Quantum Hall effect in momentum space

Ozawa, Tomoki; Price, Hannah M.; Carusotto, Iacopo

doi:10.1103/physrevb.93.195201

Cited by 11 publications

(7 citation statements)

References 32 publications

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“…If the duality leads to a simple free-particle model with well-studied solution, the trace of quantum metric can be understood under a direct inference. Similar concept of position-momentum duality was proposed previously in the context of harmonic traps [72][73][74] and fractional Chern insulators [61,62]. These works adopted the confinement potential V (r) ∼ |r| 2 to insert the band-geometry contribution into the actual electronic energy.…”

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confidence: 85%

Band geometry from position-momentum duality at topological band crossings

Lin,

Hsiao

2021

Preprint

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We show that the position-momentum duality offers a transparent interpretation of the band geometry at the topological band crossings. Under this duality, the band geometry with Berry connection is dual to the free-electron motion under gauge field. This identifies the trace of quantum metric as the dual energy in momentum space. The band crossings with Berry defects thus induce the dual energy quantization in the trace of quantum metric. For a Z nodal-point or nodal-surface semimetal, a dual Landau level quantization occurs owing to the Berry charge. Meanwhile, a nodalloop semimetal exhibits a Berry vortex line, leading to the quantized dual rotational energy about the nodal loop. A Z2 monopole brings about another dual rotational energy, which originates from the link with an additional nodal line. Nontrivial band geometry generically induces finite spread in the Wannier functions. While the spread manifests a quantized lower bound in a Z nodal-point or nodal-surface semimetal, logarithmic divergence occurs in a nodal-loop semimetal. The band geometry at the band crossings may be probed experimentally by a periodic-drive measurement.

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confidence: 85%

Band geometry from position-momentum duality at topological band crossings

Lin,

Hsiao

2021

Preprint

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“…In physics, there is a powerful duality between position and momentum, such that a physical system can be equivalently described in either position space or momentum space. If the roles of position and momentum are reversed, this can lead to momentum-space topological physics, ranging from momentum-space lattices 70,71 , to momentum-space Landau levels 72,73 , and momentumspace integer 74 and fractional 75 quantum Hall effects. In the context of synthetic dimensions, the positionmomentum duality is reflected in a re-interpretation of a discrete set of free-particle momentum states as the set of lattice sites 38,39,43 .…”

Section: B Momentum Statesmentioning

confidence: 99%

Topological quantum matter in synthetic dimensions

Ozawa¹,

2019

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In the field of quantum simulation of condensed matter phenomena by artificially engineering the Hamiltonian of an atomic, molecular or optical system, the concept of 'synthetic dimensions' has recently emerged as a powerful way to emulate phenomena such as topological phases of matter, which are now of great interest across many areas of physics. The main idea of a synthetic dimension is to couple together suitable degrees of freedom, such as a set of internal atomic states, in order to mimic the motion of a particle along an extra spatial dimension. This approach provides a way to engineer lattice Hamiltonians and enables the realisation of higher-dimensional topological models in platforms with lower dimensionality. We give an overview of the recent progress in studying topological matter in synthetic dimensions. After reviewing proposals and realizations in various setups, we discuss future prospects in many-body physics, applications, and topological effects in three or more spatial dimensions.

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“…This is the signature of position-momentum duality in quantum Hall states as recently emphasized in refs. [28,68]. For these reasons, we consider the Maxwell algebra, which is identified by the following anti-commuting relations:…”

Section: Bf Theory From the Maxwell Algebramentioning

confidence: 99%

“…where the constant β = 1, which plays the role for momentum-space magnetic length [68], has been fixed for simplicity. By replacing the zweibein with the metric tensor, it is then possible to rewrite the above action in the second-order formalism (up to a total derivative) [41] S[φ,…”

Section: Bf Theory From the Maxwell Algebramentioning

confidence: 99%

Momentum-space cigar geometry in topological phases

Palumbo

2018

Eur. Phys. J. Plus

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Abstract. In this paper, we stress the importance of momentum-space geometry in the understanding of two-dimensional topological phases of matter. We focus, for simplicity, on the gapped boundary of threedimensional topological insulators in class AII, which are described by a massive Dirac Hamiltonian and characterized by an half-integer Chern number. The gap is induced by introducing a magnetic perturbation, such as an external Zeeman field or a ferromagnet on the surface. The quantum Bures metric acquires a central role in our discussion and identifies a cigar geometry. We first derive the Chern number from the cigar geometry and we then show that the quantum metric can be seen as a solution of two-dimensional non-Abelian BF theory in momentum space. The gauge connection for this model is associated to the Maxwell algebra, which takes into account the Lorentz symmetries related to the Dirac theory and the momentum-space magnetic translations connected to the magnetic perturbation. The Witten black-hole metric is a solution of this gauge theory and coincides with the Bures metric. This allows us to calculate the corresponding momentum-space entanglement entropy that surprisingly carries information about the real-space conformal field theory describing the defect lines that can be created on the gapped boundary.

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Quantum Hall effect in momentum space

Cited by 11 publications

References 32 publications

Band geometry from position-momentum duality at topological band crossings

Band geometry from position-momentum duality at topological band crossings

Topological quantum matter in synthetic dimensions

Momentum-space cigar geometry in topological phases

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