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Topological phenomena in condensed matter — where mathematics meets materials science

Direct observation of topological properties of materials is now possible in experiments on mesoscopic samples, including measurement of “winding numbers” — which were until now considered rather abstract properties of the electronic band structure.

In a collaboration with physicists at USC we showed that the winding numbers in the Fibonacci chain, a one-dimensional quasicrystal, are manifested in oscillations of the superconducting order parameter induced by the proximity effect. These topological numbers are related to the Fibonacci numbers — a mathematical curiosity introduced in the 18th century by Leonardo of Pisa, with connections to the golden mean τ=(1+ √5 )/2 known as the “most irrational number”. Fibonacci sequences can be made with atoms by requiring that the distance between neighboring atoms have two values, L (for long) and S (for short) arranged according to a simple rule [1] in a type of ordering termed “quasiperiodic” : LSLLSLSLL….. Rather incredibly, the interatomic distances in some real quasicrystals have been seen to follow this type of sequence (see the copper ad-atoms on a AlMnPd surface in Fig.1b). We propose an experiment which consists of placing a Fibonacci quasicrystal next to superconducting material such as Pb in a ring geometry (fig.1a). The quasicrystal will develop superconductivity due to the proximity effect, and one can probe the gap at each of the sites of the chain with a scanning tunneling microscope tip. Fig. 1c shows the theoretically computed oscillations of the order parameter at the midpoint of two chains of different lengths as a function of a variable ɸ (for phase angle) [2]. When these oscillations are Fourier analyzed, one finds the integer periods: 17,4,9... These are precisely the winding numbers of edge states predicted by theory and they are related to the Fibonacci numbers in a simple way (for details see article). Our explanation of the observed oscillations is: each time an edge state sweeps through the quasicrystal – which it does q times per cycle – it leads to an increase of the superconducting order parameter. As the DOS in Fig. 1d shows, there are many gaps and corresponding edge states near E=EF which all contribute, leading to the complex oscillatory behavior seen in Fig.1c.

Figure 1. a) A ring formed from a Fibonacci chain and a superconductor b) STM image of copper adatoms showing Fibonacci sequence (McGrath et al, https://doi.org/10.1098/rsta.2011.0220) c) Oscillations of the superconducting order parameter computed on the central site of two Fibonacci chains d) Plot of the integrated density of states, showing the main gaps (plateaus) near the Fermi energy (E=0) and their gap labels

[1] Two short chains Cn-1 and Cn-2 can be placed together (concatenated) to form a longer chain Cn = Cn-1 ◦ Cn-2 : if we take C0=L and C1=L, then one gets the series of chains L, S, LS, LSL, LSLLS etc. The chain lengths are 1,1,2,3,5 … and are the Fibonacci numbers computed by the recursion formula Fn=Fn-1+Fn-2
[2] By varying the phase angle between 0 and 2π it is possible to generate all the N allowed Fibonacci sequences of a given number of elements N. In a pioneering experiment on edge states in a photonic Fibonacci chain by Tanese et al (Phys.Rev.Lett. 112 146404 (2014)) a complete set of such chains were fabricated for N=233.


Anuradha Jagannathan


Proximity effect in a superconductor-quasicrystal hybrid ring
Gautam Rai, Stephan Haas, Anuradha Jagannathan
Physical Review B 100, 165121 (2019)