Nikolic presented his work with Sachdev dealing with quantum nature of vortices and their interactions with fermionic degrees of freedom. He was seeking to understand several STM experiments on vortex cores, their immediate neighborhood as well as the observed 4x4 checkerboard modulation in underdoped cuprates. He was also interested in understanding the shape of the “Nernst dome” on the underdoped side of the phase diagram.

The basic calculation is as follows: take a single vortex. Let it fluctuate in quantum time from its equilibrium position and compute its generic quantum action. Focus on the quasiparticle contributions to various terms. This is very sensible since vortex cores in cuprates are very small and basically gapped so we do not expect much contribution from the conventional Bardeen-Stephen effect. Instead, the major effect must come from nodal quasiparticles which are present even for H = 0.

Nikolic makes FT transformation to reduce the effective Hamiltonian for nodal quasiparticles to a Dirac problem with two gauge fields describing the influence of vortex on quasiparticle motion. He then computes the response of this Hamiltonian to small quantum fluctuations in vortex position. The results are the following: first, there is no Ohmic dissipation. The reason is that the nodal fermion density of states vanishes at the Fermi level. This can also be understood by a general scaling argument using the fact that massless Dirac fermions represent a nice quantum critical system (with z = 1). Second, they compute the vortex mass and find it to be very low, of the order of only a few electron masses. These two results confirm the speaker’s expectation that vortices in underdoped cuprates are highly quantum objects and that they can be used as such to construct useful effective theories of cuprates.

Using the above results, Nikolic, Sachdev and collaborators argued that the STM structure within vortex cores (the minigaps observed at ~ 7 meV or so) can be understood as arising from quantum fluctuations in vortex core position within a vortex lattice. Furthermore, strong quantum vortex-antivortex fluctuations in zero field can be used to construct the so-called dual theory of fluctuating superconductors. This theory takes the form of a dual Hofstadter-Abrikosov problem with vortex-antivortex (bosonic) “particles” moving on a lattice and in a dual magnetic field fixed by the density of original electrons. The condensation of such dual bosons naturally leads to suppression of superconductivity in favor of a charge density wave of Cooper pairs. Such tendency is the strongest near x=1/8 where pairs of holes (electrons) can arrange themselves neatly into 4x4 checkerboards.

Several questions were raised by members of the audience. Millis inquired about dissipation and how much of it was observed in actual experiments. Norman pointed out that the STM minigaps in vortex cores were found to scale with Delta (Fischer et al). Nikolic responded that their scaling was somewhat different. Auerbach stressed the importance of the “hydrodynamic” contribution to the mass (as opposed to the quasiparticle one) and a lively discussion ensued which was brutally interrupted by a lunch truck (see Comment by Sachdev on how the hydrodynamic mass is dealt with in their approach).

Much of the physics Nikolic spoke about can be found at www.arxiv.org/abs/cond-mat/0511298 and in references therein.

## Thursday, August 16, 2007

### Predrag Nikolic (Harvard): "The role of vortices in unconventional properties of d-wave superconductors"

Posted by Zlatko Tesanovic at 3:30 PM

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## 15 comments:

On the issue of the hydrodynamic mass:

The value of the mass depends crucially on the degrees of freedom in the low energy theory into which the mass is being input. There are two approaches which have been taken:

(1) A low energy theory for the vortices alone

(2) A low energy theory for vortices coupled to a "dual" U(1) gauge field -- this gauge field mediates the logarithmic interaction between the vortices, and its dynamics represent the phase fluctuations (or "spin waves"). The coupling constants in the gauge field action are known, being related to the superfluid density and the Coulomb interactions.

In the computation by Nikolic and myself, we are using the theory (2). In this case, we claim, there is no logarithmic mass divergence. The log divergence appears only if we integrate out the U(1) gauge field, leading to a theory of the form (1). However, in the analysis of any experiment, we believe it is much simpler to keep the U(1) gauge field as a dynamic degree of freedom.

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