arXiv subjects

Anouar Moustaj

Publications and source records attributed to Anouar Moustaj.

12 recordsLinked to original sources

arXiv2026

One-dimensional quasicrystals with tensor-network finite-state automata

Quasicrystals occupy a distinctive position between the translational order of crystals and the disordered amorphous matter. Simulating this physics has remained challenging, since quasiperiodic structures lack the translational symmetry exploited for crystals and generally require costly diagonalization of large finite approximants. Quasicrystalline order admits two equivalent descriptions, a cut-and-project scheme from a higher-dimensional periodic crystal, and a discrete set of substitution rules acting on a finite alphabet. We show that the latter, written in a numeration system adapted to the substitution, defines a deterministic finite automaton with output, the digits of a site index are fed, and the automaton returns the letter occupying that site. We further exploit another equivalence to a different construction, the transition matrices are exactly the tensors of a matrix product state, whose bond dimension is the number of automaton states and is independent of system size. This allows efficient representation of extremely large tight-binding Hamiltonians in the tensor-train language, thereby yielding an exact matrix product operator for the quasicrystal Hamiltonian at any system size. We show how this framework works for two families of one-dimensional quasicrystals, the metallic-mean and $k$-bonacci families and we explicitly construct the Fibonacci, silver-mean, and Tribonacci quasicrystals. By leveraging efficient tensor-network compression and the kernel polynomial method, we compute spectral densities for chains with more than $10^9$ sites and directly resolve the hierarchical structure of the spectrum.

cond-mat.mes-hall↗

arXiv2026

Tensor network solvers for ultra-large tight-binding Hamiltonians: algorithms and applications

Understanding quantum materials at meso and even macroscopic scales requires tight-binding calculations on system sizes where explicit matrix representations become prohibitively costly. This represents a major bottleneck to rationalize phenomena in moir\'e and super-moir\'e heterostructures and quasicrystals. Here, we present a unified tensor-network methodology to solve tight-binding problems at exceptionally large scales, by mapping a system of $N = 2^L$ sites onto a many-body problem of $L$ pseudospin sites, which is subsequently solved with tensor network algorithms. For Hamiltonians with compressible real-space structure, the tensor network bond dimension remains modest, typically of order a few tens, independent of $N$. Tensor network representations of arbitrary hopping functions including long-range, spatially modulated, and twisted-layer couplings are built with quantics tensor cross interpolation, and all physical observables are evaluated entirely with tensor network algebra without explicit matrix storage or diagonalization. We demonstrate applications to spectral functions, momentum-space spectra via the tensor-network quantum Fourier transform, real-space topological invariants, real-time dynamics, correlation induced symmetry breaking with self-consistent mean-field calculations, non-Hermitian phenomena, and excitonic many-body physics. Our methodology enables routinely solving systems with billions of sites, by leveraging the tensor network compressibility of real-space structures, and establishing a flexible framework to study quantum matter at ultra-large length scales. The methodology is implemented in the open-source Julia package TensorBinding.

cond-mat.str-el↗

arXiv2026

Tensor-network methodology for real-space super-moir\'e excitons

Computing excitonic spectra in quasicrystal and super-moir\'e systems constitutes a formidable challenge due to the exceptional size of the excitonic Hilbert space. Here, we demonstrate a tensor-network method for the real-space Bethe-Salpeter Hamiltonian, allowing us to access the spectra of an excitonic $10^{18}$-dimensional Hamiltonian, and enabling the direct computation of bound-exciton spectral functions for systems exceeding one billion lattice sites, several orders of magnitude beyond the capabilities of conventional approaches. Our method combines a tensor-network encoding of the real-space Bethe-Salpeter Hamiltonian with a Chebyshev tensor network algorithm. This strategy bypasses explicit storage of the Hamiltonian while preserving full real-space resolution across widely different length scales. We demonstrate our methodology for one- and two-dimensional super-moir\'e systems, achieving the simultaneous resolution of atomistic and mesoscopic structures in the excitonic spectra in billion-size systems, showing exciton miniband formation and moir\'e-induced spatial confinement. Our results establish a real-space methodology enabling the simulation of excitonic physics in large-scale quasicrystal and super-moir\'e quantum matter.

cond-mat.str-el↗

arXiv2026

Parity-dependent double degeneracy and spectral statistics in the projected dice lattice

We investigate the spectral statistics of an interacting fermionic system derived by projecting the Hubbard interaction onto the two lowest-energy, degenerate flat bands of the dice lattice subjected to a $\pi$-flux. Surprisingly, the distributions of level spacings and gap ratios correspond to distinct Gaussian ensembles, depending on the parity of the particle number. For an even number of particles, the spectra conform to the Gaussian Orthogonal Ensemble, as expected for a time-reversal-symmetric Hamiltonian. In stark contrast, the odd-parity sector exhibits exact double degeneracy of all eigenstates even after resolving all known symmetries, and the Gaussian Unitary Ensemble accurately describes the spacing distribution between these doublets. The simultaneous emergence of two different random-matrix ensembles within a single physical system constitutes an unprecedented finding, opening new avenues for both random matrix theory and flat-band physics.

cond-mat.str-el↗

arXiv2025

Tensor network approach to momentum-resolved spectroscopy in non-periodic super-moir\'e systems

Computing spectral functions in large, non-periodic super-moir\'e systems remains an open problem due to the exceptionally large system size that must be considered. Here, we establish a tensor network methodology that allows computing momentum-resolved spectral functions of non-interacting and interacting super-moir\'e systems at an atomistic level. Our methodology relies on encoding an exponentially large tight-binding problem as an auxiliary quantum many-body problem, solved with a many-body kernel polynomial tensor network algorithm combined with a quantum Fourier transform tensor network. We demonstrate the method for one and two-dimensional super-moir\'e systems, including super-moir\'e with non-uniform strain, interactions treated at the mean-field level, and quasicrystalline super-moir\'e patterns. Furthermore, we demonstrate that our methodology allows us to compute momentum-resolved spectral functions restricted to selected regions of a super-moir\'e, enabling direct imaging of position-dependent electronic structure and minigaps in super-moir\'e systems with non-uniform strain. Our results establish a powerful methodology to compute momentum-resolved spectral functions in exceptionally large super-moir\'e systems, providing a tool to directly model quantum twisting microscope experiments in twisted van der Waals heterostructures.

cond-mat.str-el↗

arXiv2025

Latent Haldane Models

Latent symmetries, which materialize after performing isospectral reductions, have recently been shown to be instrumental in revealing novel topological phases in one-dimensional systems, among many other applications. In this work, we explore how to construct a family of seemingly complicated two-dimensional models that result in energy-dependent Haldane models upon performing an isospectral reduction. In these models, we find energy-dependent latent Semenoff masses without introducing a staggered on-site potential. In addition, energy-dependent latent Haldane masses also emerge in decorated lattices with nearest-neighbor complex hoppings. Using the Haldane model's properties, we then predict the location of the topological gaps in the aforementioned family of models and construct phase diagrams to determine where the topological phases lie in parameter space. This idea yielded, for instance, useful insights in the case of a modified version of $α$-graphyne and hexagonal plaquettes with additional decorations, where the gap-closing energies can be calculated using the ISR to predict topological phase transitions.

cond-mat.mes-hall↗

arXiv2025

Haldane model on the Sierpiński gasket

We investigate the topological phases of the Haldane model on the Sierpiński gasket. As a consequence of the fractal geometry, multiple fractal gaps arise. Additionally, a flat band appears, and due to a complex next-nearest neighbour hopping, this band splits and multiple topological flux-induced gaps emerge. Owing to the fractal nature of the model, conventional momentum-space topological invariants cannot be used. Therefore, we characterise the system's topology in terms of a real-space Chern number. In addition, we verify the robustness of the topological states to disorder. Finally, we present phase diagrams for both a fractal gap and a flux-induced gap. Previous work on a similar system claims that fractality "squeezes" the well-known Haldane phase diagram. However, this result arises because a doubled system was considered with two Sierpiński gaskets glued together. We consider only a single copy of the Sierpiński gasket, keeping global self-similarity. In contrast with these previous results, we find intricate and complex patterns in the phase diagram of this single fractal. Our work shows that the fractality of the model greatly influences the phase space of these structures, and can drive topological phases in the multitude of fractal and flux-induced gaps, providing a richer platform than a conventional integer dimensional geometry.

cond-mat.mes-hall↗

arXiv2023

Emergent non-Hermitian models

The Hatano-Nelson and the non-Hermitian Su-Schrieffer-Heeger model are paradigmatic examples of non-Hermitian systems that host non-trivial boundary phenomena. In this work, we use recently developed graph-theoretical tools to design systems whose isospectral reduction -- akin to an effective Hamiltonian -- has the form of either of these two models. In the reduced version, the couplings and on-site potentials become energy-dependent. We show that this leads to interesting phenomena such as an energy-dependent non-Hermitian skin effect, where eigenstates can simultaneously localize on either ends of the systems, with different localization lengths. Moreover, we predict the existence of various topological edge states, pinned at non-zero energies, with different exponential envelopes, depending on their energy. Overall, our work sheds new light on the nature of topological phases and the non-Hermitian skin effect in one-dimensional systems.

quant-ph↗

arXiv2023

Spectral Properties of Two Coupled Fibonacci Chains

The Fibonacci chain, i.e., a tight-binding model where couplings and/or on-site potentials can take only two different values distributed according to the Fibonacci word, is a classical example of a one-dimensional quasicrystal. With its many intriguing properties, such as a fractal eigenvalue spectrum, the Fibonacci chain offers a rich platform to investigate many of the effects that occur in three-dimensional quasicrystals. In this work, we study the eigenvalues and eigenstates of two identical Fibonacci chains coupled to each other in different ways. We find that this setup allows for a rich variety of effects. Depending on the coupling scheme used, the resulting system (i) possesses an eigenvalue spectrum featuring a richer hierarchical structure compared to the spectrum of a single Fibonacci chain, (ii) shows a coexistence of Bloch and critical eigenstates, or (iii) possesses a large number of degenerate eigenstates, each of which is perfectly localized on only four sites of the system. If additionally, the system is infinitely extended, the macroscopic number of perfectly localized eigenstates induces a perfectly flat quasi band. Especially the second case is interesting from an application perspective, since eigenstates that are of Bloch or of critical character feature largely different transport properties. At the same time, the proposed setup allows for an experimental realization, e.g., with evanescently coupled waveguides, electric circuits, or by patterning an anti-lattice with adatoms on a metallic substrate.

cond-mat.str-el↗

arXiv2023

Multifractal Properties of Tribonacci Chains

We introduce two 1D tight-binding models based on the Tribonacci substitution, the hopping and on-site Tribonacci chains, which generalize the Fibonacci chain. For both hopping and on-site models, a perturbative real-space renormalization procedure is developed. We show that the two models are equivalent at the fixed point of the renormalization group flow, and that the renormalization procedure naturally gives the Local Resonator Modes. Additionally, the Rauzy fractal, inherent to the Tribonacci substitution, is shown to serve as the analog of conumbering for the Tribonacci chain. The renormalization procedure is used to repeatedly subdivide the Rauzy fractal into copies of itself, which can be used to describe the eigenstates in terms of Local Resonator Modes. Finally, the multifractal dimensions of the energy spectrum and eigenstates of the hopping Tribonacci chain are computed, from which it can be concluded that the Tribonacci chains are critical.

cond-mat.dis-nn↗

arXiv2022

Field Theoretical Study of Disorder in Non-Hermitian Topological Models

Non-Hermitian systems have provided a rich platform to study unconventional topological phases.These phases are usually robust against external perturbations that respect certain symmetries of thesystem. In this work, we provide a new method to analytically study the effect of disorder, usingtools from quantum field theory applied to discrete models around phase-transition points. Weinvestigate two different one-dimensional models, the paradigmatic non-Hermitian SSH model andas-wave superconductor with imbalanced pairing. These analytic results are compared to numericalsimulations in the discrete models. An universal behavior is found for the two investigated models,namely that the systems are driven from a topological to a trivial phase for disorder strengths equalto about four times the energy scale of the model.

cond-mat.dis-nn↗

arXiv2021

Effects of Disorder in the Fibonacci Quasicrystal

We study the properties of the one-dimensional Fibonacci chain, subjected to the placement of on-site impurities. The resulting disruption of quasiperiodicity can be classified in terms of the renormalization path of the site at which the impurity is placed, which greatly reduces the possible amount of disordered behavior that impurities can induce. Moreover, it is found that, to some extent, the addition of multiple, weak impurities can be treated by superposing the individual contributions together and ignoring nonlinear effects. This means that a transition regime between quasiperiodic order and disorder exists, in which some parts of the system still exhibit quasiperiodicity, while other parts start to be characterized by different localisation behaviours of the wavefunctions. This is manifested through a symmetry in the wavefunction amplitude map, expressed in terms of conumbers, and through the inverse participation ratio. For the latter, we find that its average of states can also be grouped in terms of the renormalization path of the site at which the impurity has been placed.

cond-mat.dis-nn↗