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arXiv:1910.05931 · Journal of Physics A Mathematical and Theoretical

Multifractality in aperiodic quantum spin chains

Dimitrios Voliotis

Published Oct 7, 2019• 2 citations
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Abstract

Abstract Recently, it has been investigated that the ground-state wavefunction of the one dimensional quantum spin-1/2 chain models is multifractal in general with non-trivial fractal dimension. We are studying this phenomena for the quantum Ising chain with aperiodic perturbation. By performing a block real-space renormalization approach, we obtain the ground-state wave function and we extract the generalized multifractal dimension and the multifractal spectrum. For a spin chain with negative wandering exponent the multifractal quantities have the same behavior with the unperturbed chain while for a spin chain with a vanishing wandering exponent, are dependent on the coupling ratio. Finally, for a spin chain with a positive wandering exponent, the multifractal quantities present a different non-linear behavior.

DOI
10.1088/1751-8121/ab4b63

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Abstract Recently, it has been investigated that the ground-state wavefunction of the one dimensional quantum spin-1/2 chain models is multifractal in general with non-trivial fractal dimension. We are studying this phenomena for the quantum Ising chain with aperiodic perturbation. By performing a block real-space renormalization approach, we obtain the ground-state wave function and we extract the generalized multifractal dimension and the multifractal spectrum. For a spin chain with negative wandering exponent the multifractal quantities have the same behavior with the unperturbed chain while for a spin chain with a vanishing wandering exponent, are dependent on the coupling ratio. Finally, for a spin chain with a positive wandering exponent, the multifractal quantities present a different non-linear behavior.
We study the entanglement entropy of blocks of contiguous spins in non-periodic (quasi-periodic or more generally aperiodic) critical Heisenberg, XX and quantum Ising spin chains, e.g. in Fibonacci chains. For marginal and relevant aperiodic modulations, the entanglement entropy is found to be a logarithmic function of the block size with log-periodic oscillations. The effective central charge, c eff , defined through the constant in front of the logarithm may depend on the ratio of couplings and can even exceed the corresponding value in the homogeneous system. In the strong modulation limit, the ground state is constructed by a renormalization group method and the limiting value of c eff is exactly calculated. Keeping the ratio of the block size and the system size constant, the entanglement entropy exhibits a scaling property, however, the corresponding scaling function may be nonanalytic.
In this thesis we present a first study of the multifractal properties of the ground state of the one-dimensional Bose-Hubbard model in Fock space using two different bases, Fock and momentum basis. In the beginning we give a brief overview of the properties of the Bose-Hubbard model, introduce the concept of multifractality, and discuss exact diagonalisation and quantum Monte Carlo, the numerical methods we employ throughout the present work. Then we treat the limits of vanishing and infinite bosonic interaction strengths analytically, and their vicinities by using perturbation theory. We find that the limit of vanishing interaction exhibits non-trivial multifractality in the Fock basis, and that the generalised fractal dimensions exhibit (dominantly) logarithmic finite-size corrections in this limit. Our observations for other values of the interaction strength suggest that this form of the corrections might be equally valid. In order to get access to the multifractal properties at arbitrary values of the interaction strength, we use exact diagonalisation (for system sizes up to L = 10 at unit filling) and quantum Monte Carlo simulations (which enable us to reach L = 30 in certain cases, corresponding to a Hilbert space of size N = 6*10^16, and therefore to one of the largest ever considered for a multifractal analysis). While we see that the generalised fractal dimensions in the Fock basis decrease monotonously when the interaction strength is increased, we observe an inverted behaviour in the momentum basis, i.e. a monotonous increase with increasing interaction strength. Furthermore, our results suggest the existence of non-trivial multifractality in the ground state for a large range of interaction values. Finally, we investigate whether it is possible to characterise the superfluid to Mott insulator transition with the help of multifractality. We find that an analysis of the generalised fractal dimensions for different densities exposes qualitatively the transition. We furthermore consider different ideas, originating from our results in combination with other approaches used in the literature, to quantitatively characterise the transition. While the investigated methods carry great potential, we confirmed that a quantitatively accurate location of the transition using these techniques, if possible, requires access to even larger system sizes.
Many-body systems with multiple emergent time scales arise in various contexts, including classical critical systems, correlated quantum materials, and ultracold atoms. We investigate such nontrivial quantum dynamics in a different setting: a spin-1 bilinear-biquadratic chain. It has a solvable entangled ground state, but a gapless excitation spectrum that is poorly understood. By using large-scale density matrix renormalization group simulations, we find that the lowest excitations have a dynamical exponent zzz that varies from 2 to 3.2 as we vary a coupling in the Hamiltonian. We find an additional gapless mode with a continuously varying exponent $2\ensuremath{\le}z<2.7$, which establishes the presence of multiple dynamics. In order to explain these striking properties, we construct a continuum wave function for the ground state, which correctly describes the correlations and entanglement properties. We also give a continuum parent Hamiltonian, but show that additional ingredients are needed to capture the excitations of the chain. By using an exact mapping to the nonequilibrium dynamics of a classical spin chain, we find that the large dynamical exponent is due to subdiffusive spin motion. Finally, we discuss the connections to other spin chains and to a family of quantum critical models in two dimensions.