Quasiperiodic Chains Research Articles

Large-deviations approach to thermalization: the case of harmonic chains with conservative noise

We investigate the possibility of characterizing the different thermalization pathways through a large-deviation approach. Specifically, we consider clean, disordered and quasi-periodic harmonic chains under energy and momentum-conserving noise. For their associated master equations, describing the dynamics of normal modes energies, we compute the fluctuations of activity and dynamical entropy in the corresponding biased ensembles. First-order dynamical phase transition are found that originates from different activity regions in action space. At the transitions, the steady-state in the biased ensembles changes from extended to localized, yielding a kind of condensation in normal-modes space. For the disordered and quasi-periodic models, we argue that the phase-diagram has a critical point at a finite value of the disorder or potential strength.

Interacting quasiperiodic spin chains in the prethermal regime

Interacting quasiperiodic spin chains in the prethermal regime

Emergent mobility edges and intermediate phases in one-dimensional quasiperiodic plasmonic chains

Anderson localization has been widely studied in low-dimensional aperiodic electronic, photonic, and acoustic systems. However, the disorder effect in the plasmonic system, where retardation and long-range couplings interact in complex ways, remains an open question. In this work, we investigate the localization properties of one-dimensional quasiperiodic plasmonic chains using the coupled dipole method and linearized Green's function. Our models, which incorporate nearest-neighbor or long-range dipole interactions, reveal localization transitions, mobility edges, and intermediate phases. It is found that long-range dipole interactions and non-Hermiticity due to retardation both play crucial roles in Anderson localization, yielding the emergence of intermediate phases with varying widths. A link between non-Hermiticity and Anderson transition is established by the mean phase rigidity, revealing strong non-Hermiticity along the phase boundary. The plasmonic model involving long-range interplay and retarded effect presents richer localization phenomena than the electronic counterpart that usually includes only nearest-neighbor coupling, laying a foundation for experimental observations of Anderson localization on plasmonic platforms. Published by the American Physical Society 2024

Quasi-Periodic Growth of One-Dimensional Copper Boride on Cu(110).

An unexplored material of copper boride has been realized recently in two-dimensional form at a (111) surface of the fcc copper crystal. Here, one-dimensional (1-D) boron growth was observed on the Cu(110) surface, as probed by atomically resolved scanning probe microscopy. The 1-D copper boride was composed of quasi-periodic atomic chains periodically aligned parallel to each other, as confirmed by Fourier transform analysis. The 1-D growth unexpectedly proceeded across surface steps in a self-assembled manner and extended over several 100 nm. The long-range formation of a 1-D quasi-periodic structure on a surface has been theoretically modeled as a 1-D quasi-crystal and the predicted conditions matched the structural parameters obtained by the experimental work here. The quasi-periodic 1-D copper boride system enabled a way to examine 1-D quasi-crystallinity on an actual material.

General approach to the critical phase with coupled quasiperiodic chains

General approach to the critical phase with coupled quasiperiodic chains

Real-space renormalization-group treatment of quadratic chains

We have recently proposed a one-dimensional nonperiodic chain with lattice positions at 02 d, 12 d, 22 d, ... with length d a constant. The spectrum is singular-continuous, and for weak potential, the states are all extended apart from a trivial set of localized states. In this study, we obtain the exact extended-state spectrum of the quadratic chain in a nearest-neighbor tight-binding model where the quadratic modulation is in the onsite matrix elements. Then, a real-space renormalization-group method (RSRG) is used by decimation to reduce the transfer matrix for the chain into self-similar matrix products. The RSRG decimation scheme is used here to organize the calculation and facilitate numerical computation. The extended-state spectrum appears as minibands broken by numerous gaps. Previous work on quadratic chains shows that the structure factor is singular-continuous and given by a dense set of states with wavevectors with scaling exponent γ(k) = 2 as in periodic and quasi-periodic chains. The origin of extended states in this nonperiodic lattice appears to arise from a type of mechanism not yet identified in deterministic nonperiodic lattices, and is based on a hidden symmetry giving rise to an energy-dependent translational invariance of the transfer matrix.

Radiative heat transfer mediated by topological phonon polaritons in a family of quasiperiodic nanoparticle chains

Topological phonon polaritons (TPhPs) are localized boundary modes arising from the combination of topological photonics and phonon polaritons in micro/nanostructures made of polar dielectrics, which are capable of mediating long-range radiative heat transfer and infrared sensing in a robust manner. This work shows TPhPs can be sustained in a family of one-dimensional quasiperiodic silicon carbide nanoparticle (NP) chains, and can significantly improve radiative heat transfer for these arrays. This family of 1D quasiperiodic lattices is a continuous interpolation between the two paradigmatic limits, viz., the off-diagonal Aubry-André-Harper (AAH) chain and the Fibonacci chain. For different interpolation parameters τ, the band structures are calculated with respect to the modulation phase ϕ as the wavenumber in a synthetic dimension, where a series of gaps and midgap edge states can be observed. These edge states are shown to be topologically protected by applying the gap-labeling theorem and the principle of bulk-boundary correspondence. By means of many-body radiative heat transfer theory for a group of electric dipoles, it is shown quasiperiodicity and topological property can have an important impact on radiative heat transfer. These results can give useful insights on the interplay among topological order, long-range quasiperiodic order, near-field electromagnetic interactions and radiative heat transfer.

Fate of the reentrant localization phenomenon in the one-dimensional dimerized quasiperiodic chain with long-range hopping

Recently, the exciting reentrant localization transition phenomenon was found in a one-dimensional dimerized lattice with staggered quasiperiodic potentials. Usually, long-range hopping is typically important in actual physical systems. In this work, we study the effect of next-nearest neighbor hopping (NNNH) on the reentrant localization phenomenon. Due to the presence of NNNH, the broken chiral symmetry is further enhanced and the localization properties of electron states in the upper and lower bands become quite different. It is found that the reentrant localization can still persist within a range of NNNH both in Hermitian and non-Hermitian cases. Eventually, the reentrant localization disappears as the strength of NNNH increases to some extent, since the increasing NNNH weakens the dimerization of the system and destroys its competition with the quasiperiodic disorder. Our work thus reveals the effect of long-range hopping in the reentrant localization phenomenon and deepens its physical understanding.

Eigenstates in gain–loss systems of metallic-mean quasiperiodic chains

The eigenvalue spectrum of a gain–loss one-dimensional quasiperiodic lattice was examined. An eigenvalue analysis was performed for golden-, silver-, and bronze–mean systems. The eigenvalues of these systems tended to coalesce in discrete small areas of a complex plane, and the eigenvalues exhibited a hierarchical spatial distribution on that plane. In addition, the effect of positional disorder in this system was investigated. The scaling property of the eigenvectors in the golden-mean system was clarified. Multifractal properties of the system were observed except for the outermost edge of the eigenvalue spectrum. For more advanced control of wave phenomena, clarification of physical properties of gain–loss systems in quasi-periodic systems is expected to lead to new technical applications in the future.

Equivalence and superposition of real and imaginary quasiperiodicities

We take non-Hermitian Aubry–André–Harper models and quasiperiodic Kitaev chains as examples to demonstrate the equivalence and superposition of real and imaginary quasiperiodic potentials (QPs) on inducing localization of single-particle states. We prove this equivalence by analytically computing Lyapunov exponents (or inverse of localization lengths) for systems with purely real and purely imaginary QPs. Moreover, when superposed and with the same frequency, real and imaginary QPs are coherent on inducing the localization, in a way which is determined by the relative phase between them. The localization induced by a coherent superposition can be simulated by the Hermitian model with an effective strength of QP, implying that models are in the same universality class. When their frequencies are different and relatively incommensurate, they are incoherent and their superposition leads to less correlation effects. Numerical results show that the localization happens earlier and there is an intermediate mixed phase lacking of mobility edge.

Critical behaviour of the quasi-periodic quantum Ising chain

The interplay of correlated spatial modulation and symmetry breaking leads to quantum critical phenomena intermediate between those of the clean and randomly disordered cases. By performing a detailed analytic and numerical case study of the quasi-periodically (QP) modulated transverse field Ising chain, we provide evidence for the conjectures of reference (Crowley et al 2018 Phys. Rev. Lett. 120 175702) regarding the QP-Ising universality class. In the generic case, we confirm that the logarithmic wandering coefficient w governs both the macroscopic critical exponents and the energy-dependent localisation length of the critical excitations. However, for special values of the phase difference Δ between the exchange and transverse field couplings, the QP-Ising transition has different properties. For Δ = 0, a generalised Aubry–André duality prevents the finite energy excitations from localising despite the presence of logarithmic wandering. For Δ such that the fields and couplings are related by a lattice shift, the wandering coefficient w vanishes. Nonetheless, the presence of small couplings leads to non-trivial exponents and localised excitations. Our results add to the rich menagerie of quantum Ising transitions in the presence of spatial modulation.

Topological Anderson insulators with different bulk states in quasiperiodic chains

We investigate the topology and localization of one-dimensional Hermitian and non-Hermitian Su-Schrieffer-Heeger chains with quasiperiodic hopping modulations. In the Hermitian case, phase diagrams are obtained by numerically and analytically calculating various topological and localization characters. We show the presence of topological extended, intermediate, and localized phases due to the coexistence of independent topological and localization phase transitions driven by the quasiperiodic disorder. Unlike the gapless and localized TAI phase in one-dimensional random disordered systems, we uncover three types of quasiperiodic-disorder-induced gapped topological Anderson insulators (TAIs) with extended, intermediate (with mobility edges), and localized bulk states in this chiral chain. Moreover, we study the non-Hermitian effects on the TAIs by considering two kinds of non-Hermiticities from the non-conjugate complex hopping phase and asymmetric hopping strength, respectively. We demonstrate that three types of TAIs preserve under the non-Hermitian perturbations with some unique localization and topological properties, such as the non-Hermitian real-complex and localization transitions and their topological nature. Our work demonstrates that the disorder-induced TAIs in Hermitian and non-Hermitian quasiperiodic systems are not tied to Anderson transitions and have various localization properties.

Enhanced superconductivity and various edge modes in modulated t−J chains

We numerically investigate the ground state of the extended $t$-$J$ Hamiltonian with periodic local modulations in one dimension by using the density-matrix renormalization group method. Examining charge and spin excitation gaps, as well as the pair binding energy, with extrapolated results to the thermodynamic limit, we obtain a rich ground-state phase diagram consisting of the metallic state, the superconducting state, the phase separation, and insulating states at commensurate fillings. Compared to the homogeneous 1D $t$-$J$ model, the superconductivity is greatly enhanced and stabilized by the flat-band structure. This superconducting state in quasi-periodic chains shares similar properties with ladder systems: significant negative pair binding energy occurs, and the singlet pairing correlation function dominates with the algebraic decay while the single-particle Green's function and spin correlation function decay exponentially. On the other hand, quasi-periodicity leads to nontrivial topological nature in insulating states, characterized by different integer Chern numbers at different fillings. Due to the interplay among the topology, the interaction, and the 1D confinement, gapless edge modes show strong spin-charge separation and in different regions can relate to different collective modes, which are the charge of a single fermion, the magnon, and the singlet-pair. We also find two interaction driven topological transitions: i) at particle filling $\rho=1/2$, the low-energy edge excitations change from the magnon to singlet-pair, accompanied with pair formation in bulk; and ii) at $\rho=3/4$, while the gapless edge mode remains the charge of a single fermion, there is a gap-closing point and a $\pi$-phase shift in the quasi-particle spectrum.

Conformal properties of hyperinvariant tensor networks

Hyperinvariant tensor networks (hyMERA) were introduced as a way to combine the successes of perfect tensor networks (HaPPY) and the multiscale entanglement renormalization ansatz (MERA) in simulations of the AdS/CFT correspondence. Although this new class of tensor network shows much potential for simulating conformal field theories arising from hyperbolic bulk manifolds with quasiperiodic boundaries, many issues are unresolved. In this manuscript we analyze the challenges related to optimizing tensors in a hyMERA with respect to some quasiperiodic critical spin chain, and compare with standard approaches in MERA. Additionally, we show two new sets of tensor decompositions which exhibit different properties from the original construction, implying that the multitensor constraints are neither unique, nor difficult to find, and that a generalization of the analytical tensor forms used up until now may exist. Lastly, we perform randomized trials using a descending superoperator with several of the investigated tensor decompositions, and find that the constraints imposed on the spectra of local descending superoperators in hyMERA are compatible with the operator spectra of several minimial model CFTs.

Topological gaps in quasiperiodic spin chains: A numerical and K-theoretic analysis

Topological phases supported by quasi-periodic spin-chain models and their bulk-boundary principles are investigated by numerical and K-theoretic methods. We show that, for both the un-correlated and correlated phases, the operator algebras that generate the Hamiltonians are non-commutative tori, hence the quasi-periodic chains display physics akin to the quantum Hall effect in two and higher dimensions. The robust topological edge modes are found to be strongly shaped by the interaction and, generically, they have hybrid edge-localized and chain-delocalized structures. Our findings lay the foundations for topological spin pumping using the phason of a quasi-periodic pattern as an adiabatic parameter, where selectively chosen quantized bits of magnetization can be transferred from one edge of the chain to the other.

Finite-size scaling analysis of the many-body localization transition in quasiperiodic spin chains

We analyze the finite-size scaling of the average gap-ratio and the entanglement entropy across the many-body localization (MBL) transition in one dimensional Heisenberg spin-chain with quasi-periodic (QP) potential. By using the recently introduced cost-function approach, we compare different scenarios for the transition using exact diagonalization of systems up to 22 lattice sites. Our findings suggest that the MBL transition in the QP Heisenberg chain belongs to the class of Berezinskii-Kosterlitz-Thouless (BKT) transition, the same as in the case of uniformly disordered systems as advocated in recent studies. Moreover, we observe that the critical disorder strength shows a clear sub-linear drift with the system-size as compared to the linear drift seen in random disordered models, suggesting that the finite-size effects in the MBL transition for the QP systems are less severe than that in the random disordered scenario. Moreover, deep in the ergodic regime, we find an unexpected double-peak structure of distribution of on-site magnetizations that can be traced back to the strong correlations present in the QP potential.

Dephasing enhanced transport in boundary-driven quasiperiodic chains

We study dephasing-enhanced transport in boundary-driven quasi-periodic systems. Specifically we consider dephasing modelled by current preserving Lindblad dissipators acting on the non-interacting Aubry-Andr\'e-Harper (AAH) and Fibonacci bulk systems. The former is known to undergo a critical localization transition with a suppression of ballistic transport above a critical value of the potential. At the critical point, the presence of non-ergodic extended states yields anomalous sub-diffusion. The Fibonacci model, on the other hand, yields anomalous transport with a continuously varying exponent depending on the potential strength. By computing the covariance matrix in the non-equilibrium steady-state, we show that sufficiently strong dephasing always renders the transport diffusive. The interplay between dephasing and quasi-periodicity gives rise to a maximum of the diffusion coefficient for finite dephasing, which suggests the combination of quasi-periodic geometries and dephasing can be used to control noise-enhanced transport.

Localization in quasiperiodic chains: A theory based on convergence of local propagators

Quasiperiodic systems serve as fertile ground for studying localization, due to their propensity already in one dimension to exhibit rich phase diagrams with mobility edges. The deterministic and strongly correlated nature of the quasiperiodic potential nevertheless offers challenges distinct from disordered systems. Motivated by this, we present a theory of localization in quasiperiodic chains with nearest-neighbor hoppings, based on the convergence of local propagators; exploiting the fact that the imaginary part of the associated self-energy acts as a probabilistic order parameter for localization transitions and, importantly, admits a continued-fraction representation. Analyzing the convergence of these continued fractions, localization or its absence can be determined, yielding in turn the critical points and mobility edges. Interestingly, we find anomalous scalings of the order parameter with system size at the critical points, consistent with the fractal character of critical eigenstates. Self-consistent theories at high orders are also considered, shown to be conceptually connected to the theory based on continued fractions, and found in practice to converge to the same result. Results are exemplified by analyzing the theory for three quasiperiodic models covering a range of behavior.

Cascade of the delocalization transition in a non-Hermitian interpolating Aubry-André-Fibonacci chain

In this paper, the interplay of the non-Herimiticity and the cascade of delocalization transition in the quasi-periodic chain is studied. The study is applied in a non-Hermitian interpolating Aubry-Andr{\'e}-Fibonacci (IAAF) model, which combines the non-Hermitian Aubry-Andr{\'e} (AA) model and the non-Hermitian Fibonacci model through a varying parameter, and the non-Hermiticity in this model is introduced by the non-reciprocal hopping. In the non-Hermitian AA limit, the system undergoes a delocalization transition by tuning the potential strength. At the critical point, the spatial distribution of the critical state shows a self-similar structure with the relative distance between the peaks being the Fibonacci sequence, and the finite-size scaling of the inverse participation ratios $({\rm IPRs})$ of the critical ground state with lattice size $L$ shows that ${\rm IPR}_g\propto L^{-0.1189}$. In the non-Hermitian Fibonacci limit, we find that the system is always in the extended phase. Along the continuous deformation from the non-Hermitian AA model into the non-Hermitian Fibonacci model in the IAAF model, the cascade of the delocalization transition is found, but only a few plateaux appear. Moreover, the self-similar structure of spatial distribution for the critical modes along the cascade transition is also found. In addition, we find that the delocalization transition and the real-complex transition for the excited states happen at almost the same parameter. Our results show that the non-Hermiticity provides an additional knob to control the cascade of the delocalization transition besides the on-site potential.

Analysis of Photonic Bandgap Extension in One-Dimensional Quasiperiodic Superconducting Photonic Slabs

One-dimensional superconducting photonic slabs made of alternated layers of isotropic dielectric (SiO2) and a high-Tc superconductor (YBCO) are theoretically investigated by the Gorter Casimir two fluid model (GCTFM) and the transfer matrix method (TMM). The proposed photonic quasicrystals are built according to the inflation rule of quasiperiodic sequences. The enhancement of the photonic band gap is theoretically investigated. Extending photonic band gap (E-PBG) is achieved for suitable quasiperiodic chains. The bandwidth of the photonic band gap can be enhanced by changing the temperature of superconductor, parameter of quasiperiodic sequence. An overall photonic band gap is obtained in the visible frequency range and can be enhanced by applying a typical thickness’ deformation along the superconducting photonic sample.