Topological States and Adiabatic Pumping in Quasicrystals

Left to right: Kobi E. Kraus, Oded Zilberberg, Yoav Lahini, Zohar Ringel and Mor Verbin

Authors: 
Yaacov E. Kraus¹, Yoav Lahini^2,3, Zohar Ringel¹, Mor Verbin², Oded Zilberberg¹

Affiliation:
¹Dept. of Condensed Matter Physics, Weizmann Institute of Science, Israel
²Dept. of Physics of Complex Systems, Weizmann Institute of Science, Israel
³Department of Physics, Massachusetts Institute of Technology, USA

The materials that make up our world have a variety of electrical properties. Some materials, such as metals, conduct electricity extremely well, while others are insulators, and are very efficient as shields from electric currents.

Recently, a new discovery revolutionized the prevailing paradigm of electrical properties of materials, when a new type of material was discovered [1, 2]. These materials are termed “topological insulators”, and have very unique electrical properties. For example, electricity would flow smoothly on the surface of a topological insulator, while the interior will be completely insulating. Interestingly, if one would cut this material in half, the new surface that is created, which was previously buried within the insulating interior, will suddenly become conducting. If the material is cut repeatedly, the same will happen each time.

Past 2Physics article by Yoav Lahini:
November 07, 2010: "Hanbury Brown and Twiss Interferometry with Interacting Photons"
by Yoav Lahini, Yaron Bromberg, Eran Small and Yaron Silberberg

In addition to this peculiar property, the electrical behavior on the surface itself reveals unique phenomena that are even expected to simulate bizarre new particle excitations [3]. As a result, these fascinating materials generated much activity in the condensed matter physics community, in an attempt to find new topological materials and to study their intriguing properties.

In a recent paper [4], we found that other unconventional materials, known as quasicrystals, are in fact also members of the topological materials family. Moreover, the topological behavior that they exhibit is similar to that of usual topological materials in some aspects, but differs from them in others.

Quasicrystals are materials in which the atoms are arranged in a distinct way. In most solid materials, the atoms are arranged in space either periodically or in a completely random fashion. Quasicrystals are an intermediate type of solid - they are neither periodic nor random. Rather, there is some non-repeating (i.e. non periodic) but well defined rule to the arrangement of their atoms [5, 6]. Despite the fact that quasicrystals have been experimentally observed already in 1982 [5], for a long time there was a debate between crystallographers whether they exist at all, as it was assumed that all crystalline materials are necessarily periodic. The conclusion that quasicrystals are a new type of solid revolutionized material science, updated the physical definition of what is a crystal, and culminated in the awarding of the Nobel Prize in Chemistry to its discoverer, Dan Shechtman from the Technion - Israel Institute of Technology [7].

Yet, many of the physical properties of quasicrystals, such as their electrical conductance, are not fully understood. The work recently published by our group in Physical Review Letters [4], discusses the electrical properties of surfaces of quasicrystals, and finds a new and surprising connection between quasicrystals and topological states of matter. Specifically, we show that a one-dimensional quasicrystal behaves, to some extent, like two-dimensional topological matter known as quantum Hall systems. We prove this claim theoretically and measure it experimentally.

The experiments were done on a novel type of quasicrystals, known as photonic quasicrystals [8, 9]. These systems are made of quasi-periodic arrangements of transparent materials, rather than atoms. In these systems, one studies the optical properties, rather than the electrical, but the underlying physics is very much the same. A major advantage of using photonic quasicrystals is the ability to fabricate one-dimensional materials, and to directly image the propagation of light within them.

In our experiments, we have realized a one-dimensional photonic quasicrystal, and measured the boundary (the surface of a one-dimensional system) properties of these quasicrystals. We found that the photonic states that reside at the boundary are localized -- meaning that light that is injected to that boundary will stay there. This is analogous to the electric currents on the surface of topological matter, which do not penetrate the interior of the material, but remain confined to the surface. This finding was surprising, as common wisdom was that -- generally, such a behavior is not supposed to occur in one-dimensional systems.

Our theory explains how that becomes possible in quasicrystals. In brief, the arrangement of atoms in a quasicrystal can be mathematically described as some type of projection of a periodic system on a system of lower dimension – for example, projection of a two-dimensional square lattice onto a one-dimensional line [10]. Note that this description defines the position of the atoms of the quasicrystal, but do not imply the properties of any electrons (or photons) moving through it. In our case, the one-dimensional quasicrystalline models we worked with can be described as another type of one-dimensional projection of a quantum Hall system, known as “dimensional reduction” [3]. Most importantly, the novel projection used to define our one-dimensional quasicrystals preserves the topological properties! Thus, we find that beyond their mere structure, quasicrystals can, in some sense, also “inherit” nontrivial topological properties from their higher-dimensional periodic “ancestors”.

Taking things a step forward, we have shown that the boundary states observed in the experiments indeed possess nontrivial topological properties, by demonstrating a topological “pumping” of light from one side of the quasicrystal to the other [4].

Figure 1: Experimental observation of adiabatic pumping via topologically protected boundary states in a photonic quasicrystal. (a) An illustration of the adiabatically modulated photonic quasicrystal, constructed by slowly varying the spacing between the waveguides along the propagation axis z. Consequently, the injected light is pumped across the sample. (b) Experimental results: Light was injected into the rightmost waveguide. The measured intensity distributions as a function of the position are presented at different stages of the adiabatic evolution, i.e., different propagation distances. It is evident that along the adiabatic evolution the light crossed the lattice from right to left.

This fascinating discovery appears to be just the beginning. Our results suggest that additional quasicrystals should exhibit topological states [11, 12], and that these states will always be linked to systems of a higher dimension. This approach might mean that three-dimensional quasicrystalline materials -- either photonic or electronic -- would exhibit strange surface properties, which can be explained as originating from a six-dimensional topological system. These subjects are currently under active investigation.

References:
[1] “Colloquium: Topological Insulators”, M.Z. Hasan and C.L. Kane, Reviews of Modern Physics, 82, 3045 (2010). Abstract.
[2] “Topological insulators and superconductors”, Xiao-Liang Qi and Shou-Cheng Zhang , Reviews of Modern Physics, 83, 1057 (2011). Abstract.
[3] “Topological field theory of time-reversal invariant insulators”, Xiao-Liang Qi, Taylor L. Hughes, and Shou-Cheng Zhang, Physical Review B 78, 195424 (2008). Abstract.
[4] "Topological States and Adiabatic Pumping in Quasicrystals”, Yaacov E. Kraus, Yoav Lahini, Zohar Ringel, Mor Verbin, and Oded Zilberberg, Physical Review Letters, 109, 106402 (2012). Abstract.
[5] “Metallic Phase with Long-Range Orientational Order and No Translations Symmetry”, D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn, Physical Review Letters, 53, 1951 (1984). Abstract.
[6] "Quasicrystals: A New Class of Ordered Structures", Dov Levine and Paul Joseph Steinhardt, Physical Review Letters, 53, 2477 (1984). Abstract.
[7] See http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/press.html .
[8] “Wave and defect dynamics in nonlinear photonic quasicrystals”, Barak Freedman, Guy Bartal, Mordechai Segev, Ron Lifshitz, Demetrios N. Christodoulides and Jason W. Fleischer, Nature, 440, 1166 (2006). Abstract.
[9] “Observation of a Localization Transition in Quasiperiodic Photonic Lattices”, Y. Lahini, R. Pugatch, F. Pozzi, M. Sorel, R. Morandotti, N. Davidson and Y. Silberberg, Physical Review Letters, 103, 013901 (2009). Abstract.
[10] “Algebraic theory of Penrose's non-periodic tilings of the plane”, N.G. de Bruijn, Kon. Nederl. Akad. Wetensch. Proc. Ser. A (1981).
[11] “Topological Equivalence Between The Fibonacci Quasicrystal and The Harper Model”, Yaacov E. Kraus and Oded Zilberberg, Physical Review Letters, 109, 116404 (2012). Abstract.
[12] “Observation of Topological Phase Transitions in One-Dimensional Photonic Quasicrystals”, M. Verbin, Y. E. Kraus, O. Zilberberg, Y. Lahini and Y. Silberberg, in preparation.