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HomeJournalsAnnals of Quantum and Condensed Matter PhysicsVolume 3, Issue 110.58303/20ww4n34
Annals of Quantum and Condensed Matter Physics
Article

Quantum-enhanced spectroscopy of kagome metals with mixed input states

Felix Strand1*, Jakub Dąbrowski2, Valentina Rossi3

1School of Public Health, Baltic Sea Research University, Finland

2School of Physics, Meridian National University, Singapore

3Institute for Data Science, Hanseatic University of Applied Sciences, Germany

* Corresponding author.

Ann. Quantum Condens. Matter Phys. 2023, 3(1); https://doi.org/10.58303/20ww4n34

Abstract

Understanding moiré superlattices is of central importance, yet existing approaches rely on restrictive assumptions. We observe a sharp enhancement of the signal-to-noise ratio. The analysis uses a combination of rigorous arguments and extensive numerical or empirical checks. We observe a crossover between two distinct regimes controlled by the coupling strength, and we show that the result is stable under a range of perturbations of the assumptions. Together, these results contribute to a more complete picture and open the way to further applications.

Keywords:superconductivitycondensed matternumerical simulationspectroscopy

Introduction

Let Ω ⊂ R d be a bounded domain with Lipschitz boundary and let p ∈ ( 1 , ∞ ) . We consider the boundary value problem1

− div ⁡ ( | ∇ u | p − 2 ∇ u ) = f in  Ω , u = 0 on  ∂ Ω . (1)

Existence of weak solutions in W 0 1 , p ( Ω ) follows from the direct method of the calculus of variations. Our interest here is in sharp quantitative bounds that do not depend on the dimension.

Main result

Theorem 1. Suppose f ∈ L q ( Ω ) with q > d / p . Then there is a constant C = C ( d , p , q ) such that

‖ u ‖ L ∞ ( Ω ) ≤ C | Ω | 1 d − 1 p + 1 q ‖ f ‖ L q ( Ω ) 1 p − 1 . (2)

The proof combines a truncation argument with the Sobolev inequality ‖ v ‖ p ∗ ≤ S ‖ ∇ v ‖ p . A second estimate, for the gradient, is given by

∫ Ω | ∇ u | p d x ; ≤ ; ∑ k = 1 ∞ 2 − k λ k − 1 ‖ f ‖ q p ′ , p ′ = p p − 1 . (3)

Numerical illustration

Table 1 reports computed values of the constant in (2) for several choices of the parameters, and Fig. 1 shows the decay of the error under mesh refinement.

Table 1 Computed values of the constant C ( d , p , q )

dpqC (computed)C (bound)
2220.0810.125
2340.1470.210
3230.0520.083
3450.2090.341
Fig. 1

Fig. 1 Error versus mesh size on a logarithmic scale; the dashed line has slope 2.

Discussion

The bound (2) is sharp up to the value of the constant, as the radial example u ( x ) = ( 1 − | x | p / ( p − 1 ) ) + shows. Extensions to systems and to variable exponents will be treated elsewhere.

Here the divergence and gradient are taken in the sense of distributions.

References

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Author Contributions

  • Felix Strand: Data curation, Software

Declarations

Funding

The authors acknowledge support from the Wellcome Trust (grant 215801/Z/19/Z) and the National Institute for Health and Care Research.

Data Availability

The datasets generated and analyzed during the study are available from the corresponding author on reasonable request.

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