arXiv subjects

Michael R. Peterson

Publications and source records attributed to Michael R. Peterson.

At least 19 recordsLinked to original sources

arXiv2024

Capstone Experiences in Developing Augmented Reality Tables for Community Organizations

This paper examines two senior capstone experiences developed as augmented reality tables over the past two years. Both projects were public facing efforts that required working implementations. The first project was deployed at an astronomy center and focused on interactions between land use and ecological aspects of Hawaii Island while the second project focused more on historical sites on the same island. Both projects leveraged brownfield development and existing code bases to allow for student success in spite of the impacts of the COVID19 pandemic.

cs.CY↗

arXiv2022

Fractional quantum Hall effect at the filling factor $ν=5/2$

The fractional quantum Hall (FQH) effect at the filling factor $ν=5/2$ was discovered in GaAs heterostructures more than 35 years ago. Various topological orders have been proposed as possible candidates to describe this FQH state. Some of them possess non-Abelian anyon excitations, an entirely new type of quasiparticle with fascinating properties. If observed, non-Abelian anyons could offer fundamental building blocks of a topological quantum computer. Nevertheless, the nature of the FQH state at $ν=5/2$ is still under debate. In this chapter, we provide an overview of the theoretical background, numerical results, and experimental measurements pertaining to this special FQH state. Furthermore, we review some recent developments and their possible interpretations. Possible future directions toward resolving the nature of the $5/2$ state are also discussed.

cond-mat.mes-hall↗

arXiv2021

Composite fermion mass enhancement and particle-hole symmetry of fractional quantum Hall states in the lowest Landau level under realistic conditions

Particle-hole symmetry breaking in the fractional quantum Hall effect has recently been studied both theoretically and experimentally with most works focusing on non-Abelian states in the second electronic Landau level. In this work, we theoretically investigate particle-hole symmetry breaking of incompressible fractional quantum Hall states in the lowest Landau level under the influence of the realistic effect of a finite magnetic field strength. A finite magnetic field induces Landau level and sub-band mixing which are known to break particle-hole symmetry at the level of the Hamiltonian. We analyze the Haldane pseudopotentials, energy spectra and energy gaps, and the wave functions themselves, under realistic conditions. We find that particle-hole symmetry is broken, as determined by energy gaps, between states related via particle-hole conjugation, however, we find that particle-hole symmetry is largely maintained as determined by the effective mass of composite fermions. Finally, we comment and make connection to recent experimental observations regarding particle-hole symmetry in the lowest Landau level fractional quantum Hall effect [Pan et al. Phys. Rev. Lett. 124, 156801 (2020)]

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arXiv2019

A Particle-Hole-Symmetric Model for a Paired Fractional Quantum Hall State in a Half-filled Landau Level

The fractional quantum Hall effect (FQHE) observed at half filling of the second Landau level is believed to be caused by a pairing of composite fermions captured by the Moore-Read Pfaffian wave function. The generating Hamiltonian for the Moore-Read Pfaffian is a purely three-body model that breaks particle-hole symmetry and lacks other properties, such as dominate two-body repulsive interactions, expected from a physical model of the FQHE. We use exact diagonalization to study the low energy states of a more physical two-body generator model derived from the three-body model. We find that the two-body model exhibits the essential features expected from the Moore-Read Pfaffian: pairing, non-Abelian anyon excitations, and a neutral fermion mode. The model also satisfies constraints expected for a physical model of the FQHE at half-filling because it is: short range, spatially decaying, particle-hole symmetric, and supports a roton mode with a robust spectral gap in the thermodynamic limit. Hence, this two-body model offers a bridge between artificial three-body generator models for paired states and the physical Coulomb interaction and can be used to further explore properties of non-Abelian physics in the FQHE.

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arXiv2016

The enigmatic 12/5 fractional quantum Hall effect

We numerically study the fractional quantum Hall effect at filling factors $ν=12/5$ and 13/5 (the particle-hole conjugate of 12/5) in high-quality two-dimensional GaAs heterostructures via exact diagonalization including finite well width and Landau level mixing. We find that Landau level mixing suppresses $ν=13/5$ fractional quantum Hall effect relative to $ν=12/5$. By contrast, we find both $ν=2/5$ and (its particle-hole conjugate) $ν=3/5$ fractional quantum Hall effects in the lowest Landau level to be robust under Landau level mixing and finite well-width corrections. Our results provide a possible explanation for the experimental absence of the 13/5 fractional quantum Hall state as caused by Landau level mixing effects.

cond-mat.str-el↗

arXiv2016

Landau level quantization for massless Dirac fermions in the spherical geometry: graphene fractional quantum Hall effect on the Haldane sphere

We derive the single-particle eigenenergies and eigenfunctions for massless Dirac fermions confined to the surface of a sphere in the presence of a magnetic monopole, i.e., we solve the Landau level problem for electrons in graphene on the Haldane sphere. With the single-particle eigenfunctions and eigenenergies we calculate the Haldane pseudopotentials for the Coulomb interaction in the second Landau level and calculate the effective pseudopotentials characterizing an effective Landau level mixing Hamiltonian entirely in the spherical geometry to be used in theoretical studies of the fractional quantum Hall effect in graphene. Our treatment is analogous to the formalism in the planar geometry and reduces to the planar results in the thermodynamic limit.

cond-mat.str-el↗

arXiv2015

Abelian and Non-Abelian States in $ν=2/3$ Bilayer Fractional Quantum Hall Systems

There are several possible theoretically allowed non-Abelian fractional quantum Hall (FQH) states that could potentially be realized in one- and two- component FQH systems at total filling fraction $ν= n+ 2/3$, for integer $n$. Some of these states even possess quasiparticles with non-Abelian statistics that are powerful enough for universal topological quantum computation, and are thus of particular interest. Here, we initiate a systematic numerical study, using both exact diagonalization and variational Monte Carlo, to investigate the phase diagram of FQH systems at total filling fraction $ν= n+2/3$, including in particular the possibility of the non-Abelian $Z_4$ parafermion state. In $ν= 2/3$ bilayers, we determine the phase diagram as a function of interlayer tunneling and repulsion, finding only three competing Abelian states, without the $Z_4$ state. On the other hand, in single-component systems at $ν= 8/3$, we find that the $Z_4$ parafermion state has significantly higher overlap with the exact ground state than the Laughlin state, together with a larger gap, suggesting that the experimentally observed $ν= 8/3$ state may be non-Abelian. Our results from the two complementary numerical techniques agree well with each other qualitatively.

cond-mat.str-el↗

arXiv2014

The Phase Diagram of the $ν=5/2$ Fractional Quantum Hall Effect: Effects of Landau Level Mixing and Non-Zero Width

Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the anti-Pfaffian, offer candidate descriptions of the $ν= 5/2$ fractional quantum Hall state. But the significant controversy surrounding the nature of the $ν= 5/2$ state has been hampered by the fact that the competition between these and other states is affected by small parameter changes. To study the phase diagram of the $ν= 5/2$ state we numerically diagonalize a comprehensive effective Hamiltonian describing the fractional quantum Hall effect of electrons under realistic conditions in GaAs semiconductors. The effective Hamiltonian takes Landau level mixing into account to lowest-order perturbatively in $κ$, the ratio of the Coulomb energy scale to the cyclotron gap. We also incorporate non-zero width $w$ of the quantum well and sub-band mixing. We find the ground state in both the torus and spherical geometries as a function of $κ$ and $w$. To sort out the non-trivial competition between candidate ground states we analyze the following 4 criteria: its overlap with trial wave functions; the magnitude of energy gaps; the sign of the expectation value of an order parameter for particle-hole symmetry breaking; and the entanglement spectrum. We conclude that the ground state is in the universality class of the Moore-Read Pfaffian state, rather than the anti-Pfaffian, for $κ< {κ_c}(w)$, where ${κ_c}(w)$ is a $w$-dependent critical value $0.6 \lesssim{κ_c}(w)\lesssim 1$. We observe that both Landau level mixing and non-zero width suppress the excitation gap, but Landau level mixing has a larger effect in this regard. Our findings have important implications for the identification of non-Abelian fractional quantum Hall states.

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arXiv2014

Identifying topological order in the Shastry-Sutherland model via entanglement entropy

It is known that for a topologically ordered state the area law for the entanglement entropy shows a negative universal additive constant contribution, $-γ$, called the topological entanglement entropy. We theoretically study the entanglement entropy of the two-dimensional Shastry-Sutherland quantum antiferromagnet using exact diagonalization on clusters of 16 and 24 spins. By utilizing the Kitaev-Preskill construction [A. Kitaev and J. Preskill, Phys. Rev. Lett. {\bf 96}, 110404 (2006)] we extract a finite topological term, $-γ$, in the region of bond-strength parameter space corresponding to high geometrical frustration. Thus, we provide strong evidence for the existence of an exotic topologically ordered state and shed light on the nature of this model's strongly frustrated, and long controversial, intermediate phase.

cond-mat.str-el↗

arXiv2014

The Effects of Landau level mixing on the fractional quantum Hall effect in monolayer Graphene

We report results of exact diagonalization studies of the spin- and valley-polarized fractional quantum Hall effect in the $N=0$ and 1 Landau levels in graphene. We use an effective model that incorporates Landau level mixing to lowest-order in the parameter $κ= \frac{e^2/ε\ell}{\hbar v_F/\ell}=\frac{e^2}{εv_F\hbar}$ which is magnetic field independent and can only be varied through the choice of substrate. We find Landau level mixing effects are negligible in the $N=0$ Landau level for $κ\lesssim 2$. In fact, the lowest Landau level projected Coulomb Hamiltonian is a better approximation to the real Hamiltonian for graphene than it is for semiconductor based quantum wells. Consequently, the principal fractional quantum Hall states are expected in the $N=0$ Landau level over this range of $κ$. In the $N=1$ Landau level, fractional quantum Hall states are expected for a smaller range of $κ$ and Landau level mixing strongly breaks particle-hole symmetry producing qualitatively different results compared to the $N=0$ Landau level. At half-filling of the $N=1$ Landau level, we predict the anti-Pfaffian state will occur for $κ\sim 0.25$-$0.75$.

cond-mat.mes-hall↗

arXiv2013

More Realistic Hamiltonians for the Fractional Quantum Hall Regime in GaAs and Graphene

We construct an effective Hamiltonian for electrons in the fractional quantum Hall regime for GaAs and graphene that takes into account Landau level mixing (for both GaAs and graphene) and subband mixing (for GaAs, due to the nonzero width of the quantum well). This mixing has the important qualitative effect of breaking particle-hole symmetry as well as renormalizing the strength of the interparticle interactions. Both effects could have important consequences for the prospect that the fractional quantum Hall effect at $ν=5/2$ is described by states that support non-Abelian excitations such as the Moore-Read Pfaffian or anti-Pfaffian states. For GaAs, Landau level and subband mixing break particle-hole symmetry in all Landau levels and subband mixing, due to finite thickness, causes additional short-distance softening of the Coulomb interaction, further renormalizing the Hamiltonian; additionally, the Landau level and subband energy spacings are comparable so it is crucial to consider both effects simultaneously. We find that in graphene, Landau level mixing only breaks particle-hole symmetry outside of the lowest Landau level ($N\neq0$). Landau level mixing is likely to be especially important in graphene since the Landau level mixing parameter is independent of the external magnetic field and is of order one. Our realistic Hamiltonians will serve as starting points for future numerical studies.

cond-mat.mes-hall↗

arXiv2013

Variational Monte Carlo study of spin polarization stability of fractional quantum Hall states against realistic effects in half-filled Landau levels

We compare ground state energies by variational Monte Carlo of the spin unpolarized Halperin 331 and the spin polarized Moore-Read (MR) Pfaffian fractional quantum Hall states at half-filling of the lowest Landau level (LLL) and the second Landau level (SLL) as a function of small deviations around the Coulomb point via the finite thickness effect and direct alterations to the the first two Haldane pseudopotentials. In the comparison we find that in the LLL, either the 331 state or the MR Pfaffian may be lower in energy depending on the deviations. In the SLL, however, the MR Pfaffian is consistently lower in energy except for large deviations. These results suggest that even under moderate deviations in the interaction potential (through various physical processes such as finite thickness, Landau level mixing, etc.), the MR Pfaffian description is more energetically favorable than the Halperin 331 state in the half-filled SLL (i.e. $ν= 5/2$), consistent with recent experimental investigations.

cond-mat.mes-hall↗

arXiv2011

Entanglement Measures for Quasi-Two-Dimensional Fractional Quantum Hall States

We theoretically examine entanglement in fractional quantum hall states, explicitly taking into account and emphasizing the quasi-two-dimensional nature of experimental quantum Hall systems. In particular, we study the entanglement entropy and the entanglement spectrum as a function of the finite layer thickness $d$ of the quasi-two-dimensional system for a number of filling fractions $ν$ in the lowest and the second Landau levels: $ν$ = 1/3, 7/3, 1/2, and 5/2. We observe that the entanglement measures are dependent on which Landau level the electrons fractionally occupy, and find that filling factions 1/3 and 7/3, which are considered to be Laughlin states, weaken with $d$ in the lowest Landau level ($ν$=1/3) and strengthen with $d$ in the second Landau level ($ν$=7/3). For the enigmatic even-denominator $ν=5/2$ state, we find that entanglement in the ground state is consistent with that of the non-Abelian Moore-Read Pfaffian state at an optimal thickness $d$. We also find that the single-layer $ν= 1/2$ system is not a fractional quantum Hall state consistent with the experimental observation. In general, our theoretical findings based on entanglement considerations are completely consistent with the results based on wavefunction overlap calculations.

cond-mat.str-el↗

arXiv2010

The Kelvin Formula for Thermopower

Thermoelectrics are important in physics, engineering, and material science due to their useful applications and inherent theoretical difficulty, especially in strongly correlated materials. Here we reexamine the framework for calculating the thermopower, inspired by ideas of Lord Kelvin from 1854. We find an approximate but concise expression, which we term as the Kelvin formula for the the Seebeck coefficient. According to this formula, the Seebeck coefficient is given as the particle number $N$ derivative of the entropy $Σ$, at constant volume $V$ and temperature $T$, $S_{\text{Kelvin}}=\frac{1}{q_e}\{\frac{\partial Σ}{\partial N} \}_{V,T}$. This formula is shown to be competitive compared to other approximations in various contexts including strongly correlated systems. We finally connect to a recent thermopower calculation for non-Abelian fractional quantum Hall states, where we point out that the Kelvin formula is exact.

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arXiv2010

Fractional quantum Hall effects in bilayers in the presence of inter-layer tunneling and charge imbalance

Two-component fractional quantum Hall systems are providing a major motivation for a large section of the physics community. Here we study two-component fractional quantum Hall systems in the spin-polarized half-filled lowest Landau level (filling factor 1/2) and second Landau level (filling factor 5/2) with exact diagonalization utilizing both the spherical and torus geometries. The two distinct two-component systems we consider are the true bilayer and effective bilayers (wide-quantum-well). In each model (bilayer and wide-quantum-well) we completely take into account inter-layer tunneling and charge imbalancing terms. We find that in the half-filled lowest Landau level, the FQHE is described by the two-component Abelian Halperin 331 state which is remarkably robust to charge imbalancing. In the half-filled second Landau, we find that the FQHE is likely described by the non-Abelian Moore-Read Pfaffian state which is also quite robust to charge imbalancing. Furthermore, we suggest the possibility of experimentally tuning from an Abelian to non-Abelian FQHE state in the second Landau level, and comment on recent experimental studies of FQHE in wide quantum well structures.

cond-mat.mes-hall↗

arXiv2009

Half-filled 2D bilayers in a strong magnetic field: Revisiting the $ν=1/2$ fractional quantum Hall effect

We examine the quantum phase diagram of the fractional quantum Hall effect in the lowest Landau level in half-filled bilayer structures as a function of tunneling strength and layer separation. Using numerical exact diagonalization we investigate the important question of whether this system supports a fractional quantum Hall effect described by the non-Abelian Moore-Read Pfaffian state in the strong tunneling regime. We find that, although it is in principle possible, it is unlikely that the non-Abelian FQHE exists in the lowest Landau level. We establish that all so far observed FQHE states in half-filled lowest Landau level bilayers are most likely described by the Abelian Halperin 331 state.

cond-mat.mes-hall↗

arXiv2009

Quantum Hall Phase Diagram of Second Landau-level Half-filled Bilayers: Abelian versus Non-Abelian States

The quantum Hall phase diagram of the half-filled bilayer system in the second Landau level is studied as a function of tunneling and layer separation using exact diagonalization. We make the striking prediction that bilayer structures would manifest two distinct branches of incompressible fractional quantum Hall effect (FQHE) corresponding to the Abelian 331 state (at moderate to low tunneling and large layer separation) and the non-Abelian Pfaffian state (at large tunneling and small layer separation). The observation of these two FQHE branches and the quantum phase transition between them will be compelling evidence supporting the existence of the non-Abelian Pfaffian state in the second Landau level.

cond-mat.mes-hall↗

arXiv2008

Realizing the strongly correlated $d$-Mott state in a fermionic cold atom optical lattice

We show that a new state of matter, the d-wave Mott-insulator state (d-Mott state) (introduced recently by [H. Yao, W. F. Tsai, and S. A. Kivelson, Phys. Rev. B 76, 161104 (2007)]), which is characterized by a non-zero expectation value of a local plaquette operator embedded in an insulating state, can be engineered using ultra-cold atomic fermions in two-dimensional double-well optical lattices. We characterize and analyze the parameter regime where the $d$-Mott state is stable. We predict the testable signatures of the state in the time-of-flight measurements.

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