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Teaching Documents by Leo P B Goutte

An introduction to the simplest form of a general topological invariant. Intended as a teaching d... more An introduction to the simplest form of a general topological invariant. Intended as a teaching document.

The strange physics of quantum me- chanics would theoretically allow a parti- cle to "tunnel" thr... more The strange physics of quantum me- chanics would theoretically allow a parti- cle to "tunnel" through a classically im- passable barrier. A new experiment out of the University of Toronto not only con- firms this phenomenon, but also deter- mines how fast particles can tunnel.

** Completed within the scope of PHYS 321 -- Observational astrophysics and data analysis ** The... more ** Completed within the scope of PHYS 321 -- Observational astrophysics and data analysis **

The 21-cm signal from neutral hydrogen is one of the most promising ways of exploring the dark ages of the universe. Using this signal to understand the universe has gained lots of attention and there exists a wide range of literature and theories about the signal. In 2018, the EDGES collaboration reported the first detection of the 21-cm signal and, surprisingly, their detection was outside the range of the predicted measurements. If confirmed, current models for the dark ages of the universe must be revised to account for this detection. In this paper, we use a Nested Sampling algorithm (PolyChord, Handley Handley (2015)) to perform Bayesian model selection and determine if the data presented supports discarding the current models. We find decisive evidence against current models; they cannot explain the data presented by EDGES.

Fast radio bursts (FRBs) are millisecond long transients with extra-galactic dispersive delays. W... more Fast radio bursts (FRBs) are millisecond long transients with extra-galactic dispersive delays. While some have been found to repeat, most are single-burst events similar to individual pulsar bursts, though with much greater characteristic energy. Despite a limited observing time, the catalogue of verified FRBs is growing rapidly, and with it the inventory of progenitor theories. One such theory, that single-burst FRBs emanate from catastrophic event origins, posits that they would be detectable not only in the radio spectrum, but in the gamma-ray spectrum as well. Here, we conduct a simple analysis of FRB sources in the gamma-ray spectrum using Fermi-LAT data and develop 4 analysis tools - lightcurve, burstsearch, temporal stacking and spatial stacking - to investigate the behaviour of these bursts. Since it is already known that FRBs release a great amount of energy, if we were to find a gamma burst correlation, they could possibly be candidates to most high-energy events in the Universe, providing a powerful tool to probe its extremes and the intervening unknown medium.

Drafts by Leo P B Goutte

We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken ... more We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken Weyl semimetal (WSM). When coupled to a non-magnetic parabolic band, the WSM's chiral arc state lowers in energy and forms, together with a previously extended state, a noticeable spindependent asymmetry in the interface spectrum in the vicinity of the Weyl nodes. We study these effects with a lattice model which we solve numerically on a finite sample and analytically through using an ansatz on an infinite sample. Our ansatz agrees very well with the numerical simulation as it accurately describes the behaviour of the chiral state, from its energy asymmetry to the spin canting at the interface. We find that the tunnelling effectively increases the Fermi arc length, allowing for the presence of interface states beyond the bare Weyl nodes. These additional states may carry current along the interface and their contribution can be detected in the conductance. Across the interface, the spin-independent conductance reproduces the results of an electron tunnelling experiment to reveal the WSM's density of states. Besides conductivity, the effect of tunneling between the WSM and the metallic band can be seen in quantum oscillations experiments which we briefly comment about.

We construct a tool for the calculation and visualization of Moiré bands belonging to the broad f... more We construct a tool for the calculation and visualization of Moiré bands belonging to the broad family of arbitrarily stacked twisted N + M multilayer graphene. We present the underlying k • p model of twisted bilayer graphene and extend it to twisted N + M multilayer

We present and study the Bernevig, Hughes, and Zhang (BHZ) model of a 2-dimensional first- order ... more We present and study the Bernevig, Hughes, and Zhang (BHZ) model of a 2-dimensional first- order topological insulator under an external magnetic field B. The applied gauge field is accounted for by performing a Peierls substitution which modifies the motion of the electrons on the two- dimensional square lattice. We discuss how this model can be used to describe the surface of a three-dimensional, second-order topological insulator (SOTI). We derive an effective continuum theory for the Landau level (LL) energies and wavefunctions in the BHZ model and compare it to numerical lattice calculations. The results are then compared to the SOTI’s continuum surface theory in an attempt to explain three inconsistencies: the deviation of the lowest LL from the surface gap; the asymmetry of ±nth LLs around zero and; the fact that the SOTI’s negative LLs are missing. Moreover, we quantify the effect of an increasing magnetic field on the system’s energies via Hofstadter’s butterfly, considering the various regimes of validity of our previous continuum theories. It is determined that the discrepancies between surface theory and lattice model simulations can be resolved by keeping terms up to O(k2) in the SOTI’s surface theory, which can be thought of as a momentum-dependent mass term.

We obtain and analyze Hofstadter's butterfly for the bulk and the edges of the second-order topol... more We obtain and analyze Hofstadter's butterfly for the bulk and the edges of the second-order topological insulator (SOTI) subject to an external magnetic field. The applied gauge field is accounted for by performing a Peierls substitution t → texp(i A · dr), modifying the motion of the electrons on the lattice. We compare and contrast the lowest Landau level energy predicted by the continuum surface theory (E = ∆ surf) to the one obtained through numerical simulation. We find that as the applied magnetic flux increases, or equivalently, as the magnetic length decreases to the order of lattice lengths, the magnitude of these Landau levels approaches 0. This is confirmed by a clear signature within the surface gap of Hofstadter's butterfly for the hinge modes: the lowest Landau levels which lie on the sides of the cone emerging from the addition of all possible kz start from E = ∆ surf for Φ = 0 and linearly move towards E = 0, eventually joining the bulk of the sample. This strongly suggests that the physics underlying Hofstadter's butterfly are indeed responsible for the continuum-to-lattice discrepancy.

Papers by Leo P B Goutte

arXiv (Cornell University), Dec 12, 2022

We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken ... more We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken Weyl semimetal (WSM). When coupled to a non-magnetic parabolic band, the WSM's chiral arc state lowers in energy and forms, together with a previously extended state, a noticeable spindependent asymmetry in the interface spectrum in the vicinity of the Weyl nodes. We study these effects with a lattice model which we solve numerically on a finite sample and analytically through using an ansatz on an infinite sample. Our ansatz agrees very well with the numerical simulation as it accurately describes the behaviour of the chiral state, from its energy asymmetry to the spin canting at the interface. We find that the tunnelling effectively increases the Fermi arc length, allowing for the presence of interface states beyond the bare Weyl nodes. These additional states may carry current along the interface and their contribution can be detected in the conductance. Across the interface, the spin-independent conductance reproduces the results of an electron tunnelling experiment to reveal the WSM's density of states. Besides conductivity, the effect of tunneling between the WSM and the metallic band can be seen in quantum oscillations experiments which we briefly comment about.

Physical Review B, 2021

We study the surface states and chiral hinge states of a three-dimensional second-order topologic... more We study the surface states and chiral hinge states of a three-dimensional second-order topological insulator in the presence of an external magnetic gauge field. Surfaces pierced by flux host Landau levels, while surfaces parallel to the applied field are not significantly affected. The chiral hinge modes mediate spectral flow between neighboring surfaces. As the magnetic field strength is increased, the surface Landau quantization deviates from that of a massive Dirac cone. Quantitatively, the n = 0 Landau level falls inside the surface Dirac gap and not at the gap edge. The n = 0 levels exhibit a further, qualitative discrepancy: while the massive Dirac cone is expected to produce pairs of levels (±n) which are symmetric around zero energy, the n and −n levels become asymmetric in our lattice model-one of the pair may even be absent from the spectrum or hybridized with the continuum. In order to resolve the issue, we extend the standard two-dimensional (2D) massive Dirac surface theory by including additional Hamiltonian terms at O(k^2). While these terms do not break particle-hole symmetry in the absence of magnetic field, they lead to the aforementioned Landau level asymmetry once the magnetic field is applied. We argue that similar O(k^2) correction terms are generically expected in lattice models containing gapped Dirac fermions, using the BHZ model of a 2D topological insulator as an example.

An introduction to the simplest form of a general topological invariant. Intended as a teaching d... more An introduction to the simplest form of a general topological invariant. Intended as a teaching document.

The strange physics of quantum me- chanics would theoretically allow a parti- cle to "tunnel" thr... more The strange physics of quantum me- chanics would theoretically allow a parti- cle to "tunnel" through a classically im- passable barrier. A new experiment out of the University of Toronto not only con- firms this phenomenon, but also deter- mines how fast particles can tunnel.

** Completed within the scope of PHYS 321 -- Observational astrophysics and data analysis ** The... more ** Completed within the scope of PHYS 321 -- Observational astrophysics and data analysis **

The 21-cm signal from neutral hydrogen is one of the most promising ways of exploring the dark ages of the universe. Using this signal to understand the universe has gained lots of attention and there exists a wide range of literature and theories about the signal. In 2018, the EDGES collaboration reported the first detection of the 21-cm signal and, surprisingly, their detection was outside the range of the predicted measurements. If confirmed, current models for the dark ages of the universe must be revised to account for this detection. In this paper, we use a Nested Sampling algorithm (PolyChord, Handley Handley (2015)) to perform Bayesian model selection and determine if the data presented supports discarding the current models. We find decisive evidence against current models; they cannot explain the data presented by EDGES.

Fast radio bursts (FRBs) are millisecond long transients with extra-galactic dispersive delays. W... more Fast radio bursts (FRBs) are millisecond long transients with extra-galactic dispersive delays. While some have been found to repeat, most are single-burst events similar to individual pulsar bursts, though with much greater characteristic energy. Despite a limited observing time, the catalogue of verified FRBs is growing rapidly, and with it the inventory of progenitor theories. One such theory, that single-burst FRBs emanate from catastrophic event origins, posits that they would be detectable not only in the radio spectrum, but in the gamma-ray spectrum as well. Here, we conduct a simple analysis of FRB sources in the gamma-ray spectrum using Fermi-LAT data and develop 4 analysis tools - lightcurve, burstsearch, temporal stacking and spatial stacking - to investigate the behaviour of these bursts. Since it is already known that FRBs release a great amount of energy, if we were to find a gamma burst correlation, they could possibly be candidates to most high-energy events in the Universe, providing a powerful tool to probe its extremes and the intervening unknown medium.

We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken ... more We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken Weyl semimetal (WSM). When coupled to a non-magnetic parabolic band, the WSM's chiral arc state lowers in energy and forms, together with a previously extended state, a noticeable spindependent asymmetry in the interface spectrum in the vicinity of the Weyl nodes. We study these effects with a lattice model which we solve numerically on a finite sample and analytically through using an ansatz on an infinite sample. Our ansatz agrees very well with the numerical simulation as it accurately describes the behaviour of the chiral state, from its energy asymmetry to the spin canting at the interface. We find that the tunnelling effectively increases the Fermi arc length, allowing for the presence of interface states beyond the bare Weyl nodes. These additional states may carry current along the interface and their contribution can be detected in the conductance. Across the interface, the spin-independent conductance reproduces the results of an electron tunnelling experiment to reveal the WSM's density of states. Besides conductivity, the effect of tunneling between the WSM and the metallic band can be seen in quantum oscillations experiments which we briefly comment about.

We construct a tool for the calculation and visualization of Moiré bands belonging to the broad f... more We construct a tool for the calculation and visualization of Moiré bands belonging to the broad family of arbitrarily stacked twisted N + M multilayer graphene. We present the underlying k • p model of twisted bilayer graphene and extend it to twisted N + M multilayer

We present and study the Bernevig, Hughes, and Zhang (BHZ) model of a 2-dimensional first- order ... more We present and study the Bernevig, Hughes, and Zhang (BHZ) model of a 2-dimensional first- order topological insulator under an external magnetic field B. The applied gauge field is accounted for by performing a Peierls substitution which modifies the motion of the electrons on the two- dimensional square lattice. We discuss how this model can be used to describe the surface of a three-dimensional, second-order topological insulator (SOTI). We derive an effective continuum theory for the Landau level (LL) energies and wavefunctions in the BHZ model and compare it to numerical lattice calculations. The results are then compared to the SOTI’s continuum surface theory in an attempt to explain three inconsistencies: the deviation of the lowest LL from the surface gap; the asymmetry of ±nth LLs around zero and; the fact that the SOTI’s negative LLs are missing. Moreover, we quantify the effect of an increasing magnetic field on the system’s energies via Hofstadter’s butterfly, considering the various regimes of validity of our previous continuum theories. It is determined that the discrepancies between surface theory and lattice model simulations can be resolved by keeping terms up to O(k2) in the SOTI’s surface theory, which can be thought of as a momentum-dependent mass term.

We obtain and analyze Hofstadter's butterfly for the bulk and the edges of the second-order topol... more We obtain and analyze Hofstadter's butterfly for the bulk and the edges of the second-order topological insulator (SOTI) subject to an external magnetic field. The applied gauge field is accounted for by performing a Peierls substitution t → texp(i A · dr), modifying the motion of the electrons on the lattice. We compare and contrast the lowest Landau level energy predicted by the continuum surface theory (E = ∆ surf) to the one obtained through numerical simulation. We find that as the applied magnetic flux increases, or equivalently, as the magnetic length decreases to the order of lattice lengths, the magnitude of these Landau levels approaches 0. This is confirmed by a clear signature within the surface gap of Hofstadter's butterfly for the hinge modes: the lowest Landau levels which lie on the sides of the cone emerging from the addition of all possible kz start from E = ∆ surf for Φ = 0 and linearly move towards E = 0, eventually joining the bulk of the sample. This strongly suggests that the physics underlying Hofstadter's butterfly are indeed responsible for the continuum-to-lattice discrepancy.

arXiv (Cornell University), Dec 12, 2022

We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken ... more We study the effects of tunnelling on the band structure and Fermi arc of a time-reversal broken Weyl semimetal (WSM). When coupled to a non-magnetic parabolic band, the WSM's chiral arc state lowers in energy and forms, together with a previously extended state, a noticeable spindependent asymmetry in the interface spectrum in the vicinity of the Weyl nodes. We study these effects with a lattice model which we solve numerically on a finite sample and analytically through using an ansatz on an infinite sample. Our ansatz agrees very well with the numerical simulation as it accurately describes the behaviour of the chiral state, from its energy asymmetry to the spin canting at the interface. We find that the tunnelling effectively increases the Fermi arc length, allowing for the presence of interface states beyond the bare Weyl nodes. These additional states may carry current along the interface and their contribution can be detected in the conductance. Across the interface, the spin-independent conductance reproduces the results of an electron tunnelling experiment to reveal the WSM's density of states. Besides conductivity, the effect of tunneling between the WSM and the metallic band can be seen in quantum oscillations experiments which we briefly comment about.

Physical Review B, 2021

We study the surface states and chiral hinge states of a three-dimensional second-order topologic... more We study the surface states and chiral hinge states of a three-dimensional second-order topological insulator in the presence of an external magnetic gauge field. Surfaces pierced by flux host Landau levels, while surfaces parallel to the applied field are not significantly affected. The chiral hinge modes mediate spectral flow between neighboring surfaces. As the magnetic field strength is increased, the surface Landau quantization deviates from that of a massive Dirac cone. Quantitatively, the n = 0 Landau level falls inside the surface Dirac gap and not at the gap edge. The n = 0 levels exhibit a further, qualitative discrepancy: while the massive Dirac cone is expected to produce pairs of levels (±n) which are symmetric around zero energy, the n and −n levels become asymmetric in our lattice model-one of the pair may even be absent from the spectrum or hybridized with the continuum. In order to resolve the issue, we extend the standard two-dimensional (2D) massive Dirac surface theory by including additional Hamiltonian terms at O(k^2). While these terms do not break particle-hole symmetry in the absence of magnetic field, they lead to the aforementioned Landau level asymmetry once the magnetic field is applied. We argue that similar O(k^2) correction terms are generically expected in lattice models containing gapped Dirac fermions, using the BHZ model of a 2D topological insulator as an example.