Scaling of von Neumann entropy at the Anderson transition

Abstract

Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two---a metal-insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.

arXiv comment15 pages, 3 figures, submitted as a chapter in the book "50 years of Anderson localization"

References in corpus (13)

• Anderson Transitions
• Entanglement in a simple quantum phase transition
• Scaling Theory of the Integer Quantum Hall Effect
• Quantum Entanglement in Fermionic Lattices
• Wigner Glass, Spin-liquids, and the Metal-Insulator Transition
• Reconciling Conductance Fluctuations and the Scaling Theory of Localization
• Criticality and entanglement in random quantum systems
• Multifractal analysis of the metal-insulator transition in the 3D Anderson model I: Symmetry relation under typical averaging
• Entanglement entropy and multifractality at localization transitions
• Replacing energy by von Neumann entropy in quantum phase transitions
• Multiparticle ring exchange in the Wigner glass and its possible relevance to strongly interacting two-dimensional electron systems in the presence of disorder
• Quantum Hall plateau transition in the lowest Landau level of disordered graphene
• Glassy states in fermionic systems with strong disorder and interactions

Cited by in corpus (10)

• Area laws in a many-body localized state and its implications for topological order
• Entanglement entropy in one-dimensional disordered interacting system: The role of localization
• The Renyi Entropy and the Multifractal Spectrum of Systems Near the Localization Transition
• Universal Behavior of Entanglement in 2D Quantum Critical Dimer Models
• Localised zero-energy modes in the Kitaev model with vacancy-disorder
• Many-body Localization Transition in Rokhsar-Kivelson-type wave functions
• Large-scale behaviour of local and entanglement entropy of the free Fermi gas at any temperature
• Resolution of the exponent puzzle for the Anderson transition in doped semiconductors
• Sensitivity of the entanglement spectrum to boundary conditions as a characterization of the phase transition from delocalization to localization
• Directional Localization in Disordered 2D Tight-Binding Systems: Insights from Single Particle Entanglement Measures