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Register a new userLocalisation, delocalisation, and topological transitions in disordered 2D quantum walks
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We investigate time-independent disorder on several two-dimensional discrete-time quantum walks. We find numerically that, contrary to claims in the literature, random onsite phase disorder, spin-dependent or otherwise, cannot localise the Hadamard quantum walk; rather, it induces diffusive spreading of the walker. In contrast, split-step quantum walks are generically localised by phase disorder. We explain this difference by showing that the Hadamard walk is a special case of the split-step quantum walk, with parameters tuned to a critical point at a topological phase transition. We show that the topological phase transition can also be reached by introducing strong disorder in the rotation angles. We determine the critical exponent for the divergence of the localisation length at the topological phase transition, and find $ u=2.6$, in both cases. This places the two-dimensional split-step quantum walk in the universality class of the quantum Hall effect.
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We study the effect of electrostatic disorder on the conductivity of a three-dimensional antiferromagnetic insulator (a stack of quantum anomalous Hall layers with staggered magnetization). The phase diagram contains regions where the increase of disorder first causes the appearance of surface conduction (via a topological phase transition), followed by the appearance of bulk conduction (via a metal-insulator transition). The conducting surface states are stabilized by an effective time-reversal symmetry that is broken locally by the disorder but restored on long length scales. A simple self-consistent Born approximation reliably locates the boundaries of this socalled statistical topological phase.
In this paper we consider Schr{o}dinger operators on $M times mathbb{Z}^{d_2}$, with $M={M_{1}, ldots, M_{2}}^{d_1}$ (`quantum wave guides) with a `$Gamma$-trimmed random potential, namely a potential which vanishes outside a subset $Gamma$ which is periodic with respect to a sub lattice. We prove that (under appropriate assumptions) for strong disorder these operators have emph{pure point spectrum } outside the set $Sigma_{0}=sigma(H_{0,Gamma^{c}})$ where $H_{0,Gamma^{c}} $ is the free (discrete) Laplacian on the complement $Gamma^{c} $ of $Gamma $. We also prove that the operators have some emph{absolutely continuous spectrum} in an energy region $mathcal{E}subsetSigma_{0}$. Consequently, there is a mobility edge for such models. We also consider the case $-M_{1}=M_{2}=infty$, i.~e.~ $Gamma $-trimmed operators on $mathbb{Z}^{d}=mathbb{Z}^{d_1}timesmathbb{Z}^{d_2}$. Again, we prove localisation outside $Sigma_{0} $ by showing exponential decay of the Green function $G_{E+ieta}(x,y) $ uniformly in $eta>0 $. For emph{all} energies $Einmathcal{E}$ we prove that the Greens function $G_{E+ieta} $ is emph{not} (uniformly) in $ell^{1}$ as $eta$ approaches $0$. This implies that neither the fractional moment method nor multi scale analysis emph{can} be applied here.
We investigate disorder-driven topological phase transitions in quantized electric quadrupole insulators. We show that chiral symmetry can protect the quantization of the quadrupole moment $q_{xy}$, such that the higher-order topological invariant is well-defined even when disorder has broken all crystalline symmetries. Moreover, nonvanishing $q_{xy}$ and consequent corner modes can be induced from a trivial insulating phase by disorder that preserves chiral symmetry. The critical points of such topological phase transitions are marked by the occurrence of extended boundary states even in the presence of strong disorder. We provide a systematic characterization of these disorder-driven topological phase transitions from both bulk and boundary descriptions.
Quantum walks constitute a versatile platform for simulating transport phenomena on discrete graphs including topological material properties while providing a high control over the relevant parameters at the same time. To experimentally access and directly measure the topological invariants of quantum walks we implement the scattering scheme proposed by Tarasinski et al.[Phys. Rev. A 89, 042327 (2014)] in a photonic time multiplexed quantum walk experiment. The tunable coin operation provides opportunity to reach distinct topological phases, and accordingly to observe the corresponding topological phase transitions. The ability to read-out the position and the coin state distribution, complemented by explicit interferometric sign measurements, allowed the reconstruction of the scattered reflection amplitudes and thus the computation of the associated bulk topological invariants. As predicted we also find localised states at the edges between two bulks belonging to different topological phases. In order to analyse the impact of disorder we have measured invariants of two different types of disordered samples in large ensemble measurements, demonstrating their constancy in one disorder regime and a continuous transition with increasing disorder strength for the second disorder sample.
Uncertainty relations are studied for a characterization of topological-band insulator transitions in 2D gapped Dirac materials isostructural with graphene. We show that the relative or Kullback-Leibler entropy in position and momentum spaces, and the standard variance-based uncertainty relation, give sharp signatures of topological phase transitions in these systems.
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