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Entanglement spectrum of graphene systems

Sonja Predin

Published Jan 1, 2017• 0 citations
Publisher

Abstract

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.

DOI
10.5283/epub.36119

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\n\nThe concept of entanglement spectrum, which was previously associated with gapped topological phases, is extended to the gapless Weyl semimetal phase. By dimensional reduction, we numerically investigate entanglement spectra of Weyl semimetal phases in two three-dimensional (3D) lattice models consisting of coupled layers of 2D quantum anomalous Hall insulators. It is found that for the gapless phase, there still exists a correspondence between the entanglement surface state and the physical surface state; and the “trace index”, defined as the discontinuity in the trace of the single-particle entanglement spectrum, remains equivalent to the Chern number of the 2D insulator layers.\n

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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.
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.
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.
In this paper we consider the interaction of electrons in bilayer graphene with a constant homogeneous magnetic field which is orthogonal to the bilayer surface. Departing from the energy eigenstates of the effective Hamiltonian, the corresponding coherent states will be constructed. For doing this, first we will determine appropriate creation and annihilation operators in order to subsequently derive the coherent states as eigenstates of the annihilation operator with complex eigenvalue. Then, we will calculate some physical quantities, as the Heisenberg uncertainty relation, the probabilities and current density as well as the mean energy value. Finally, we will explore the time evolution for these states and we will compare it with the corresponding evolution for monolayer graphene coherent states.
In this paper we consider the interaction of electrons in bilayer graphene with a constant homogeneous magnetic field which is orthogonal to the bilayer surface. Departing from the energy eigenstates of the effective Hamiltonian, the corresponding coherent states will be constructed. For doing this, first we will determine appropriate creation and annihilation operators in order to subsequently derive the coherent states as eigenstates of the annihilation operator with complex eigenvalue. Then, we will calculate some physical quantities, as the Heisenberg uncertainty relation, the probabilities and current density as well as the mean energy value. Finally, we will explore the time evolution for these states and we will compare it with the corresponding evolution for monolayer graphene coherent states.
The evolution of single particle excitations of bilayer graphene under effects of non-Markovian noise is described with focus on the decoherence process of lattice-layer (LL) maximally entangled states. Once the noiseless dynamics of an arbitrary initial state is identified by the correspondence between the tight-binding Hamiltonian for the AB-stacked bilayer graphene and the Dirac equation---which includes pseudovectorlike and tensorlike field interactions---the noisy environment is described as random fluctuations on bias voltage and mass terms. The inclusion of noisy dynamics reproduces the Ornstein-Uhlenbeck processes: A non-Markovian noise model with a well-defined Markovian limit. Considering that an initial amount of entanglement shall be dissipated by the noise, two profiles of dissipation are identified. On one hand, for eigenstates of the noiseless Hamiltonian, deaths and revivals of entanglement are identified along the oscillation pattern for long interaction periods. On the other hand, for departing LL Werner and Cat states, the entanglement is suppressed although, for both cases, some identified memory effects compete with the pure noise-induced decoherence in order to preserve the the overall profile of a given initial state.
\n\nThe concept of entanglement spectrum, which was previously associated with gapped topological phases, is extended to the gapless Weyl semimetal phase. By dimensional reduction, we numerically investigate entanglement spectra of Weyl semimetal phases in two three-dimensional (3D) lattice models consisting of coupled layers of 2D quantum anomalous Hall insulators. It is found that for the gapless phase, there still exists a correspondence between the entanglement surface state and the physical surface state; and the “trace index”, defined as the discontinuity in the trace of the single-particle entanglement spectrum, remains equivalent to the Chern number of the 2D insulator layers.\n