Quasi-resonant diffusion of wave packets in one-dimensional disordered mosaic lattices

Ba Phi Nguyen; Duy Khuong Phung; Kihong Kim

Home/Archive/arXiv

arXiv · 2207.13906

Quasi-resonant diffusion of wave packets in one-dimensional disordered mosaic lattices

Ba Phi Nguyen·Duy Khuong Phung·Kihong Kim

View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate numerically the time evolution of wave packets incident on one-dimensional semi-infinite lattices with mosaic modulated random on-site potentials, which are characterized by the integer-valued modulation period $κ$ and the disorder strength $W$. For Gaussian wave packets with the central energy $E_0$ and a small spectral width, we perform extensive numerical calculations of the disorder-averaged time-dependent reflectance, $\langle R(t)\rangle$, for various values of $E_0$, $κ$, and $W$. We find that the long-time behavior of $\langle R(t)\rangle$ obeys a power law of the form $t^{-γ}$ in all cases. In the presence of the mosaic modulation, $γ$ is equal to 2 for almost all values of $E_0$, implying the onset of the Anderson localization, while at a finite number of discrete values of $E_0$ dependent on $κ$, $γ$ approaches 3/2, implying the onset of the classical diffusion. This phenomenon is independent of the disorder strength and arises in a quasi-resonant manner such that $γ$ varies rapidly from 3/2 to 2 in a narrow energy range as $E_0$ varies away from the quasi-resonance values. We deduce a simple analytical formula for the quasi-resonance energies and provide an explanation of the delocalization phenomenon based on the interplay between randomness and band structure and the node structure of the wave functions. We explore the nature of the states at the quasi-resonance energies using a finite-size scaling analysis of the average participation ratio and find that the states are neither extended nor exponentially localized, but ciritical states..

Explore related subjects

cond-mat.dis-nn physics.optics

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Ba Phi Nguyen, Duy Khuong Phung, Kihong Kim. 2022-09-15. Quasi-resonant diffusion of wave packets in one-dimensional disordered mosaic lattices. https://arxiv.org/abs/2207.13906

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Sampling at intermediate temperatures is optimal for training large language models in protein structure prediction

Using a statistical mechanics framework, we investigate the parameter space of transformer models trained on protein sequence data. We sample the loss landscape at varying temperatures using Langevin dynamics to characterize the low-loss manifold, and to understand the mechanisms underlying transformers' superior performance in protein structure prediction. We find that, at variance with networks not based on the attention mechanism, the lack of a first--order--like transition in the loss of the transformer produces a range of intermediate temperatures with good learning properties; this is true both for synthetic and natural protein sequences. We also show that the parameters of most layers are highly conserved at these temperatures if the dimension of the embedding is optimal, and we provide an operative way to find this dimension. Additionally, we show that the attention matrix is more predictive of the contact maps of the protein at higher temperatures and for higher dimensions of the embedding than those optimal for learning. Finally, we showed that the models sampled at intermediate temperatures can predict the free-energy variation upon mutation, better than models obtained through standard optimization techniques.

cond-mat.dis-nn↗

arXiv2026

The Cross-Substrate Access Assay: What an Indicator Test Must Declare to Travel from Brain to Language Model

Testing an artificial system for a property linked to consciousness means applying a measurement developed on brains to a system that is not one. Such a transfer must re-examine five parts of the procedure: the competing statistical models, how they are fitted, the unit the inference generalizes over, the quantity the uncertainty interval is about, and the rule that turns a result into a verdict. The Cross-Substrate Access Assay declares all five. Because brain and model signals share no physical scale, every model is scored by the cross-entropy it assigns to held-out data, in nats per trial. The test case is the global neuronal workspace theory, which predicts that near threshold a stimulus either enters a capacity-limited workspace or does not, so that single-trial responses form a mixture of two states. A published test of this prediction on twenty people's electroencephalograms partly reproduces in a re-implementation: the first crossing and the broad ordering over time match, the window-by-window agreement does not. On 12,000 synthetic datasets generated with a single graded state, all of them members of the families the procedure fits and none within 0.0067 nat per trial of the decision boundary, the two models of that test carried over unchanged reported two states in 989 and the expanded families in none; on 600 datasets carrying a mixture the expanded procedure reported two states in 599. Its nominal 95% interval contained the procedure's mean result less often than the required 90% at six of twelve graded settings. No claim about experience is made.

cond-mat.dis-nn↗

arXiv2026

Measure-zero delocalization in the complex plane: exact mobility arcs in a non-Hermitian off-diagonal quasiperiodic lattice

We investigate Anderson localization in a one-dimensional lattice with non-Hermitian off-diagonal quasiperiodic disorder, extending a recently studied Hermitian mosaic model to the non-Hermitian regime. Using Avila's global theory, we derive the exact Lyapunov exponent and the complete phase diagram in the complex energy plane. This work contains two central findings. First, we discover mobility arcs---open curved segments in the complex plane---as a new class of mobility edges and the generic form of open mobility edges, which coexist with closed mobility rings in a complementary parameter regime. These arcs share the same localization physics as the previously reported mobility lines: eigenstates are delocalized if and only if their energies lie exactly on these sets; any deviation yields localized states. This constitutes a striking measure-zero delocalization phenomenon: delocalized states occupy only zero-measure sets (arcs or lines) in the complex plane, in sharp contrast to the mobility rings, which enclose a finite-area region of delocalized states. Second, we reveal that mobility rings, arcs, and lines all share a common mathematical origin in the generalized Joukowski transformation $P(E) = \frac{1}{2}(u - w^2/u)$, rooted in the algebraic structure of the underlying polynomial: the preimage of the boundary of an elliptical region under the polynomial map $P(E)$ gives the rings, while the branch cut inside this ellipse gives rise to the mobility arcs and lines in the complementary parameter regime.

cond-mat.dis-nn↗