Multiple correlation lengths and type-1.5 superconductivity in $U(1)$ superconductors due to hidden competition between irreducible representations of nonlocal pairing

Abstract

A fundamental characteristic of a superconducting state is the coherence length $ξ$. Multicomponent superconductors, particularly ones breaking multiple symmetries, are characterized by multiple coherence lengths. Here we show that even, nominally $single$-component superconductors under certain conditions are characterized by multiple coherence lengths. We consider nearest-neighbor pairing interactions on a square lattice that leads to $s$-wave and $d$-wave representations of link superconducting order parameter. We show that even if the subdominant order parameter is completely suppressed in the ground state, it results in multiple correlation lengths with nontrivial hierarchy, resulting in important physical consequences in inhomogeneous solutions. Under certain conditions, this leads to type-1.5 superconductivity, where magnetic field penetration length falls between two coherence lengths, leading to vortex clustering in an external magnetic field.

Figures (8)

• Figure 1: The phase diagram of the microscopic Hamiltonian Eq. (\ref{['eq:mean-field_Hamiltonian']}) in chemical potential $\mu$ [or band filling at $U(1)$ critical temperature $n(T_c^{U(1)})$], temperature $T$ coordinates. The region for type-1.5 superconductivity is estimated based on simple approximations described in the text. The green, black, and red points correspond to ($\mu=1.665$, $T=2.1\cdot10^{-4}$), ($\mu=1.673$, $T=2.48\cdot10^{-4}$), and ($\mu=1.665$, $T=2.4\cdot10^{-4}$), respectively. In our microscopic calculations, the parameters we employed are: energy cutoff $\omega_D =0.1$, nearest-neighbor interaction strength $V=2$. Ginzburg--Landau calculation parameters: Unit cell size $a=0.4 \,$ nm, interlayer distance $L_z=1.3 \,$ nm, hopping parameter $t_{xy} = 0.01\,$ eV.
• Figure 2: Temperature dependence of estimates of coherence lengths $\xi_{1(2)} (\varphi)$ for amplitude fluctuation in different directions defined by the angle $\varphi$ from the $x$ axis, coherence length $\xi_\theta$ for phase difference and magnetic field penetration length $\lambda$. Coherence length $\xi_\theta$ and magnetic field penetration length $\lambda$ are independent on the direction in the assumption of no mode mixing. A coherence length for amplitude fluctuation diverges at superconducting phase transition, another divergent length scale occurs at time-reversal symmetry breaking transition. This divergence occurs at both sides of the phase transitions, even when subdominant order parameter is suppressed. Type-1.5 superconductivity regime arises when magnetic field penetration length is the intermediate length scale, i.e. when $\lambda$ (black line) is in between $\xi$'s (colored lines). Chemical potential $\mu=1.665$, other parameters are identical to Fig. \ref{['fig:s+id phase diagram wD=0.1']}. Calculations of coherence length for phase difference variations $\xi_\theta$ are presented in Appendix \ref{['app:coherence length']}.
• Figure 3: Two flux quanta vortex cluster solutions in Ginzburg--Landau model for three distinct ground states: (a) $s+id$ state, (b) pure $s$-wave, and (c) pure $d$-wave. The ground states (GSs) correspond to green, black, and red points in Fig. \ref{['fig:s+id phase diagram wD=0.1']}. Panels 1-4 show amplitudes $|\Delta_s|$, $|\Delta_d|$, relative phase difference $\theta=\arg (\Delta_s \Delta_d^*)$, and magnetic field $B_z$. Vortex interaction energy is negative that can be seen in Fig. \ref{['fig: Binding energies']}. The total energy density qualitatively follows the magnetic field shown in panels 4. Boundary conditions correspond to no current flowing through the boundary, and the external magnetic field is zero. Gap amplitudes are presented in units $t_{xy} \sqrt{|\alpha_1|/\beta_1}$, magnetic field $B_z$ in $2 \hbar |\alpha_1|/(e a^2 \gamma_1)$. Model parameters are presented in Table \ref{['tab:model parameters']}.
• Figure 4: Plot of the pseudo-spin texture $\vec{\phi} = \frac{1}{|\Delta_s|^2+|\Delta_d|^2}\left( \Delta_s^*\Delta_d + \Delta_s\Delta_d^*, \, i(\Delta_s^*\Delta_d - \Delta_s\Delta_d^*), \, |\Delta_d|^2-|\Delta_s|^2 \right)$ in the $s+id$ regime [Fig. \ref{['fig: type-1.5']}(a)]. It shows that the double-quanta vortex is a skyrmion with topological charge $\mathcal{Q}=2$. In the type-1.5 regime it is a bound state of two $\mathcal{Q}=1$ skyrmions. The skyrmions are (iso-)rotated relative to one another such that the interaction energy between them is minimized. The skyrmion coloring is the standard Runge coloring scheme Leask_2022.
• Figure 5: The phase diagram of the microscopic Hamiltonian Eq. (\ref{['eq:mean-field_Hamiltonian']}) in chemical potential $\mu$, temperature $T$ coordinates. (a) energy cutoff $\omega_D =0.1$, (b) $\omega_D \rightarrow \infty$ (full Brillouin zone contributes). Orange and blue dashed lines for $T_s$ and $T_d$ do not correspond to phase transitions. Nearest-neighbor interaction strength $V$ is 2.