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the abrikosov vortex lattice melts into an extended vortex - liquid phase in high - temperature superconductors subject to an external magnetic field oriented perpendicular to the conducting copper - oxygen planes that make them up@xcite@xcite . the large size in temperature and magnetic field of the vortex - liquid phase can be attributed to such layer anisotropy@xcite@xcite@xcite . a cross - over from a vortex - line liquid at temperatures just above the melting point of the abrikosov vortex lattice to a decoupled vortex liquid at higher temperature that shows negligible correlations of the superconducting order parameter across layers is predicted if the vortex lattice in isolated layers melts through a continuous or a weakly first - order phase transition@xcite . such dimensional cross - over is observed experimentally in electronic transport studies of the vortex - liquid phase in moderately anisotropic high - temperature superconductors@xcite . the abrikosov vortex lattice is predicted to sublimate directly into a decoupled vortex liquid at large enough layer anisotropy , on the other hand , if the vortex lattice in isolated layers melts through a first - order phase transition@xcite . electronic transport studies of the mixed phase in extremely layered high - temperature superconductors are consistent with the last sublimation scenario@xcite . an anomalous nernst effect is also observed in the vortex - liquid phase of high - temperature superconductors@xcite . in particular , a gradient in temperature along the copper - oxygen planes generates an electric field perpendicular to it along the copper - oxygen planes as well . the low - temperature onset of the anomalous nernst signal coincides with the melting point of the abrikosov vortex lattice , while the high - temperature onset can lie above the critical temperature of the superconducting state at zero field . the authors of ref . @xcite argue that this effect is principally due to vortex excitations in the mixed phase of high - temperature superconductors . it is then tempting to identify the cross - over between three - dimensional ( 3d ) and two - dimensional ( 2d ) vortex - liquid behavior that is predicted for layered superconductors in certain instances@xcite with the peak in the nernst signal . the fact that anomalous nernst signals are also observed in the vortex - liquid phase of extremely layered high - temperature superconductors that do not show the former dimensional cross - over@xcite@xcite rules out that interpretation , however .    the anomalous nernst effect observed in the vortex - liquid phase of high - temperature superconductors may instead be principally due to vortex excitations in copper - oxygen planes that are virtually isolated from one another@xcite . in this letter , the theoretical consequences of that proposal are examined through a duality analysis of the uniformly frustrated @xmath0 model for the mixed phase of extremely type - ii superconductors@xcite@xcite . we find first that weak collective pinning of the vortex lattice results in a melting / decoupling temperature that does _ not _ extrapolate to the mean - field transition in zero field . instead , a relatively big region of vortex liquid that is stabilized by random pinning centers is predicted to exist at temperatures below the mean - field transition . second , a high - temperature expansion of the uniformly frustrated @xmath0 model yields linear diamagnetism at temperatures just below the mean - field transition . the temperature dependence of the predicted equilibrium magnetization is found to agree quantitatively with recent experimental reports of a diamagnetic signal extracted from the vortex - liquid phase of high - temperature superconductors@xcite . last , we emphasize that an anomalous nernst effect is generally expected inside of the vortex liquid phase@xcite , where it tracks the temperature dependence shown by the diamagnetism in the vicinity of the mean - field phase transition . the @xmath0 model with uniform frustration is the minimum theoretical description of vortex matter in extremely type - ii superconductors . both fluctuations of the magnetic induction and of the magnitude of the superconducting order parameter are neglected within this approximation . the model hence is valid deep inside the interior of the mixed phase . its thermodynamics is determined by the superfluid kinetic energy @xmath3|_{r } , \label{3dxy}\ ] ] which is a functional of the phase of the superconducting order parameter , @xmath4 , over the cubic lattice , @xmath5 . here , @xmath6 and @xmath7 denote the local phase rigidities over nearest - neighbor links within layers . these are equal and constant , except over links in the vicinity of a pinning center . the josephson coupling across adjacent layers , @xmath8 , shall be assumed to be constant and weak . it can be parameterized by @xmath9 , where @xmath10 is the gaussian stiffness of the @xmath0 model for each layer in isolation , and where @xmath11 is the model anisotropy parameter . the vector potential @xmath12 represents the magnetic induction oriented perpendicular to the layers , @xmath13 . here @xmath14 denotes the square lattice constant , which is of order the coherence length of the cooper pairs , @xmath15 denotes the flux quantum , and @xmath16 denotes the concentration of vortices per site . the thermal / bulk average of the josephson coupling between adjacent layers is given by the expression@xcite@xcite @xmath17 } \label{cos1}\ ] ] in the decoupled vortex liquid to lowest order in the fugacity @xmath18 . here @xmath19 is the gauge - invariant phase difference across adjacent layers @xmath20 and @xmath21 , and @xmath22 is the autocorrelation function of the superconducting order parameter within layer @xmath20 in isolation ( @xmath23 ) . short - range correlations on the scale of @xmath24 following @xmath25 yields the result@xcite @xmath26 ^ 2 \label{cos2}\ ] ] for the inter - layer `` cosine '' ( [ cos1 ] ) . here , @xmath27 is a quenched disorder scale for the vortex lattices _ across _ adjacent pairs of isolated layers that appears through the autocorrelation @xmath28 \cdot   { \rm exp } [ { -i\phi_{l , l + 1}^{(0 ) } ( 2 ) } ] } = % e^{i b_{\parallel } x_{m , \bar m } }   e^{-r_{1,2}/l_{\phi}}. \label{form3}\ ] ] of the quenched inter - layer phase difference , @xmath29 . it is set by the density of dislocations quenched into the 2d vortex lattices found in each layer at zero temperature in the present case of uncorrelated pinning centers . also , above we have @xmath30 and the josephson penetration depth @xmath31 .    in the absence of inter - layer coupling , arbitrarily weak random point pins result in a stack of 2d vortex lattices with dislocations quenched in@xcite . let us assume that each 2d vortex lattice is in a _ hexatic vortex glass _ state@xcite , such that dislocations do _ not _ arrange themselves into grain boundaries . the quenched disorder scale @xmath27 that renormalizes down the interlayer josephson coupling ( [ cos2 ] ) is then set by the density of such dislocations@xcite . recent theoretical calculations find that each isolated layer shows a net superfluid density near zero temperature in the collective pinning regime , where the number of dislocations quenched into each 2d vortex lattice is small in comparison to the number of pinned vortices@xcite . application of collective pinning theory to the 2d vortex lattices found in isolated layers yields a density of quenched - in dislocations identical to the density of larkin domains@xcite @xmath32 , where @xmath33 denotes the density of pinned vortices per layer , where @xmath34 denotes the maximum pinning force , and where @xmath35 denotes the shear modulus of the 2d vortex lattice . here the critical state is assumed to be limited by plastic creep of larkin domains by an elementary burgers vector @xmath36 of the 2d vortex lattice . consider now the limit of weak pinning centers that do not crowd together : @xmath37 and @xmath38 , respectively , where @xmath39 denotes the density of pinning centers per layer , and where @xmath40 denotes the range of each pinning center . simple probabilistic considerations then yield the identity @xmath41 between the fraction of occupied pinning centers and the ratio of the effective area of each pinning center to the area per vortex , @xmath42 . this yields the result @xmath43 for the density of pinned vortices@xcite , where @xmath44 is the density of vortices per layer . finally , substitution of the estimate @xmath45 for the shear modulus@xcite yields the result @xmath46 for the density of larkin domains , which is independent of magnetic field . note , however , that all of the above is valid only in the 2d collective pinning regime that exists at perpendicular magnetic fields above the threshold @xmath47 , in which case many vortices are pinned in each layer within a larkin domain of dimensions @xmath48 @xcite . single - vortex pinning exists at magnetic field below that threshold , on the other hand , in which case each larkin domain contains only a single pinned vortex : @xmath49 . assembling the above suggests the profile for the density of pinned vortices per layer versus the density of vortices that is depicted by fig . it implies that the quenched disorder scale @xmath50 is independent of magnetic fields above the threshold @xmath51 . we are finally in a position to determine the melting / decoupling line of the 3d vortex lattice at temperatures outside of the 2d critical regime , @xmath52 , at big enough perpendicular magnetic fields such that larkin domains can be defined , @xmath53 . the identification of the separation between dislocations quenched into each 2d vortex lattice with the 2d larkin scale@xcite , @xmath50 , necessarily yields the inequality @xmath54 in such case . at temperatures lying inside of the interval @xmath55 $ ] bounded by melting of the 2d vortex - lattice and by the kosterlitz - thouless transition in isolated layers , yet lying outside of the 2d critical regime , a partial duality analysis of the pristine layered @xmath0 model with uniform frustration ( [ 3dxy ] ) finds a first - order melting / decoupling transition of the 3d vortex lattice at interlayer josephson coupling@xcite @xmath56 the first - order nature of this melting / decoupling line and its coincidence with the contour defined above is consistent both with monte carlo simulations of the same model@xcite and with elastic medium descriptions of the vortex lattice in layered superconductors@xcite . observe now that the criterion ( [ xover ] ) for first - order melting / decoupling should remain valid in the present regime of weak pinning such that @xmath57 . substitution of expression ( [ cos2 ] ) for the inter - layer `` cosine '' in the decoupled vortex liquid then yields the melting / decoupling field @xmath58 where @xmath59 is the melting / decoupling field in the pristine limit@xcite@xcite@xcite . these results are summarized by the phase diagram shown in fig . the short sections of dashed and solid lines that emanate perpendicularly from the horizontal axis originate respectively from the decoupling cross - over ( [ xover ] ) and the second - order phase transition shown by the layered @xmath0 model in the absence of uniform frustration ( cf . we conclude this section by observing that the melting / decoupling line does _ not _ extrapolate to the mean - field critical temperature at zero - field [ @xmath60 due to the presence of dislocations quenched into the weakly pinned vortex lattices found in isolated layers . the phase diagram for the mixed phase of layered superconductors shown by fig .  [ f.2 ] implies a large region of vortex liquid in the vicinity of the meanfield transition at zero magnetic field because of the effects of random point pins . in particular , the equilibrium diamagnetic susceptibility due to the emergence of cooper pairs is well defined at temperatures inside of the window @xmath61 $ ] . the former quantity can be obtained from the uniformly frustrated @xmath0 model ( [ 3dxy ] ) in the vicinity of the meanfield transition via a high - temperature expansion in powers of the fugacity @xmath62 @xcite . in particular , a duality analysis yields that the corresponding partition function is approximated by @xmath63 as @xmath64 , where @xmath65 , and where @xmath66 . here @xmath67 is the ratio between a first - order and a zero - order modified bessel function , which is approximately @xmath68 for @xmath69 . also , @xmath70 represents nearest - neighbor links within layers , and @xmath71 represents elementary plaquettes within layers . the equilibrium magnetization is given by @xmath72 in the extreme type - ii limit , where @xmath73 is the gibbs free energy , and where @xmath74 is the volume . substitution of the previous high - temperature approximation yields @xmath75 where @xmath76 , and where @xmath77 denotes the spacing between layers@xcite . the magnetization therefore varies linearly with vanishing magnetic field like @xmath78 , with a diamagnetic susceptibility @xmath79 here @xmath80 is the usual ratio of the london penetration depth to the coherence length of the cooper pairs . the former is related to the gaussian phase stiffness of each layer by @xmath81 . non - linear diamagnetism is observed experimentally in the vortex liquid phase of the extremely layered high - temperature superconductor bi@xmath82sr@xmath82cacu@xmath82o@xmath83 ( bscco ) , at temperatures just above the superconducting transition in zero field@xcite . linear diamagnetism is displayed at yet higher temperature in the same samples , on the other hand . by eq . ( [ m ] ) , the uniformly frustrated @xmath0 model ( [ 3dxy ] ) predicts such linear diamagnetism , @xmath84 , at perpendicular magnetic fields that are small compared to the upper - critical scale @xmath85 , at temperatures just below the mean - field phase transition . the corresponding diamagnetic susceptibility predicted by the @xmath0 model is given by eq . ( [ chi1 ] ) . use of the relation quoted previously between the gaussian phase rigidity within planes and the london penetration length in conjunction with physical parameters @xmath86 , @xmath87 , and @xmath88 nm appropriate for bscco@xcite yields the estimate @xmath89 ^ 3 ( j_{4}/j_0)^4 ( a/\xi)^2 \ , { \rm a / tm } \label{chi2}\ ] ] for the diamagnetic susceptibility of that material in the vicinity of the mean - field phase transition . here @xmath90 is the superfluid fraction . the mean - field superfluid density expected from a pristine @xmath77-wave state in 2d is approximately @xmath91 the corresponding @xmath92-wave result in the vicinity of the meanfield transition at zero field@xcite ; i.e. , @xmath93 , where @xmath94 is the reduced temperature . equation ( [ chi2 ] ) then implies that the diamagnetic susceptibility vanishes like @xmath95 with temperature as it approaches the mean - field transition . figure  [ f.3 ] displays the cube - root of the diamagnetic signal extracted experimentally in ref . @xcite from an underdoped sample of bscco with @xmath96 k , in perpendicular magnetic field @xmath97 t , as a function of temperature . the solid line is a fit to the linear diamagnetism , @xmath84 , predicted by the high - temperature expansion of the uniformly frustrated @xmath0 model , eq . ( [ chi2 ] ) , with @xmath98 , @xmath99 k , and with @xmath0 model parameter @xmath100 . the success of the fit indicates that the onset of the diamagnetic signal observed in the vortex liquid phase of high - temperature superconductors reflects nothing other than the mean - field phase transition at which cooper pairs emerge . the large suppression of @xmath101 compared to the meanfield transition temperature @xmath2 obtained here can be accounted for by quenched disordering of the superconducting order parameter , which could be generic to under - doped high - temperature superconductors@xcite . a gradient in temperature along the layers in the vortex liquid phase of high - temperature superconductors generates a voltage in the perpendicular direction within the layers@xcite . in particular , the nernst signal defined by the ratio @xmath102 between the electric field that is generated and the gradient in temperature peaks inside of the vortex liquid . standard transport theory yields the identity@xcite @xmath103 between the nernst signal and the product of the flux - flow electrical resistivity @xmath104 with the off - diagonal peltier coefficient @xmath105 . also , application of ginzburg - landau theory for the superconducting order parameter yields the estimate@xcite @xmath106 for the peltier coefficient near the mean - field transition , where @xmath107 is of order @xmath108 . observe now that the flux - flow resistance increases with temperature in the vortex liquid , while the equilibrium magnetization decreases with temperature there [ cf . ( [ chi1 ] ) ] . substitution of the estimate ( [ alpha ] ) into the identity ( [ e_y ] ) then yields ( _ i _ ) that the low - temperature onset of the anomalous nernst signal is given by the melting / decoupling temperature of the vortex lattice . also , the linear diamagnetism ( [ chi1 ] ) extracted from the high - temperature regime of the frustrated @xmath0 model implies ( _ ii _ ) that the anomalous nernst signal vanishes with temperature at the mean - field transition as @xmath1 . where exactly the nernst signal peaks inside of the vortex - liquid phase depends on how pinning affects the flux - flow resistance@xcite , which is beyond the scope of the paper .    in conclusion , a high - temperature analysis of the layered @xmath0 model with uniform frustration finds that the simultaneous onset of linear diamagnetism and of an anomalous nernst effect in the normal phase of high - temperature superconductors@xcite can be identified with the mean - field transition for cooper pairing . the low - temperature onset of the anomalous nernst signal at the melting / decoupling line of the vortex lattice was also found to be depressed substantially by the presence of dislocations quenched into the vortex lattice in isolated layers . the author thanks louis taillefer for discussions and p.w . anderson for correspondence . this work was supported in part by the us air force office of scientific research under grant no . fa9550 - 06 - 1 - 0479 . a natural worry is that the cubic dependence of the magnetization with the fugacity @xmath109 in eq . ( [ m ] ) is an artifact of the square - lattice grid . the same cubic dependence is obtained if a triangular grid is used , however . and although a honeycomb ( graphene ) grid yields a @xmath110 dependence for the magnetization in the high - temperature limit , the energy functional of that @xmath0 model ( [ 3dxy ] ) is _ not _ the superfluid kinetic energy @xmath111 at long wavelength , but a massless dirac functional instead .
linear diamagnetism is predicted in the vortex - liquid phase of layered superconductors at temperatures just below the mean - field phase transition on the basis of a high - temperature analysis of the corresponding frustrated @xmath0 model . 
 the diamagnetic susceptibility , and the nernst signal by implication , is found to vanish with temperature as @xmath1 in the vicinity of the meanfield transition at @xmath2 . 
 quantitative agreement with recent experimental observations of a diamagnetic signal in the vortex - liquid phase of high - temperature superconductors is obtained .