Author Correction: Axionic charge-density wave in the Weyl semimetal (TaSe4)2I.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Unhinging the Surfaces of Higher-Order Topological Insulators and Superconductors.

We show that the chiral Dirac and Majorana hinge modes in three-dimensional higher-order topological insulators (HOTIs) and superconductors (HOTSCs) can be gapped while preserving the protecting C_{2n}T symmetry upon the introduction of non-Abelian surface topological order. In both cases, the topological order on a single side surface breaks time-reversal symmetry, but appears with its time-reversal conjugate on alternating sides in a C_{2n}T preserving pattern. In the absence of the HOTI/HOTSC bulk, such a pattern necessarily involves gapless chiral modes on hinges between C_{2n}T-conjugate domains. However, using a combination of K-matrix and anyon condensation arguments, we show that on the boundary of a 3D HOTI/HOTSC these topological orders are fully gapped and hence "anomalous." Our results suggest that new patterns of surface and hinge states can be engineered by selectively introducing topological order only on specific surfaces.

Axionic charge-density wave in the Weyl semimetal (TaSe4)2I.

An axion insulator is a correlated topological phase, which is predicted to arise from the formation of a charge-density wave in a Weyl semimetal1,2-that is, a material in which electrons behave as massless chiral fermions. The accompanying sliding mode in the charge-density-wave phase-the phason-is an axion3,4 and is expected to cause anomalous magnetoelectric transport effects. However, this axionic charge-density wave has not yet been experimentally detected. Here we report the observation of a large positive contribution to the magnetoconductance in the sliding mode of the charge-density-wave Weyl semimetal (TaSe4)2I for collinear electric and magnetic fields. The positive contribution to the magnetoconductance originates from the anomalous axionic contribution of the chiral anomaly to the phason current, and is locked to the parallel alignment of the electric and magnetic fields. By rotating the magnetic field, we show that the angular dependence of the magnetoconductance is consistent with the anomalous transport of an axionic charge-density wave. Our results show that it is possible to find experimental evidence for axions in strongly correlated topological condensed matter systems, which have so far been elusive in any other context.

Topological Lifshitz transitions and Fermi arc manipulation in Weyl semimetal NbAs.

Surface Fermi arcs (SFAs), the unique open Fermi-surfaces (FSs) discovered recently in topological Weyl semimetals (TWSs), are unlike closed FSs in conventional materials and can give rise to many exotic phenomena, such as anomalous SFA-mediated quantum oscillations, chiral magnetic effects, three-dimensional quantum Hall effect, non-local voltage generation and anomalous electromagnetic wave transmission. Here, by using in-situ surface decoration, we demonstrate successful manipulation of the shape, size and even the connections of SFAs in a model TWS, NbAs, and observe their evolution that leads to an unusual topological Lifshitz transition not caused by the change of the carrier concentration. The phase transition teleports the SFAs between different parts of the surface Brillouin zone. Despite the dramatic surface evolution, the existence of SFAs is robust and each SFA remains tied to a pair of Weyl points of opposite chirality, as dictated by the bulk topology.

A gap-protected zero-Hall effect state in the quantum limit of the non-symmorphic metal KHgSb.

A recurring theme in topological matter is the protection of unusual electronic states by symmetry, for example, protection of the surface states in Z2 topological insulators by time-reversal symmetry1-3. Recently, interest has turned to unusual surface states in the large class of non-symmorphic materials4-12. In particular, KHgSb is predicted to exhibit double quantum spin Hall states10. Here we report measurements of the Hall conductivity in KHgSb in a strong magnetic field B. In the quantum limit, the Hall conductivity is observed to fall exponentially to zero, but the diagonal conductivity is finite. A large gap protects this unusual zero-Hall state. We theoretically propose that, in this quantum limit, the chemical potential drops into the bulk gap, intersecting equal numbers of right- and left-moving quantum spin Hall surface modes to produce the zero-Hall state. The zero-Hall state illustrates how topological protection in a non-symmorphic material with glide symmetry may lead to highly unusual transport phenomena.

The chiral anomaly and thermopower of Weyl fermions in the half-Heusler GdPtBi.

The Dirac and Weyl semimetals are unusual materials in which the nodes of the bulk states are protected against gap formation by crystalline symmetry. The chiral anomaly, predicted to occur in both systems, was recently observed as a negative longitudinal magnetoresistance (LMR) in Na3Bi (ref. ) and in TaAs (ref. ). An important issue is whether Weyl physics appears in a broader class of materials. We report evidence for the chiral anomaly in the half-Heusler GdPtBi. In zero field, GdPtBi is a zero-gap semiconductor with quadratic bands. In a magnetic field, the Zeeman energy leads to Weyl nodes. We have observed a large negative LMR with the field-steering properties specific to the chiral anomaly. The chiral anomaly also induces strong suppression of the thermopower. We report a detailed study of the thermoelectric response function αxx of Weyl fermions. The scheme of creating Weyl nodes from quadratic bands suggests that the chiral anomaly may be observable in a broad class of semimetals.

Correlation lengths and topological entanglement entropies of unitary and nonunitary fractional quantum Hall wave functions.

Using the newly developed matrix product state formalism for non-Abelian fractional quantum Hall (FQH) states, we address the question of whether a FQH trial wave function written as a correlation function in a nonunitary conformal field theory (CFT) can describe the bulk of a gapped FQH phase. We show that the nonunitary Gaffnian state exhibits clear signatures of a pathological behavior. As a benchmark we compute the correlation length of a Moore-Read state and find it to be finite in the thermodynamic limit. By contrast, the Gaffnian state has an infinite correlation length in (at least) the non-Abelian sector, and is therefore gapless. We also compute the topological entanglement entropy of several non-Abelian states with and without quasiholes. For the first time in the FQH effect the results are in excellent agreement in all topological sectors with the CFT prediction for unitary states. For the nonunitary Gaffnian state in finite size systems, the topological entanglement entropy seems to behave like that of the composite fermion Jain state at equal filling.

Extracting excitations from model state entanglement.

We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.

Topological entanglement in Abelian and non-Abelian excitation eigenstates.

Entanglement in topological phases of matter has so far been investigated through the perspective of their ground-state wave functions. In contrast, we demonstrate that the excitations of fractional quantum Hall (FQH) systems also contain information to identify the system's topological order. Entanglement spectrum of the FQH quasihole (QH) excitations is shown to differentiate between the conformal field theory (CFT) sectors, based on the relative position of the QH with respect to the entanglement cut. For Read-Rezayi model states, as well as Coulomb interaction eigenstates, the counting of the QH entanglement levels in the thermodynamic limit matches exactly the CFT counting, and sector changes occur as non-Abelian quasiholes successively cross the entanglement cut.

Topological entanglement and clustering of Jain hierarchy states.

We obtain several clustering properties of the Jain states at filling k/2k+1: they are a product of a Vandermonde determinant and a bosonic polynomial at filling k/k+1 which vanishes when k+1 particles cluster together. We show that all Jain states satisfy a "squeezing rule" which severely reduces the dimension of the Hilbert space necessary to generate them. We compute the topological entanglement spectrum of the Jain nu=2/5 state and compare it to both the Coulomb ground state and the nonunitary Gaffnian state. All three states have a very similar "low-energy" structure. However, the Jain state entanglement "edge" state counting matches both the Coulomb counting as well as two decoupled U(1) free bosons, whereas the Gaffnian edge counting misses some of the edge states of the Coulomb spectrum.

Localization and the Kosterlitz-Thouless transition in disordered graphene.

We investigate disordered graphene with strong long-range impurities. Contrary to the common belief that delocalization should persist in such a system against any disorder, as the system is expected to be equivalent to a disordered two-dimensional Dirac fermionic system, we find that states near the Dirac points are localized for sufficiently strong disorder (therefore inevitable intervalley scattering) and the transition between the localized and delocalized states is of Kosterlitz-Thouless type. Our results show that the transition originates from bounding and unbounding of local current vortices.

Emergence of the persistent spin helix in semiconductor quantum wells.

According to Noether's theorem, for every symmetry in nature there is a corresponding conservation law. For example, invariance with respect to spatial translation corresponds to conservation of momentum. In another well-known example, invariance with respect to rotation of the electron's spin, or SU(2) symmetry, leads to conservation of spin polarization. For electrons in a solid, this symmetry is ordinarily broken by spin-orbit coupling, allowing spin angular momentum to flow to orbital angular momentum. However, it has recently been predicted that SU(2) can be achieved in a two-dimensional electron gas, despite the presence of spin-orbit coupling. The corresponding conserved quantities include the amplitude and phase of a helical spin density wave termed the 'persistent spin helix'. SU(2) is realized, in principle, when the strengths of two dominant spin-orbit interactions, the Rashba (strength parameterized by alpha) and linear Dresselhaus (beta(1)) interactions, are equal. This symmetry is predicted to be robust against all forms of spin-independent scattering, including electron-electron interactions, but is broken by the cubic Dresselhaus term (beta(3)) and spin-dependent scattering. When these terms are negligible, the distance over which spin information can propagate is predicted to diverge as alpha approaches beta(1). Here we report experimental observation of the emergence of the persistent spin helix in GaAs quantum wells by independently tuning alpha and beta(1). Using transient spin-grating spectroscopy, we find a spin-lifetime enhancement of two orders of magnitude near the symmetry point. Excellent quantitative agreement with theory across a wide range of sample parameters allows us to obtain an absolute measure of all relevant spin-orbit terms, identifying beta(3) as the main SU(2)-violating term in our samples. The tunable suppression of spin relaxation demonstrated in this work is well suited for application to spintronics.

Spinon-holon attraction in the supersymmetric t-J model with 1/r(2) interaction.

We derive the coordinate representation of the one-spinon one-holon wave function for the supersymmetric t-J model with 1/r(2) interaction. This result allows us to show that a spinon and a holon attract each other at short distance. The attraction gets stronger as the size of the system is increased and, in the thermodynamic limit, it is responsible for the square-root singularity in the hole spectral function [Y. Kato, Phys. Rev. Lett. 81, 5402 (1998)].

Spinon attraction in spin- 1/2 antiferromagnetic chains.

We derive the representation of the two-spinon wave function for the Haldane-Shastry model in terms of the spinon coordinates. This result allows us to rigorously analyze spinon interaction and its physical effects. We show that spinons attract one another. The attraction gets stronger as the size of the system is increased and, in the thermodynamic limit, determines the power law with which the susceptibility diverges.