Gapless Topological Phases Research Articles

Effective field theory for fractional quantum Hall systems nearν=5/2

We propose an effective field theory (EFT) of fractional quantum Hall systems near the filling fraction $\nu=5/2$ that flows to pertinent IR candidate phases, including non-abelian Pfaffian, anti-Pfaffian, and particle-hole Pfaffian states (Pf, APf, and PHPf). Our EFT has a 2+1$d$ O(2)$_{2,L}$ Chern-Simons gauge theory coupled to four Majorana fermions by a discrete charge conjugation gauge field, with Gross-Neveu-Yukawa-Higgs terms. Including deformations via a Higgs condensate and fermion mass terms, we can map out a phase diagram with tunable parameters, reproducing the prediction of the recently-proposed percolation picture and its gapless topological quantum phase transitions. Our EFT captures known features of both gapless and gapped sectors of time-reversal-breaking domain walls between Pf and APf phases. Moreover, we find that Pf$\mid$APf domain walls have higher tension than domain walls in the PHPf phase. Then the former, if formed, may transition to the energetically-favored PHPf domain walls; this could, in turn, help further induce a bulk transition to PHPf.

Higher-order topological Dirac superconductors

We introduce higher-order topological Dirac superconductor (HOTDSC) as a new gapless topological phase of matter in three dimensions, which extends the notion of Dirac phase to a higher-order topological version. Topologically distinct from the traditional topological superconductors and known Dirac superconductors, a HOTDSC features helical Majorana hinge modes between adjacent surfaces, which are direct consequences of the symmetry-protected higher-order band topology manifesting in the system. Specifically, we show that rotational, spatial inversion, and time-reversal symmetries together protect the coexistence of bulk Dirac nodes and hinge Majorana modes in a seamless way. We define a set of topological indices that fully characterizes the HOTDSC. We further show that a practical way to realize the HOTDSC phase is to introduce unconventional odd-parity pairing to a three-dimensional Dirac semimetal while preserving the necessary symmetries. As a concrete demonstration of our idea, we construct a corresponding minimal lattice model for HOTDSC obeying the symmetry constraints. Our model exhibits the expected topological invariants in the bulk and the defining spectroscopic features on an open geometry, as we explicitly verify both analytically and numerically. Remarkably, the HOTDSC phase offers an example of a higher-order topological quantum critical point, which enables realizations of various higher-order topological phases under different symmetry-breaking patterns. In particular, by breaking the inversion symmetry of a HOTDSC, we arrive at a higher-order Weyl superconductor, which is yet another new gapless topological state that exhibits hybrid higher-order topology.

Induced half-metallicity and gapless chiral topological superconductivity in the CrI3−Pb interface

We study a two-dimensional heterostructure comprised of a monolayer of the magnetic insulator chromium triiodide (CrI$_3$) on a superconducting lead (Pb) substrate. Through first-principles computation and a tight-binding model, we demonstrate that charge transfer from the Pb substrate dopes the CrI$_3$ into an effective half-metal, allowing for the onset of a gapless topological superconductivity phase via the proximity effect. This phase, in which there exists a superconducting gap only in part of the Fermi surface, is shown to occur generically in 2D half-metal-superconductor heterostructures which lack two-fold in-plane rotational symmetry. However, a sufficiently large proximity-induced pairing amplitude can bring such a system into a fully-gapped topological superconducting phase. As such, these results are expected to better define the optimal 2D component materials for future proposed TSC heterostructures.

Topological Fermi-arc surface resonances in bcc iron

The topological classification of matter has been extended to include semimetallic phases characterized by the presence of topologically protected band degeneracies. In Weyl semimetals, the foundational gapless topological phase, chiral degeneracies are isolated near the Fermi level and give rise to the Fermi-arc surface states. However, it is now recognized that chiral degeneracies are ubiquitous in the band structures of systems with broken spatial inversion ($\mathcal{P}$) or time-reversal ($\mathcal{T}$) symmetry. This leads to a broadly defined notion of topological metals, which implies the presence of disconnected Fermi surface sheets characterized by non-zero Chern numbers inherited from the enclosed chiral degeneracies. Here, we address the possibility of experimentally observing surface-related signatures of chiral degeneracies in metals. As a representative system we choose bcc iron, a well-studied archetypal ferromagnetic metal with two nontrivial electron pockets. We find that the (110) surface presents arc-like resonances attached to the topologically nontrivial electron pockets. These Fermi-arc resonances are due to two different chiral degeneracies, a type-I elementary Weyl point and a type-II composite (Chern numbers $\pm 2$) Weyl point, located at slightly different energies close to the Fermi level. We further show that these surface resonances can be controlled by changing the orientation of magnetization, eventually being eliminated following a topological phase transition. Our study thus shows that the intricate Fermi-arc features can be observed in materials as simple as ferromagnetic iron, and are possibly very common in polar and magnetic materials broadly speaking. Our study also provides methodological guidelines to identifying Fermi-arc surface states and resonances, establishing their topological origin and designing control protocols.

Disentangling quantum matter with measurements

Measurements destroy entanglement. Building on ideas used to study `quantum disentangled liquids', we explore the use of this effect to characterize states of matter. We focus on systems with multiple components, such as charge and spin in a Hubbard model, or local moments and conduction electrons in a Kondo lattice model. In such systems, measurements of (a subset of) one of the components can leave behind a quantum state of the other that is easy to understand, for example in terms of scaling of entanglement entropy of subregions. We bound the outcome of this protocol, for any choice of measurement, in terms of more standard information-theoretic quantities. We apply this quantum disentangling protocol to several problems of physical interest, including gapless topological phases, heavy fermions, and scar states in Hubbard model.

Fractional Quantum Hall Effect in Weyl Semimetals.

Weyl semimetal may be thought of as a gapless topological phase protected by the chiral anomaly, where the symmetries involved in the anomaly are the U(1) charge conservation and the crystal translational symmetry. The absence of a band gap in a weakly interacting Weyl semimetal is mandated by the electronic structure topology and is guaranteed as long as the symmetries and the anomaly are intact. The nontrivial topology also manifests in the Fermi arc surface states and topological response, in particular taking the form of an anomalous Hall effect in magnetic Weyl semimetals, whose magnitude is only determined by the location of the Weyl nodes in the Brillouin zone. Here we consider the situation when the interactions are not weak and ask whether it is possible to open a gap in a magnetic Weyl semimetal while preserving its nontrivial electronic structure topology along with the translational and the charge conservation symmetries. Surprisingly, the answer turns out to be yes. The resulting topologically ordered state provides a nontrivial realization of the fractional quantum Hall effect in three spatial dimensions in the absence of an external magnetic field, which cannot be viewed as a stack of two dimensional states. Our state contains loop excitations with nontrivial braiding statistics when linked with lattice dislocations.

Manifestation of a topological gapless phase in a two-dimensional chiral symmetric system through Loschmidt echo

Unlike the edge state of a topological insulator where its energy level lives in the bulk energy gap, the edge state of a topological semimetal hides in the bulk spectrum and is difficult to be identified by the energy. We investigate the sensitivity of bulk and edge states of the gapless phase for a topological semimetal to the disordered perturbation via a concrete two-dimensional chiral symmetric lattice model. The topological gapless phase is characterized by two opposite vortices in the momentum space and nonzero winding numbers, which correspond to the edge flatband when the open boundary condition is applied. For this system, numerical results reveal that a distinguishing feature is that the robustness of the edge states against weak disorder and the flatband edge modes remain locked at zero energy in the presence of weak chiral-symmetry-preserving disorder. We employ the Loschmidt echo (LE) for both bulk and edge states to study the dynamic effect of disordered perturbation. We show that, for an initial bulk state, the LE decays exponentially, whereas it converges to a constant for an initial edge state in the presence of weak disorder. Furthermore, the convergent LE can be utilized to identify the positions of vortices as well as the phase diagram. We discuss the realization of such dynamic investigations in a topological photonic system.

Observation of a topological nodal surface and its surface-state arcs in an artificial acoustic crystal

Three-dimensional (3D) gapless topological phases can be classified by the dimensionality of the band degeneracies, including zero-dimensional (0D) nodal points, one-dimensional (1D) nodal lines, and two-dimensional (2D) nodal surfaces. Both nodal points and nodal lines have been realized recently in photonics and acoustics. However, a nodal surface has never been observed in any classical-wave system. Here, we report on the experimental observation of a twofold symmetry-enforced nodal surface in a 3D chiral acoustic crystal. In particular, the demonstrated nodal surface carries a topological charge of 2, constituting the first realization of a higher-dimensional topologically-charged band degeneracy. Using direct acoustic field measurements, we observe the projected nodal surface and its Fermi-arc-like surface states and demonstrate topologically-induced robustness of the surface states against disorders. This discovery of a higher-dimensional topologically-charged band degeneracy paves the way toward further explorations of the physics and applications of new topological semimetal phases.

Multifractality at the Weyl-semimetal–diffusive-metal transition for generic disorder

A Weyl semimetal is a three dimensional topological gapless phase. In the presence of strong enough disorder it undergoes a quantum transition towards a diffusive metal phase whose universality class depends on the range of disorder correlations. Similar to other quantum transitions driven by disorder, the critical wave functions at the semimetal-diffusive metal transition exhibit multifractality. Using renormalization group methods we study the corresponding multifractal spectrum as a function of the range of disorder correlations for generic disorder including random scalar and vector potentials. We also discuss the relation between the geometric fluctuations of critical wave functions and the broad distribution of the local density of states (DOS) at the transition. We derive a new scaling relation for the typical local DOS and argue that it holds for other disorder-driven transitions in which both the average and typical local DOS vanish on one side of the transition. As an illustration we apply it to the recently discussed unconventional quantum transition in disordered semiconductors with power-law dispersion relation near the band edge.

Majorana flat band edge modes of topological gapless phase in 2D Kitaev square lattice

We study a Kitaev model on a square lattice, which describes topologically trivial superconductor when gap opens, while supports topological gapless phase when gap closes. The degeneracy points are characterized by two vortices in momentum space, with opposite winding numbers. We show rigorously that the topological gapless phase always hosts a partial Majorana flat band edge modes in a ribbon geometry, although such a single band model has zero Chern number as a topologically trivial superconductor. The flat band disappears when the gapless phase becomes topologically trivial, associating with the mergence of two vortices. Numerical simulation indicates that the flat band is robust against the disorder.

Helical metals and insulators: Sheet singularity of the inflated Berry monopole

We study the new phases of interacting Dirac matter that host novel Berry signatures. We predict a topological Lifshitz phase transition caused by the changes of a Dirac cone intersection from a semimetalic phase to helical insulating or metallic phases. These helical phases provide the examples of gapless topological phase where spectral gap is not required for a topological protection. To realize nodal helical phases one would need to consider isotropic infinite-range inter-particle interaction. This interaction could emerge because of a momentum conserving scattering of electron from a bosonic mode. For repulsive/attractive interactions in density/pseudospin channel system undergoes a transition to helical insulator phase. For an attractive density-density interaction, a new metallic phase forms that hosts {\it nodal circle} and {\it nodal sphere} in two and three dimensions, respectively. A {\it sheet singularity} of Berry curvature is highlighted as a peculiar feature of the nodal sphere phase in 3D and represent the extension of the Berry monopole singularities into inflated monopole. To illustrate the properties of these helical phases we investigate Landau levels in both metallic and insulating phases. Our study provides an extension of the paradigm in the interacting Dirac matter and makes an interesting connection to inflated topological singularities in cosmology.

Intermediate gapless phase and topological phase transition of the Kitaev model in a uniform magnetic field

We study the Kitaev spin liquid (KSL) in a [001] magnetic field employing the mean field theory (MFT/MF) in the Majorana fermion representation. The MF Hamiltonian of the system has the Bogoliubov de-Gennes (BdG) form of a 2D Weyl superconductor. We discover a robust gapless regime in intermediate magnetic field for both gapless and gapped AFM KSL with Jx = Jy before the system is polarized in high magnetic field. A topological phase transition (PT) connecting two gapless phases with nodal lines takes place at a critical magnetic field hc1 in this regime. While the nodal lines at hc1 is protected by the mirror symmetry with Jx = Jy and disappear at Jx 6= Jy with a gap opening near the critical field, the nodal points at E = 0 can exist at intermediate fields even without mirror symmetry. We reveal that the phase evolution of the KSL in the magnetic field is driven by the competition between the magnetic field and the particle-hole (p-h) asymmetry of the normal state of the BdG Hamiltonian, which results in the robust intermediate gapless phase for AFM KSL. For FM KSL, there is no intermediate PT before polarization. The above phase diagrams are confirmed by dynamical MFT (DMFT) results.

Rashba sandwiches with topological superconducting phases

We introduce a versatile heterostructure harboring various topological superconducting phases characterized by the presence of helical, chiral, or unidirectional edge states. Changing parameters, such as an effective Zeeman field or chemical potential, one can tune between these three topological phases in the same setup. Our model relies only on conventional non-topological ingredients. The bilayer setup consists of an s-wave superconductor sandwiched between two two-dimensional electron gas layers with strong Rashba spin-orbit interaction. The interplay between two different pairing mechanisms, proximity induced direct and crossed Andreev superconducting pairings, gives rise to multiple topological phases. In particular, helical edge states occur if crossed Andreev superconducting pairing is dominant. In addition, an in-plane Zeeman field leads to a 2D gapless topological phase with unidirectional edge states, which were previously predicted to exist only in non-centrosymmetric superconductors. If the Zeeman field is tilted out of the plane, the system is in a topological phase hosting chiral edge states.

Symmetry-protected gapless Z2 spin liquids

Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical ${\mathbb{Z}}_{2}$ gauge fields. Requiring $\text{U}(1)$ spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-$\frac{1}{2}$ systems on square, triangular, and kagome lattices, we show that all gapped symmetric ${\mathbb{Z}}_{2}$ spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped ${\mathbb{Z}}_{2}$ spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.

Topological gapless phase in Kitaev model on square lattice

We study the topological feature of gapless states in the fermionic Kitaev model on a square lattice. There are two types of gapless states which are topologically trivial and nontrivial. We show that the topological gapless phase lives in a wide two-dimensional parameter region and are characterized by two vertices of an auxiliary vector field de-fined in the two-dimensional momentum space, with opposite winding numbers. The isolated band touching points, as the topological defects of the field, move, emerge, and disappear as the parameters vary. The band gap starts to open only at the merg-ing points, associated with topologically trivial gapless states. The symmetry protect-ing the topological gapless phase and the robustness under perturbations are also discussed.

Non-necessity of band inversion process in two-dimensional topological insulators for bulk gapless states and topological phase transitions

In commonly employed models for 2D topological insulators, bulk gapless states are well known to form at the band inversion points where the degeneracy of the states is protected by symmetries. It is thus sometimes quite tempting to consider this feature, the occurrence of gapless states, a result of the band inversion process under protection of the symmetries. Similarly, the band inversion process might even be perceived as necessary to induce 2D topological phase transitions. To clarify these misleading perspectives, we propose a simple model with a flexible Chern number to demonstrate that the bulk gapless states emerge at the phase boundary of topological phase transitions, despite the absence of band inversion process. Furthermore, the bulk gapless states do not need to occur at the special $k$-points protected by symmetries. Given the significance of these fundamental \textit{conceptual} issues and their wide-spread influence, our clarification should generate strong general interests and significant impacts. Furthermore, the simplicity and flexibility of our general model with an arbitrary Chern number should prove useful in a wide range of future studies of topological states of matter.

Tunable topological phases with fermionic atoms in a one-dimensional flux lattice

We present a simple scheme for implementing a one-dimensional (1D) magnetic-flux lattice of ultracold fermionic spin-$1/2$ atoms. The resulting tight-binding model supports gapped and gapless topological phases, and chiral currents for Meissner and vortex phases. Its single-particle spectra exhibit topological flat bands at small flux, and the flatness sensitively depends on hopping strength. An effective $p$-wave interaction arises in a $s$-wave paired superfluid. Treating atomic internal states as forming a synthetic dimension and balancing the interplay of magnetic flux and Zeeman field, our model describes a tunable topological Fermi superfluid, which paves the way towards experimental explorations of non-Abelian topological matter in 1D atomic quantum gases.

Nonlinear Dirac cones

Physics arising from two-dimensional~(2D) Dirac cones has been a topic of great theoretical and experimental interest to studies of gapless topological phases and to simulations of relativistic systems. Such $2$D Dirac cones are often characterized by a $\pi$ Berry phase and are destroyed by a perturbative mass term. By considering mean-field nonlinearity in a minimal two-band Chern insulator model, we obtain a novel type of Dirac cones that are robust to local perturbations without symmetry restrictions. Due to a different pseudo-spin texture, the Berry phase of the Dirac cone is no longer quantized in $\pi$, and can be continuously tuned as an order parameter. Furthermore, in an Aharonov-Bohm~(AB) interference setup to detect such Dirac cones, the adiabatic AB phase is found to be $\pi$ both theoretically and computationally, offering an observable topological invariant and a fascinating example where the Berry phase and AB phase are fundamentally different. We hence discover a nonlinearity-induced quantum phase transition from a known topological insulating phase to an unusual gapless topological phase.

Evidence for magnetic Weyl fermions in a correlated metal.

Weyl fermions have been observed as three-dimensional, gapless topological excitations in weakly correlated, inversion-symmetry-breaking semimetals. However, their realization in spontaneously time-reversal-symmetry-breaking phases of strongly correlated materials has so far remained hypothetical. Here, we report experimental evidence for magnetic Weyl fermions in Mn3Sn, a non-collinear antiferromagnet that exhibits a large anomalous Hall effect, even at room temperature. Detailed comparison between angle-resolved photoemission spectroscopy (ARPES) measurements and density functional theory (DFT) calculations reveals significant bandwidth renormalization and damping effects due to the strong correlation among Mn 3d electrons. Magnetotransport measurements provide strong evidence for the chiral anomaly of Weyl fermions-namely, the emergence of positive magnetoconductance only in the presence of parallel electric and magnetic fields. Since weak magnetic fields (approximately 10 mT) are adequate to control the distribution of Weyl points and the large fictitious fields (equivalent to approximately a few hundred T) produced by them in momentum space, our discovery lays the foundation for a new field of science and technology involving the magnetic Weyl excitations of strongly correlated electron systems such as Mn3Sn.

Interacting nodal-line semimetal: Proximity effect and spontaneous symmetry breaking

The nodal-line semimetal stands out as a paradigmatic representative of a gapless topological phase of matter that supports linearly dispersing quasiparticles around an isolated closed loop in the bulk and accommodates drumhead shaped surface states. The author demonstrates an intriguing possibility of realizing either charge or spin order on the surface of this system for arbitrarily weak Coulomb repulsion. When onsite Hubbard repulsion dominates, each surface becomes ferromagnetic with its moment pointing in opposite directions on complementary faces, which yields a global antiferromagnet. As shown numerically, surface ordering via proximity effect can in turn induce charge or spin order in the bulk at weak coupling, making the proposed scenario for spontaneous symmetry breaking relevant for materials, such as Ca${}_{3}$P${}_{2}$ and PbTaSe${}_{2}$, specifically in a thin-film geometry.