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Vortex lattice for a holographic superconductor

Kengo Maeda, Makoto Natsuume, Takashi Okamura

TL;DR

The paper addresses how a vortex lattice, akin to the Abrikosov lattice in type II superconductors, emerges in a holographic (2+1)D superconductor modeled via AdS/CFT near the second-order transition under a perpendicular magnetic field $B_{c2}$.A perturbative construction is performed by superposing leading droplet solutions in the AdS$_4$ bulk, yielding a vortex lattice characterized by lattice parameters $a_1$ and $a_2$ and a nonlocal free-energy functional. Key contributions include an explicit expression for the upper critical field $B_{c2}(T)$, a nonlocal formulation for the free energy and R-current that reduces to the Ginzburg–Landau form in the long-wavelength limit, and the demonstration that the triangular lattice minimizes the free energy. These results illuminate how holographic superconductors reproduce Abrikosov lattice physics while revealing nonlocal bulk-to-boundary structure and set the stage for dynamical studies of vortex lattices in strongly coupled systems.

Abstract

We investigate the vortex lattice solution in a (2+1)-dimensional holographic model of superconductors constructed from a charged scalar condensate. The solution is obtained perturbatively near the second-order phase transition and is a holographic realization of the Abrikosov lattice. Below a critical value of magnetic field, the solution has a lower free energy than the normal state. Both the free energy density and the superconducting current are expressed by nonlocal functions, but they reduce to the expressions in the Ginzburg-Landau (GL) theory at long wavelength. As a result, a triangular lattice becomes the most favorable solution thermodynamically as in the GL theory of type II superconductors.

Vortex lattice for a holographic superconductor

TL;DR

The paper addresses how a vortex lattice, akin to the Abrikosov lattice in type II superconductors, emerges in a holographic (2+1)D superconductor modeled via AdS/CFT near the second-order transition under a perpendicular magnetic field $B_{c2}$.A perturbative construction is performed by superposing leading droplet solutions in the AdS$_4$ bulk, yielding a vortex lattice characterized by lattice parameters $a_1$ and $a_2$ and a nonlocal free-energy functional. Key contributions include an explicit expression for the upper critical field $B_{c2}(T)$, a nonlocal formulation for the free energy and R-current that reduces to the Ginzburg–Landau form in the long-wavelength limit, and the demonstration that the triangular lattice minimizes the free energy. These results illuminate how holographic superconductors reproduce Abrikosov lattice physics while revealing nonlocal bulk-to-boundary structure and set the stage for dynamical studies of vortex lattices in strongly coupled systems.

Abstract

We investigate the vortex lattice solution in a (2+1)-dimensional holographic model of superconductors constructed from a charged scalar condensate. The solution is obtained perturbatively near the second-order phase transition and is a holographic realization of the Abrikosov lattice. Below a critical value of magnetic field, the solution has a lower free energy than the normal state. Both the free energy density and the superconducting current are expressed by nonlocal functions, but they reduce to the expressions in the Ginzburg-Landau (GL) theory at long wavelength. As a result, a triangular lattice becomes the most favorable solution thermodynamically as in the GL theory of type II superconductors.

Paper Structure

This paper contains 13 sections, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Phase diagram and solutions
  3. "Droplet" solution
  4. Vortex lattice solution
  5. Free energy and R-current
  6. Preliminaries
  7. First order solution of $A_\mu$
  8. Free energy and R-current
  9. The long-wavelength limit and the triangular lattice
  10. A representation of Green functions
  11. The long-wavelength limit
  12. Conclusions and discussion
  13. "Orthogonality condition"

Figures (2)

  • Figure 1: (color online). $B_{c2}$($+$) as a function of $T/T_c$. Dashed line: $B/\mu^2 = (1/8) (T/T_c)^2$ (See Sec. \ref{['sec:low_energy_limit']}).
  • Figure 2: The vortex lattice structure for the triangular lattice in the $(x,y)$-plane. The vertical line represents $\sigma = |\gamma_L|^2$ and vortex cores are located at $|\gamma_L|=0$.