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arXiv:1205.4541 · Physical Review E

Multifractality of eigenfunctions in spin chains

Y. Y. Atas, E. Bogomolny

Published Aug 3, 2012• 77 citations
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Abstract

We investigate different one-dimensional quantum spin-1/2 chain models, and by combining analytical and numerical calculations we prove that their ground state wave functions in the natural spin basis are multifractals with, in general, nontrivial fractal dimensions.
DOI
10.1103/physreve.86.021104

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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.
Multifractal dimensions allow for characterizing the localization properties of states in complex quantum systems. For ergodic states the finite-size versions of fractal dimensions converge to unity in the limit of large system size. However, the approach to the limiting behavior is remarkably slow. Thus, an understanding of the scaling and finite-size properties of fractal dimensions is essential. We present such a study for random matrix ensembles, and compare with two chaotic quantum systems-the kicked rotor and a spin chain. For random matrix ensembles we analytically obtain the finite-size dependence of the mean behavior of the multifractal dimensions, which provides a lower bound to the typical (logarithmic) averages. We show that finite statistics has remarkably strong effects, so that even random matrix computations deviate from analytic results (and show strong sample-to-sample variation), such that restoring agreement requires exponentially large sample sizes. For the quantized standard map (kicked rotor) the multifractal dimensions are found to follow the random matrix predictions closely, with the same finite statistics effects. For a XXZ spin-chain we find significant deviations from the random matrix prediction-the large-size scaling follows a system-specific path towards unity. This suggests that local many-body Hamiltonians are "weakly ergodic," in the sense that their eigenfunction statistics deviate from random matrix theory.
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.