**Author details**

### Loder Florian

18 Superconductors

was estimated that for a ring of a cuprate superconductor with a circumference of ∼ 1 μm,

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

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

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

*iqθ*

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

Δ*q*=Δ<sup>0</sup>

−−−−→ Δ(*ϕ*)*e*

<sup>−</sup>*iqθ*/2 cos

 *q* 2 *θ* 

, (23)

Δ0(*ϕ*) + Δ*q*(*ϕ*)*e*

this ratio is about 1% and should be observable experimentally.

complex context of the self consistent calculations on the annulus.

**3. Conclusion**

was absent.

<sup>Δ</sup>˜(*θ*, *<sup>ϕ</sup>*) = <sup>1</sup>

2  *Experimental Physics VI & Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany*

### Kampf Arno P.

*Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany*

### Kopp Thilo

*Experimental Physics VI, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg, 86135 Augsburg, Germany*
