We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimensions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.
We relate the entropy of entanglement of ensembles of random vectors to their generalized fractal dimensions. Expanding the von Neumann entropy around its maximum we show that the first order only depends on the participation ratio, while higher orders involve other multifractal exponents. These results can be applied to entanglement behavior near the Anderson transition.
In this work, we establish a relation between entanglement entropy and fractal dimension D of generic many-body wave functions, by generalizing the result of Page [Phys. Rev. Lett. 71, 1291 (1993)PRLTAO0031-900710.1103/PhysRevLett.71.1291] to the case of sparse random pure states (SRPS). These SRPS living in a Hilbert space of size N are defined as normalized vectors with only N^{D} (0≤D≤1) random nonzero elements. For D=1, these states used by Page represent ergodic states at an infinite temperature. However, for 0<D<1, the SRPS are nonergodic and fractal, as they are confined in a vanishing ratio N^{D}/N of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy S_{1}(A) of a subsystem A, with Hilbert space dimension N_{A}, scales as S_{1}[over ¯](A)∼DlnN for small fractal dimensions D, N^{D}<N_{A}. Remarkably, S_{1}[over ¯](A) saturates at its thermal (Page) value at an infinite temperature, S_{1}[over ¯](A)∼lnN_{A} at larger D. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly nonergodic. Finally, we generalize our results to Renyi entropies S_{q}(A) with q>1 and to genuine multifractal states and also show that their fluctuations have ergodic behavior in a narrower vicinity of the ergodic state D=1.
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.
We study the fractal properties of single-particle eigenmodes of an entanglement Hamiltonian in free-fermion models. One of these modes that has the highest entanglement information and is thus called the maximally entangled mode (MEM) is especially considered. In free-fermion models with Anderson localization, the fractality of the MEM is obtained numerically and compared with the fractality of the Hamiltonian eigenmode at the Fermi level. We show that both eigenmodes have similar fractal properties: both have the same single-fractal dimension in the delocalized phase which equals the dimension of the system, and both show multifractality at the phase transition point. Therefore, we conclude that fractal behavior of the MEM, in addition to the fractal behavior of the Hamiltonian eigenmode, can be used as a quantum phase transition characterization.
We use the heat kernel method to calculate the entanglement entropy for a given entangling region on a fractal. The leading divergent term of the entropy is obtained as a function of the fractal dimension as well as the walk dimension. The power of the UV cutoff parameter is (generally) a fractional number, which, indeed, is a certain combination of these two indices. This exponent is known as the spectral dimension. We show that there is a novel log-periodic oscillatory behavior in the expression of entropy which has root in the complex dimension of the fractal. We finally indicate that the holographic calculation in a certain hyperscaling-violating bulk geometry yields the same leading term for the entanglement entropy, if one identifies the effective dimension of the hyperscaling-violating theory with the spectral dimension of the fractal. We provide additional support by comparing the behavior of the thermal entropy in terms of the temperature, computed for two geometries, the fractal geometry and the hyperscaling-violating background.
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
Quantum systems with short-range interactions are known to respect an area law for the entanglement entropy: The von Neumann entropy S associated to a bipartition scales with the boundary p between the two parts. Here we study the case in which the boundary is a fractal. We consider the topologically ordered phase of the toric code with a magnetic field. When the field vanishes it is possible to analytically compute the entanglement entropy for both regular and fractal bipartitions (A,B) of the system and this yields an upper bound for the entire topological phase. When the $A\ensuremath{-}B$ boundary is regular we have S/p=1 for large p. When the boundary is a fractal of the Hausdorff dimension D, we show that the entanglement between the two parts scales as $S/p=\ensuremath{\gamma}\ensuremath{\leqslant}1/D$, and $\ensuremath{\gamma}$ depends on the fractal considered.