Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect

We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism.

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“…saturating the lower bound for the leading coefficient proposed in [7,8]. This follows directly from the form of the LLL wave function (37).…”

Section: Response To Variation Of Curvature and The Structure Factorsupporting

confidence: 56%

“…It captures the universal features of the phenomena. For further discussion of this and related matters, we refer to recent papers [7,8]. Comparison with ab initio numerical studies would be desirable in this respect.…”

Section: Introductionmentioning

confidence: 99%

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

2015

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“…Another effective theory of the GMP mode was considered in Ref. [30], where the Wess-Zumino-Witten action was used to construct the kinetic term. Geometric degrees of freedom and possible means of their observation were also discussed in Refs.…”

Section: Introductionmentioning

confidence: 99%

Bimetric Theory of Fractional Quantum Hall States

2017

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We present a bimetric low-energy effective theory of fractional quantum Hall (FQH) states that describes the topological properties and a gapped collective excitation, known as Girvin-Macdonald-Platzman (GMP) mode. The theory consist of a topological Chern-Simons action, coupled to a symmetric rank two tensor, and an action \`a la bimetric gravity, describing the gapped dynamics of the spin-$2$ GMP mode. The theory is formulated in curved ambient space and is spatially covariant, which allows to restrict the form of the effective action and the values of phenomenological coefficients. Using the bimetric theory we calculate the projected static structure factor up to the $k^6$ order in the momentum expansion. To provide further support for the theory, we derive the long wave limit of the GMP algebra, the dispersion relation of the GMP mode, and the Hall viscosity of FQH states. We also comment on the possible applications to fractional Chern insulators, where closely related structures arise. Finally, it is shown that the familiar FQH observables acquire a curious geometric interpretation within the bimetric formalism.Comment: 14 pages, v2: Acknowledgments updated, v3: A few presentation improvements, Published versio

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“…Newton-Cartan geometry was originally developed by Cartan to describe Newtonian gravity within a geometric framework similar to that of General Relativity [10,11] (see also [12,13]). Recently, it has been used in the condensed matter literature as the natural setting for Galilean invariant physics, with applications that include cold atoms [14], non-relativistic fluids [6,[15][16][17], the quantum Hall effect [18][19][20][21][22], as well as non-relativistic holographic systems [23][24][25][26][27]. It is well recognized in the literature that it is necessary to couple these systems to torsionful geometries to define the full suite of currents available in a non-relativistic system and to study their linear response [9,23,24,26,28].…”

Section: Bargmann Spacetimesmentioning

confidence: 99%

Physical stress, mass, and energy for non-relativistic matter

Geracie

Prabhu

Roberts

2017

J. High Energ. Phys.

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For theories of relativistic matter fields there exist two possible definitions of the stress-energy tensor, one defined by a variation of the action with the coframes at fixed connection, and the other at fixed torsion. These two stress-energy tensors do not necessarily coincide and it is the latter that corresponds to the Cauchy stress measured in the lab. In this note we discuss the corresponding issue for non-relativistic matter theories. We point out that while the physical non-relativistic stress, momentum, and mass currents are defined by a variation of the action at fixed torsion, the energy current does not admit such a description and is naturally defined at fixed connection. Any attempt to define an energy current at fixed torsion results in an ambiguity which cannot be resolved from the background spacetime data or conservation laws. We also provide computations of these quantities for some simple non-relativistic actions.

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Spectral sum rules and magneto-roton as emergent graviton in fractional quantum Hall effect

Cited by 56 publications

References 27 publications

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

Bimetric Theory of Fractional Quantum Hall States

Physical stress, mass, and energy for non-relativistic matter

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