Vortex Solid Phase with Frozen Undulations in Superconducting Josephson-Junction Arrays in External Magnetic Fields

A vortex solid with self-generated randomness is found theoretically in a frustrated Josephson-junction array under external magnetic field with anisotropic couplings. Vortices induced by the external magnetic field develop stripes parallel to the direction of weaker coupling. It is shown analytically that there is a continuous, gapless band of metastable states in which stripes are deformed randomly by transverse undulation. The vortex solid with the frozen undulation in a metastable state freely slides along the direction of stronger coupling, thereby destroying the ordering of phases even at zero temperature, but is jammed along the direction of weaker coupling.

Figure 1

Vortex patterns in an irrationally frustrated JJA under external magnetic field with anisotropic coupling. Here $\lambda =1.5$ so that the coupling is stronger along the $y$ direction. Such an anisotropic JJA can be fabricated in the laboratory by lithography technique(s) [9]. Panel (a) displays the JJA on a square lattice. A fraction $f=21/55$, which approximates an irrational number $(3-\sqrt{5})/2=0.381966\dots$, of the plaquettes is occupied by vortices with charge $1-f$ represented by filled squares. Panel (b) displays an equilibrium vortex pattern at $T=0.2$ and panel (c) displays that at a nearby energy minimum reached via an energy descent algorithm.

Figure 2

Structure factor of vortices. (a) displays the cross sections of the structure factor $S(q_x,q_y)$ ($L=55$) with thermal average (thick lines) and at the energy minimum shown in Fig. 1c (thin lines). (b) displays the amplitude of the peak of the structure factor $S_{*}=S(\pi ,2\pi f)$ with thermal average $⟨S_{*}⟩$, average over the energy minima $[S_{*}{]}_{\min .}$ and variance of the minima-to-minima fluctuation $\sqrt{[S_{*}^2{]}_{\min .}-[S_{*}{]}_{\min .}^2}$. Here the average over minima $[\dots {]}_{\min }$ is taken over 100 energy minima obtained by independent initial conditions.

Figure 3

Configuration of the gauge-invariant phase differences across Josephson junctions in energy minima. Here the anisotropy is $\lambda =1.5$. In panel (a) the original $y$ axis is “folded” to $[y]=fy-\mathrm{int}(fy)$. The symbols are data of the gauge-invariant phase differences ${\psi }_x={\psi }_{(x,y)(x+1,y)}$ (filled symbols) and ${\psi }_y={\psi }_{(x,y)(x,y+1)}$ (open symbols) across Josephson junctions parallel to the $x$ and $y$ axis. Each data set consists of data points on a “column” $(x,1),(x,2),\dots ,(x,L)$ at an arbitrary chosen $x$ ($1\le x\le L$) in an arbitrary chosen energy minima, respectively. Each data set is shifted globally by some $\alpha$ so that different data sets collapse on top of each other. The lines are analytically obtained hull functions of the ground state. In panel (b) the data on the same energy minimum ($L=55$) are plotted against the folded $x$ axis.