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Europhysics Letters (EPL)

Entanglement spectrum of topological Weyl semimetal

Huaiqiang Wang, Rui Wang, Yiming Pan, Li Sheng, B. G. Wang, D. Y. Xing

Published Aug 1, 2014• 5 citations
Publisher

Abstract

\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

DOI
10.1209/0295-5075/107/40007

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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
We investigate the entanglement spectra of topological insulators, which manifest edge states on a lattice with spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a ``trace index.'' We find that any free-fermion topological insulator that exhibits spectral flow has a nonvanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics.
We investigate the entanglement spectra of topological insulators, which manifest edge states on a lattice with spatial boundaries. In the physical energy spectrum, a subset of the edge states that intersect the Fermi level translates to discontinuities in the trace of the single-particle entanglement spectrum, which we call a ``trace index.'' We find that any free-fermion topological insulator that exhibits spectral flow has a nonvanishing trace index, which provides us with a new description of topological invariants. In addition, we identify the signatures of spectral flow in the single-particle and many-body entanglement spectrum; in the process we present new methods to extract topological invariants and establish a connection between entanglement and quantum Hall physics.
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.
Weyl semimetals are three-dimensional, topologically protected, gapless phases which show exotic phenomena such as Fermi arc surface states or negative magnetoresistance. It is an open question whether interparticle interactions can turn the topological semimetal into a topologically nontrivial Mott-insulating phase. We investigate an experimentally motivated model for Weyl physics of cold atoms in optical lattices, with the main focus on interaction effects and topological properties, by means of dynamical mean-field theory. We characterize topological phases by numerically evaluating the Chern number via the Ishikawa-Matsuyama formula for interacting phases. Within our studies, we find that the Chern numbers become trivial when interactions lead to insulating behavior. For a deeper understanding of the Weyl-semimetal-to-Mott-insulator topological phase transition, we evaluate the topological properties of quasiparticle bands as well as so-called blind bands. Finally, we consider a system with an open boundary along one spatial direction in order to study correlation effects of surface states. In a narrow regime close to the topological phase transition, we find a correlation-induced state in which the surface becomes metallic while the bulk is semimetallic.
We study the entanglement spectra of surface states of symmetry-protected topological phases. The topological bulk imprints the surface with an anomaly that does not permit it to form a trivial “vacuum” state that is gapped, unfractionalized, and symmetry preserving. Any surface wave function encodes the topology of the underlying bulk in addition to a specific surface phase. We show that the real-space entanglement spectrum of such surface wave functions are dominated by the bulk topology and do not readily permit identifying the surface phase. We thus use a modified form of entanglement spectra that incorporates the anomaly and argue that they correspond to physical edge states between different surface states. We support these arguments by explicit analytical and numerical calculations for free and interacting surfaces of three-dimensional topological insulators of electrons. Published by the American Physical Society 2024
Understanding correlation effects in topological phases of matter is at the forefront of current research in condensed matter physics. Here we try to clarify some subtleties in studying topological behaviors of interacting Weyl semimetals. It is well known that there exist two topological invariants defined to identify their topological character. One is the many-body Chern number, which can be directly linked to the Hall conductivity and thus to the two-particle correlations. The other is the topological index constructed from the single-particle Green's functions. Because the information on Green's functions is achieved more easily than the many-body wave functions, usually only the topological index is employed in the literature. However, the approach based on the single-particle Green's function can break down in the strongly correlated phase. For illustration, an exactly solvable two-orbital model with momentum-local two-body interactions is discussed, in which both topological invariants can be calculated analytically. We find that the topological index calculated from the Green's function formalism can be nonzero even for a nontopological strongly correlated phase with a vanishing many-body Chern number. In addition, we stress that the physical surface states implied by nonzero many-body Chern numbers should be the edge modes of particle-hole collective excitations, rather than those of a quasiparticle nature derived from the Green's function formalism. Our observations thus demonstrate the limitation of the validity of the Green's function formalism in the investigations of interacting topological materials.