcder-parameterphase. Nevertheless e ı changes a from phaseexponential transitiontoPrincipal Steps:temperature where the correlation funcarametere ı changes from exponential totransition. The quantum dissipative generalizatiomodel includes two dynamical terms and is given bpower law. This is the Kosterlitz-Thoulesstransition. 25,26 The quantum dissipative generalizmodel includes two dynamical terms and is giveZ = D i exp − 0= Z D i exp − 0S diss =−d0did ij,klC2 2 ij − J cos ij − klij,kl ij − kl ij,kl − ij − kl − + S diss ,e capacitance andS=R Q /R, where R Q =h/4e 2 .cs of this phase transition diss d ,=−isd0better understoodij,klin terms of the − topological defects of the system. To dndard procedure of using the Villain transform and integrating out the phase degrees of freedom. 20 Thacitance olves expanding Villain =R Transformation:the Q periodic /R, where function R Q =h/4e in terms 2 . of a periodic Gaussianthis phase transitionexp − J is better understood 1 − cos ij − kl in termsij,klm ij;klexp − of J the topological defects of ij − kl − 2m ij;kl /2 2 ,the system. Tprocedure of using the Villain transform and integrating out the phase degrees of freedom. 20ij,klre integers that live on the links of the original. We can combine the two link variables m i,j;i+1,jto one integrated two-component over. vector m i,j that lives onof the lattice see Fig. 1. We expand the quadtransform to Fourier space. Keeping the lead-diC2 2 ij − J cos ij − kl xy − x+1,y − 2m xx,y+ S diss ,2 ij − kl − ij − kl 2expanding the periodic function in terms of a periodic Gaussianexp − J ij,kl1 − cos ij − kl m ij;klexp − J ij − kl − 2m ij;kl /2 2 ,ij,klNow the model is Gaussian in theθ’swhich can be, 2 a 2 2 xx xy +4a x xy m x,yxwhere m x,y is the x component of the vector fieldthe integer m . In the absence of dissipation,egers that live on the links of the original − − 2m x 2 a 2 2 +4a m x+