Absence of a gap in the Gaffnian state

Jolicœur, Thierry; Mizusaki, T.; Lecheminant, P.

doi:10.1103/physrevb.90.075116

Cited by 18 publications

(28 citation statements)

References 40 publications

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“…This "paradox" would be resolved if the Gaffnian parent Hamiltonian had gapless energy spectrum 12 , potentially arising as a critical point between two gapped FQH phases (such as in a bilayer system with tunneling 39 ). Recent numerical tests on the sphere 40 are indeed consistent with the Gaffnian neutral gap vanishing with system size, although similar results on the torus geometry have been inconclusive 41 . Furthermore, it has been analytically established that in the so-called thin torus limit the Gaffnian state is gapped 42,43 .…”

Section: Introductionmentioning

confidence: 88%

Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

Yang

Papić

2019

Phys. Rev. B

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Topological phases of matter are distinguished by topological invariants, such as Chern numbers and topological spins, that quantize their response to electromagnetic currents and changes of ambient geometry. Intriguingly, in the ν = 2/5 fractional quantum Hall effect, prominent theoretical approaches -the composite fermion theory and conformal field theory -have constructed two distinct states, the Jain composite fermion (CF) state and the Gaffnian state, for which many of the topological indices coincide and even the microscopic ground state wave functions have high overlap with each other in system sizes accessible to numerics. At the same time, some aspects of these states are expected to be very different, e.g., their elementary excitations should have either Abelian (CF) or non-Abelian (Gaffnian) statistics. In this paper we investigate the close relationship between these two states by considering not only their ground states, but also the low-energy charged excitations. We show that the low-energy physics of both phases is spanned by the same type of quasielectrons of the neighbouring Laughlin phase. The main difference between the two states arises due to an implicit assumption of short-range interaction in the CF approach, which causes a large splitting of the variational energies of the Gaffnian excitations. We thus propose that the Jain phase emerges as an effective Abelian low-energy description of the Gaffnian phase when the Hamiltonian is dominated by two-body interactions of sufficiently short range. arXiv:1907.12572v1 [cond-mat.str-el]

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Section: Introductionmentioning

confidence: 88%

Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

Yang

Papić

2019

Phys. Rev. B

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“…Guided by a recent Exact Diagonalization (ED) study 21 , we argue that the gap closing happens at the wave vector q = 0 at the Gaffnian state. (This kind of gap closing scenario has been studied before [24][25][26] in the FQHE in various other situations.)…”

Section: Neutral Excitations In the Gaffnian Statementioning

confidence: 86%

“…The CFT for the Gaffnian state is nonunitary resulting in a nonunitary braiding among quasi-holes 20 , which is not physically sensible. There has been an argument 30 , numerical evidence in the sphere geometry 21 , and even a proof 31 that the Gaffnian state represents a gapless state, but the physical picture of how the Gaffnian state becomes gapless remains elusive so far. To understand its gaplessness better, we focus on the neutral excitations of the Gaffnian state and examine the physical mechanism of closing of the gap.…”

Section: Neutral Excitations In the Gaffnian Statementioning

confidence: 99%

“…In this paper, we focus on a model wavefunction called the Gaffnian state 11 . Unlike other model FQH states, the Gaffnian state is believed to represent a quantum critical point 11,21,31 , or possibly a gapless phase, rather than an incompressible (gapped) state. Our interest in the Gaffnian state is as a likely example of gapless matter with some remnant of topological physics, even though precisely what "topological" means in a gapless state is difficult to define.…”

Section: Introductionmentioning

confidence: 99%

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Neutral excitations in the Gaffnian state

Kang¹,

Moore²

2017

Phys. Rev. B

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We study a model fractional quantum Hall (FQH) wavefunction called the Gaffnian state, which is believed to represent a gapless, strongly correlated state that is very different from conventional metals. To understand this exotic gapless state better, we provide a representation based on work of Halperin in which the pairing structure of the Gaffnian state becomes more explicit. We employ the single-mode approximation introduced by Girvin, MacDonald, and Platzman (GMP), here extended to three-body interactions, in order to treat a neutral collective exitation mode in order to clarify the physical origin of the gaplessness of the Gaffnian state. We discuss approaches to extract systematically the relevant physics in the long-distance, large-electron-number limit of FQH states using numerical calculations with relatively few electrons. In an appendix, we provide second quantized expressions for many-body Haldane pseudopotentials in various geometries including the plane, sphere, cylinder, and the torus based on the proper definition of the relative angular momentum.

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“…[20] that FQH model WFs built on a non-unitary CFT generically describe compressible states, and thus could not capture the physics of a Hall quantized conductance plateau. For instance the Gaffnian [23], which relies on a non-unitary CFT, was shown to be critical [13,30]. The HR state is suspected to follow the same behavior [8,25].…”

Section: A Overview Of the Hr Statementioning

confidence: 99%

Matrix product state description and gaplessness of the Haldane-Rezayi state

2019

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We derive an exact matrix product state representation of the Haldane-Rezayi state on both the cylinder and torus geometry. Our derivation is based on the description of the Haldane-Rezayi state as a correlator in a non-unitary logarithmic conformal field theory. This construction faithfully captures the ten degenerate ground states of this model state on the torus. Using the cylinder geometry, we probe the gapless nature of the phase by extracting the correlation length, which diverges in the thermodynamic limit. The numerically extracted topological entanglement entropies seem to only probe the Abelian part of the theory, which is reminiscent of the Gaffnian state, another model state deriving from a non-unitary conformal field theory. arXiv:1905.05192v2 [cond-mat.str-el]

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Absence of a gap in the Gaffnian state

Cited by 18 publications

References 40 publications

Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

Effective Abelian theory from a non-Abelian topological order in the ν=2/5 fractional quantum Hall effect

Neutral excitations in the Gaffnian state

Matrix product state description and gaplessness of the Haldane-Rezayi state

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