On the spectrum of 1D quantum Ising quasicrystal
arXiv:1110.6894
Abstract
We consider one dimensional quantum Ising spin-1/2 chains with two-valued nearest neighbor couplings arranged in a quasi-periodic sequence, with uniform, transverse magnetic field. By employing the Jordan-Wigner transformation of the spin operators to spinless fermions, the energy spectrum can be computed exactly on a finite lattice. By employing the transfer matrix technique and investigating the dynamics of the corresponding trace map, we show that in the thermodynamic limit the energy spectrum is a Cantor set of zero Lebesgue measure. Moreover, we show that local Hausdorff dimension is continuous and nonconstant over the spectrum. This forms a rigorous counterpart of numerous numerical studies.
45 pages, 84 references, 14 figures. Final version. To appear in Annal. H. Poincare
References in corpus (5)
- The Fractal Dimension of the Spectrum of the Fibonacci Hamiltonian
- Entanglement entropy of aperiodic quantum spin chains
- Hyperbolicity of the Trace Map for the Weakly Coupled Fibonacci Hamiltonian
- Spectral analysis of tridiagonal Fibonacci Hamiltonians
- Properties of 1D classical and quantum Ising quasicrystals: rigorous results