Classification of ground states and normal modes for phase-frustrated multicomponent superconductors

We classify ground states and normal modes for $n$-component superconductors with frustrated intercomponent Josephson couplings, focusing on $n=4$. The results should be relevant not only to multiband superconductors, but also to Josephson-coupled multilayers and Josephson-junction arrays. It was recently discussed that three-component superconductors can break time-reversal symmetry as a consequence of phase frustration. We discuss how to classify frustrated superconductors with an arbitrary number of components. Although already for the four-component case there are a large number of different combinations of phase-locking and phase-antilocking Josephson couplings, we establish that there are a much smaller number of equivalence classes where properties of frustrated multicomponent superconductors can be mapped to each other. This classification is related to the graph-theoretical concept of Seidel switching. Numerically, we calculate ground states, normal modes, and characteristic length scales for the four-component case. We report conditions of appearance of new accidental continuous ground-state degeneracies.

Figure 1

Switching classes of complete graphs on four vertices with edges colored red (dashed lines, attractive coupling) and blue (solid lines, repulsive coupling). The uppermost switching class corresponds to the unfrustrated signatures, the central class to the singly frustrated signatures, and the lowermost class to the multiply frustrated signatures.

Figure 2

The ground-state value of the (generalized) coordinate $x_i$ as a (potentially multivalued) function of the parameter $\alpha$. By assumed continuity, we have that ${\partial }_i^{2}U=0$ at the critical point. Thus, there exists a massless mode at this point.

Figure 3

Examples of ground-state phase configurations for singly frustrated signatures with discriminatory couplings somewhat stronger than the critical values. There is ${\mathbb{Z}}_2$ degeneracy corresponding to complex conjugation of the ${\psi }_i$. The corresponding singly frustrated phase configurations are obtained by reducing the marked angles to zero.

Figure 4

Ground states (cf. Fig. 3), (inverse) length scales, and normal modes for signature 7 with ${\alpha }_i=-1$, ${\beta }_i=1$, and $|{\eta }_{ij}|=1$ for $ij\ne 12$. ${\eta }_{12}$ is plotted on the $x$ axes. Note that there is a second massless mode at the critical point ${\eta }_{12}={\eta }_{12}^{\mathrm{c}}=1.21$. Also, several of the modes are mixed phase-density modes when ${\eta }_{12}>{\eta }_{12}^{\mathrm{c}}$, whereas none of the modes are mixed phase-density modes when ${\eta }_{12}<{\eta }_{12}^{\mathrm{c}}$.

Figure 5

Parametrizations of the three relevant degrees of freedom in the phases.

Figure 6

Examples of ground-state phase configurations for multiply frustrated signatures with complete intercomponent symmetry. The phases are pairwise locked or antilocked. The phase rotations corresponding to alteration of the angle $\gamma$ are energetically free.

Figure 7

Classification of ground states for signature 11 under the assumption of pairwise ground-state equivalence of phases. In region A there is no TRSB, in region B there is TRSB, and in region C there are energetically free phase rotations and thus degeneracy between states with and without TRSB. Representative ground states are shown for the three regions. We set ${\tilde{\eta }}_{13}={\tilde{\eta }}_{14}={\tilde{\eta }}_{23}={\tilde{\eta }}_{24}=1$. Along the curve ${\tilde{\eta }}_{12}{\tilde{\eta }}_{34}=1$, the situation is more complicated than this classification suggests (Fig. 11).

Figure 8

Ground states, (inverse) length scales, and normal modes for signature 11 with ${\alpha }_i=-1$, ${\beta }_i=1$, and ${\eta }_{ij}=-1$. The rotational angle $\gamma$ (Fig. 6) is plotted on the $x$ axes. Note that there is a third massless mode at the points $\gamma =0$ and $\pi$. Also, these points are the only points for which there are no mixed phase-density modes.

Figure 9

Illustration of ground-state phase configurations for signature 11 with complete intercomponent symmetry. We assume that ${\varphi }_1=0$. There are three possible pairwise antilockings, corresponding to the three connected circles. In each circle, the displayed phase configuration corresponds to the black dot; other equivalent possibilities are given by the other dots. Points marked with squares coincide. Permutation of the three unfixed phases corresponds either to moving within the set of black and gray (dark) dots, or to moving within the set of white dots. Complex conjugation corresponds to moving between the set of dark dots and the set of white dots. There are two types of special cases in which the number of equivalent states is smaller than in the typical case.

Figure 10

Ground states, (inverse) length scales, and normal modes with parameters as in Fig. 8 except that ${\eta }_{12}=-2$. The rotational angle $\gamma$ (Fig. 6) is plotted on the $x$ axes. Note that there is no third massless mode at the points $\gamma =0,\pi$. Also, the modes corresponding to pure density excitations (modes 5 and 7) do not have the same length scales.

Figure 11

Sizes of the ranges over which ground-state phase differences can be varied for signature 11 under the assumption that ${\varphi }_1$ and ${\varphi }_2$ are equivalent in the ground states (${\tilde{\eta }}_{13}={\tilde{\eta }}_{23}=:{\tilde{\eta }}_1$ and ${\tilde{\eta }}_{14}={\tilde{\eta }}_{24}=:{\tilde{\eta }}_2$), and that ${\tilde{\eta }}_1{\tilde{\eta }}_2-{\tilde{\eta }}_{12}{\tilde{\eta }}_{34}=0$. We set ${\tilde{\eta }}_2=1$. The curve ${\tilde{\eta }}_{12}{\tilde{\eta }}_{34}=1$ coincides with the corresponding curve in Fig. 7.

Figure 12

Ground states, (inverse) length scales, and normal modes for signature 11 with ${\alpha }_i=-1$, ${\beta }_i=1$, and ${\eta }_{23}={\eta }_{24}={\eta }_{34}=-1$. The magnitude of the parameter $\eta :={\eta }_{12}={\eta }_{13}={\eta }_{14}$ is plotted on the $x$ axes. There are continuous ground-state degeneracies for a range of values of $\eta$. Note that for $\eta =-1$ the situation illustrated here is equivalent to that for $\gamma =\pi /2$ in Fig. 8; for this value of $\eta$, there are energetically free phase rotations. For some $\eta \ne -1$ there are other types of continuous degeneracies; $\eta =-0.5$ corresponds to Fig. 13 and $\eta =-1.5$ corresponds to Fig. 14.

Figure 13

Ground states, (inverse) length scales, and normal modes with ${\alpha }_i=-1$, ${\beta }_i=1$, ${\eta }_{23}={\eta }_{24}={\eta }_{34}=-1$, and $\eta :={\eta }_{12}={\eta }_{13}={\eta }_{14}=-0.5$ (cf. Fig. 12). The phase difference $-{\varphi }_{12}$ is plotted on the $x$ axes. This phase difference can change by $2\pi$ without cost in potential energy. This can be seen from Fig. 11 via either of the relabelings $1\leftrightarrow 3$ or $1\leftrightarrow 4$. Note that the state with ${\varphi }_1={\varphi }_2$ is equivalent to that with ${\varphi }_1={\varphi }_4$; thus, we could have narrowed the displayed range of ${\varphi }_{12}$ without loss of information.

Figure 14

Ground states, (inverse) length scales, and normal modes with ${\alpha }_i=-1$, ${\beta }_i=1$, ${\eta }_{23}={\eta }_{24}={\eta }_{34}=-1$, and $\eta :={\eta }_{12}={\eta }_{13}={\eta }_{14}=-1.5$ (cf. Fig. 12). The phase difference $-{\varphi }_{12}$ is plotted on the $x$ axes. This phase difference can vary from $1.62$ to $4.67$ (the displayed range) without cost in potential energy. This can be seen (approximately) from Fig. 11 via either of the relabelings $1\leftrightarrow 3$ or $1\leftrightarrow 4$. Note that, due to the equivalence of ${\psi }_2,{\psi }_3$, and ${\psi }_4$, the above graphs consist of three equivalent segments (e.g., the state with ${\varphi }_2={\varphi }_3$ is equivalent to that with ${\varphi }_2={\varphi }_4$).

Figure 15

Ground states, (inverse) length scales, and normal modes with ${\alpha }_i=-1$, ${\beta }_i=1$, and ${\eta }_{12}=1$ in (23). We plot $-{\kappa }_{12}$ on the $x$ axes. Note that for ${\kappa }_{12}<-0.2$ there is phase-density mode mixing.