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arXiv:1506.00775 · Physical Review B

Manifestations of topological band crossings in bulk entanglement spectrum: An analytical study for integer quantum Hall states

Chi-Ken Lu, Dah-Wei Chiou, Feng-Li Lin

Published Aug 20, 2015• 3 citations
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Abstract

We consider integer quantum Hall states and calculate the bulk entanglement spectrum by formulating the correlation matrix in the guiding center representation. Our analytical approach is based on the strategy of redefining the inner product of states in the Hilbert space, via a projection operator, to take care of the restriction imposed by the (rectangle-pixeled) checkerboard partition. The resultant correlation matrix contains various couplings between states of different guiding centers parameterized by the magnetic length and the pixel size. Given a fixed magnetic field, we find various patterns of band crossings by tuning the pixel size (quantified by the flux $\mathrm{\ensuremath{\Phi}}$ threading each pixel) and by changing the filling factor $\ensuremath{\nu}\ensuremath{\in}\mathbb{N}$ (determined by the Fermi level). When $\ensuremath{\nu}=1$ and $\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}$, or $\ensuremath{\nu}=2$ and $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$, one Dirac band crossing is found. For $\ensuremath{\nu}=1$ and $\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}$, the band crossings are in the form of a nodal line, enclosing the Brillouin zone. As for $\ensuremath{\nu}=2$ and $\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}$, the doubled Dirac point, or the quadratic point, is seen. Additionally, we infer that the quadratic point is protected by the ${\mathrm{C}}_{4}$ symmetry of the pixel since it evolves into two separate Dirac points when the symmetry is lowered to ${\mathrm{C}}_{2}$. We also identify the emerging symmetries responsible for the symmetric bulk entanglement spectra, which are absent in the underlying quantum Hall states.

DOI
10.1103/physrevb.92.075130

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We consider integer quantum Hall states and calculate the bulk entanglement spectrum by formulating the correlation matrix in the guiding center representation. Our analytical approach is based on the strategy of redefining the inner product of states in the Hilbert space, via a projection operator, to take care of the restriction imposed by the (rectangle-pixeled) checkerboard partition. The resultant correlation matrix contains various couplings between states of different guiding centers parameterized by the magnetic length and the pixel size. Given a fixed magnetic field, we find various patterns of band crossings by tuning the pixel size (quantified by the flux \ensuremathΦ\mathrm{\ensuremath{\Phi}}\ensuremathΦ threading each pixel) and by changing the filling factor \ensuremathν\ensuremath∈N\ensuremath{\nu}\ensuremath{\in}\mathbb{N}\ensuremathν\ensuremath∈N (determined by the Fermi level). When \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 and \ensuremathΦ=2\ensuremathπ\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}\ensuremathΦ=2\ensuremathπ, or \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 and \ensuremathΦ=\ensuremathπ\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}\ensuremathΦ=\ensuremathπ, one Dirac band crossing is found. For \ensuremathν=1\ensuremath{\nu}=1\ensuremathν=1 and \ensuremathΦ=\ensuremathπ\mathrm{\ensuremath{\Phi}}=\ensuremath{\pi}\ensuremathΦ=\ensuremathπ, the band crossings are in the form of a nodal line, enclosing the Brillouin zone. As for \ensuremathν=2\ensuremath{\nu}=2\ensuremathν=2 and \ensuremathΦ=2\ensuremathπ\mathrm{\ensuremath{\Phi}}=2\ensuremath{\pi}\ensuremathΦ=2\ensuremathπ, the doubled Dirac point, or the quadratic point, is seen. Additionally, we infer that the quadratic point is protected by the C4{\mathrm{C}}_{4}C4​ symmetry of the pixel since it evolves into two separate Dirac points when the symmetry is lowered to C2{\mathrm{C}}_{2}C2​. We also identify the emerging symmetries responsible for the symmetric bulk entanglement spectra, which are absent in the underlying quantum Hall states.
We have studied extensively the band-crossing patterns of the bulk entanglement spectrum (BES) for various lattice Chern insulators. We find that only partitions with dual symmetry can have either stable nodal lines or nodal points in the BES when the system is in the topological phase of a nonzero Chern number. By deforming the Hamiltonian to lift the accidental symmetry, one can see that only nodal points are robust. They thus should bear certain topological characteristics of the BES. By studying the band-crossing patterns in detail, we conclude that the topological characteristics of the BES are inherited from the topological order of the underlying Chern insulators, and the former can have more refined topological structures. We then conjecture that the sum of the vorticities in the BES in a properly chosen reduced Brillouin zone equals the Chern number of the underlying Chern insulator. This relation is beyond the usual classification scheme of topological insulators/superconductors.
The critical theories for the topological phase transitions of integer quantum Hall states to a trivial insulating state with the same symmetry can be obtained by calculating the ground state entanglement spectrum under a symmetric checkerboard bipartition. In contrast to the gapless edge excitations under the left-right bipartition, a quantum network with bulk gapless excitations naturally emerges without fine tuning. On a large finite lattice, the resulting critical theory for the $\ensuremath{\nu}=1$ state is the (2 + 1)-dimensional relativistic quantum field theory characterized by a single Dirac cone spectrum and a pair of fractionalized zero-energy states, while for the $\ensuremath{\nu}=2$ state the critical theory exhibits a parabolic spectrum and no sign of fractionalization in the zero-energy states. A triangular correspondence is established among the bulk topological theory, gapless edge theory, and the critical theory via the ground state entanglement spectrum.
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.
We analyze translationally invariant insulators with inversion symmetry that fall outside the current established classification of topological insulators. These insulators exhibit no edge or surface modes in the energy spectrum and hence they are not edge metals when the Fermi level is in the bulk gap. However, they do exhibit protected modes in the entanglement spectrum localized on the cut between two entangled regions. Their entanglement entropy cannot be made to vanish adiabatically, and hence the insulators can be called topological. There is a direct connection between the inversion eigenvalues of the Hamiltonian band structure and the midgap states in the entanglement spectrum. The classification of protected entanglement levels is given by an integer N\mathcal{N}N, which is the difference between the negative inversion eigenvalues at inversion symmetric points in the Brillouin zone, taken in sets of 2. When the Hamiltonian describes a Chern insulator or a nontrivial time-reversal invariant topological insulator, the entirety of the entanglement spectrum exhibits spectral flow. If the Chern number is zero for the former, or time reversal is broken in the latter, the entanglement spectrum does not have spectral flow, but, depending on the inversion eigenvalues, can still exhibit protected midgap bands similar to impurity bands in normal semiconductors. Although spectral flow is broken (implying the absence of real edge or surface modes in the original Hamiltonian), the midgap entanglement bands cannot be adiabatically removed, and the insulator is ``topological.'' We analyze the linear response of these insulators and provide proofs and examples of when the inversion eigenvalues determine a nontrivial charge polarization, a quantum Hall effect, an anisotropic three-dimensional (3D) quantum Hall effect, or a magnetoelectric polarization. In one dimension, we establish a link between the product of the inversion eigenvalues of all occupied bands at all inversion symmetric points and charge polarization. In two dimensions, we prove a link between the product of the inversion eigenvalues and the parity of the Chern number of the occupied bands. In three dimensions, we find a topological constraint on the product of the inversion eigenvalues thereby showing that some 333D materials are protected topological metals; we show the link between the inversion eigenvalues and the 333D Quantum Hall Effect, and analyze the magnetoelectric polarization ($\ensuremath{\theta}$ vacuum) in the absence of time-reversal symmetry.
The discovery of integer quantum Hall effect and its subsequent theoretical formulation heralded a new paradigm of thinking in condensed matter physics, which has by now blossomed into the rapidly growing field of topological phases. In this work we investigate several mutually related topics in the framework of topological band theory. In Chapter 2, we study solutions to boundary states on a lattice and see how they are related to the bulk topology. To elicit a real space manifestation of the non-trivial topology, the presence of a physical edge is not strictly necessary. We study two other possibilities, namely the entanglement spectrum associated with an imaginary spatial boundary, and the localization centers of Wannier functions, in Chapters 3,4, and 5. Topological classification through discrete indices is so far possible only for systems described by pure quantum states - in the existing scheme, quantization is lost for systems in mixed states. In Chapter 6, we present a program through which discrete topological indices can be defined for topological band systems at finite temperature, based on Uhlmann's parallel transport of density matrices. The potential of topologocal insulators in realistic applications lies in the existence of Dirac nodes on its surface spectrum. Dirac physics, however, is not exclusive to TI surfaces. In a recently discovered class of materials known as Weyl semimetals, energy nodes which emit linear dispersions also occur in the bulk material. In Chapter 7, we study the possibility of resonance states induced by localized impurities near the nodal energy in Weyl semimetals, which will help us in understanding the stability of density-of-state suppression at the energy nodes. Finally, in Chapter 8, we apply the topological characterization developed for noninteracting particles to a class of interacting spin models in 3D, which are generalizations of Kitaev's honeycomb model, and identify several exotic quantum phases such as spin metals and spin semimetals
In a recent work by Gong and Ueda [Phys. Rev. Lett. 121, 250601 (2018)], the classification of (1+1)-dimensional quench dynamics for the ten Altland-Zirnbauer classes is achieved, and entanglement-spectrum crossings of the time-dependent states for the topological classes (AIII, DIII, CII, BDI, and D) are discovered as a consequence of the bulk-edge correspondence. We note that their classification scheme for the (1+1)-dimensional bulk topology as well as the adoption of entanglement-spectrum crossings as a diagnosis of topological edge states focus on the limit that the spectrum of the postquench Hamiltonian is flat. When band dispersion of the postquench Hamiltonian is considered, even though the bulk topology is still stable, the entanglement-spectrum crossing at the edge might be unstable due to the fact that the band dispersion can be regarded as a ``symmetry-breaking disorder'' in the frequency domain. Such a symmetry-breaking disorder lowers the symmetry for the edge side, and the conventional bulk-edge correspondence could not be applied directly for the quench dynamics. We show that because of the reduction of symmetry by finite energy dispersion the gapless entanglement-spectrum crossing in the flatband limit in classes AIII, DIII, and CII is unstable and could be gapped without closing the bulk gap. The entanglement-spectrum crossing in classes BDI and D is still stable against energy dispersion. We show that the quench process for classes BDI and D can be understood as a Z2{\mathbb{Z}}_{2}Z2​ fermion parity pump, and the entanglement-spectrum crossing for this case is protected by the conservation of fermion parity.