Marcel Franz


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Graduate Student Supervision

Doctoral Student Supervision (Jan 2008 - May 2021)
Dirac materials and the response to elastic lattice deformation (2019)

Dirac materials have formed a thriving and prosperous direction in modern condensed matter physics. Their bulk bands can linearly attach at discrete points or along curves, leading to arc or drumhead surface states. The candidate Dirac materials are exemplified by Dirac/Weyl semimetals, Dirac/Weyl superconductors, and Dirac/Weyl magnets. Owing to the relativistic band structure, these materials have unique responses to the applied elastic crystalline lattice deformation, which can induce pseudo-magnetic and pseudo-electric fields near the band crossings and produce transport distinguished from that caused by ordinary magnetic and electric fields. In this dissertation, I will demonstrate the exotic transport due to the strain-induced gauge field in Weyl semimetals, Weyl superconductors, and Weyl ferromagnets. I will first elucidate that a simple bend deformation can induce a pseudo-magnetic field that can give rise to the Shubnikov-de Haas oscillation in Weyl semimetals. Then I will elaborate that strain can Landau quantize charge neutral Bogoliubov quasiparticles as well and result in thermal conductivity quantum oscillation in Weyl superconductors. Lastly, I will consider the strain-induced gauge field beyond the fermionic paradigm and explain various quantum anomalies of magnons in Weyl ferromagnets.

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Realizing high-energy physics in topological semimetals (2019)

The discovery of topological phases of matter has brought high-energy and condensed matter communities together by giving us shared interests and challenges. One fruitful outcome is the broadened range of possibilities to study high-energy physics in cost-effective table-top experiments. I have investigated scenarios in which influential high-energy ideas emerge in solid-state systems built from topological semimetals – gapless topological phases which have drawn intense research efforts in recent years. My Thesis details three proposals for realizing Majorana fermions, Adler-Bell-Jackiw anomaly, and holographic black holes in superconductor-Weyl-semimetal heterostructures, mechanically strained Weyl semimetal nanowires/films, and graphene flakes subject to strong magnetic fields, respectively. By analyzing the effects of realistic experimental conditions, I wish to demonstrate that these proposals are experimentally tangible with existing technologies.

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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 - 2020)
Majorana bound states on a periodic superconducting vortex lattice (2020)

Majorana quasi-particles can exist as zero-energy excitations bound to vortices present on the surface of a topological insulator that is proximity-coupled to a type-II superconductor. Such a system finds its natural realisation on the surface of the iron-based superconductor FeTe₀.₅₅Se₀.₄₅ which has been identified as a potential topological superconductor and is expected to host Majorana quasi-particles as zero-energy excitations on vortices. This thesis aims to explain the occurrence of Majorana vortex bound states present on such materials by constructing a model for a vortex lattice on periodic manifolds. A two-dimensional square lattice model is developed to capture the low energy physics of the surface states of a topological insulator proximity-coupled to an s-wave superconductor. Using the Franz-Tesanovic singular gauge transformation, multiple vortices can be incorporated in the system by circumventing the problem of branch cuts. To construct finite vortex lattice on periodic manifolds (torus in two dimensions), we need to account for the fluxes through the non-contractible loops of the torus, leading to certain correction terms appearing in the singular gauge transformation. This thesis provides the details of the construction of vortex lattice on a periodic manifold on the surface of topological insulator proximity-coupled to a superconductor and investigates the Majorana zero modes present on the vortices. The theory developed for periodic vortex lattice on the topological insulator-superconductor heterostructure is successful in replicating the experimental observations of Majorana vortex bound states on the surface of the iron-based superconductor FeTe₀.₅₅Se₀.₄₅.

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Chern insulators for electromagnetic waves in electrical circuit networks (2019)

Periodic networks composed of capacitors and inductors have been demonstrated to possess topological properties with respect to incident electromagnetic waves. In this thesis, we develop an analogy between the mathematical description of waves propagating in such networks and models of Majorana fermions hopping on a lattice. Using this analogy we propose simple electrical network architectures that realize Chern insulating phases for electromagnetic waves. Such Chern insulating networks have a bulk gap for a range of signal frequencies that is easily tunable and exhibit topologically protected chiral edge modes that traverse the gap and are robust to perturbations. The requisite time reversal symmetry breaking is achieved by including a class of weakly dissipative Hall resistor elements whose physical implementation we describe in detail.

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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)

The main subject of this work is the axion term in the effective electromagnetic action of topological insulators, which is responsible for the special electromagnetic properties of these materials. The axion term is characterized by a parameter θ, which can only take the values of 0, for regular insulators, or π, for topological insulators, respecting at least one of the time reversal or parity symmetries. A non zero axion term leads to a variety of measurable phenomena, generally referred to as the magneto-electric effects.We focus our interest on the value the parameter θ takes for a topological insulator, when both time reversal and parity are broken. In this case θ no longer must be quantized to 0 or π. We use a lattice model for a topological insulator, and introduce a symmetry breaking term in the Hamiltonian. We numerically find the value of θ in this case using calculations of the magnitude of various magneto-electric effects. The results are compared to the theoretical prediction. We find that θ is no longer quantized when a specific symmetry breaking term is introduced.

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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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Current Students & Alumni

This is a small sample of students and/or alumni that have been supervised by this researcher. It is not meant as a comprehensive list.

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