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Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice

Andreas Schaefer

TL;DR

The paper develops a rigorous framework for dynamical localization of coined quantum walks on the hexagonal lattice under strong disorder introduced by random phases in the coin at each site. It combines a six-dimensional fractional-moment analysis with a finite-volume method to prove exponential decay of resolvent matrix elements, yielding dynamical localization in the decorrelated setting and a quantitative bound $\mathbb{E}|\langle x^{(i)}|(U_\omega(C)-z)^{-1}|y^{(j)}\rangle|^s \le c e^{-g|x-y|}$ for $s\in(0,1/3)$. The work also extends to translation-invariant cases, where a band-structure analysis via $V(k)$ and its trace informs Dirac-point behavior, and to transport properties through a topological index for certain coin matrices, linking spectral properties to robust transport. The methods accommodate correlated randomness inherent in the triplet-phase construction and show how topological criteria can guarantee extended states despite disorder, offering insights into graphene-like lattices and quantum information transport in complex networks.

Abstract

We study coined Random Quantum Walks on the hexagonal lattice, where the strength of disorder is monitored by the coin matrix. Each lattice site is equipped with an i.i.d. random variable that is uniformly distributed on the torus and acts as a random phase in every step of the QW. We show exponential decay of the fractional moments of the Green function in the regime of strong disorder, that is whenever the coin matrix is sufficiently close to the fully localized case, using a fractional moment criterion and a finite volume method. In the decorrelated case, we deduce dynamical localization. Moreover, we adapt a topological index to our model and thereby obtain transport for some coin matrices.

Dynamical Localization and Transport properties of Quantum Walks on the hexagonal lattice

TL;DR

The paper develops a rigorous framework for dynamical localization of coined quantum walks on the hexagonal lattice under strong disorder introduced by random phases in the coin at each site. It combines a six-dimensional fractional-moment analysis with a finite-volume method to prove exponential decay of resolvent matrix elements, yielding dynamical localization in the decorrelated setting and a quantitative bound for . The work also extends to translation-invariant cases, where a band-structure analysis via and its trace informs Dirac-point behavior, and to transport properties through a topological index for certain coin matrices, linking spectral properties to robust transport. The methods accommodate correlated randomness inherent in the triplet-phase construction and show how topological criteria can guarantee extended states despite disorder, offering insights into graphene-like lattices and quantum information transport in complex networks.

Abstract

We study coined Random Quantum Walks on the hexagonal lattice, where the strength of disorder is monitored by the coin matrix. Each lattice site is equipped with an i.i.d. random variable that is uniformly distributed on the torus and acts as a random phase in every step of the QW. We show exponential decay of the fractional moments of the Green function in the regime of strong disorder, that is whenever the coin matrix is sufficiently close to the fully localized case, using a fractional moment criterion and a finite volume method. In the decorrelated case, we deduce dynamical localization. Moreover, we adapt a topological index to our model and thereby obtain transport for some coin matrices.

Paper Structure

This paper contains 21 sections, 13 theorems, 126 equations, 14 figures.

Table of Contents

  1. Introduction
  2. The model and main result
  3. The band structure of translation-invariant QWs
  4. Exponential decay of the fractional moments
  5. Fractional moment bound
  6. The boundary of a box
  7. The transition operator
  8. Spectral gaps of the fully localized operator
  9. Polynomial decay of the boxed resolvent
  10. The iterative step
  11. Finite volume method
  12. Transport using a topological index
  13. The scattering Quantum Walk
  14. The index
  15. Relation to the coefficients of the coin matrix
  16. ...and 6 more sections

Key Result

Theorem 1

For any $s \in (0,\frac{1}{3})$ there exists $\varphi, c, g > 0$ such that for any $C \in U(3)$ with $\| C - C_0 \|_\infty \leq \varphi$ and all lattice sites $x, y \in \Gamma$, coin states $i,j \in \{1,2,3\}$, $z \in \mathbb{C}$ with $|z|<1$:

Figures (14)

  • Figure 1: The action of the shift operators $S_j$
  • Figure 2: The invariant subspace $\mathcal{H}^{j,k}$ of $U_\omega(C_0)$
  • Figure 3: The coin matrices are changed from $C$ to $C_0$ on all green lattice sites. The subspace $\mathcal{H}_L$ for $L=(3,3)$ is illustrated as the inner blue area, whereas $\mathcal{H}_L^C$ is the exterior red area. Note that sites in $\Gamma_{C_0}^{(L)}$ are split depending on the coin state.
  • Figure 4: Labeling of the random variables $\omega_x$ for a fixed subspace $\mathcal{H}^{j,k}$. We note that if the random variable $\omega_0$ is at site $\ket{j,k} \otimes f_1$, then $\omega_1$ is at $\ket{j,k-1}\otimes f_2$, $\omega_2$ is at $\ket{j+1,k-1} \otimes f_2$ and $\omega_3$ is at $\ket{j,k} \otimes f_2$.
  • Figure 5: The boundaries of $\Lambda_L$ and $\Lambda_L^C$.
  • ...and 9 more figures

Theorems & Definitions (34)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Theorem 2
  • proof
  • Lemma 1
  • ...and 24 more