Marcel Franz

Professor

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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2019)
Exotic phenomena in topological states of matter (2014)

Electronic states in band insulators and semimetals can form nontrivial topological structures which can be classified by introducing a set of well defined topological invariants. There are interesting experimentally observable phenomena tied to these topological invariants which are robust as long as the invariants remain well-defined. One important class manifesting these topological phenomena in the bulk and at the edges is the time reversal invariant topological band insulators first discovered in HgTe in 2007. Since then, there have been enormous efforts from both the experimental and the theoretical sides to discover new topological materials and explore their robust physical signatures. In this thesis, we study one important aspect, i.e., the electromagnetic response in the bulk and at the spatial boundaries. First we show how the topological action, which arises in a time reversal invariant three dimensional band insulator with nontrivial topology, is quantized for open and periodic boundary conditions. This confirms the Z2 nature of the strong topological invariant required to classify time-reversal invariant insulators. Next, we introduce an experimentally observable signature in the response of electronic spins on the surface of these materials to the perpendicular magnetic field. We proceed by considering electromagnetic response in the bulk of topological Weyl semimetals in a systematic way by considering a lattice model and we address important questions on the existence or absence of the Chiral anomaly. In the end, we show how a topological phase in a one dimensional system can be an energetically favourable state of matter and introduce the notion of self-organized topological state by proposing an experimentally feasible setup.

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A study of topological insulators in three dimensions (2012)

In this work we study four interesting effects in the field of topological insulators: the Witten effect, the Wormhole effect, topological Anderson insulators and the absence of bulk magnetic order in magnetically doped topological insulators. According to the Witten effect, a unit magnetic monopole placed inside a medium with non zero θ is predicted to bind a fractional electric charge. We conduct a first test of the Witten effect based on the recently established fact that the electromagnetic response of a topological insulator is given by the axion term θ(e²/2πh)B·E, and that θ=π for strong topological insulators. We establish the Wormhole effect, in which a strong topological insulator, with an infinitely thin solenoid of magnetic half flux quantum carries protected gapless and spin filtered one-dimensional fermionic modes, which represent a distinct bulk manifestation of the topologically non-trivial insulator.We demonstrate that not only are strong topological insulators robust to disorder but, remarkably, under certain conditions disorder can become fundamentally responsible for their existence. We show that disorder, when sufficiently strong, can transform an ordinary metal with strong spin-orbit coupling into a strong topological ‘Anderson’ insulator, a new topological phase of matter in three dimensions.Finally, we lay out the hypothesis that a temperature window exists in which the surface of magnetically doped topological insulators is magnetically ordered but the bulk is not. We present a simple and intuitive argument why this is so, and a mean-field calculation for two simple tight binding topological insulator models which shows that indeed a sizeable regime such as described above could exist. This indicates a possible physical explanation for the results seen in recent experiments.

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A study of topological insulators in two and three dimensions (2012)

In this work we introduce a new model for a topological insulator in both two and three dimensions, and then discussthe possibility of creating a fractional topological insulator in the three dimensional version. We also explore the possibilityof engineering a quantum spin Hall phase in Graphene through the application of heavy metal adatoms.We show that the two dimensional model on the Lieb lattice, and its three dimensional counterpart, the so called edge centered cubic or Perovskite lattice, possess non-trivial Z₂ invariants, and gapless edge modes, which are the signatures of the topological insulating state.Having established that these lattices can become topological insulators, we then tune several short range hoppings in the model and showthat it is possible to flatten the lowest energy bands in each case. After flattening the bands we add in a Hubbard term and then use a mean field decoupling to show that there is a portion of phase space where the system remains non-magnetic, and then conjecture that the many body ground state in three dimensions could become a fractional topological insulator.For the model on graphene, we start by using density functional theory (DFT) to find a pair of suitable heavy elements that are non-magnetic, have a strong spin orbit coupling and prefer to sit at the center of the hexagonal lattice. We then establish that for adatoms distributed in a periodic configuration, again with DFT, that a gap will open at the Dirac point in the presence of spin orbit coupling. To prove the gap is topologically non-trivial, we show that it is possible to adiabatically connect this model to the original Kane-Mele model, a known topological insulator. Lastly, we show that for adatoms distributed randomly the effect survives.

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Master's Student Supervision (2010 - 2018)
Charge transport in a graphene flake realization of the Sachdev-Ye-Kitaev model (2018)

We address the transport properties of a mesoscopic realization of the Sachdev-Ye-Kitaev (SYK) model which is an exactly solvable system of interacting spinless fermions connected to the black hole physics through the holographic principle. Starting with a recent proposal for simulating the SYK model in a graphene flake in an external magnetic field and extending it by considering leads attached to it, we model a realistic transport experiment and calculate directly measurable quantities featuring non-Fermi liquid signatures of the SYK physics. We show that the graphene flake realization is robust in the presence of leads and that measuring the tunneling current across the leads one can experimentally observe a non-Fermi liquid - Fermi liquid transition by tuning the external magnetic field threading the flake. After establishing the transport signatures of the SYK model near equilibrium using linear response framework, we then derive a formula to extend our results for tunneling current using Keldysh formalism to explore the effects of finite bias voltage across the leads, going beyond equilibrium.

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Tricritical Ising edge modes in a Majorana-Ising ladder (2017)

While Majorana fermions remain at large as fundamental particles, they emerge in condensed matter systems with peculiar properties. Grover et al. proposed a Majorana-Ising chain model, or the GSV model, where the system undergoes a tricritical Ising transition by tuning just one parameter. In this work, we generalize this model to a ladder with inter-chain Majorana couplings. From a mean field analysis, we argue that the tricritical Ising transition will also occur with inter-chain couplings that allow the system to be gapless in the non-interacting case. More crucially, based on analysis of the interacting chain model and the non-interacting ladder model, we expect the tricritical Ising modes to appear on the edges, a feature that might persist when going to 2d. We carry out extensive DMRG calculations to verify the theory in the ladder model. Finally, we discuss possible numerical probes of a 2d model.

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Majorana bands in topological superconductors (2015)

Majorana fermions can exist in condensed matter systems as quasi-particle excitations called Majorana bands. The details of Majorana bands will be the central concern of this thesis. In the thesis, Majorana bands are studiedanalytically and numerically in two square lattice systems with vortices. The p+ip superconductor, containing two vortices in each magnetic unit cell, exhibits slightly dispersing Majorana bands in the middle of the superconducting gap. With the same vortex geometry, the Fu-Kane model shows similar Majorana bands, which, however, can become completely flat when chemical potential is tuned to coincide with the Dirac point. By comparison to a tight binding model of vortex lattice, it is clear that the dispersion is mainly contributed by first and second nearest neighbor hoppings of Majorana fermions bound in vortices. The hoppings, which are extracted from numerical diagonalization, are not quite identical to the existing analytical prediction. Therefore, we built two simple equations that show the phenomenologically correct trends of the hoppings.

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Axion term in topological insulators with broke time reversal and parity (2014)

No abstract available.

Emergence and stability of Majorana fermions in proximity-coupled topological insulator nanowires (2012)

A finite-length topologicalinsulator nanowire, proximity-coupled to an ordinary bulk s-wavesuperconductor and subject to a longitudinal applied magnetic field is shown to realize a one-dimensional topological superconductor with an unpairedMajorana fermion localized at each end of the nanowire. Here we also showthat the unpaired Majorana fermions persist in this system for anyvalue of the chemical potential inside the bulk band gap of order 300meV in Bi₂Se₃ by computing the Majorana number. From this calculation, wealso show that the unpaired Majorana fermions persist when themagnetic flux through the nanowire cross-section deviatessignificantly from half flux quantum. Lastly, we demonstrate that theunpaired Majorana fermions persist in strongly disordered wires withfluctuations in the on-site potential ranging in magnitude up to thesize of the bulk band gap. These results suggest this solid-statesystem should exhibit unpaired Majorana fermions under accessible conditions likely important for experimental study or future applications.

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