The Quantum Hall Effects

MacDonald, Allan H.

doi:10.1007/978-1-4899-3698-1_13

Cited by 5 publications

(4 citation statements)

References 59 publications

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“…Interestingly, topology in driven systems may directly give rise to quantization of time-averaged observables, such as the pumped current in the Thouless pump [34]. Similarly, the response of magnetization density to changes of chemical potential in a quantum Hall system is quantized when the chemical potential lies in an energy gap [35][36][37]. In contrast, here we find quantization of the magnetization density itself.…”

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confidence: 51%

Quantized Magnetization Density in Periodically Driven Systems

et al. 2017

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We study micromotion in two-dimensional periodically driven systems in which all bulk Floquet eigenstates are localized by disorder. We show that this micromotion gives rise to a quantized timeaveraged orbital magnetization density in any region completely filled with fermions. The quantization of magnetization density has a topological origin, and reveals the physical nature of the new phase identified in P. Titum, E. Berg, M. S. Rudner, G. Refael, and N. H. Lindner [Phys. Rev. X 6, 021013 (2016)]. We thus establish that the topological index of this phase can be accessed directly in bulk measurements, and propose an experimental protocol to do so using interferometry in cold-atom-based realizations. DOI: 10.1103/PhysRevLett.119.186801 Periodic driving was recently introduced as a means for achieving topological phenomena in a wide variety of quantum systems. Beyond providing new ways to obtain topologically nontrivial band structures [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15], periodic driving can give rise to wholly new types of topological phenomena without analogues in equilibrium [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32].In a periodically driven system, the unitary Floquet operator acts as a generator of discrete time evolution over each full driving period. As in nondriven systems, the spectrum and eigenstates of the Floquet operator can be classified according to topology [2,4,16]. However, in addition to the stroboscopic evolution of the system, the micromotion that takes place within each driving period is crucial for the topological classification of periodically driven systems [17][18][19][20][21][24][25][26][27][28].Here we uncover a new type of topological quantization phenomenon associated with the micromotion of periodically driven quantum systems. We focus on periodically driven two-dimensional (2D) lattice systems in which all bulk Floquet eigenstates are localized by disorder (see Fig. 1). We show that, within a region where all states are occupied, the time-averaged orbital magnetization density ⟪m⟫ is quantized, ⟪m⟫ ¼ ν=T, where ν is an integer and T is the driving period. The bulk observable ⟪m⟫ thus serves as a topological order parameter, characterizing the topologically distinct fully localized phases found in Ref. [22]. We propose a bulk interference measurement to probe this invariant in cold-atom systems.Topological invariants are often associated with quantized response functions. Famously, the Hall conductivity of an insulator is proportional to the Chern number [33]. Interestingly, topology in driven systems may directly give rise to quantization of time-averaged observables, such as the pumped current in the Thouless pump [34]. Similarly, the response of magnetization density to changes of chemical potential in a quantum Hall system is quantized when the chemical potential lies in an energy gap [35][36][37]. In contrast, here we find quantization of the magnetization density itself.For concreteness, we consider a periodically driven two-dimen...

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confidence: 51%

Quantized Magnetization Density in Periodically Driven Systems

et al. 2017

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“…In this section, we present a general formalism for establishing the second-quantized frustration-free Hamiltonians of interacting electrons confined to two spatial dimensions in the presence of an applied (perpendicular to the plane) magnetic field. As long known [50], under the influence of such a magnetic field, electrons occupy LL orbitals. Strong interactions among electrons may, however, effectively lead to the occupation of multiple LLs that Jain denominated as Λ-levels [51].…”

Section: Frustration-free Qh Hamiltoniansmentioning

confidence: 92%

“…( 22) may be considered as an expansion of the interaction potential in powers of its range (magnetic length) ℓ. This can be seen by noting that, for a ground state of H int with filling fraction ν, a relevant correlation length is [50] ∼ ℓ/ ν, proportional to the Wigner-Seitz radius. Thus, for a short range two-body interaction, it is typically sufficient to keep the first few pseudopotentials.…”

Section: Projected Two-body Hamiltoniansmentioning

confidence: 99%

Partons as unique ground states of quantum Hall parent Hamiltonians: The case of Fibonacci anyons

et al. 2023

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We present microscopic, multiple Landau level, (frustration-free and positive semi-definite) parent Hamiltonians whose ground states, realizing different quantum Hall fluids, are parton-like and whose excitations display either Abelian or non-Abelian braiding statistics. We prove ground state energy monotonicity theorems for systems with different particle numbers in multiple Landau levels, demonstrate S-duality in the case of toroidal geometry, and establish complete sets of zero modes of special Hamiltonians stabilizing parton-like states, specifically at filling factor \nu=2/3ν=2/3. The emergent Entangled Pauli Principle (EPP), introduced in [Phys. Rev. B 98, 161118(R) (2018)] and which defines the “DNA” of the quantum Hall fluid, is behind the exact determination of the topological characteristics of the fluid, including charge and braiding statistics of excitations, and effective edge theory descriptions. When the closed-shell condition is satisfied, the densest (i.e., the highest density and lowest total angular momentum) zero-energy mode is a unique parton state. We conjecture that parton-like states generally span the subspace of many-body wave functions with the two-body MM-clustering property within any given number of Landau levels, that is, wave functions with MMth-order coincidence plane zeroes and both holomorphic and anti-holomorphic dependence on variables. General arguments are supplemented by rigorous considerations for the M=3M=3 case of fermions in four Landau levels. For this case, we establish that the zero mode counting can be done by enumerating certain patterns consistent with an underlying EPP. We apply the coherent state approach of [Phys. Rev. X 1, 021015 (2011)] to show that the elementary (localized) bulk excitations are Fibonacci anyons. This demonstrates that the DNA associated with fractional quantum Hall states encodes all universal properties. Specifically, for parton-like states, we establish a link with tensor network structures of finite bond dimension that emerge via root level entanglement.

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“…The parameters of the theory are phenomenological and are extracted from experiments. In 1d, this Fermi-liquid picture breaks down [33], [35][36]. For example, the correlation functions have algebraic behaviour with anomalous exponents [10].…”

Section: Connections Between Orbifold Edge Theories and Generalized Lmentioning

confidence: 99%

Orbifold duality symmetries and quantum Hall systems

Skoulakis

Thomas

1999

Nuclear Physics B

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We consider the possible role that chiral orbifold conformal field theories may play in describing the edge state theories of quantum Hall systems. This is a generalization of work that already exists in the literature, where it has been shown that 1+1 chiral bosons living on a n-dimensional torus, and which couple to a U 1 gauge field, give rise to anomalous electric currents, the anomaly being related to the Hall conductivity.The well known O(n, n; Z) duality group associated with such toroidal conformal field theories transforms the edge states and Hall conductivities in a way which makes interesting connections between different theories, e.g. between systems exhibiting the integer and fractional quantum Hall effect. In this paper we try to explore the extension of these constructions to the case where such bosons live on a n-dimensional orbifold. We give a general formalism for discussing the relevant quantities like the Hall conductance and their transformation under the duality groups present in orbifold compactifications. We illustrate these ideas by presenting a detailed analysis of a toy model based on the two-dimensional Z 3 orbifold. In this model we obtain new classes of filling fractions, which generally correspond to fermionic edge states carrying fractional electric charge. We also consider the relation between orbifold edge theories and Luttinger liquids (LL's), which in the past have provided important insights into the physics of quantum Hall systems.

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The Quantum Hall Effects

Cited by 5 publications

References 59 publications

Quantized Magnetization Density in Periodically Driven Systems

Quantized Magnetization Density in Periodically Driven Systems

Partons as unique ground states of quantum Hall parent Hamiltonians: The case of Fibonacci anyons

Orbifold duality symmetries and quantum Hall systems

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