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Phase transition in a periodic tubular structure

Alexander V. Kiselev, Kirill Ryadovkin

Abstract

We consider an $\varepsilon$-periodic ($\varepsilon\to 0$) tubular structure, modelled as a magnetic Laplacian on a metric graph, which is periodic along a single axis. We show that the corresponding Hamiltonian admits norm-resolvent convergence to an ODE on $\mathbb{R}$ which is fourth order at a discrete set of values of the magnetic potential (\emph{critical points}) and second-order generically. In a vicinity of critical points we establish a mixed-order asymptotics. The rate of convergence is also estimated. This represents a physically viable model of a phase transition as the strength of the (constant) magnetic field increases.

Phase transition in a periodic tubular structure

Abstract

We consider an -periodic () tubular structure, modelled as a magnetic Laplacian on a metric graph, which is periodic along a single axis. We show that the corresponding Hamiltonian admits norm-resolvent convergence to an ODE on which is fourth order at a discrete set of values of the magnetic potential (\emph{critical points}) and second-order generically. In a vicinity of critical points we establish a mixed-order asymptotics. The rate of convergence is also estimated. This represents a physically viable model of a phase transition as the strength of the (constant) magnetic field increases.
Paper Structure (11 sections, 10 theorems, 108 equations, 2 figures)

This paper contains 11 sections, 10 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Periodic graphs
  4. The operator $\Delta(A)$
  5. Fibre representation
  6. The boundary triple
  7. The graph $\mathcal{G}_1$
  8. Rescaling
  9. Degeneracy of band edges
  10. Analysis in vicinity of a critical value of magnetic potential
  11. Homogenisation of the $\varepsilon$-periodic tubular structure

Key Result

Theorem 1

There exists a periodic sequence $\mathscr A$ of values of the magnetic potential which are critical in the following sense. Let $A\in \mathscr A$, $A'=A+\delta$ and $z'_\varepsilon$ be the lower edge of the spectrum of the self-adjoint magnetic Laplacian ${\Delta}_\varepsilon(A'/\varepsilon)$ in th respectively, where $\kappa_2$ and $\kappa_4$ are explicitly computed real constants. Then the foll

Figures (2)

  • Figure 1: The graph $\mathcal{G}_1$ (left) and its fundamental graph $\widetilde{\mathcal{G}}_1$ (right). Both solid and dotted lines represent graph edges.
  • Figure 2: The graph of the function $f_+(t)$ for (a) $A=\frac{4\pi}{9}$; (b) $A=\frac{\pi}{9}$.

Theorems & Definitions (24)

  • Theorem
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 1: Kochubej
  • Definition 2
  • Definition 3
  • Proposition 4.1: Version of the Kreı n formula of DM
  • Lemma 4.2
  • ...and 14 more