scholarly journals Linking topological features of the Hofstadter model to optical diffraction figures

2021 ◽  
Author(s):  
Francesco Di Colandrea ◽  
Alessio D'Errico ◽  
Maria Maffei ◽  
Hannah Price ◽  
Maciej Lewenstein ◽  
...  
Keyword(s):  
Diophantine Equations ◽  
Quantum Hall ◽  
Structural Disorder ◽  
Optical Diffraction ◽  
Bragg Condition ◽  
Energy Bands ◽  
Topological Features ◽  
Chern Numbers ◽  
Spatial Dimensions ◽  
Diffraction Peaks

Abstract In two, three and even four spatial dimensions, the transverse responses experienced by a charged particle on a lattice in a uniform magnetic field are fully controlled by topological invariants called Chern numbers, which characterize the energy bands of the underlying Hofstadter Hamiltonian. These remarkable features, solely arising from the magnetic translational symmetry, are captured by Diophantine equations which relate the fraction of occupied states, the magnetic flux and the Chern numbers of the system bands. Here we investigate the close analogy between the topological properties of Hofstadter Hamiltonians and the diffraction figures resulting from optical gratings. In particular, we show that there is a one-to-one relation between the above mentioned Diophantine equation and the Bragg condition determining the far-field positions of the optical diffraction peaks. As an interesting consequence of this mapping, we discuss how the robustness of diffraction figures to structural disorder in the grating is a direct analogy of the robustness of transverse conductance in the Quantum Hall effect.

scholarly journals Topological gaps by twisting

2021 ◽  
Vol 4 (1) ◽  
Author(s):  
Matheus I. N. Rosa ◽  
Massimo Ruzzene ◽  
Emil Prodan
Keyword(s):  
Magnetic Field ◽  
Quantum Hall Effect ◽  
Twist Angle ◽  
Quantum Hall ◽  
Topological Phases ◽  
Edge States ◽  
Layered Systems ◽  
Virtual Laboratories ◽  
Higher Dimensional ◽  
Chern Numbers

AbstractTwisted bilayered systems such as bilayered graphene exhibit remarkable properties such as superconductivity at magic angles and topological insulating phases. For generic twist angles, the bilayers are truly quasiperiodic, a fact that is often overlooked and that has consequences which are largely unexplored. Herein, we uncover that twisted n-layers host intrinsic higher dimensional topological phases, and that those characterized by second Chern numbers can be found in twisted bi-layers. We employ phononic lattices with interactions modulated by a second twisted lattice and reveal Hofstadter-like spectral butterflies in terms of the twist angle, which acts as a pseudo magnetic field. The phason provided by the sliding of the layers lives on 2n-tori and can be used to access and manipulate the edge states. Our work demonstrates how multi-layered systems are virtual laboratories for studying the physics of higher dimensional quantum Hall effect, and can be employed to engineer topological pumps via simple twisting and sliding.

scholarly journals Measuring Chern numbers in Hofstadter strips

2017 ◽  
Vol 3 (2) ◽  
Author(s):  
Samuel Mugel ◽  
Alexandre Dauphin ◽  
Pietro Massignan ◽  
Leticia Tarruell ◽  
Maciej Lewenstein ◽  
...  
Keyword(s):  
Boundary Conditions ◽  
Periodic Boundary Conditions ◽  
Chern Number ◽  
Open Boundary ◽  
Topological Invariants ◽  
Transverse Displacement ◽  
Open Boundary Conditions ◽  
Open Questions ◽  
Chern Numbers ◽  
Spatial Dimensions

Topologically non-trivial Hamiltonians with periodic boundary conditions are characterized by strictly quantized invariants. Open questions and fundamental challenges concern their existence, and the possibility of measuring them in systems with open boundary conditions and limited spatial extension. Here, we consider transport in Hofstadter strips, that is, two-dimensional lattices pierced by a uniform magnetic flux which extend over few sites in one of the spatial dimensions. As we show, an atomic wave packet exhibits a transverse displacement under the action of a weak constant force. After one Bloch oscillation, this displacement approaches the quantized Chern number of the periodic system in the limit of vanishing tunneling along the transverse direction. We further demonstrate that this scheme is able to map out the Chern number of ground and excited bands, and we investigate the robustness of the method in presence of both disorder and harmonic trapping. Our results prove that topological invariants can be measured in Hofstadter strips with open boundary conditions and as few as three sites along one direction.

Organic Conductors and Superconductors: New Directions in the Solid State

MRS Bulletin ◽  
1993 ◽  
Vol 18 (8) ◽  
pp. 29-37 ◽  
Author(s):  
J.S. Brooks
Keyword(s):  
Magnetic Field ◽  
Quantum Hall ◽  
Energy Scale ◽  
Energy Bands ◽  
Organic Salts ◽  
Semiconductor Physics ◽  
Low Dimensions ◽  
New Directions ◽  
One Step ◽  
Present Understanding

In single-crystal organic salts, we find a keen competition between superconducting, magnetic, insulating, and metallic states. The physics of these materials is further enriched by the sensitivity of these states to pressure, temperature, chemical formulation, and magnetic field. A growing international community of scientists have turned their attention to these materials, and are applying the techniques and theories of metal and semiconductor physics to probe these new systems. In this article we will explore these materials. We will discover that these materials have given us many new things: a renaissance in fermiology, new high-magnetic-field states of matter, a bulk quantum Hall effect, new challenges in the calculation of energy bands on a small energy scale, and elusive behavior which seems one step away from our present understanding of physics in low dimensions. Electron correlations probably play an important role in determining the phenomena, and should be considered in more microscopic theoretical treatments of these systems.

scholarly journals Lattice Clifford fractons and their Chern-Simons-like theory

2021 ◽  
Vol 4 (2) ◽  
Author(s):  
Weslei Fontana ◽  
Pedro Gomes ◽  
Claudio Chamon
Keyword(s):  
Clifford Algebra ◽  
Degrees Of Freedom ◽  
Quantum Hall ◽  
Effective Theory ◽  
Matrix Representations ◽  
Quantum Hall States ◽  
Dirac Matrix ◽  
Spatial Dimensions ◽  
K Matrix ◽  
Chern Simons

We use Dirac matrix representations of the Clifford algebra to build fracton models on the lattice and their effective Chern-Simons-like theory. As an example, we build lattice fractons in odd D spatial dimensions and their (D+1) spacetime dimensional effective theory. The model possesses an anti-symmetric K matrix resembling that of hierarchical quantum Hall states. The gauge charges are conserved in sub-dimensional manifolds which ensures the fractonic behavior. The construction extends to any lattice fracton model built from commuting projectors and with tensor products of spin-1/2 degrees of freedom at the sites.

EDGE CURRENT CHANNELS AND CHERN NUMBERS IN THE INTEGER QUANTUM HALL EFFECT

2002 ◽  
Vol 14 (01) ◽  
pp. 87-119 ◽  
Author(s):  
J. KELLENDONK ◽  
T. RICHTER ◽  
H. SCHULZ-BALDES
Keyword(s):  
Exact Sequence ◽  
Half Plane ◽  
Quantum Hall Effect ◽  
Duality Theorem ◽  
Quantum Hall ◽  
Edge Channel ◽  
Integer Quantum Hall Effect ◽  
Edge Current ◽  
Chern Numbers ◽  
Integer Quantum

A quantization theorem for the edge currents is proven for discrete magnetic half-plane operators. Hence the edge channel number is a valid concept also in presence of a disordered potential. Under a gap condition on the corresponding planar model, this quantum number is shown to be equal to the quantized Hall conductivity as given by the Kubo–Chern formula. For the proof of this equality, we consider an exact sequence of C*-algebras (the Toeplitz extension) linking the half-plane and the planar problem, and use a duality theorem for the pairings of K-groups with cyclic cohomology.

Diffraction anomalous fine structure study of iron/iron oxide nanoparticles

2009 ◽  
Vol 42 (4) ◽  
pp. 642-648 ◽  
Author(s):  
Carlo Meneghini ◽  
Federico Boscherini ◽  
Luca Pasquini ◽  
Hubert Renevier
Keyword(s):  
Fine Structure ◽  
Local Structure ◽  
Structural Disorder ◽  
Structure Study ◽  
Nanocrystalline Powders ◽  
Oxide Shell ◽  
Iron Iron ◽  
Diffraction Peaks ◽  
Diffraction Patterns ◽  
High Degree

Diffraction anomalous fine structure is a recently developed technique which can provide a measurement of the local structure of a given element in a particular phase or crystallographic site. Most previous investigations have applied the technique to bulk solids, thin films, and nano-dots or wires on crystalline substrates. In this paper, the technique is applied to highly disordered nanometre-sized Fe/Fe oxide core–shell nanocrystalline powders, the diffraction patterns of which exhibit weak and greatly broadened diffraction peaks. Focusing on the oxide shell diffraction peaks, a qualitative analysis of the near-edge spectral region and a quantitative analysis of the extended energy region are provided; in particular, good quality fittings of the extended range spectra are obtained. The local structure is selectively probed around the tetrahedral and octahedral sites of the oxide shell, finding the presence of a high degree of structural disorder. This study demonstrates that diffraction anomalous fine structure can now be fruitfully applied to nanocrystalline powders.

scholarly journals Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Bruno Mera ◽  
Anwei Zhang ◽  
Nathan Goldman
Keyword(s):  
Quantum Physics ◽  
Berry Phase ◽  
Broad Class ◽  
Topological States Of Matter ◽  
Quantum Geometry ◽  
Special Cases ◽  
Chern Numbers ◽  
Spatial Dimensions ◽  
Topological States ◽  
Engineered Systems

Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles:~the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.

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