Four-dimensional topological lattices through connectivity

Thanks to recent advances, the 4D quantum Hall (QH) effect is becoming experimentally accessible in various engineered set-ups. In this paper, we propose a new type of 4D topological system that, unlike other 2D and 4D QH models, does not require complicated (artificial) gauge fields and/or time-reversal symmetry breaking. Instead, we show that there are 4D QH systems that can be engineered for spinless particles by designing the lattice connectivity with real-valued hopping amplitudes, and we explain how this physics can be intuitively understood in analogy with the 2D Haldane model. We illustrate our discussion with a specific 4D lattice proposal, inspired by the widely-studied 2D honeycomb and brickwall lattice geometries. This also provides a minimal model for a topological system in Class AI, which supports nontrivial topological band invariants only in four spatial dimensions or higher

Momentum-space Harper-Hofstadter model

We show how the weakly trapped Harper-Hofstadter model can be mapped onto a Harper-Hofstadter model in momentum space. In this momentum-space model, the band dispersion plays the role of the periodic potential, the Berry curvature plays the role of an effective magnetic field, the real-space harmonic trap provides the momentum-space kinetic energy responsible for the hopping, and the trap position sets the boundary conditions around the magnetic Brillouin zone. Spatially local interactions translate into nonlocal interactions in momentum space: within a mean-field approximation, we show that increasing interparticle interactions leads to a structural change of the ground state, from a single rotationally symmetric ground state to degenerate ground states that spontaneously break rotational symmetry.Comment: 10 pages, 7 figure

Quantum Mechanics with a Momentum-Space Artificial Magnetic Field

The Berry curvature is a geometrical property of an energy band which acts as a momentum space magnetic field in the effective Hamiltonian describing single-particle quantum dynamics. We show how this perspective may be exploited to study systems directly relevant to ultracold gases and photonics. Given the exchanged roles of momentum and position, we demonstrate that the global topology of momentum space is crucially important. We propose an experiment to study the Harper-Hofstadter Hamiltonian with a harmonic trap that will illustrate the advantages of this approach and that will also constitute the first realization of magnetism on a torus

The 6D quantum Hall effect and 3D topological pumps

Modern technological advances allow for the study of systems with additional synthetic dimensions. Using such approaches, higher-dimensional physics that was previously deemed to be of purely theoretical interest has now become an active field of research. In this work, we derive from first principles using a semiclassical equation of motions approach, the bulk response of a six-dimensional Chern insulator. We find that in such a system a quantized bulk response appears with a quantization originating from a six-dimensional topological index -- the 3rd Chern number. Alongside this novel six-dimensional response, we rigorously describe the lower even-dimensional Chern-like responses that can occur due to nonvanishing 1st and 2nd Chern numbers in sub-spaces of the six-dimensional space. Last, we propose how to realize such a bulk response using three-dimensional topological charge pumps in cold atomic systems.Comment: 12 pages + 13 pages of supporting material, 2 figures, published versio

An evaluation of the economic impact of broadband in Lincolnshire: updated final report

The Lincolnshire Broadband Initiative, ‘onlincolnshire’, was launched in 2003 to bring a range of broadband supply and demand stimulation activities to businesses across the county. The initiative has used £15 million of European funding, together with matched funding from Lincolnshire County Council, to support a series of significant Information and Communication Technology (ICT) interventions to provide support and financial assistance to eligible Lincolnshire businesses. The ‘onlincolnshire’ initiative has four long term strategic objectives. By 2010: • Lincolnshire will be the foremost rural County in the UK, with regards to ICT usage and skills and will have a commercial environment that embraces ICT; • The main employment sites and premises will have attracted increased and more diverse investment; • ICT will have made a major contribution to business competitiveness, expansion and diversification of the economy – measured through an increase in ICT related employment and a range of ICT based activities; • To have engaged individuals and employers in improving ICT skills to increase local competitiveness, raise the standards, participation and achievement in ICT throughout the County

Understanding take-up of broadband by small and micro-enterprises: a case study

The paper examines patterns of broadband take-up, implementation and effects within small and micro-sized enterprises. The research focuses on the ‘onlincolnshire’ initiative, a local government-led programme which aims to encourage broadband take-up through provision of connection subsidies to SMEs in rural areas of Lincolnshire

Floquet topological system based on frequency-modulated classical coupled harmonic oscillators

We theoretically propose how to observe topological effects in a generic classical system of coupled harmonic oscillators, such as classical pendula or lumped-element electric circuits, whose oscillation frequency is modulated fast in time. Making use of Floquet theory in the high frequency limit, we identify a regime in which the system is accurately described by a Harper-Hofstadter model where the synthetic magnetic field can be externally tuned via the phase of the frequency-modulation of the different oscillators. We illustrate how the topologically-protected chiral edge states, as well as the Hofstadter butterfly of bulk bands, can be observed in the driven-dissipative steady state under a monochromatic drive. In analogy with the integer quantum Hall effect, we show how the topological Chern numbers of the bands can be extracted from the mean transverse shift of the steady-state oscillation amplitude distribution. Finally we discuss the regime where the analogy with the Harper-Hofstadter model breaks down.Comment: 15 pages, 9 figure

Propagating edge states in strained honeycomb lattices

We investigate the helically-propagating edge states associated with pseudo-Landau levels in strained honeycomb lattices. We exploit chiral symmetry to derive a general criterion for the existence of these propagating edge states in the presence of only nearest-neighbour hoppings and we verify our criterion using numerical simulations of both uni-axially and trigonally strained honeycomb lattices. We show that the propagation of the helical edge state can be controlled by engineering the shape of the edges. Sensitivity to chiral-symmetry-breaking next-nearest-neighbour hoppings is assessed. Our result opens up an avenue toward the precise control of edge modes through manipulation of the edge shape

How to directly observe Landau levels in driven-dissipative strained honeycomb lattices

We study the driven-dissipative steady-state of a coherently-driven Bose field in a honeycomb lattice geometry. In the presence of a suitable spatial modulation of the hopping amplitudes, a valley-dependent artificial magnetic field appears and the low-energy eigenmodes have the form of relativistic Landau levels. We show how the main properties of the Landau levels can be extracted by observing the peaks in the absorption spectrum of the system and the corresponding spatial intensity distribution. Finally, quantitative predictions for realistic lattices based on photonic or microwave technologies are discussed.Comment: Special Issue Article: Focus on Artificial Graphen