**3. Conclusion**

We analyzed the crossover from the Φ<sup>0</sup> periodic persistent currents as a function of magnetic flux in a metallic loop to the Φ0/2 periodic supercurrent in the groundstate of the loop. We considered conventional *s*-wave pairing in a one-dimensional as well as in a multi-channel annulus. Although a one-dimensional superconducting ring is a rather idealized system, it proves valuable for discussing the physics of this crossover, which includes the emergence of a new minimum in the free energy for odd center-of-mass angular momenta *q* of the Cooper pairs and the restoration of the flux periodicity of the free energy. The physical concepts, which we illustrated in a simplified form in section 2.2, remain thereby valid even in the more complex context of the self consistent calculations on the annulus.

In the superconducting state, a distinguished minimum in the free energy develops at *ϕ* = *q*/2. Choosing the proper value for *q* at each flux value leads to a series of minima at integer and half-integer flux values which, however, differ in energy for finite systems. In rings with a radius smaller than half the superconducting coherence length, the two electrons forming a Cooper pair are not forced to circulate the ring as a pair, and the supercurrent shows a Φ<sup>0</sup> periodicity. Only if the order parameter Δ is larger than the maximal Doppler shift *�*D, the supercurrent is Φ0/2 periodic. This is equivalent to the condition that the maximum flux induced current is smaller than the critical current *J*c. Assuming that the relations obtained from the one-dimensional model remain valid on a ring with finite thickness *R*<sup>2</sup> − *R*<sup>1</sup> � *R*1, as indeed suggested by the multi-channel model, the critical radius to observe Φ0/2 periodicity, *R*<sup>c</sup> = *at*/2Δ, would be of the order of 1 μm for aluminum rings. Within the temperature controlled crossover upon cooling through *T*c, Φ<sup>0</sup> periodicity might by difficult to observe since the differences in the energy spectra for integer and half-integer flux values are exponentially suppressed by temperature. Φ<sup>0</sup> periodicity is therefore only observable if a normal persistent current would be observable at the same temperature if superconductivity was absent.

In the introduction we referred to experiments where flux oscillations with "fractional periodicities", i.e., fractions of Φ0/2, were observed. Among various suggested origins, there is one particularly elegant approach based on a standard two-electron interaction. Consider the order parameter Δ*q*(*ϕ*) for electron pairs with center-of-mass angular momentum *q*. In real space, *q* describes the phase winding of the order parameter Δ(*θ*, *ϕ*) = Δ(*ϕ*)*eiq<sup>θ</sup>* , where *θ* is the angular coordinate in the ring and Δ(*ϕ*) is real. To ensure that Δ(*θ*, *ϕ*) is a single valued and continuous function, *q* must be an integer number. If, however, Δ(*θ*, *ϕ*) is zero somewhere on the ring, it can change sign. Such a sign changing order parameter is modeled as

$$\tilde{\Delta}(\theta,\varphi) = \frac{1}{2} \left[ \Delta\_0(\varphi) + \Delta\_q(\varphi) e^{iq\theta} \right] \xrightarrow{\Delta\_{\tilde{q}} = \Delta\_0} \Delta(\varphi) e^{-iq\theta/2} \cos\left(\frac{q}{2}\theta\right),\tag{23}$$

which displays a phase-winding number *q*/2 if Δ*<sup>q</sup>* = Δ0, and consequently a vanishing supercurrent at the fractional flux value *q*Φ0/4. In momentum space, Δ˜(*θ*, *ϕ*) is represented by the two-component order parameter {Δ0(*ϕ*), Δ*q*(*ϕ*)}. Such a superconducting state is typically referred to as a "pair-density wave" (PDW) state [3], since the real-space order parameter is periodically modulated and therefore *q* can no longer be interpreted as an angular momentum. Agterberg and Tsunetsugu showed within a Ginzburg-Landau approach, that a PDW superconductor indeed allows for vortices carrying a Φ0/4 flux quantum [2, 3].

Based on a microscopic model, it was shown in reference [33] that a PDW state cannot result from an on-site pairing interaction. However, the PDW state can be a stable alternative for unconventional superconductors with gap nodes, specifically for *d*-wave superconductivity as realized in the high-*T*c cuprate superconductors. More work is, however, needed to analyze under which conditions the PDW state indeed develops an energy minimum at multiples Φ0/4, and these minima become degenerate in the limit of large loops. The notion of the absence of coexistence of Cooper pairs with different center-of-mass momenta for conventional superconductors (as, e.g., in the PDW state) justifies the reduction of the sum over *q* in the Hamiltonian (6) in section 2.2 to one specific *q* in order to derive the periodicity crossover in superconducting rings.

Here we mention a further system where a similar mechanism as described above may lead to Φ0/4 flux periodicity: Sr2RuO4. Experimental evidence exists that Sr2RuO4 is a spin triplet *p*-wave superconductor. A triplet superconductor can be represented by the two-component order parameter {Δ↑↑ *<sup>q</sup>*<sup>1</sup> (*ϕ*), Δ↓↓ *<sup>q</sup>*<sup>2</sup> (*ϕ*)}, with the center-of-mass momenta *q*<sup>1</sup> and *q*<sup>2</sup> for the *sz* = 1 and the *sz* = −1 condensates. This realizes a similar situation as for the PDW state where Φ0/4 periodicity is possible [46]. Indeed Jang *et al*. observed recently that the flux through a microscopic Sr2RuO4 ring is quantized in units of Φ0/4 [24].
