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arXiv:1205.5276 · Physical Review B

Rényi entropy and the multifractal spectra of systems near the localization transition

Xiao Chen, Benjamin Hsu, Taylor L. Hughes, Eduardo Fradkin

Published Oct 3, 2012• 25 citations
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Abstract

We show that the R\'enyi entropies of single particle, critical wave functions for disordered systems contain information about the multifractal spectrum. It is shown for moments of the R\'enyi entropy, ${S}_{n}$, where $|n|<1$, it is possible to extract universal information about the multifractility of such systems. This is shown through a generic calculation and then illustrated through two example models. We find good agreement between our analytic formula and numerical simulations of the two test models. Our formalism is easily extendable to generic noninteracting fermion models. It is also suggested that recent experimental advances in measuring the multifractal spectrum might allow some moments of the R\'enyi entropy to be measured.

DOI
10.1103/physrevb.86.134201

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We show that the R'enyi entropies of single particle, critical wave functions for disordered systems contain information about the multifractal spectrum. It is shown for moments of the R'enyi entropy, Sn{S}_{n}Sn​, where ∣n∣<1|n|<1∣n∣<1, it is possible to extract universal information about the multifractility of such systems. This is shown through a generic calculation and then illustrated through two example models. We find good agreement between our analytic formula and numerical simulations of the two test models. Our formalism is easily extendable to generic noninteracting fermion models. It is also suggested that recent experimental advances in measuring the multifractal spectrum might allow some moments of the R'enyi entropy to be measured.
The von Neumann entanglement entropy is a useful measure to characterize a quantum phase transition. We investigate the nonanalyticity of this entropy at disorder-dominated quantum phase transitions in noninteracting electronic systems. At these critical points, the von Neumann entropy is determined by the single particle wave function intensity, which exhibits complex scale invariant fluctuations. We find that the concept of multifractality is naturally suited for studying von Neumann entropy of the critical wave functions. Our numerical simulations of the three dimensional Anderson localization transition and the integer quantum Hall plateau transition show that the entanglement at these transitions is well described using multifractal analysis.
How many states of a configuration space contribute to a wave function? Attempts to answer this ubiquitous question have a long history in physics and are keys to understanding, e.g., localization phenomena. Beyond single-particle physics, a quantitative study of the ground state complexity for interacting many-body quantum systems is notoriously difficult, mainly due to the exponential growth of the configuration (Hilbert) space with the number of particles. Here we develop quantum Monte Carlo schemes to overcome this issue, focusing on Shannon-Rényi entropies of ground states of large quantum many-body systems. Our simulations reveal a generic multifractal behavior while the very nature of quantum phases of matter and associated transitions is captured by universal subleading terms in these entropies.
The thermodynamic formalism of multifractals assigns a mass exponent τ(q) and a singularity spectrum f(α) to any normalised distribution. Applied to the eigenvalues of a reduced density matrix, it turns Rényi entropies into generalised dimensions and the refined Rényi entropy into f(α_q). We use this to separate two notions that are often conflated: the multifractality of single-particle wave functions at localisation transitions, and the multifractal structure of many-body entanglement spectra. For Gaussian fermionic states we prove that the refined Rényi entropy is exactly the entropy of an ideal Fermi gas at temperature 1/q whose single-particle levels are the entanglement energies; an analogous ideal Bose-gas result holds for bosonic Gaussian states. Consequently, the entanglement singularity spectrum is always non-negative and concave. Its dependence on wave-function multifractality enters only through the density of entanglement energies, which is controlled by overlaps between different eigenstates rather than by single-state moments. The identity is verified to 5×10⁻¹⁰ on critical power-law random banded matrices. For tensor-network cascades with correlations between layers, modelled by a Markov chain, the mass exponent is set by a Perron root. Its first-order correction is linear in the subleading eigenvalue λ₂, and it is real-analytic for λ₂ < 1. A first-order (Maxwell) transition appears only in the reducible limit λ₂ = 1. At the quantum Hall transition, Weyl-group invariance makes the anomalous exponents functions of u = q(1−q). The measured non-parabolicity is therefore the coefficient of the u² (higher-Casimir) term. This term appears at four loops in the 2+ε expansion with the observed sign, and its non-vanishing excludes theories in which the multifractal operators are current-algebra (Sugawara) primaries. For Gauss–Bonnet cosmic branes in d = 4, we confirm that the second Taylor coefficient of τ(q) tracks C_T/a, and we find that the refined Rényi entropy turns negative at finite q for positive Gauss–Bonnet coupling. We also explain why no correspondence between holographic couplings and the quantum Hall non-parabolicity should be expected. Companion paper: R. Chen, Holographic Entanglement of Fractal Regions, DOI 10.5281/zenodo.23189575. Code, data and figures: https://github.com/Ruqing1963/multifractal-quantum-entanglement
We discuss some properties of the generalized entropies, called Rényi entropies, and their application to the case of continuous distributions. In particular, it is shown that these measures of complexity can be divergent; however, their differences are free from these divergences, thus enabling them to be good candidates for the description of the extension and the shape of continuous distributions. We apply this formalism to the projection of wave functions onto the coherent state basis, i.e., to the Husimi representation. We also show how the localization properties of the Husimi distribution on average can be reconstructed from its marginal distributions that are calculated in position and momentum space in the case when the phase space has no structure, i.e., no classical limit can be defined. Numerical simulations on a one-dimensional disordered system corroborate our expectations.
We investigate the structure of many-body wave functions of 1D quantum circuits with local measurements employing the participation entropies. The leading term in system size dependence of participation entropy indicates a model-dependent multifractal scaling of the wave functions at any nonzero measurement rate. The subleading term contains universal information about measurement-induced phase transitions and plays the role of an order parameter, being constant nonzero in the error-correcting phase and vanishing in the quantum Zeno phase. We provide robust numerical evidence investigating a variety of quantum many-body systems and provide an analytical interpretation of this behavior expressing the participation entropy in terms of partition functions of classical statistical models in 2D.
Disordered systems present multifractal properties at criticality. In particular, as discovered by Ludwig [A.W.W. Ludwig, Nucl. Phys. B 330, 639 (1990)] in the case of a diluted two-dimensional Potts model, the moments rho(q) (r) of the local order parameter rho(r) scale with a set x(q) of nontrivial exponents x(q) not = qx(1). We reexamine these ideas to incorporate more recent findings: (i) whenever a multifractal measure w(r) normalized over space sum(r) w(r) = 1 occurs in a random system, it is crucial to distinguish between the typical values and the disorder-averaged values of the generalized moments Y(q) = sum(r) w(q) (r), since they may scale with different generalized dimensions D(q) and D(q), and (ii), as discovered by Wiseman and Domany [S. Wiseman and E. Domany, Phys. Rev. E 52, 3469 (1995)], the presence of an infinite correlation length induces a lack of self-averaging at critical points for thermodynamic observables, in particular for the order parameter. After this general discussion, valid for any random critical point, we apply these ideas to random polymer models that can be studied numerically for large sizes and good statistics over the samples. We study the bidimensional wetting or the Poland-Scheraga DNA model with loop exponents c = 1.5 (marginal disorder) and c = 1.75 (relevant disorder). Finally, we argue that the presence of finite Griffiths-ordered clusters at criticality determines the asymptotic value x(q-->infinity) = d and the minimal value alpha(min) = D(q-->infinity) = d - x(1) of the typical multifractal spectrum f(alpha).