The anisotropic quantum XY spin chain in a modulating magnetic field incommensurate with the periodicity of the chain is studied. Using Jordan-Wigner transformation, the model is mapped to a tight-binding model in fermions. In the absence of symmetry-breaking terms, the fermion model is identical with the well-known Harper equation exhibiting extended and localized phases with the onset of transition being a critical point with fractal spectrum and wave functions. The effect of O(2) symmetry breaking is to fatten the critical point resulting in a critical phase with power-law localization existing in a finite window of size determined by the anisotropy. In this three-phase spectral diagram, the transition from critical to localized phase is accompanied by the magnetic transition to long-range order. Furthermore, the scaling properties of the fractal spectrum vary in the critical phase. This leads to interesting consequences on the low-temperature thermodynamical properties of the system.The anisotropic quantum XY spin chain in a modulating magnetic field incommensurate with the periodicity of the chain is studied. Using Jordan-Wigner transformation, the model is mapped to a tight-binding model in fermions. In the absence of symmetry-breaking terms, the fermion model is identical with the well-known Harper equation exhibiting extended and localized phases with the onset of transition being a critical point with fractal spectrum and wave functions. The effect of O(2) symmetry breaking is to fatten the critical point resulting in a critical phase with power-law localization existing in a finite window of size determined by the anisotropy. In this three-phase spectral diagram, the transition from critical to localized phase is accompanied by the magnetic transition to long-range order. Furthermore, the scaling properties of the fractal spectrum vary in the critical phase. This leads to interesting consequences on the low-temperature thermodynamical properties of the system.

Physical review. B, Condensed matterFeb 1, 199417 citations

DYNAMICS AND MULTIFRACTALITY OF QUANTUM DISORDERED SYSTEMS

Weitao Chen

Aug 1, 20250 citations

State Change, Complexity and Fractal in Quantum Systems

Masanori Ohya

Springer eBooksJan 1, 199526 citations

Critical quantum states and hierarchical spectral statistics in a Cantor potential

F. Iwase

We study the spectral statistics and wave-function properties of a one-dimensional quantum system subject to a Cantor-type fractal potential. By analyzing the nearest-neighbor level spacings, inverse participation ratio (IPR), and the scaling behavior of the integrated density of states (IDS), we demonstrate how the self-similar geometry of the potential is imprinted on the quantum spectrum. The energy-resolved level spacings form a hierarchical, filamentary structure, in sharp contrast to those of periodic and random systems. The normalized level-spacing distribution exhibits a bimodal structure, reflecting the deterministic recurrence of spectral gaps. A multifractal analysis of eigenstates reveals critical behavior: the generalized fractal dimensions $D_q$ lie strictly between the limits of extended and localized states, exhibiting a distinct $q$ -dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.We study the spectral statistics and wave-function properties of a one-dimensional quantum system subject to a Cantor-type fractal potential. By analyzing the nearest-neighbor level spacings, inverse participation ratio (IPR), and the scaling behavior of the integrated density of states (IDS), we demonstrate how the self-similar geometry of the potential is imprinted on the quantum spectrum. The energy-resolved level spacings form a hierarchical, filamentary structure, in sharp contrast to those of periodic and random systems. The normalized level-spacing distribution exhibits a bimodal structure, reflecting the deterministic recurrence of spectral gaps. A multifractal analysis of eigenstates reveals critical behavior: the generalized fractal dimensions $D_q$ lie strictly between the limits of extended and localized states, exhibiting a distinct $q$-dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.

cond-mat.dis-nnCross-listedarXiv:2601.08324Jan 14, 2026

Incommensurability enabled quasi-fractal order in 1D narrow-band moiré systems

Miguel Gonçalves, Bruno Amorim, Flavio Elias Riche +2 more

Abstract We demonstrate that quasiperiodicity can radically change the ground state properties of 1D moiré systems with respect to their periodic counterparts. By studying an illustrative example we show that while narrow bands play a significant role in enhancing interactions both for commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the quasiperiodic-enabled state has contributions from infinitely many wave vectors. This quasi-factal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and signal out multifractal non-interacting phases, recently found in 2D incommensurate moiré systems, as particularly promising parent states.Abstract We demonstrate that quasiperiodicity can radically change the ground state properties of 1D moiré systems with respect to their periodic counterparts. By studying an illustrative example we show that while narrow bands play a significant role in enhancing interactions both for commensurate and incommensurate structures, only quasiperiodicity is able to extend the ordered phase down to an infinitesimal interaction strength. In this regime, the quasiperiodic-enabled state has contributions from infinitely many wave vectors. This quasi-factal regime cannot be stabilized in the commensurate case even in the presence of a narrow band. These findings suggest that quasiperiodicity may be a critical factor in stabilizing non-trivial ordered phases in interacting moiré structures and signal out multifractal non-interacting phases, recently found in 2D incommensurate moiré systems, as particularly promising parent states.

arXiv:2305.03800Nature PhysicsOct 11, 202425 citations

Fractals and quantum criticality

Anonymous

PhysicsJul 21, 20080 citations

D_q

lie strictly between the limits of extended and localized states, exhibiting a distinct

q

-dependence. Consistently, the IPR indicates the coexistence of quasi-extended and localized features, characteristic of critical wave functions. The IDS shows anomalous power-law scaling at low energies, with an exponent close to the Hausdorff dimension of the underlying Cantor set, indicating that the geometric fractality governs the spectral dimensionality. At higher energies, this scaling crosses over to the semiclassical Weyl law. Our results establish a direct connection between deterministic fractal geometry, hierarchical spectral statistics, and quantum criticality.