Chiral classical states in a rhombus and a rhombi chain of Josephson junctions with two-band superconducting elements

We present a study of Josephson junctions arrays with two-band superconducting elements in the high-capacitance limit. We consider two particular geometries for these arrays: a single rhombus and a rhombi chain with two-band superconducting elements at the spinal positions. We show that the rhombus shaped JJ circuit and the rhombi chain can be mapped onto a triangular JJ circuit and a JJ two-leg ladder, respectively, with zero effective magnetic flux, but with Josephson couplings that are magnetic flux dependent. If the two-band superconductors are in a sign-reversed pairing state, one observes transitions to or from chiral phase configurations in the mapped superconducting arrays when magnetic flux or temperature are varied. The phase diagram for these chiral configurations is discussed. When half-flux quantum threads each rhombus plaquette, new phase configurations of the rhombi chain appear that are characterized by the doubling of the periodicity of the energy density along the chain, with every other two-band superconductor locked in a sign-reversed state. In the case of identical Josephson couplings, the energy of these phase configurations becomes independent of the inner flux in the rhombi chain and the supercurrent along the rhombi chain is zero.

Figure 1

Schematic diagram of a rhombi JJ chain with periodic boundary conditions, where TBSs at the spinal positions alternate with OBSs at the edge positions. The crosses indicate JJs, single lines indicate OBSs, and double lines indicate TBSs. ${\Phi }_f$ is the magnetic flux per plaquette and ${\Phi }_i$ is the inner flux in the chain.

Figure 2

The rhombi chain displayed in Fig. 1 can be considered as equivalent to a JJ two-leg ladder of one-band superconductors with diagonal tunnellings, if one interprets the two bands of the two-band superconductors as two sites of the JJ ladder. In this figure, the dots indicate the superconducting elements, the two-band superconductors are shown as blue dashed boxes (one dot per band) and the segments indicate the Josephson tunnellings.

Figure 3

Superconducting phases in a rhombus circuit of OBSs given a $2\varphi$ phase difference between spinal positions.

Figure 4

(a) Evolution of the normalized energy $\Delta E/\Delta E_{\text{max}}=(E-E_{\text{min}})/(E_{\text{max}}-E_{\text{min}})$ of the symmetric JJ rhombi array (with six rhombi) of OBSs as a function of magnetic flux ${\Phi }_1$ for several values of the frustrating magnetic field ${\Phi }_f$. (b) Example of supercurrent distribution in the rhombi chain for ${\Phi }_f=0.97\pi$ and ${\Phi }_1=\pi$. (c) Energies $E_i^{\delta }$ for $f={\Phi }_f/\left(2\pi \right)=0.482$ and $J=1$ in the case of a chain with four rhombi. (d) Amplitude of the energy oscillations as ${\varphi }_f$ is varied ($E_{\text{max}}-E_{\text{min}}$ on the left axis), and minimum value of the energy ($E_{\text{min}}$ on the right axis) for the chain of six rhombi.

Figure 5

The energy of a rhombus with spinal TBSs can be written as a sum of two terms, which involve only the first or the second band of the TBSs, and a third term with only the interband energy contribution.

Figure 6

Examples of nonchiral superconducting states in a rhombus JJ circuit with TBSs with zero interband coupling at the spinal nodes in the absence of magnetic flux. If the interband coupling is zero, all the Josephson energies can be simultaneously minimized and no frustration occurs. Here, we show the respective relative superconducting phase configurations, mapping the superconducting phases into XY spin angles.

Figure 7

Possible phase configurations for the minimum energy state of a rhombus JJ circuit with TBSs at the spinal nodes in the presence of applied magnetic flux. The arrows exemplify a possible current configuration. The actual current directions depend on the values of $\varphi$, $\alpha$, and $a$. Additional configurations can be obtained from these applying symmetry transformations.

Figure 8

Chiral superconducting behavior can be observed in the absence of magnetic field in the rhombus JJ circuit with TBSs at the spinal sites. In (a) and (b), the superconducting phases $\varphi$ and $\alpha$, respectively, are shown as functions of the couplings ratios $(J_1/J_{12})$ and$(J_2/J_{12})$ for negative $J_{12}$. The region of chirality corresponds to finite interband current as shown in (c). In (d), the respective energy of the rhombus circuit is shown for $J_{12}=-1$.

Figure 9

In (a), the region of chirality of the rhombus JJ circuit in the $(J_1/J_{12})$-$(J_2/J_{12})$ plane in the absence of magnetic flux is shown in red (dark region), for negative $J_{12}$. The boundaries of this region give second-order phase transitions to nonchiral states. As flux is applied, the region delimited by these transition curves expands up to ${\Phi }_f=\pi$ when it becomes the region delimited by the dashed blue curve. This transformation is reversed increasing further the magnetic flux and the phase diagram becomes the initial one for ${\Phi }_f=2\pi$. In (b) and (c), the superconducting phase $\alpha$ and the energy as functions of the magnetic flux are shown.

Figure 10

In (a), $E\left({\Phi }_1\right)$ plots are shown for the following sequence of magnetic flux values through the plaquettes: ${\Phi }_f=0$; ${\Phi }_f=0.9\pi$ or $1.1\pi$; ${\Phi }_f=0.996\pi$ or $1.004\pi$; ${\Phi }_f=\pi$. The set of couplings $J_1=0.6$, $J_2=0.55$ and $J_{12}=-2$ corresponds to a point within the chiral phase in Fig. 9. In (c), $E\left({\Phi }_1\right)$ plots are displayed when ${\Phi }_f=\pi$, $J_{12}=-2$, and for several values of Josephson couplings [corresponding to points along the diagonal of the phase diagram of Fig. 9]: $J_1=J_2=0.9$; $J_1=J_2=1.1$; $J_1=J_2=1.2$; $J_1=J_2=1.3$. In (b) and (d), we show the supercurrent plots corresponding to the energy plots shown in (a) and (c). (e) Example of a current configuration in the rhombi chain with two-band spinal elements, where a doubling of periodicity is observed. Parameters: $J_1=0.6$, $J_2=0.6$, $J_{12}=-2$, ${\Phi }_f=\pi$, and ${\Phi }_i=0$.

Figure 11

Amplitude of the energy-phase oscilations as a function of ${\Phi }_f$ ($E_{\max }-E_{\min }$ on the left axis), and minimum value of the energy as a function of ${\Phi }_f$ ($E_{\min }$ on the right axis) of the rhombi chain with TBSs at the spinal sites. The amplitude decreases much more [compared with Fig. 4] at ${\Phi }_f=\pi$ due to the existence of new solutions, which break the translation symmetry of the energy distribution along the rhombi chain. Parameters: $J_1=0.6$, $J_2=0.55$, $J_{12}=-2$.

Figure 12

Temperature dependence of the energy gaps of a TBS for several values of the interband coupling $V_{12}$. A single critical temperature is present even in the limit of very small interband tunneling. Besides a global enhancement of the superconducting gaps with increasing interband coupling, one can observe a clear change of the behavior of the lower superconducting gap near the critical temperature.

Figure 13

Temperature evolution of (a) gap functions ${\Delta }_1$, ${\Delta }_2$, and ${\Delta }_3$ (neglecting the contribution of the Josephson tunnellings); (b) interband and Josephson couplings; and (c) superconducting phases $\varphi$ and $\alpha$ of the rhombus circuit with TBSs at the spinal positions and in the absence of magnetic field. In (a), the two-band gaps ${\Delta }_1$ and ${\Delta }_2$ vanish at the same critical temperature $T_{c,12}$ which we chose larger than the critical temperature of the OBSs, $T_{c3}\approx 0.77T_{c,12}$ (indicated by the vertical dashed line). With increasing temperature, second-order phase transitions are observed in the superconducting phases plot, first from a nonchiral state to a chiral state and second from a chiral state to a nonchiral state. In these plots, the pairing constants for the TBSs are $V_{12}=.01$ and $V_{22}=.88$ in units of $V_{11}$ and for the OBSs $V_3=.95V_{11}$. The Josephson couplings between the two-band and the one-band superconductors are such that $J_1\left(0\right)\approx 0.08J_{12}\left(0\right)$ and $J_2\left(0\right)\approx 0.12J_{12}\left(0\right)$.