The Ophiuridae / by C. F. Lütken and Th. Mortensen

Lütken, Chr. Fr.; Mortensen, Th.

doi:10.5962/bhl.title.65637

Cited by 10 publications

(27 citation statements)

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“…In both cases we reproduce the result, that η = ±1/q, [12] and [13]. Note in passing that the transition from bosonic to fermionic conductivities given by equation (14) of reference [4] is implemented by the action of an element of Γ(1) which is not in Γ 0 (2).…”

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confidence: 70%

“…The above discussion gives the explicit connection between the Chern-Simons analysis of [4] and the group theory analysis of [2]. We make a final comment about the 'semi-circle' law of reference [14] - [16].…”

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confidence: 91%

“…The purpose of this letter is to explore the consequences of the proposal, made in [1] and examined further in [2] [3], that the hierarchical structure of the zero temperature integer and fractional quantum Hall effects can be interpreted in terms of the properties of a subgroup of the modular group, Sl(2, Z) := Γ(1) -specifically the subgroup which consists of elements of Γ(1) whose bottom left entry is even, sometimes denotd Γ 0 (2) in the mathematical literature. This group acts on the upperhalf complex plane, parameterised by the complex conductivity, σ = σ xy + iσ xx , in units of e 2 h , and is generated by two operations, T : σ → σ + 1 and X : σ → σ 2σ+1 . If γ = a b 2c d ∈ Γ 0 (2), with a, b, c, and d ∈ Z and ad − 2cb = 1, then γ(σ) = aσ+b 2cσ+d .…”

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confidence: 99%

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Duality and the modular group in the quantum Hall effect

Dolan¹

1999

J. Phys. A: Math. Gen.

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We explore the consequences of introducing a complex conductivity into the quantum Hall effect. This leads naturally to an action of the modular group on the upper-half complex conductivity plane. Assuming that the action of a certain subgroup, compatible with the law of corresponding states, commutes with the renormalisation group flow, we derive many properties of both the integer and fractional quantum Hall effects including: universality; the selection rule |p1q2 − p2q1| = 1 for transitions between quantum Hall states characterised by filling factors ν1 = p1/q1 and ν2 = p2/q2; critical values of the conductivity tensor; and Farey sequences of transitions. Extra assumptions about the form of the renormalisation group flow lead to the semi-circle rule for transitions between Hall plateaux.

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confidence: 91%

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Duality and the modular group in the quantum Hall effect

Dolan¹

1999

J. Phys. A: Math. Gen.

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“…A common feature of these two systems is the emergence of modular symmetry (strictly a sub-group of the modular group) in the infra-red regime. That the modular group may be relevant to the quantum Hall effect was first suggested in [1], though the correct subgroup was only found later [2].…”

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confidence: 99%

N = 2 Supersymmetric Yang–mills and the Quantum Hall Effect

Dolan

2006

Int. J. Mod. Phys. A

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It is argued that there are strong similarities between the infra-red physics of N=2 supersymmetric Yang-Mills and that of the quantum Hall effect, both systems exhibit a hierarchy of vacua with a sub-group of the modular group mapping between them. The coupling flow for pure SU (2) N = 2 supersymmetric Yang-Mills in 4-dimensions is reexamined and an earlier suggestion in the literature, that was singular at strong coupling, is modified to a form that is well behaved at both weak and strong coupling and describes the crossover in an analytic fashion. Similarities between the phase diagram and the flow of SUSY Yang-Mills and that of the quantum Hall effect are then described, with the Hall conductivity in the latter playing the role of the θparameter in the former. Hall plateaux, with odd denominator filling fractions, are analogous to fixed points at strong coupling in N=2 SUSY Yang-Mills, where the massless degrees of freedom carry an odd monopole charge.

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“…To date we are not aware of any experimental study of another plateau-insulator transition that probes the duality relations in the quantum regime where the temperatures are low enough for the fixed point structure to be reached [8]. Given that the duality predictions following from modular symmetry are rigid and precise, such measurements are capable of providing a definitive test of the emergent modular symmetry in the quantum Hall system.…”

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On the origin of duality in the quantum Hall system

Lütken

Ross

2010

Physics Letters A

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We discuss the possible origin of the duality observed in the quantum Hall current-voltage characteristics. We clarify the difference between "particle-vortex" (complex modular) duality, which acts on the full transport tensor, and "charge-flux" ("real") duality, which acts directly on the filling factor. Comparison with experiment strongly favors the form of duality which descends from the modular symmetry group acting holomorphically on the compexified conductivity. The dissipative IV-characteristics V x (B, I) = R xx (B, I) · I, on the other hand, are extremely non-linear in both phases, degenerating to a linear (Ohmic) relation only when B → B c (Fig. 1, top inset).Shahar et al.[1] discovered that to each B there exists a dual field value B d , such that the dissipative IV-curve V (B, I) (suppressing the now superfluous subscript on V x ) after reflection in the diagonal V = I is virtually identical to the dual IV-curve V d (B d , I d ) in the opposite phase. This is implies thata remarkable relation providing unambiguous evidence of a duality symmetry in the QH system, for a particularly simple quantum phase transition. It reveals a profound1 The Hall bar is a high mobility (µ = 5.5 × 10 5 cm 2 /sV ), low density (n = 6.5 × 10 10 /cm 2 ) GaAs/AlGaAs hetero-structure of size Lx × Ly, aligned with a current I in the x-direction so that R * x = (L * /Ly)ρ * x. The Hall resistance R H = ρ H = ρyx is quantized in the fundamental unit of resistance, h/e 2 ≈ 25.8 kΩ, while the dissipative resistance R D = ρxx/ is rescaled by the aspect ratio = Ly/Lx. connection between the transport mechanisms deep inside the quantum liquid and insulator phases, a symmetry which holds to great accuracy even far from the quantum critical point found at:consistent with the self-dual value of eq. (1). We are only aware of two (related) theoretical frameworks that aspire to account for these data (and which motivated the experimental investigation of duality). The first [4] is based on an effective description of the macroscopic theory that posesses a holomorphic modular symmetry relating the complexified response functionsin different QH phases. It successfully predicts the full phase diagram of the QH system, both integer and fractional, including the position of the quantum critical points governing the scaling behaviour of transitions between QH levels [4]. A "microscopic" 3 interpretation of the symmetry as "particle-vortex duality" was given in ref. [5].The second framework is provided by a "microscopic" model [6] in which a so-called "flux attachment transformation" maps the two-dimensional electron system in a magnetic field onto a bosonic system in a different "effective" field. This "charge-flux duality" resulted in a set of rules, known collectively as "the law of corresponding states", which relates QH states carrying different filling factors ν. It also determines the topology of the phase diagram, but neither the location of quantum critical points 3 Our timid use of the term "microscopic" signals that a rigorous derivat...

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The Ophiuridae / by C. F. Lütken and Th. Mortensen

Cited by 10 publications

References 1 publication

Duality and the modular group in the quantum Hall effect

Duality and the modular group in the quantum Hall effect

N = 2 Supersymmetric Yang–mills and the Quantum Hall Effect

On the origin of duality in the quantum Hall system

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