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"Kane, Charles L"
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Topological Multipartite Entanglement in a Fermi Liquid
We show that the topology of the Fermi sea of aD-dimensional Fermi gas is reflected in the multipartite entanglement characterizingD+1regions that meet at a point. For oddDwe introduce the multipartite mutual information and show that it exhibits alogDLdivergence as a function of system sizeLwith a universal coefficient that is proportional to the Euler characteristicχFof the Fermi sea. This provides a generalization, for a Fermi gas, of the well-known result forD=1that expresses thelogLdivergence of the bipartite entanglement entropy in terms of the central chargeccharacterizing a conformal field theory. For evenDwe introduce a charge-weighted entanglement entropy that is manifestly odd under a particle-hole transformation. We show that the corresponding charge-weighted mutual information exhibits a similarlogDLdivergence proportional toχF. Our analysis relates the universal behavior of the multipartite mutual information in the absence of interactions to theD+1order equal-time density correlation function, which we show exhibits a universal behavior in the long wavelength limit proportional toχF. Our analytic results are based on the replica method. In addition, we perform a numerical study of the charge-weighted mutual information forD=2that confirms several aspects of the analytic theory. Finally, we consider the effect of interactions perturbatively within the replica theory. We show that forD=3thelog3Ldivergence of the topological mutual information is not perturbed by weak short-ranged interactions, though forD=2the charge-weighted mutual information is perturbed. Thus, forD=3the multipartite mutual information provides a robust classification that distinguishes distinct topological Fermi liquid phases.
Journal Article
Mechanical graphene
We present a model of a mechanical system with a vibrational mode spectrum identical to the spectrum of electronic excitations in a tight-binding model of graphene. The model consists of point masses connected by elastic couplings, called 'tri-bonds', that implement certain three-body interactions, which can be tuned by varying parameters that correspond to the relative hopping amplitudes on the different bond directions in graphene. In the mechanical model, this is accomplished by varying the location of a pivot point that determines the allowed rigid rotations of a single tri-bond. The infinite system constitutes a Maxwell lattice, with the number of degrees of freedom equal to the number of constraints imposed by the tri-bonds. We construct the equilibrium and compatibility matrices and analyze the model's phase diagram, which includes spectra with Weyl points for some placements of the pivot and topologically polarized phases for others. We then discuss the edge modes and associated states of self stress for strips cut from the periodic lattice. Finally, we suggest a physical realization of the tri-bond, which allows access to parameter regimes not available to experiments on (strained) graphene and may be used to create other two-dimensional mechanical metamaterials with different spectral features.
Journal Article
Pairing in Luttinger Liquids and Quantum Hall States
We study spinless electrons in a single-channel quantum wire interacting through attractive interaction, and the quantum Hall states that may be constructed by an array of such wires. For a single wire, the electrons may form two phases, the Luttinger liquid and the strongly paired phase. The Luttinger liquid is gapless to one- and two-electron excitations, while the strongly paired state is gapped to the former and gapless to the latter. In contrast to the case in which the wire is proximity coupled to an external superconductor, for an isolated wire there is no separate phase of a topological, weakly paired superconductor. Rather, this phase is adiabatically connected to the Luttinger liquid phase. The properties of the one-dimensional topological superconductor emerge when the number of channels in the wire becomes large. The quantum Hall states that may be formed by an array of single-channel wires depend on the Landau-level filling factors. For odd-denominator fillings ν=1/(2n+1) , wires at the Luttinger phase form Laughlin states, while wires in the strongly paired phase form a bosonic fractional quantum Hall state of strongly bound pairs at a filling of 1/(8n+4) . The transition between the two is of the universality class of Ising transitions in three dimensions. For even-denominator fractions ν=1/2n , the two single-wire phases translate into four quantum Hall states. Two of those states are bosonic fractional quantum Hall states of weakly and strongly bound pairs of electrons. The other two are non-Abelian quantum Hall states, which originate from coupling wires close to their critical point. One of these non-Abelian states is the Moore-Read state. The transitions between all of these states are of the universality class of Majorana transitions. We point out some of the properties that characterize the different phases and the phase transitions.
Journal Article
Topological Classification of Crystalline Insulators through Band Structure Combinatorics
We present a method for efficiently enumerating all allowed, topologically distinct, electronic band structures within a given crystal structure in all physically relevant dimensions. The algorithm applies to crystals without time-reversal, particle-hole, chiral, or any other anticommuting or anti-unitary symmetries. The results presented match the mathematical structure underlying the topological classification of these crystals in terms of K-theory and therefore elucidate this abstract mathematical framework from a simple combinatorial perspective. Using a straightforward counting procedure, we classify all allowed topological phases of spinless particles in crystals in class A. Employing this classification, we study transitions between topological phases within class A that are driven by band inversions at high-symmetry points in the first Brillouin zone. This enables us to list all possible types of phase transitions within a given crystal structure and to identify whether or not they give rise to intermediate Weyl semimetallic phases.
Journal Article
Condensed matter: An insulator with a twist
The insulating state is the most basic electronic phase of matter. Experiment has now proved the existence of the predicted three-dimensional topological insulator in the semiconducting alloy Bi1xSbx.
Journal Article
A New Spin on the Insulating State
Electrical insulators are usually appreciated for their ability to do nothing. Such materials either trap or restrict the motion of free charges in matter. This is useful in all kinds of applications, ranging from the wiring in your home to directing the flow of electrons in the tiny circuits of your iPod. Now, on page 1757 of this issue, Bernevig et al. have proposed a new kind of two-dimensional insulator, which permits the flow of charge only at its edges. This may lead to the development of a new kind of solid-state electronic device.
Journal Article
Erasing electron mass
Two-dimensional graphite, which is the form of carbon found in pencil lead, leaves its mark thanks to weakly coupled layers of atoms that slide easily over one another and could be useful in carbon-based electronic devices. Kane discusses how electrons move in these structures, which seems best described by relativistic quantum physics, modeling them as if they have no mass at all.
Journal Article
Erasing electron mass
Two-dimensional graphite could be useful in carbon-based electronic devices. How electrons move in these structures seems best described by relativistic quantum physics, modelling them as if they have no mass at all.
Journal Article