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arXiv:1406.7214 · Journal of Statistical Mechanics Theory and Experiment

Entanglement spectrum and entanglement Hamiltonian of a Chern insulator with open boundaries

Maria Hermanns, Yann Salimi, Masudul Haque, Lars Fritz

Published Oct 21, 2014• 19 citations
arXivHTMLPDF

Abstract

We study the entanglement spectrum of a Chern insulator on a cylinder geometry, with the cut separating the two partitions taken parallel to the cylinder edge, at varying distances from the edge. In contrast to similar studies on a torus, there is only one cut, and hence only one virtual edge mode in the entanglement spectrum. The entanglement spectrum has a gap when the cut is close enough to the physical edge of the cylinder such that the edge mode spatially extends over the cut. This effect is suppressed for parameter choices where the edge mode is sharply localized at the edge. In the extreme case of a perfectly localized edge mode, the entanglement spectrum is gapless even if the smaller partition consists of a single edge row. For the single-row cut, we construct the corresponding entanglement Hamiltonian, which is a one-dimensional tight-binding Hamiltonian with complex long-range hopping and interesting properties. We also study and explain the effect of two different schemes of flux insertion through a ring described by such an entanglement Hamiltonian.

DOI
10.1088/1742-5468/2014/10/p10030

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We study the entanglement spectrum of a Chern insulator on a cylinder geometry, with the cut separating the two partitions taken parallel to the cylinder edge, at varying distances from the edge. In contrast to similar studies on a torus, there is only one cut, and hence only one virtual edge mode in the entanglement spectrum. The entanglement spectrum has a gap when the cut is close enough to the physical edge of the cylinder such that the edge mode spatially extends over the cut. This effect is suppressed for parameter choices where the edge mode is sharply localized at the edge. In the extreme case of a perfectly localized edge mode, the entanglement spectrum is gapless even if the smaller partition consists of a single edge row. For the single-row cut, we construct the corresponding entanglement Hamiltonian, which is a one-dimensional tight-binding Hamiltonian with complex long-range hopping and interesting properties. We also study and explain the effect of two different schemes of flux insertion through a ring described by such an entanglement Hamiltonian.
We study the correspondence between boundary spectrum of non-chiral topological orders on an open manifold M\mathcal{M}M with gapped boundaries and the entanglement spectrum in the bulk of gapped topological orders on a closed manifold. The closed manifold is bipartitioned into two subsystems, one of which has the same topology as M\mathcal{M}M. Specifically, we focus on the case of generalized string-net models and discuss the cases where M\mathcal{M}M is a disk or a cylinder. The correspondence turns out to be very different from that of the chiral topological orders. When M\mathcal{M}M has the topology of a cylinder, different boundary conditions of the cylinder will correspond to different entanglement cuts on the torus. Charge (smooth) and flux (rough) boundaries of toric code, more general ZN\mathbb{Z}_NZN​ models, and the simplest non-abelian case of doubled Fibonacci are demonstrated.
We study two a priori unrelated constructions: the spectrum of edge modes in a band topological insulator or superconductor with a physical edge, and the ground state entanglement spectrum in an extended system where an edge is simulated by an entanglement bipartition. We prove an exact relation between the ground state entanglement spectrum of such a system and the spectrum edge modes of the corresponding spectrally flattened Hamiltonian. In particular, we show that gapless edge modes result in degeneracies of the 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.
It has been recently realized that strong interactions in topological Bloch bands give rise to the appearance of novel states of matter. Here we study connections between these systems---fractional Chern insulators and the fractional quantum Hall states---via generalization of a gauge-fixed Wannier-Qi construction in the cylinder geometry. Our setup offers a number of important advantages compared to the earlier exact diagonalization studies on a torus. Most notably, it gives access to edge states and to a single-cut orbital entanglement spectrum, hence to the physics of bulk-edge correspondence. It is also readily implemented in the state-of-the-art density matrix renormalization group method that allows for numerical simulations of significantly larger systems. We demonstrate our general approach on examples of flat-band models on ruby and kagome lattices at bosonic filling fractions $\ensuremath{\nu}=1/2$ and $\ensuremath{\nu}=1$, which show the signatures of (non)-Abelian phases, and establish the correspondence between the physics of edge states and the entanglement in the bulk. Notably, we find that the non-Abelian $\ensuremath{\nu}=1$ phase can be stabilized by purely on-site interactions in the presence of a confining potential.
We study the quantum entanglement of the spin and orbital degrees of freedom in the one-dimensional Kugel-Khomskii model, which includes both gapless and gapped phases, using analytical techniques and exact diagonalization with up to 16 sites. We compute the entanglement entropy and the entanglement spectra using a variety of partitions or ``cuts'' of the Hilbert space, including two distinct real-space cuts and a momentum-space cut. Our results show that the Kugel-Khomski model possesses a number of new features not previously encountered in studies of the entanglement spectra. Notably, we find robust gaps in the entanglement spectra for both gapped and gapless phases with the orbital partition, and show these are not connected to each other. The counting of the low-lying entanglement eigenvalues shows that the ``virtual edge'' picture, which equates the low-energy Hamiltonian of a virtual edge, here one gapless leg of a two-leg ladder, to the ``low-energy'' entanglement Hamiltonian, breaks down for this model, even though the equivalence has been shown to hold for a similar cut in a large class of closely related models. In addition, we show that a momentum space cut in the gapless phase leads to qualitative differences in the entanglement spectrum when compared with the same cut in the gapless spin-1/2 Heisenberg spin chain. We emphasize the new information content in the entanglement spectra compared to the entanglement entropy, and using quantum entanglement, we present a refined phase diagram of the model. Using analytical arguments, exploiting various symmetries of the model, and applying arguments of adiabatic continuity from two exactly solvable points of the model, we are also able to prove several results regarding the structure of the low-lying entanglement eigenvalues.
Results are presented for the entanglement entropy and spectrum of half-filled graphene following the switch on of a circularly polarized laser. The laser parameters are chosen to correspond to several different Floquet-Chern insulator phases. The entanglement properties of the unitarily evolved wave functions are compared with the state where one of the Floquet bands is completely occupied. The true states show a volume law for the entanglement, whereas the Floquet states show an area law. Qualitative differences are found in the entanglement properties of the off-resonant and on-resonant laser. Edge states are found in the entanglement spectrum corresponding to certain physical edge states expected in a Chern insulator. However, some edge states that would be expected from the Floquet band structure are missing from the entanglement spectrum. An analytic theory is developed for the long-time structure of the entanglement spectrum. It is argued that only edge states corresponding to off-resonant processes appear in the entanglement spectrum.