Non-Abelian generalizations of the Hofstadter model: spin–orbit-coupled butterfly pairs

Abstract The Hofstadter model, well known for its fractal butterfly spectrum, describes two-dimensional electrons under a perpendicular magnetic field, which gives rise to the integer quantum Hall effect. Inspired by the real-space building blocks of non-Abelian gauge fields from a recent experiment, we introduce and theoretically study two non-Abelian generalizations of the Hofstadter model. Each model describes two pairs of Hofstadter butterflies that are spin–orbit coupled. In contrast to the original Hofstadter model that can be equivalently studied in the Landau and symmetric gauges, the corresponding non-Abelian generalizations exhibit distinct spectra due to the non-commutativity of the gauge fields. We derive the genuine (necessary and sufficient) non-Abelian condition for the two models from the commutativity of their arbitrary loop operators. At zero energy, the models are gapless and host Weyl and Dirac points protected by internal and crystalline symmetries. Double (8-fold), triple (12-fold), and quadrupole (16-fold) Dirac points also emerge, especially under equal hopping phases of the non-Abelian potentials. At other fillings, the gapped phases of the models give rise to topological insulators. We conclude by discussing possible schemes for experimental realization of the models on photonic platforms.

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Synthesis and observation of non-Abelian gauge fields in real space

Science ◽

10.1126/science.aay3183 ◽

2019 ◽

Vol 365 (6457) ◽

pp. 1021-1025 ◽

Cited By ~ 9

Author(s):

Yi Yang ◽

Chao Peng ◽

Di Zhu ◽

Hrvoje Buljan ◽

John D. Joannopoulos ◽

...

Keyword(s):

Gauge Field ◽

Building Blocks ◽

Real Space ◽

Gauge Fields ◽

Final States ◽

Abelian Gauge ◽

Classical Waves ◽

Break Time ◽

Bohm Effect ◽

Aharonov Bohm

Particles placed inside an Abelian (commutative) gauge field can acquire different phases when traveling along the same path in opposite directions, as is evident from the Aharonov-Bohm effect. Such behaviors can get significantly enriched for a non-Abelian gauge field, where even the ordering of different paths cannot be switched. So far, real-space realizations of gauge fields have been limited to Abelian ones. We report an experimental synthesis of non-Abelian gauge fields in real space and the observation of the non-Abelian Aharonov-Bohm effect with classical waves and classical fluxes. On the basis of optical mode degeneracy, we break time-reversal symmetry in different manners, via temporal modulation and the Faraday effect, to synthesize tunable non-Abelian gauge fields. The Sagnac interference of two final states, obtained by reversely ordered path integrals, demonstrates the noncommutativity of the gauge fields. Our work introduces real-space building blocks for non-Abelian gauge fields, relevant for classical and quantum exotic topological phenomena.

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INTERACTING QUANTUM TOPOLOGIES AND THE QUANTUM HALL EFFECT

International Journal of Modern Physics A ◽

10.1142/s0217751x08039888 ◽

2008 ◽

Vol 23 (09) ◽

pp. 1327-1336 ◽

Cited By ~ 4

Author(s):

A. P. BALACHANDRAN ◽

KUMAR S. GUPTA ◽

SEÇKIN KÜRKÇÜOǦLU

Keyword(s):

Hall Effect ◽

Quantum Hall Effect ◽

Quantum Hall ◽

Wave Functions ◽

Gauge Fields ◽

Representation Space ◽

Covariant Formulation ◽

Integer Quantum Hall Effect ◽

Background Magnetic Field ◽

Integer Quantum

The algebra of observables of planar electrons subject to a constant background magnetic field B is given by [Formula: see text], the product of two mutually commuting Moyal algebras. It describes the free Hamiltonian and the guiding center coordinates. We argue that [Formula: see text] itself furnishes a representation space for the actions of these two Moyal algebras, and suggest physical arguments for this choice of the representation space. We give the proper setup to couple the matter fields based on [Formula: see text] to electromagnetic fields which are described by the Abelian commutative gauge group [Formula: see text], i.e. gauge fields based on [Formula: see text]. This enables us to give a manifestly gauge covariant formulation of integer quantum Hall effect (IQHE). Thus, we can view IQHE as an elementary example of interacting quantum topologies, where matter and gauge fields based on algebras [Formula: see text] with different θ′ appear. Two-particle wave functions in this approach are based on [Formula: see text]. We find that the full symmetry group in IQHE, which is the semidirect product [Formula: see text] acts on this tensor product using the twisted coproduct Δθ. Consequently, as we show, many particle sectors of each Landau level have twisted statistics. As an example, we find the twisted two particle Laughlin wave functions.

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Experimental realization of non-Abelian gauge field in circuit system

10.21203/rs.3.rs-85833/v1 ◽

2020 ◽

Author(s):

Rui Yu ◽

Ziyin Song ◽

Tianyu Wu ◽

Wenquan Wu

Keyword(s):

Gauge Field ◽

Building Blocks ◽

The Novel ◽

Spin Orbit ◽

Edge States ◽

Abelian Gauge ◽

Experimental Realization ◽

Wide Range ◽

Exotic Physics ◽

Spin Texture

Abstract Synthetic gauge field, especially the non-Abelian gauge field, has emerged as a new way to explore exotic physics in a wide range of materials and platforms. Here we present the building blocks, consisting of capacitors and inductors, to implement the non-Abelian tunneling matrices and show that circuit system is an appropriate choice to realize the non-Abelian gauge field. To demonstrate the novel physics enabled by the non-Abelian gauge field, we provide a simple and modular scheme to design the Rashba-Dresselhaus spin-orbit interaction and topological Chern state in circuits. By measuring the spin texture and chiral edge states of the resonant frequency band structures, we confirm the spin-orbit effect and topological Chern state in circuits. Our schemes open a broad avenue to study non-Abelian gauge field and related physics in circuit platform.

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REAL-SPACE RENORMALIZATION-GROUP APPROACH TO THE INTEGER QUANTUM HALL EFFECT

International Journal of Modern Physics B ◽

10.1142/s0217979205029742 ◽

2005 ◽

Vol 19 (13) ◽

pp. 2085-2119 ◽

Cited By ~ 5

Author(s):

PHILIPP CAIN ◽

RUDOLF A. RÖMER

Keyword(s):

Renormalization Group ◽

Large Scale ◽

Transport Coefficients ◽

Quantum Hall ◽

Real Space ◽

Hall Resistance ◽

Spacing Distribution ◽

Integer Quantum Hall Effect ◽

Real Space Renormalization Group ◽

Integer Quantum

We review recent results based on an application of the real-space renormalization group (RG) approach to a network model for the integer quantum Hall (QH) transition. We demonstrate that this RG approach reproduces the critical distribution of the power transmission coefficients, i.e., two-terminal conductances, P c (G), with very high accuracy. The RG flow of P(G) at energies away from the transition yields a value of the critical exponent ν that agrees with most accurate large-size lattice simulations. A description of how to obtain other relevant transport coefficients such as R L and R H is given. From the non-trivial fixed point of the RG flow we extract the critical level-spacing distribution (LSD). This distribution is close, but distinctively different from the earlier large-scale simulations. We find that the LSD obeys scaling behavior around the QH transition with ν = 2.37±0.02. Away from the transition it crosses over towards the Poisson distribution. We next investigate the plateau-to-insulator transition at strong magnetic fields. For a fully quantum coherent situation, we find a quantized Hall insulator with R H ≈h/e2 up to R L ~20h/e2 when interpreting the results in terms of most probable value of the distribution function P(R H ). Upon further increasing R L →∞, the Hall insulator with diverging Hall resistance [Formula: see text] is seen. The crossover between these two regimes depends on the precise nature of the averaging procedure for the distributions P(R L ) and P(R H ). We also study the effect of long-ranged inhomogeneities on the critical properties of the QH transition. Inhomogeneities are modeled by a smooth random potential with a correlator which falls off with distance as a power law r-α. Similar to the classical percolation, we observe an enhancement of ν with decreasing α. These results exemplify the surprising fact that a small RG unit, containing only five nodes, accurately captures most of the correlations responsible for the localization-delocalization transition.

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Localization and conductance fluctuations in the integer quantum Hall effect: Real-space renormalization-group approach

Physical Review B ◽

10.1103/physrevb.56.1422 ◽

1997 ◽

Vol 56 (3) ◽

pp. 1422-1429 ◽

Cited By ~ 45

Author(s):

A. G. Galstyan ◽

M. E. Raikh

Keyword(s):

Renormalization Group ◽

Quantum Hall Effect ◽

Quantum Hall ◽

Real Space ◽

Group Approach ◽

Integer Quantum Hall Effect ◽

Real Space Renormalization Group ◽

Conductance Fluctuations ◽

Renormalization Group Approach ◽

Integer Quantum

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An all-epitaxial nitride heterostructure with concurrent quantum Hall effect and superconductivity

Science Advances ◽

10.1126/sciadv.abf1388 ◽

2021 ◽

Vol 7 (8) ◽

pp. eabf1388

Author(s):

Phillip Dang ◽

Guru Khalsa ◽

Celesta S. Chang ◽

D. Scott Katzer ◽

Neeraj Nepal ◽

...

Keyword(s):

Hall Effect ◽

Quantum Hall Effect ◽

Electron Gas ◽

Quantum Hall ◽

High Electron Mobility Transistor ◽

High Electron ◽

2D Electron Gas ◽

Integer Quantum Hall Effect ◽

Topological Quantum ◽

Integer Quantum

Creating seamless heterostructures that exhibit the quantum Hall effect and superconductivity is highly desirable for future electronics based on topological quantum computing. However, the two topologically robust electronic phases are typically incompatible owing to conflicting magnetic field requirements. Combined advances in the epitaxial growth of a nitride superconductor with a high critical temperature and a subsequent nitride semiconductor heterostructure of metal polarity enable the observation of clean integer quantum Hall effect in the polarization-induced two-dimensional (2D) electron gas of the high-electron mobility transistor. Through individual magnetotransport measurements of the spatially separated GaN 2D electron gas and superconducting NbN layers, we find a small window of magnetic fields and temperatures in which the epitaxial layers retain their respective quantum Hall and superconducting properties. Its analysis indicates that in epitaxial nitride superconductor/semiconductor heterostructures, this window can be significantly expanded, creating an industrially viable platform for robust quantum devices that exploit topologically protected transport.

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Wiedemann-Franz laws and Sl(2, ℤ) duality in AdS/CMT holographic duals and one-dimensional effective actions for them

Journal of High Energy Physics ◽

10.1007/jhep05(2021)092 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Dmitry Melnikov ◽

Horatiu Nastase

Keyword(s):

Quantum Hall Effect ◽

Effective Action ◽

Transport Coefficients ◽

Quantum Hall ◽

Dual Theory ◽

One Dimensional ◽

Integer Quantum Hall Effect ◽

Weakly Coupled ◽

Holographic Duals ◽

Integer Quantum

Abstract In this paper we study the Wiedemann-Franz laws for transport in 2+1 dimensions, and the action of Sl(2, ℤ) on this transport, for theories with an AdS/CMT dual. We find that Sl(2, ℤ) restricts the RG-like flow of conductivities and that the Wiedemann-Franz law is $$ \overline{L}=\overline{\kappa}/\left( T\sigma \right)={cg}_4^2\uppi /3 $$ L ¯ = κ ¯ / Tσ = cg 4 2 π / 3 , from the weakly coupled gravity dual. In a self-dual theory this value is also the value of L = κ/(Tσ) in the weakly coupled field theory description. Using the formalism of a 0+1 dimensional effective action for both generalized SY Kq models and the AdS4 gravity dual, we calculate the transport coefficients and show how they can be matched at large q. We construct a generalization of this effective action that is invariant under Sl(2, ℤ) and can describe vortex conduction and integer quantum Hall effect.

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