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arXiv:1003.2434

Highly entangled quantum systems in 3+1 dimensions

Brian G. Swingle

Published Mar 11, 2010• 5 citations
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Abstract

Many systems exhibit boundary law scaling for entanglement entropy in more than one spatial dimension. Here I describe three systems in 3+1 dimensions that violate the boundary law for entanglement entropy. The first is free Weyl fermions in a magnetic field, the second is a holographic strong coupling generalization of the Weyl fermion system, and the third is a strong topological insulator in the presence of dislocations. These systems are unified by the presence of a low energy description that includes many gapless 1+1 dimensional modes. I conclude with some comments on the search for highly entangled states of quantum matter and some potential experimental signatures.

DOI
10.48550/arxiv.1003.2434

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We study fermionic non-local magic in higher-dimensional free-fermion systems through the lens of the entanglement spectrum. For translationally invariant Gaussian states in strip geometries, dimensional reduction maps the problem to independent lower-dimensional momentum sectors, leading in two dimensions to a multiplicative logarithmic violation of the boundary law analogous to Gioev-Klich-Widom scaling. We further show that the second Rényi fermionic non-local magic is bounded by the capacity of entanglement and introduce an entanglement-temperature deformation that probes the low-energy structure of the entanglement Hamiltonian. Applied to the matter-Majorana sector of the Kitaev model, this response is exponentially suppressed in the Abelian phase and algebraic in the non-Abelian phase, where its ratio with the capacity approaches a universal constant. Out of equilibrium, we extend the quasi-particle description to fermionic non-local magic in higher-dimensional quenches and show that random Gaussian circuits exhibit diffusive spreading in two spatial dimensions.
The Fermi surface may be usefully viewed as a collection of (1+11+11+1)-dimensional chiral conformal field theories. This approach permits straightforward calculation of many anomalous ground-state properties of the Fermi gas, including entanglement entropy and number fluctuations. The (1+11+11+1)-dimensional picture also generalizes to finite temperature and the presence of interactions. We argue that the low-energy entanglement structure of Fermi liquid theory is universal, depending only on the geometry of the interacting Fermi surface. We also describe three additional systems in 3+13+13+1 dimensions where a similar mechanism leads to a violation of the boundary law for entanglement entropy.
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Topological phenomena in physical systems are determined by topological structures and are thus universal and protected against perturbations. We theoretically establish exotic topological phenomena and their consequences for experiments in crystalline systems such as topological insulators and topologically ordered phases. We show that protected one-dimensional fermionic modes may be associated with the line defects such as dislocations in three-dimensional topological insulators, and strong electron-electron repulsion may lead to topological Mott insulators via spontaneous spin-orbit correlations in three dimensions. We also predict anomalous Aharonov Bohm conductance oscillations maximized at half integer multiples of a flux quantum in a topological insulator nanowire with strong surface disorder, arising from surface curvature induced Berry phase. In addition, we classify three-dimensional inversion symmetric insulators and their quantized responses.Quantum entanglement provides a promising probe to the properties of many-body systems, especially topological phases not captured by local order parameters. We present a characterization of topological insulators using entanglement spectrum based only on bulk ground-state wave function. Further, by studying entanglement of trivial partitions, we establish topological order in candidate Gutzwiller projected wave functions for gapped spin liquids and Laughlin states; and with entanglement's dependence on the ground states for bipartition of a torus into two cylinders, we demonstrate a method to extract the modular matrices and statistics and braiding of quasiparticle excitations. Our method helps to determine the topological order with only the set of ground-state wave functions on a torus. Our variational Monte Carlo calculations of topological entanglement entropy agree well with theory. We also find a violation of the boundary law for a critical spin liquid of Gutzwiller projected Fermi sea on the triangular lattice, where the entanglement entropy's enhancement by a logarithmic factor reflects the presence of emergent fermions in a bosonic wave function.
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