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arXiv:1512.06556 · Physical review. B./Physical review. B

Trigonal warping in bilayer graphene: Energy versus entanglement spectrum

Sonja Predin, Paul Wenk, John Schliemann

Published Mar 2, 2016• 28 citations
arXivHTMLPDF

Abstract

We present a mainly analytical study of the entanglement spectrum of Bernal-stacked graphene bilayers in the presence of trigonal warping in the energy spectrum. Upon tracing out one layer, the entanglement spectrum shows qualitative geometric differences to the energy spectrum of a graphene monolayer. However, topological quantities such as Berry-phase-type contributions to Chern numbers agree. The latter analysis involves not only the eigenvalues of the entanglement Hamiltonian but also its eigenvectors. We also discuss the entanglement spectra resulting from tracing out other sublattices. As a technical basis of our analysis, we provide closed analytical expressions for the full eigensystem of bilayer graphene in the entire Brillouin zone with a trigonally warped spectrum.

DOI
10.1103/physrevb.93.115106

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We present a mainly analytical study of the entanglement spectrum of Bernal-stacked graphene bilayers in the presence of trigonal warping in the energy spectrum. Upon tracing out one layer, the entanglement spectrum shows qualitative geometric differences to the energy spectrum of a graphene monolayer. However, topological quantities such as Berry-phase-type contributions to Chern numbers agree. The latter analysis involves not only the eigenvalues of the entanglement Hamiltonian but also its eigenvectors. We also discuss the entanglement spectra resulting from tracing out other sublattices. As a technical basis of our analysis, we provide closed analytical expressions for the full eigensystem of bilayer graphene in the entire Brillouin zone with a trigonally warped spectrum.
In the present thesis, the properties of the entanglement spectra of the ground state of graphene bilayers Bernal-stacked is analytically investigated. In addition, the entanglement spectra of the ground state of the honeycomb lattice in the presence of superconductivity instabilities are analytically studied. We consider not only the eigenvalues of the reduced density matrix that form the entanglement spectrum, but also, its eigenstates. From these eigenstates, we construct topological quantities, such as the Berry phase and the Chern number, in order to investigate the topological properties of the entanglement spectrum. In the first part of this thesis, we present an analytical study of the graphene bilayers, mainly focusing on the effects of the trigonal warping. When the term causing the trigonal warping is neglected, the entanglement Hamiltonian obtained by tracing out one layer shows a proportionality with the energetic Hamiltonian of the remaining monolayer graphene, in the limit of strongly coupled layers. We demonstrate that this proportionality leads to an agreement of the topological quantities of these Hamiltonians. Moreover, we demonstrate that the entanglement spectrum of graphene bilayers with a trigonal warping spectrum clearly differs in the geometric shape from the energy spectrum of the remaining monolayer graphene. However, there is an agreement of the topological quantities such as Berry phase contributions to Chern numbers. In the second part of this thesis, we give a detailed study of graphene in the presence of superconductivity instabilities, mainly considering the s-wave and the chiral dx2−y2 +idxy state.We investigate the relationship between the entanglement and energy spectrum, making use of the concepts of the Chern number constructed from eigenstates of the entanglement Hamiltonian. We demonstrate that the entanglement and remaining subsystem Hamiltonians can have different topologies. These findings are illustrated by considering the entanglement Hamiltonian of the ground state of graphene with dx2−y2 + idxy superconductivity obtained by tracing out one spin direction. Our investigations are based on closed analytical expressions for the full eigensystem in the entire Brillouin zone of bilayer graphene with a trigonally warped spectrum and graphene with superconductivity instabilities.
Abstract We perform an analytical study of the energy and entanglement spectrum of non-interacting fermionic bilayer honeycomb lattices in the presence of trigonal warping in the energy spectrum, on-site energy difference and uniform magnetic field. Employing single particle correlation functions, we present an explicit form for a layer–layer entanglement Hamiltonian whose spectrum is the entanglement spectrum. We demonstrate that in the absence of trigonal warping, at zero on-site energy difference exact correspondence is established between the entanglement spectrum and energy spectrum of a monolayer which means that the entanglement spectrum perfectly reflects the edge state properties of the bilayer. We also show that trigonal warping breaks down such a perfect correspondence, however, in Γ -K direction in the hexagonal Brillouin zone, their behaviors are remarkably the same for particular relevances of hopping parameters. In the presence of an on-site energy difference the symmetry of the entanglement spectrum is broken with opening an indirect entanglement gap. We also study the effects of a perpendicular magnetic field on both energy and the entanglement spectrum of the bilayer in the presence of trigonal warping and on-site energy difference. We demonstrate that the entanglement spectrum versus magnetic flux has a self similar fractal structure, known as the Hofstadter butterfly. Our results also show that the on-site energy difference causes a transition from the Hofstadter butterfly to a tree-like picture.
Closing the gap between ray tracing simulations and experimentally observed electron jetting in bilayer graphene (BLG), we study all-electronic, gate-defined BLG cavities using tight-binding simulations and semiclassical equations of motion. Such cavities offer a rich playground to investigate anisotropic electron transport due to the trigonally warped Fermi surfaces. In this work, we achieve two things: First, we verify the existence of triangular modes (as predicted by classical ray tracing calculations) in the quantum solutions of closed circular BLG cavities. Then, we explore signatures of said triangular modes in transport through open BLG cavities connected to leads. We show that the triangular symmetry translates into anisotropic transport and present an optimal setup for experimental detection of the triangular modes as well as for controlled modulation of transport in preferred directions.
We study Haldane's honeycomb lattice model and a bilayer generalization thereof from the perspective of edge states, entanglement spectra, and Wannier function behavior. For the monolayer model, we obtain the zigzag edge states analytically, and identify the edge state crossing point kck_ckc​ with where the f=1/2f = 1/2f=1/2 entanglement occupancy and the half-odd-integer Wannier centers occur. A continuous interpolation between the entanglement states and the Wannier states is introduced. We then construct a bilayer model by Bernal stacking two monolayers coupled by interlayer hopping. We analyze a particular limit of this model where an extended symmetry, related to inversion, is present. The band topology then reveals a break-down of the correspondence between edge and entanglement spectrum, which is analyzed in detail, and compared with the inversion-symmetric Z2 topological insulators which show a similar phenomenon.
We map out the possible ordered states in bilayer graphene at the neutrality point by extending the previous renormalization group treatment of many-body instabilities to finite temperature, trigonal warping and externally applied perpendicular electric field. We were able to analytically determine all outcomes of the RG flow equations for the nine four-fermion coupling constants. While the full phase diagram exhibits a rich structure, we confirm that when forward scattering dominates, the only ordering tendency with divergent susceptibility at finite temperature is the nematic. At finite temperature, this result is stable with respect to small back and layer imbalance scattering; further increasing their strength leads to the layer antiferromagnet. We also determine conditions for other ordered states to appear and compare our results to the special cases of attractive and repulsive Hubbard models where exact results are available.
We study the entanglement spectrum of noninteracting band insulators, which can be computed from the two-point correlation function, when restricted to one part of the system. In particular, we analyze a type of partitioning of the system that maintains its full translational symmetry, by tracing over a subset of local degrees of freedom, such as sublattice sites or spin orientations. The corresponding single-particle entanglement spectrum is the band structure of an entanglement Hamiltonian in the Brillouin zone. We find that the hallmark of a nontrivial topological phase is a gapless entanglement spectrum with an ``entanglement Fermi surface.'' Furthermore, we derive a relation between the entanglement spectrum and the quantum geometry of Bloch states contained in the Fubini-Study metric. The results are illustrated with lattice models of Chern insulators and Z2{\mathbb{Z}}_{2}Z2​ topological insulators.