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Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions.

We analyze the effects of Robin-like boundary conditions on different quantum field theories of spin 0, 1/2 and 1 on manifolds with boundaries. In particular, we show that these conditions often lead to the appearance of edge states. These states play a significant role in physical phenomena like quantum Hall effect and topological insulators. We prove in a rigorous way the existence of spectral lower bounds on the kinetic term of different Hamiltonians, even in the case of Abelian gauge fields where it is a non-elliptic differential operator. This guarantees the stability and consistency of massive field theories with masses larger than the lower bound of the kinetic term. Moreover, we find an upper bound for the deepest edge state. In the case of Abelian gauge theories, we analyze a generalization of Robin boundary conditions. For Dirac fermions, we analyze the cases of Atiyah–Patodi–Singer and chiral bag boundary conditions. The explicit dependence of the bounds on the boundary conditions and the size of the system is derived under general assumptions.

Establishing the (non)existence of a spectral gap above the ground state in the thermodynamic limit is one of the fundamental steps for characterizing the topological phase of a quantum lattice model. This is particularly challenging when a model is expected to have low-lying edge excitations, but nevertheless a positive bulk gap. We review the bulk gap strategy introduced in [S. Warzel and A. Young, The spectral gap of a fractional quantum Hall system on a thin torus, J. Math. Phys.63 (2022) 041901; S. Warzel and A. Young, A bulk spectral gap in the presence of edge states for a truncated pseudopotential, Ann. Henri Poincaré24 (2023) 133–178], while studying truncated Haldane pseudopotentials. This approach is able to avoid low-lying edge modes by separating the ground states and edge states into different invariant subspaces before applying spectral gap bounding techniques. The approach is stated in a general context, and we reformulate specific spectral gap methods in an invariant subspace context to illustrate the necessary conditions for combining them with the bulk gap strategy. We then review its application to a truncation of the 1/3-filled Haldane pseudopotential in the cylinder geometry.

We present a brief review of the fuzzy disc, the finite algebra approximating functions on a disc, which we have introduced earlier. We also present a comparison with recent papers of Balachandran, Gupta and Kürkçüoǧlu, and of Pinzul and Stern, aimed at the discussion of edge states of a Chern-Simons thoery.

The question "What lies beyond the Quantized String or Superstring Theory?" and the question "What lies beyond Quantum Mechanics itself?" might have one common answer: a discretized, classical version of string theory, which lives on a lattice in Minkowski space. The size a of the meshes on this lattice in Minkowski space is determined by the string slope parameter, α′.

Ground states of quadratic Hamiltonians for fermionic systems can be characterized in terms of orthogonal complex structures. The standard way in which such Hamiltonians are diagonalized makes use of a certain “doubling” of the Hilbert space. In this work, we show that this redundancy in the Hilbert space can be completely lifted if the relevant orthogonal structure is taken into account. Such an approach allows for a treatment of Majorana fermions which is both physically and mathematically transparent. Furthermore, an explicit connection between orthogonal complex structures and the topological ${\mathbb{Z}}_2$-invariant is given.

We analyze the global theory of boundary conditions for a constrained quantum system with classical configuration space a compact Riemannian manifold M with regular boundary Γ=∂M. The space ℳ of self-adjoint extensions of the covariant Laplacian on M is shown to have interesting geometrical and topological properties which are related to the different topological closures of M. In this sense, the change of topology of M is connected with the nontrivial structure of ℳ. The space ℳ itself can be identified with the unitary group of the Hilbert space of boundary data . This description, is shown to be equivalent to the classical von Neumann's description in terms of deficiency index subspaces, but it is more efficient and explicit because it is given only in terms of the boundary data, which are the natural external inputs of the system. A particularly interesting family of boundary conditions, identified as the set of unitary operators which are singular under the Cayley transform, (the Cayley manifold), turns out to play a relevant role in topology change phenomena. The singularity of the Cayley transform implies that some energy levels, usually associated with edge states, acquire an infinity energy when by an adiabatic change the boundary conditions reaches the Cayley submanifold 𝒞_. In this sense topological transitions require an infinite amount of quantum energy to occur, although the description of the topological transition in the space ℳ is smooth. This fact has relevant implications in string theory for possible scenarios with joint descriptions of open and closed strings. In the particular case of elliptic self-adjoint boundary conditions, the space 𝒞_ can be identified with a Lagrangian submanifold of the infinite dimensional Grassmannian. The corresponding Cayley manifold 𝒞_ is dual of the Maslov class of ℳ. The phenomena are illustrated with some simple low dimensional examples.

In a $(2+1)$-dimensional Maxwell–Chern–Simons theory coupled with a fermion and a scalar, which has ${𝒩}=2$ SUSY in the absence of the boundary, supersymmetry is broken on the insertion of a spatial boundary. We show that only a subset of the boundary conditions allowed by the self-adjointness of the Hamiltonian can preserve partial (${𝒩}=1$) supersymmetry, while for the remaining boundary conditions SUSY is completely broken. In the latter case, we demonstrate two distinct SUSY-breaking mechanisms. In one scenario, the SUSY-breaking boundary conditions are not consistent with the supersymmetry transformations. In another scenario, despite the boundary conditions being consistent with the SUSY transformations, unpaired fermionic edge states in the domain of the Hamiltonian leads to the breaking of the supersymmetry.

We report on tunneling experiment between two quantum Hall droplets separated by a nearly ideal tunnel barrier. The device is produced by cleaved edge overgrowth that laterally juxtaposes two two-dimensional electron systems across a high quality semi-conductor barrier. The dramatic evolution of the tunneling characteristics is consistent with the magnetic field-dependent tunneling between the coupled edge states of the quantum Hall droplets. We identify a series of quantum critical points between successive strong and weak tunneling regimes that are reminiscent of the plateau-transitions in quantum Hall effect. Scaling analysis shows that the conductance near the critical magnetic fields B_c is a function of a single scaling argument |B-B_c|T^-κ, where the exponent κ=0.42. This puzzling resemblance to a quantum Hall-insulator transition points to the significance of interedge correlation in the lateral tunneling of quantum Hall droplets.

The recent quantum Hall experiments in graphene have confirmed the theoretically well-understood picture of the quantum Hall (QH) conductance in fermion systems with continuum Dirac spectrum. In this paper we take into account the lattice and perform an exact diagonalization of the Landau problem on the hexagonal lattice. At very large magnetic fields the Dirac argument fails completely and the Hall conductance, given by the number of edge states present in the gaps of the spectrum, is dominated by lattice effects. As the field is lowered, the experimentally observed situation is recovered through a phenomenon which we call band collapse. As a corollary, for low magnetic fields, graphene will exhibit two qualitatively different QHE's: at low filling, the QHE will be dominated by the "relativistic" Dirac spectrum and the Hall conductance will be odd-integer; above a certain filling, the QHE will be dominated by a non-relativistic spectrum, and the Hall conductance will span all integers, even and odd.

We investigate the transport of heat in the integer quantized Hall regime. We make use of quantum point contacts (QPC's) positioned along the edge of a large quantum Hall droplet to both locally heat and locally detect temperature rises at the edge of the droplet. The detection scheme is thermoelectric, in essence identical to one introduced by Molenkamp, et al.¹ in the early 1990's for heat transport experiments at zero magnetic field.

At zero magnetic field we find that heat moves away from the heater QPC more or less isotropically. As expected from the Mott formula, we find a close connection between the detector QPC's thermoelectric response and the derivative, with respect to gate voltage, of its conductance.

At high magnetic field our results show, not surprisingly, that heat transport is chiral in the quantum Hall regime. At total filling factor ν = 1 we inject a hot distribution of electrons into the edge with one of three QPC's. We observe a thermoelectric voltage at the other QPC's only if they are "downstream" from the heater. No signals are detected in the upstream direction. The magnitude of the detected thermal response is dependent upon the distance between the heater and detector QPC's.

Additional measurements, in which a second QPC, between the heater and the detector, is used to drain away a portion of the injected heat, strongly suggest that the chiral heat transport we observe is indeed confined to the edge of the Hall droplet.

Experiments are underway in the fractional quantum Hall regime to search for "upstream" heat propagation. Theory has suggested that such anti-chiral transport should exist at certain fractions, notably ν = 2/3, owing to backward-propagating neutral modes.

Note from Publisher: This article contains the abstract only.

At not-too-low temperatures, ħω_c ≫ k_BT ≫ ħv_g/2ℓ₀ where v_g is the group velocity of the edge states and ℓ₀ the magnetic length, we obtain new edge magnetoplasmons (EMPs). For ν = 1(2) and very strong dissipation, we find that all EMPs are very strongly damped, due to the dissipative processes at the edge states, but a new weakly damped mode termed edge helicon. We also study the effect of metallic gate on the helicon spectrum. For ν = 4 we obtain fundamental modes of the n = 0, 1 Landau levels and their renormalization due to the Coulomb coupling and the effect of a metallic gate on them.

We suggest that exotic sphere fibrations can be mapped to band topologies in condensed matter systems. These fibrations can correspond to geometric phases of two double bands or state vector bases with second Chern numbers m+n and -n, respectively. They can be related to topological insulators, magnetoelectric effects, and photonic crystals with special edge states. We also consider time-reversal symmetry breaking perturbations of topological insulator, and heterostructures of topological insulators with normal insulators and with superconductors. We consider periodic TI/NI/TI/NI′ heterostructures, and periodic TI/SC/TI/SC′ heterostructures. They also give rise to models of Weyl semimetals which have thermal and electrical transports.

The electronic structure of silicene supported by monolayer of different monochalcogenide MX (GaS, GaSe, GaTe and InSe) substrates has been investigated by first principle density functional theory. By calculating the formation energies and phonons, it has been seen that silicene supported by monolayer of MX remains stable. The systems retain their almost $2D$ planner configurations with small buckling as that of the free standing silicene and also the Bader charge analysis shows that silicene hardly interacts with any of the MX substrates. The Dirac cone with a small gap ($\sim 30–50$ meV) has been observed in each of the cases. All the systems show quantum spin Hall effect and the quantum spin Hall conductivities have been estimated to be within the range $\sim 2-5\frac{\hslash }{e}$S/cm, which are larger than that of the free standing silicene. Our calculations show that even if the systems have bulk band gaps but the edge states are conducting in nature.

Frequently, it is argued that the microstates responsible for the Bekenstein–Hawking entropy should arise from some physical degrees of freedom located near or on the black hole horizon. In this essay, we elucidate that instead entropy may emerge from the conversion of physical degrees of freedom, attached to a generic boundary, into unobservable gauge degrees of freedom attached to the horizon. By constructing the reduced phase space, it can be demonstrated that such a transmutation indeed takes place for a large class of black holes, including Schwarzschild.

It is known that a two-spin system with four energy levels can be used to realize a two-qubit quantum gate. A feasible realization of quantum gates should rely on stable quantum mechanical states. An example of such states are edge states which arise around regions with high potential in a strong magnetic field. In this paper we show that certain edge states around a pair of antidots may be suitable for quantum gate implementation.

The dynamics of quantum field theories on bounded domains requires the introduction of boundary conditions on the quantum fields. We address the problem from a very general perspective by using charge conservation as a fundamental principle for scalar and fermionic quantum field theories. Unitarity arises as a consequence of the choice of charge preserving boundary conditions. This provides a powerful framework for the analysis of global geometrical and topological properties of the space of physical boundary conditions. Boundary conditions which allow the existence of edge states can only arise in theories with a mass gap which is also a physical requirement for topological insulators.

Spin waves (SWs) have been studied experimentally and by simulations in 1000 nm side equilateral triangular Permalloy dots in the Buckle state (B, with in-plane field along the triangle base) and the Y state (Y, with in-plane field perpendicular to the base). The excess of exchange energy at the triangles edges creates channels that allow effective spin wave propagation along the edges in the B state. These quasi one-dimensional SWs emitted by the vertex magnetic charges gradually transform from propagating to standing due to interference and (as pointed out by simulations) are weakly affected by small variations of the aspect ratio (from equilateral to isosceles dots) or by interdot dipolar interaction present in our dot arrays. SWs excited in the Y state have mainly a two-dimensional character. Propagation of the SWs along the edge states in triangular dots opens possibilities for creation of new and versatile spintronic devices.

We report on tunneling experiment between two quantum Hall droplets separated by a nearly ideal tunnel barrier. The device is produced by cleaved edge overgrowth that laterally juxtaposes two two-dimensional electron systems across a high quality semiconductor barrier. The dramatic evolution of the tunneling characteristics is consistent with the magnetic field-dependent tunneling between the coupled edge states of the quantum Hall droplets. We identify a series of quantum critical points between successive strong and weak tunneling regimes that are reminiscent of the plateau-transitions in quantum Hall effect. Scaling analysis shows that the conductance near the critical magnetic fields B_c is a function of a single scaling argument |B − B_c|T^-κ, where the exponent κ = 0.42. This puzzling resemblance to a quantum Hall-insulator transition points to the significance of interedge correlation in the lateral tunneling of quantum Hall droplets.