W-infinity symmetry in the quantum hall effect beyond the edge

Cappelli, Andrea; Maffi, Lorenzo

doi:10.1007/jhep05(2021)120

Cited by 6 publications

(5 citation statements)

References 53 publications

(118 reference statements)

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“…Abstractly, one can also think of (18) as the logarithmic time derivative of the path of deformations g t (ϕ). One can then plug (18) in the Lie algebra operator (12) and use the Berry phase…”

Section: Berry Phases From Circle Deformationsmentioning

confidence: 99%

“…In the case of 1D deformations, we exhibited this with eq. ( 19), involving the 1D vector field (18) and its flow g t . It is therefore essential to become acquainted with the 2D version of these objects, i.e.…”

Section: Divergence-free Vector Fields and Incompressible Flowsmentioning

confidence: 99%

“…As usual for Lie groups, the map that sends Lie algebra elements on group elements is the exponential. The latter is really a flow in the case of diffeomorphism groups: given a vector field v t , say even time-dependent, consider the time-dependent diffeomorphisms g t given by the 2D version of the logarithmic derivative (18),…”

Section: Divergence-free Vector Fields and Incompressible Flowsmentioning

confidence: 99%

“…The key actors for these lines of thought are area-preserving deformations spanning the Girvin-MacDonald-Platzmann algebra [9], i.e. a w 1+∞ algebra [11][12][13][14][15][16][17][18]. In particular, a staple of the seminal works [12][13][14] was the sketch of a relation between bulk deformations and conformal transformations of gapless edge modes.…”

Section: Introductionmentioning

confidence: 99%

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Adiabatic deformations of quantum Hall droplets

Oblak,

Estienne

2023

SciPost Phys.

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We consider area-preserving deformations of the plane, acting on electronic wave functions through “quantomorphisms” that change both the underlying metric and the confining potential. We show that adiabatic sequences of such transformations produce Berry phases that can be written in closed form in terms of the many-body current and density, even in the presence of interactions. For a large class of deformations that generalize squeezing and shearing, the leading piece of the phase is a super-extensive Aharonov-Bohm term (\propto N^2∝N2 for NN electrons) in the thermodynamic limit. Its gauge-invariant subleading partner only measures the current, whose dominant contribution to the phase stems from a jump at the edge in the limit of strong magnetic fields. This results in a finite Berry curvature per unit area, reminiscent of the Hall viscosity. We show that the latter is in fact included in our formalism, bypassing its standard derivation on a torus and suggesting realistic experimental setups for its observation in quantum simulators.

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Section: Berry Phases From Circle Deformationsmentioning

confidence: 99%

Section: Divergence-free Vector Fields and Incompressible Flowsmentioning

confidence: 99%

Section: Divergence-free Vector Fields and Incompressible Flowsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Adiabatic deformations of quantum Hall droplets

Oblak,

Estienne

2023

SciPost Phys.

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“…Richer algebraic structures in QH systems other than the Heisenberg algebra of the ladder operators were proposed by Cappelli, Trugenberger, and Zemba in 1993, who found that the product of ladder operators could form a W ∞ -algebra, and the similar algebra was generalized to FQH afterward [103][104][105][106][107]. In fact, the GMP algebra of density operators is isomorphic to a W ∞ -algebra as well, which reveals their nature as area-preserving deformations of the incompressible phase and partially explains why there can be analytic expressions of physical quantities for the SMA wave function including those derived in this thesis (more details can be found in Appendix.B).…”

Section: Harvest Time: 1993 -2003mentioning

confidence: 99%

Graviton modes in fractional quantum hall liquids

Wang¹

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The fractional quantum Hall fluid is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures. Neutral excitations in a fractional quantum Hall droplet define the incompressibility gap of the topological phase. Among these states, there are some specific modes in the long-wavelength limit that can be understood as "spin-2 gravitons". In this thesis, we will introduce a set of analytical results for the energy gap of the graviton modes for model Hamiltonians in the thermodynamic limit, which is governed by a well-defined and universal characteristic tensor. These results can help to construct model Hamiltonians for the graviton modes of different FQH phases and elucidate a hierarchical structure of conformal Hilbert spaces (null spaces of model Hamiltonians) containing the graviton modes and their corresponding ground states. An isomorphism can be defined for these conformal Hilbert spaces, and the mapping between them can be regarded as a more rigorous and general reinterpretation of the composite fermionization of FQH states, with naturally emergent composite fermions (each consisting of one electron and an even number of fluxes). The results from exact diagonalization will be shown, which confirm that for gapped phases, low-lying neutral excitations can undergo a phase transition even when the ground state remains in the same phase. Furthermore, the gaplessness of the Gaffnian state could be testified based on this formalism with further numerical experiments. Recently there has been numerical evidence implying the signature of multiple graviton modes. We will introduce the microscopic theory for the emergence of multiple gravitons in fractional quantum Hall droplets based on composite fermionization and the well-defined particle-hole conjugate within a specific conformal Hilbert space. This reveals the dynamical nature of this phenomenon and provides theoretical insights into the chirality and the "merging and splitting" behaviors of the graviton modes. The experimental relevance of multiple graviton modes will be discussed. The microscopic theory of gravitons can also provide valuable insights into the field-theoretical approaches.

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Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

Aizawa,

Sato

2023

Nuclear Physics B

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W-infinity symmetry in the quantum hall effect beyond the edge

Cited by 6 publications

References 53 publications

Adiabatic deformations of quantum Hall droplets

Adiabatic deformations of quantum Hall droplets

Graviton modes in fractional quantum hall liquids

Hom-Lie-Virasoro symmetries in Bloch electron systems and quantum plane in tight binding models

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