arXiv subjects

C. A. Lütken

Publications and source records attributed to C. A. Lütken.

8 recordsLinked to original sources

arXiv2026

Elliptic mirror of the quantum Hall effect

Toroidal sigma models of magneto-transport are analyzed, in which integer and fractional quantum Hall effects automatically are unified by a {holomorphic modular symmetry}. By exploiting a quantum equivalence called \emph{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 %(arguably the simplest and most natural one) has a critical delocalization exponent $ν_{\rm tor} = 18 \ln 2 /(π^2 G^4) = 2.6051\dots$ ($G$ is Gauss' constant) that is in excellent agreement with the value $ν_{\rm num} = 2.607\pm\,.004$ 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 $ν_{\rm exp} = 2.3\pm 0.2$. The modular model suggests how these theoretical and experimental results may be reconciled, but in order to determine if these theoretical models really are in the quantum Hall universality class, improved finite size scaling experiments are urgently needed.

cond-mat.str-el↗

arXiv2025

Designing toroidal cavities for quantum computation

Toroidal microwave cavities are investigated for potential use in quantum information storage and computation. Since exact analytical results are not available for this geometry, extensive numerical simulation has been used to develop a universal phenomenological model ("spectral flow diagram"). This model is needed to guide the non-trivial design of toroidal resonators. A host of new modes that do not exist in cylindrical cavities are classified, including novel counter-intuitive ground states, and "dark nodal modes" that are decoupled from the environment in the absence of antennae. Numerical results are found to be in good agreement with experimental data. The existence of dark nodal modes in a shallow smooth cavity geometry that offers easy access for high quality surface treatment, suggests that high-Q toroidal cavities may be exploited for long-term storage of quantum information used in quantum processors.

quant-ph↗

arXiv2012

The quantum Hall curve

We show how the modular symmetries that have been found to be consistent with most available scaling data from quantum Hall systems, derive from a rigid family of algebraic curves of the elliptic type. The complicated special functions needed to describe scaling data arise in a simple and transparent way from the group theory and geometry of these \emph{quantum Hall curves}. The renormalization-group potential therefore emerges naturally in a geometric context that complements the phenomenology found in our companion paper [Phys. Rev. B 85, 155123 (2012)]. We show how the algebraic geometry of elliptic curves is an efficient way to analyze specific scaling data, extract the modular symmetries of the transport coefficients, and use this information to fit the given system into the one-dimensional (real) family of curves that may model all universal properties of quantum Hall systems.

cond-mat.str-el↗

arXiv2012

An RG potential for the quantum Hall effects

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 $β$-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 $β$-function given by a saddle point of the RG potential. The data are consistent with $a \in \mathbb{R}$, which together with the symmetry leads to a generalized semi-circle law.

cond-mat.str-el↗

arXiv2010

Experimental probes of emergent symmetries in the quantum Hall system

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 \emph{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.

cond-mat.str-el↗

arXiv2010

On the origin of duality in the quantum Hall system

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.

cond-mat.str-el↗

arXiv2004

Non-linear Yang-Mills instantons from strings are $π$-stable D-branes

We show that B-type $Π$-stable D-branes do not in general reduce to the (Gieseker-) stable holomorphic vector bundles used in mathematics to construct moduli spaces. We show that solutions of the almost Hermitian Yang--Mills equations for the non-linear deformations of Yang--Mills instantons that appear in the low-energy geometric limit of strings exist iff they are $π$-stable, a geometric large volume version of $Π$-stability. This shows that $π$-stability is the correct physical stability concept. We speculate that this string-canonical choice of stable objects, which is encoded in and derived from the central charge of the string-\emph{algebra}, should find applications to algebraic geometry where there is no canonical choice of stable \emph{geometrical} objects.

hep-th↗

arXiv1994

Bosonization in Higher Dimensions

Using the recently discovered connection between bosonization and duality transformations (hep-th/9401105 and hep-th/9403173), we give an explicit path-integral representation for the bosonization of a massive fermion coupled to a U(1) gauge potential (such as electromagnetism) in d space (D=d+1 spacetime) dimensions. The bosonic theory is described by a rank d-1 antisymmetric Kalb-Ramond-type gauge potential. We construct the bosonized lagrangian explicitly in the limit of large fermion mass. We find that the resulting action is local for d=2 (and given by a Chern-Simons action), but nonlocal for d larger than 3. By coupling to a statistical Chern-Simons field for d=2, we obtain a bosonized formulation of anyons. The bosonic theory may be further dualized to a theory involving purely scalars, for any d, and we show this to be governed by a higher-derivative lagrangian for which the scalar decouples from the U(1) gauge potential.

hep-th↗