In this Blog, every month I will comment about the physics paper that attacheted particularly my attention.

A WORLD PREMIER ON quantum fractal

Fractals, first named by mathematician Benoit Mandelbrot [1] in 1975, are never-ending complex patterns that look the same whether seen from far or close. These non-regular patterns show the same degree of non- regularity on all scales. The Romanesco broccoli, the nautilus shell or the network of veins are perfect natural examples that show the self-repeating nature of fractals. Although nothing in nature is really infinite in that way, we can use fractals to model them. Now, reporting in Nature Physics, Kempkes at al. [2] present the first spec- troscopic characterisation of the electronic wave function inside a confined structure with fractal geometry. 

The Lieb lattice goes condensed matter

Verifying condensed matter theoretical predictions requires the full experimental control and the tunability of the system parameters. However, in condensed matter, a universal simulator does not exist. An alternative strat- egy is to mimic the system with cold atoms trapped in a periodic optical potential [1]. This approach realizes Richard Feynman’s pioneering idea of "quantum simulation" — using one quantum system to investigate another one [2].

Last movember on the arXiv appered to inretesting articles where a complex lattice structures were realized in a completely condensed matter systems; the two reports are by Robert Drost et al. [3] and Marlou Slot et al. [4]. Both describe two different approaches for realizing a fermionic Lieb lattice; this is a bi-partite face-centered square lattice characterized by three sites in the unit cell with unequal connectivity. This structure leads to a peculiar property to the spectrum of the system — a Dirac cone crossed by a flat band at zero energy. We can link the flat band to the topology of the lattice; in analogy to the "airport" system, the central lattice site has four next neighbors (Hub site) whereas the remain- ing two sites have only two next neighbor (Rim sites). This topology does not allow to move directly between rims without passing through a hub. This structure allows for a zero energy solution associated with a wave function that is finite and has opposite phase on the two rim sites and zero on the hub one; besides this peculiar localized solution is not related to the disorder. Contrary to the Dirac cones of graphene, a three-dimensional sys- tem characterized by a Dirac cone and a flat band in its middles represents a system of particles that has no equivalent in high-energy physics [5]. So far, the Lieb lattice was realized only in photonic lattices [6, 7].

In the contribution by Robert Drost et al. [3], the Lieb lattice is realized by using a new technique developed by the group of Sander Otte at TU Delft [8]. Here, they consider a chlorine lattice placed on top of a Cu(100) surface. Vacancies in the chlorine structure act as lattice points [3]. In addition to the Leib lattice, with the same technique Drost et al. have also realized an SSH model [9] as a test case of the method [3]. This model is the simplest of the topological models that we know in condensed matter. Initially, the SSH model was developed to account for solitons in polyacetylene [9]. A first simulation of the SSH model without polyacetylene was obtained in the framework of one-dimensional optical lattices [10]. In this respect, results presented by Drost et al. [3] represents the first fully condensed matter simulation of this model.

In the case of the work by Slot et al. [4], the Lieb lattice is achieved with the so-called molecular-graphene technique [11]. Here, CO molecules are placed on top of a two-dimensional electron gas (2DEG) formed in the Cu(111) surface. The molecules are arranged at lattice sites of the dual lattice of the planned structure. The electrons in the Cu(111) 2DEG see an effective electrostatic potential created by the CO molecules. This approach makes it easy to realize lattices that are convex uniform tilings, i.e. showing a dual lattice. We can obtain a dual lattice of one structure by creating a new structure, which has its sites placed at the center of each plaquette of the initial lattice system. For example, the honeycomb and the triangular lattices are one the dual of the other [11]. Although the Lieb lattice does not have a dual, Slot et al. [4] obtained an optimal geometrical configuration of the CO molecules so to resemble a suitable electrostatic potential profile as close as possible to the Lieb lattice. The complexity of this geometrical configuration is increased by the different symmetry of the Cu(111) substrate and the Lieb lattice: triangular and square, respectively.

A unique feature of the flat band state is that the wave function is localized to the rim sites and is zero on the hub one [12]. In both the experiments discussed here, the tip of a scanning tunneling microscope (STM) is used for realizing the Lieb lattices and also for measuring the wave function on the different lattice sites at various energies. As predicted by the theory, in the case of the Dirac cone, the wave function is finite on all three lattice sites composing the system unit cell. On the contrary, the wave function of the flat band is finite only on the two rim sites and zero on the hub one. The two teams have verified these properties in both experimental approaches.

For future research two main points can be envisioned: Firstly, when the coherence length is long enough, com- plex junctions will make it possible to observe super-Klein tunneling and super-collimation predicted for this type of systems [13, 14]. Secondly, the use of a superconducting host material in place of Cu(111) or Cu (100) could pave the way to study lattice systems of Cooper pairs. In this way, Feynman’s vision could be entirely accomplished in condensed matter systems.

[1] M. Greiner and S. Folling, Nature 453, 736 (2008).
[2] R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982).
[3]  R. Drost, T. Ojanen, A. Harju, and P. Liljeroth,  (2016).
[4]  M. Slot, et al., (2016).
[5]  B. Bradlyn, et al., Science 355, 5037 (2016).
[6]  R. Vicencio, et al., Phys. Rev. Lett.
114, 245503 (2015).
[7]  S. Mukherjee, et al., Phys. Rev. Lett.
114, 245504 (2015).
[8]  F. E. Kalff, et al., Nat. Nanotechnol. ,
926 (2016).
[9]  W. P. Su, J. R. Schrieffer, and A. J. Heeger, Phys. Rev. Lett.
42, 1698 (1979).
[10] M. Atala, et al., Nature Phys.
9, 795 (2013).
[11] K. K. Gomes, W. Mar, W. Ko, F. Guinea, and H. C. Manoharan, Nature
483, 306 (2012).
[12] N. Goldman, D. F. Urban, and D. Bercioux, Phys. Rev. A 83, 063601 (2011).
[13] D. F. Urban, D. Bercioux, M. Wimmer, and W. Häusler, Phys. Rev. B 84, 115136 (2011).
[14] A. Fang, Z. Q. Zhang, S. G. Louie, and C. T. Chan, Phys. Rev. B
93, 035422 (2016). 

© Dario Bercioux 2018