Josephson-frustrated superconductors in a magnetic field

We study the effect of an externally imposed rotation or magnetic field on frustrated multiband superconductors and superfluids. The frustration originates with multiple superconducting bands crossing the Fermi surface in conjunction with interband Josephson couplings with a positive sign. These couplings tend to frustrate the phases of the various components of the superconducting order parameter. This in turn leads to an effective description in terms of a $U\left(1\right)\times Z_2$-symmetric system, where essentially only the $U\left(1\right)$ sector couples to the gauge field representing the rotation or magnetic field. By imposing a large enough net vorticity on the system at low temperatures, one may therefore reveal a resistive vortex-liquid state which will feature an unusual additional phase transition in the $Z_2$ sector. At low enough vorticity there is a corresponding vortex-lattice phase featuring a $Z_2$ phase transition. We argue that this transition phase should be readily observable in experiments.

Figure 1

The arrows in panel (a) () correspond to $({\theta }_1,{\theta }_2,{\theta }_3)$. Panels (b) and (c) show examples of phase configurations for the two $Z_2$ symmetry classes of the ground states, shown on a $2\times 2$ lattice of selected points of a planar slice of the system. Here $g_{12}>g_{23}>g_{13}>0$. The spatial contribution to the energy is minimized by making the spatial gradient zero [hence breaking the global $U\left(1\right)$ symmetry]. Then there are two classes of phase configurations, one with chirality $+1$ and one with chirality $-1$, minimizing the energy associated with the interband interaction. The chirality is defined as $+1$ if the phases (modulo $2\pi$) are cyclically ordered ${\theta }_1<{\theta }_2<{\theta }_3$ and $-1$ if not.

Figure 2

One of the two $Z_2$ phase configurations in the $g_{\alpha {\alpha }^{\prime }}\to \infty$ limit when $n=3$. ${\varphi }_{\alpha }$ is the phase difference between the first and the $\alpha \mathrm{th}$ component, a constant.

Figure 3

Simulation results for the full model with $a$ as indicated and $g=(5,5,5),f=\frac{1}{32}$, and $L=64$. The $Z_2$ sector is monitored by the Binder cumulant for the $Z_2$ magnetization, $U_2$ (blue). The vortex-lattice ordering (in each component) is monitored by the value of the structure function at the first Bragg peak, $S\left(\mathit{K}\right)$ (red). Note how the order of the phase transitions changes as the anisotropy of the system $a_3/a_{1,2}$ is varied. In the top panel, the system features a normal metallic state (vortex liquid), with broken $Z_2$ symmetry being restored at $\beta \approx 0.37$. In the bottom panel, the $Z_2$ symmetry is restored inside the superconducting (vortex-lattice) phase. In the middle panel, the transitions occur roughly simultaneously.

Figure 4

Simulation results for the reduced model with $K_1=0.99,f=\frac{1}{32},L=128$, and $K_2=0,0.058,0.173,0.289,0.404$. The choice of the parameter $K_1$ is explained in the text. The $Z_2$ sector is monitored by the Binder cumulant for the $Z_2$ magnetization, $U_2$ (blue). The vortex-lattice ordering is monitored by the helicity modulus ${\rm Y}$ (red) in the various directions, where $z$ is parallel to the external field.

Figure 5

Simulation results for the reduced model with $K_1=0.99,K_2=0.0,L=128$, and $f=0,\frac{1}{128},\frac{1}{64},\frac{1}{32},\frac{1}{16}$. The $Z_2$ sector is monitored by the Binder cumulant for the $Z_2$ magnetization, $U_2$ (blue). The vortex lattice ordering is monitored by the helicity modulus ${\rm Y}$ (red) in the various directions, where $z$ is parallel to the external field. In zero external field $\left(f=0\right)$ the system is isotropic.

Figure 6

Specific-heat capacity of the reduced model with $K_1=0.99,f=\frac{1}{32},L=128$, and $K_2=0,0.289,0.404$, associated with Fig. 4.

Figure 7

Specific heat of the reduced model with $K_1=0.99,K_2=0.0,L=128$, and $f=0,\frac{1}{128},\frac{1}{64},\frac{1}{32},\frac{1}{16}$. These results correspond to those shown in Fig. 5. Note that the $Z_2$ anomaly is considerably more prominent than the sharp peak associated with the melting of the vortex lattice.