Toroidal sigma models of magneto-transport are analyzed, in which integer and fractional quantum Hall effects automatically are unified by a holomorphic modular symmetry, whose group structure is ...determined by the spin structure of the toroidal target space (an elliptic curve). Hall quantization is protected by the topology of stable holomorphic vector bundles V on this space, and plateau values σH⊕=μ∈Q of the Hall conductivity are rational because such bundles are classified by their slope μ(V)=deg(V)/rk(V), where deg(V) is the degree and rk(V) is the rank of V. By exploiting a quantum equivalence called mirror symmetry, these models are mapped to tractable mirror models (also elliptic), in which topological protection is provided by more familiar winding numbers. Phase diagrams and scaling properties of elliptic models are compared to some of the experimental and numerical data accumulated over the past three decades. The geometry of scaling flows extracted from quantum Hall experiments is in good agreement with modular predictions, including the location of many quantum critical points. One conspicuous model has a critical delocalization exponent νtor=18ln2/(π2G4)=2.6051⋯ (G is Gauss' constant) that is in excellent agreement with the value νnum=2.607±.004 calculated in the numerical Chalker-Coddington model, suggesting that these models are in the same universality class. The real delocalization exponent may be disentangled from other scaling exponents in finite size scaling experiments, giving an experimental value νexp=2.3±0.2. The modular model suggests how these theoretical and experimental results may be reconciled, but to determine if these theoretical models really are in the quantum Hall universality class, improved finite-size scaling experiments are urgently needed.
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 complexified conductivity.
The phenomenological analysis of fully spin-polarized quantum Hall systems, based on holomorphic modular symmetries of the renormalization group (RG) flow, is generalized to more complicated ...situations where the spin or other "flavors" of charge carriers are relevant, and where the symmetry is different. We make the simplest possible ansatz for a family of RG potentials that can interpolate between these symmetries. It is parametrized by a single number a and we show that this suffices to account for almost all scaling data obtained to date. The potential is always symmetric under the main congruence group at level two, and when a takes certain values, this symmetry is enhanced to one of the maximal subgroups of the modular group. We compute the covariant RG beta function, which is a holomorphic vector field derived from the potential, and compare the geometry of this gradient flow with available temperature driven scaling data. The value of a is determined from experiment by finding the location of a quantum critical point, i.e., an unstable zero of the beta function given by a saddle point of the RG potential. The data are consistent with a setmembership R, which together with the symmetry leads to a generalized semicircle law.
Experiments studying renormalization group flows in the quantum Hall system provide significant evidence for the existence of an emergent holomorphic modular symmetry
Γ
0
(
2
)
. We briefly review ...this evidence and show that, for the lowest temperatures, the experimental determination of the position of the quantum critical points agrees to the parts
per mille level with the prediction from
Γ
0
(
2
)
. We present evidence that experiments giving results that deviate substantially from the symmetry predictions are not cold enough to be in the quantum critical domain. We show how the modular symmetry extended by a non-holomorphic particle–hole duality leads to an extensive web of dualities related to those in plateau–insulator transitions, and we derive a formula relating dual pairs
(
B
,
B
d
)
of magnetic field strengths across any transition. The experimental data obtained for the transition studied so far is in excellent agreement with the duality relations following from this emergent symmetry, and rule out the duality rule derived from the “law of corresponding states”. Comparing these generalized duality predictions with future experiments on other transitions should provide stringent tests of modular duality deep in the non-linear domain far from the quantum critical points.
The
Campanula
genus comprises several economically important ornamental plants species. Wide hybridisation is a method to increase phenotypic variability, but is limited due to interspecies ...hybridisation barriers. In this study we investigated whether ovule culture could be used to increase the success rate of interspecific hybridisation between
C. portenschlagiana
×
C. poscharskyana
and
C. medium
×
C. formanekiana
. The effect of different ovule isolation times on ovule germination in vitro was examined. In general, the number of collectible ovules and ovule germination was low. Interspecific hybrids between
C. medium
and
C. formanekiana
exhibited an increased number of viable ovules with later isolation time, but with different ovule germination rates. A parent-of-origin effect on both the number of collectible ovules and ovule germination was observed for
C. medium
×
C. formanekiana
. Histological analysis of embryo and endosperm development in collectible ovules isolated at different time points from interspecific crosses showed that the vast majority of ovules did not contain an embryo. When present, embryo development only progressed with ovule collection time in the
C. medium
and
C. formanekiana
crosses. The occurrence of miscoloured seedlings in interspecific crosses indicated incompatibilities between the parental lines that could not be prevented by reciprocal crossing. The low number of collectible ovules and germination rates might be inhibited due to fertilisation barriers. With this study, a protocol for ovule culture was established and the usefulness of ovule culture to obtain interspecific hybrids of selected
Campanula
species was demonstrated.
Previous work has reported that children creatively make syntactic errors that are ungrammatical in their target language, but are grammatical in another language. One of the most well‐known examples ...is medial wh‐question errors in English‐speaking children's wh‐questions (e.g., What do you think who the cat chased? from Thornton, 1990). The evidence for this non‐target‐like structure in both production and comprehension has been taken to support the existence of innate, syntactic parameters that define all possible grammatical variation, which serve as a top‐down constraint guiding children's language acquisition process. The present study reports new story‐based production and comprehension experiments that challenge this interpretation. While we replicated previous observations of medial wh‐question errors in children's sentence production (Experiment 1), we saw a reduction in evidence indicating that English‐speaking children assign interpretations that conform to the medial wh‐question pattern (Experiment 2). Crucially, we found no correlation between production and comprehension errors (Experiment 3). We suggest that these errors are the result of children's immature sentence production mechanisms rather than immature grammatical knowledge.
The new states of matter and concomitant quantum critical phenomena revealed by the quantum Hall effect appear to be accompanied by an emergent modular symmetry. The extreme rigidity of this infinite ...symmetry makes it easy to falsify, but two decades of experiments have failed to do so, and the location of quantum critical points predicted by the symmetry is in increasingly accurate agreement with scaling experiments. The symmetry severely constrains the structure of the effective quantum field theory that encodes the low energy limit of quantum electrodynamics of 10 super(10) charges in two dirty dimensions. If this is a non-linear sigma-model the target space is a torus, rather than the more familiar sphere. One of the simplest toroidal models gives a critical (correlation length) exponent that agrees with the value obtained from numerical simulations of the quantum Hall effect.
We analyze experimental quantum Hall data from a wide range of different materials, including semiconducting heterojunctions, thin films, surface layers, graphene, mercury telluride, bismuth ...antimonide, and black phosphorus. The fact that these materials have little in common, except that charge transport is effectively two-dimensional, shows how robust and universal the quantum Hall phenomenon is. The scaling and fixed point data we analyzed appear to show that magnetotransport in two dimensions is governed by a small number of universality classes that are classified by modular symmetries, which are infinite discrete symmetries not previously seen in nature. The Hall plateaux are (infrared) stable fixed points of the scaling-flow, and quantum critical points (where the wave function is delocalized) are unstable fixed points of scaling. Modular symmetries are so rigid that they in some cases fix the global geometry of the scaling flow, and therefore predict the exact location of quantum critical points, as well as the shape of flow lines anywhere in the phase diagram. We show that most available experimental quantum Hall scaling data are in good agreement with these predictions.
The transitions between neighbouring plateaux in the quantum Hall system are observed to follow “anti-holomorphic” scaling with “superuniversal” scaling exponents, showing that the system contains an ...emergent sub-modular discrete symmetry and a holomorphic structure at low energies. We identify a class of effective scaling models consistent with this data, which is parametrized by the complex structure of a torus with a special spin structure, in which only the number of fermions
(
c
)
remains undetermined. For
c
=
2
this gives the superuniversal anti-holomorphic scaling potential previously inferred from data, with scaling exponent
ν
≈
2.6
, in reasonable agreement with available scaling data.
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