Gapless Topological Phases Research Articles

Quantum phase transition and critical behavior between the gapless topological phases

Quantum phase transition and critical behavior between the gapless topological phases

Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup

Gapless symmetry-protected topological phases and generalized deconfined critical points from gauging a finite subgroup

Multiple topological phase transitions unveiling gapless topological superconductivity in magnet/unconventional superconductor hybrid platform

We propose a theoretical framework for generating gapless topological superconductivity (GTSC) hosting Majorana flat edge modes (MFEMs) in the presence of a two-dimensional (2D) array of magnetic adatoms with noncollinear spin texture deposited on top of an unconventional superconductor. Our observations reveal two distinct topological phase transitions within the emergent Shiba band depending on the exchange coupling strength (J) between magnetic adatom spins and superconducting electrons: the first one designates transition from gapless non-topological to gapless topological phase at lower J, while the second one denotes transition from gapless topological to a trivial gapped superconducting phase at higher J. The gapless topological superconducting phase survives at intermediate values of J, hosting MFEMs. Furthermore, we investigate the nature of the bulk effective pairings, which indicate that GTSC appears due to the interplay between pseudo “s-wave” and pseudo “px+py” types of pairing. Consequently, our study opens a promising avenue for the experimental realization of GTSC in 2D Shiba lattice based on d-wave superconductors as a high-temperature platform.

Characterization of gapless topological quantum phase transition via magnetocaloric effect

It has attracted much attention on the interplay between topology and criticality in topological quantum systems, i.e., gapless topological quantum phase transition (TQPT) on critical lines. It is demonstrated that the magnetocaloric effect (MCE) is able to signal the gapless TQPT in an extended Kitaev chain, which is witnessed by the unusual critical behavior of joint effect of Grüneison ration (GR) and isothermal entropy change. We uncover the conventional TQPT between gapped phases showing high symmetry (HS) and non-HS critical points with self-duality, which is signaled by the finite value of GR as T → 0 and the largest inverse MCE with T-linear relation of isothermal entropy change. Two critical lines intersect at a multicritical point, at which there is a T−1 critical divergence of GR as well as T1/2 critical behavior of isothermal entropy change. It manifests a “weak” gapless TQPT on one critical line, demonstrated by the shifting zero-crossing of GR without the largest inverse MCE. On the other critical line, it shows a “strong” gapless TQPT, which is evidenced by the negative dip of GR together with the largest inverse MCE. Our work provides a novel thermodynamic perspective on gapless TQPT.

Bulk-boundary correspondence for intrinsically gapless symmetry-protected topological phases from group cohomology

Intrinsically gapless symmetry-protected topological phases (igSPT) are gapless systems with SPT edge states with properties that could not arise in a gapped system with the same symmetry and dimensionality. igSPT states arise from gapless systems in which an anomaly in the low-energy (infrared) symmetry group emerges from an extended anomaly-free microscopic (ultraviolet) symmetry. We construct a general framework for constructing lattice models for igSPT phases with emergent anomalies classified by group cohomology, and establish a direct connection between the emergent anomaly, group-extension, and topological edge states by gauging the extending symmetry. In many examples, the edge-state protection has a physically transparent mechanism: the extending UV symmetry operations pump lower-dimensional SPTs onto the igSPT edge, tuning the edge to a (multi)critical point between different SPTs protected by the IR symmetry. In two- and three-dimensional systems, an additional possibility is that the emergent anomaly can be satisfied by an anomalous symmetry-enriched topological order, which we call a quotient-symmetry-enriched topological order (QSET) that is sharply distinguished from the nonanomalous UV SETs by an edge phase transition. We construct exactly solvable lattice models with QSET order.

Coexistence of Dirac points and nodal chains in photonic metacrystal.

Gapless topological phases, i.e. topological semimetals, come in various forms such as Weyl/Dirac semimetals, nodal line/chain semimetals, and surface-node semimetals. However, the coexistence of two or more topological phases in a single system is still rare. Here, we propose the coexistence of Dirac points and nodal chain degeneracies in a judiciously designed photonic metacrystal. The designed metacrystal exhibits nodal line degeneracies lying in perpendicular planes, which are chained together at the Brillouin zone boundary. Interestingly, the Dirac points, which are protected by nonsymmorphic symmetries, are located right at the intersection points of nodal chains. The nontrivial Z2 topology of the Dirac points is revealed by the surface states. The Dirac points and nodal chains are located in a clean frequency range. Our results provide a platform for studying the connection between different topological phases.

Topological semimetal phase and dynamical characterization in a non-Hermitian two-leg ladder model

We show the phase transition between the trivial and the topological semimetal phases in a non-Hermitian two-leg ladder model. We find that, the topological properties of this special model can be captured by a equivalent model with chiral symmetry in which the equivalent model owns the pure imaginary nonreciprocal intercell and long-range hopping simultaneously. Based on the equivalent model, we reveal that the system possesses the gapless topological semimetal phase induced by four exceptional points, which is different from the general topological semimetal phase usually owning two exceptional points. Significantly, we demonstrate that the system has the distinct transmission behaviors within the trivial and topological semimetal phases, leading the phase transition between trivial and topological semimetal phases and the boundary between left and right non-Hermitian skin effects can both be characterized dynamically via the propagation speed of the evolved initial state. Our investigation further supplies the contents on a topological semismetal phase in the non-Hermitian two-leg ladder model and may provide a possible tool to visualize the phase transition of the topological semimetal phase.

Morse theory study on the evolution of nodal lines in PT -symmetric nodal-line semimetals

A nodal-line semimetal is a topological gapless phase containing one-dimensional degeneracies called nodal lines. The nodal lines are deformed by a continuous change of the system such as pressure and they can even change their topology, but it is not systematically understood what kind of changes of topology of nodal lines are possible. In this paper, we classify the events of topology change of nodal lines by the Morse theory and reveal that only three types of topology changes of nodal lines, i.e., creation, reconnection, and annihilation, are possible in the spinless nodal-line semimetal protected by inversion and time-reversal symmetries. They are characterized by an index having the values 0, 1, and 2 for the above three types in the Morse theory. Moreover, we extend our theory to systems with rotational symmetries and mirror symmetry and disclose the possible events of topology change of nodal lines under each symmetry.

Entanglement signature of hinge arcs, Fermi arcs, and crystalline symmetry protection in higher-order Weyl semimetals

The existence of $1/2$ modes in the entanglement spectrum (ES) has been shown to be a powerful quantum-informative signature of boundary states of gapped topological phases of matter, e.g., topological insulators and topological superconductors, where the finite bulk gap allows us to establish a crystal-clear correspondence between $1/2$ modes and boundary states. Here we investigate the recently proposed higher-order Weyl semimetals (HOWSM), where bulk supports gapless higher-order Weyl nodes and boundary supports hinge arcs and Fermi arcs. We find that the aim of unambiguously identifying higher-order boundary states ultimately drives us to make full use of eigen quantities of the entanglement Hamiltonian: ES as well as Schmidt vectors (entanglement wavefunctions, abbr. EWF). We demonstrate that, while both hinge arcs and Fermi arcs contribute to $1/2$ modes, the EWFs corresponding to hinge arcs and Fermi arcs are respectively localized on the virtual hinges and surfaces of the partition. Besides, by means of various symmetry-breaking partitions, we can identify the minimal crystalline symmetries that protect boundary states. Therefore, for gapless topological phases such as HOWSMs, we can combine ES and EWF to universally identify boundary states and potential symmetry requirement. While HOWSMs are prototypical examples of gapless phases, our work sheds light on general theory of entanglement signature in gapless topological phases of matter.

Gapless symmetry-protected topological phase of quantum antiferromagnets on anisotropic triangular strip

We study a three-leg spin-1/2 ladder with geometrically frustrated interleg interactions. We call this model an anisotropic triangular-strip (ATS) model. We numerically and field-theoretically show that its ground state belongs to a gapless symmetry-protected topological (SPT) phase. The numerical approach is based on density-matrix renormalization group analyses of the entanglement entropy and the entanglement spectrum. Whereas the entanglement entropy exhibits a critical behavior, the entanglement spectrum is nontrivially degenerate. These entanglement properties imply that the ground state is a gapless topological phase. We investigate the ATS model using a quantum field theory to support the numerical findings. When the frustrated interchain interaction is deemed a perturbation acting on the three spin chains, the frustrated interchain interaction almost isolates the second chain from the other two chains. However, at the same time, the second chain mediates a ferromagnetic interaction between the first and third chains. Therefore, the ground state of the ATS model is a gapless Tomonaga-Luttinger liquid weakly coupled to a spin-1 Haldane chain with irrelevant interactions. Last but not least, we show that the gapless SPT phase of the ATS model is a symmetry-protected critical phase. We point out that the symmetry protection of criticality is essential in characterization of the gapless SPT phase.

Magnetic field switching of Weyl semimetal associated with quantum criticality in one-dimensional extended Su-Schrieffer-Heeger chain

Graphene/silicon nanoribbons (G/SiNRs) provide low-dimensional platforms to engineer the topological characters for carbon/silicon-based electronics. Herein, we propose a one-dimensional (1D) extended Su-Schrieffer-Heeger (SSH) model to describe the G/SiNRs with a superlattice consisting of two interacting dimers, in which the topological quantum phase transition (QPT) and quantum criticality are investigated. The trivial band insulator transition into a nontrivial topological insulator via a critical Dirac semimetal is revealed, the self-duality of which is manifested by the zero value of Grüneisen ratio (GR) irrelevant of temperature. In a magnetic field, the time-reversal symmetry is broken such that the Weyl semimetal appears and is switched by a critical quadratic contact point semimetal with gap closing, at which the quantum criticality is not self-dual with GR Γ∼±T−1. Furthermore, the quantum critical scaling is done to analyze the quantum criticality, which provides a new clue to detect the QPT. The low-temperature specific heat demonstrates the low-lying excitation of gapped and gapless topological quantum phases explicitly.

Crystalline Weyl semimetal phase in Quantum Spin Hall systems under magnetic fields

We investigate an unconventional topological phase transition that occurs in quantum spin Hall (QSH) systems when applying an external in-plane magnetic field. We show that this transition between QSH and trivial insulator phases is separated by a stable topological gapless phase, which is protected by the combination of particle-hole and reflection symmetries, and thus, we dub it as crystalline Weyl semimetal. We explore the stability of this new phase when particle-hole symmetry breaking terms are present. Especially, we predict a robust unconventional topological phase transition to be visible for materials described by the Kane and Mele model even if particle-hole symmetry is significantly broken.

Lattice model constructions for gapless domain walls between topological phases

Domain walls between different topological phases are one of the most interesting phenomena that reveal the non-trivial bulk properties of topological phases. Very recently, gapped domain walls between different topological phases have been intensively studied. In this paper, we systematically construct a large class of lattice models for gapless domain walls between twisted and untwisted gauge theories with arbitrary finite group $G$. As simple examples, we numerically study several finite groups(including both Abelian and non-Abelian finite group such as $S_3$) in $2$D using the state-of-the-art loop optimization of tensor network renormalization algorithm. We also propose a physical mechanism for understanding the gapless nature of these particular domain wall models. Finally, by taking advantage of the classification and construction of twisted gauge theories using group cohomology theory, we generalize such constructions into arbitrary dimensions, which might provide us a systematical way to understand gapless domain walls and topological quantum phase transitions.

Non-Abelian braiding of Weyl nodes via symmetry-constrained phase transitions

Weyl semimetals are arguably the most paradigmatic form of a gapless topological phase. While the stability of Weyl nodes, as quantified by their topological charge, has been extensively investigated, recent interest has shifted to the manipulation of the location of these Weyl nodes for non-Abelian braiding. To accomplish this braiding it is necessary to drive significant Weyl node motion using realistic experimental parameter changes. We show that a family of phase transitions characterized by certain symmetry constraints impose that the Weyl nodes have to reorganise by a large amount, shifting from one high symmetry plane to another. Additionally, for a subset of pairs of nodes with nontrivial Euler class topology, this reorganization can only occur through a braiding process with adjacent nodes. As a result, the Weyl nodes are forced to move a large distance across the Brillouin zone and to braid, all driven by small temperature changes, a process we illustrate with Cd$_2$Re$_2$O$_7$. Our work opens up routes to readily manipulate Weyl nodes using only slight external parameter changes, paving the way for the practical realization of reciprocal space braiding.

Gapless Topological Phases and Symmetry-Enriched Quantum Criticality

We introduce topological invariants for gapless systems and study the associated boundary phenomena. More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants, establishing the notion of symmetry-enriched quantum criticality. The charges of nonlocal scaling operators, or more generally of symmetry defects, are topological and imply the presence of localized edge modes. We primarily focus on the $1+1d$ case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors. An example of the former is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase. An example of the latter arises between the gapped Ising and cluster phases: this symmetry-enriched Ising CFT has an edge mode with finite-size splitting $\sim 1/L^{14}$. In addition to such new cases, our formalism unifies various examples previously studied in the literature. Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs. This raises the question of classification, to which we give a partial answer -- including a complete characterization of symmetry-enriched $1+1d$ Ising CFTs. Non-trivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched $2+1d$ CFT without gapped sectors.

Intrinsically gapless topological phases

Topology in quantum matter is typically associated with gapped phases. For example, in symmetry protected topological (SPT) phases, the bulk energy gap localizes edge modes near the boundary. In this work we identify a new mechanism that leads to topological phases which are not only gapless but where the absence of a gap is essential. These `intrinsically gapless SPT phases' have no gapped counterpart and are hence also distinct from recently discovered examples of gapless SPT phases. The essential ingredient of these phases is that on-site symmetries act in an anomalous fashion at low energies. Intrinsically gapless SPT phases are found to display several unique properties including (i) protected edge modes that are impossible to realize in a gapped system with the same symmetries, (ii) string order parameters that are likewise forbidden in gapped phases, and (iii) constraints on the phase diagram obtained upon perturbing the phase. We verify predictions of the general theory in a specific realization protected by $\mathbb Z_4$ symmetry, the one dimensional Ising-Hubbard chain, using both numerical simulations and effective field theory. We also discuss extensions to higher dimensions and possible experimental realizations.

Linking structures of doubly charged nodal surfaces in centrosymmetric superconductors

In topological semimetals and nodal superconductors, band crossings between occupied and unoccupied bands form stable nodal points/lines/surfaces carrying quantized topological charges. In particular, in centrosymmetric systems, some nodal structures at the Fermi energy $E_F$ carry two distinct topological charges, and thus they are called doubly charged nodes. Here we show that doubly charged nodal surfaces of centrosymmetric superconductors in three-dimensions always develop peculiar linking structures with nodal points or lines formed between occupied bands below $E_F$. Such linking structures can naturally explain the inherent relationship between the charge of the node below $E_F$ and the two charges of the nodal surfaces at $E_F$. Based on the Altland-Zirnbauer (AZ)-type ten-fold classification of nodes with additional inversion $\mathcal{I}$ symmetry, which is called the AZ$+\mathcal{I}$ classification, we provide the complete list of linking structures of doubly charged nodes in centrosymmetric systems. The linking structures of doubly charged nodes clearly demonstrate that not only the local band structure around the node but also the global band structure play a critical role in characterizing gapless topological phases.

Realization of an ideal Weyl semimetal band in a quantum gas with 3D spin-orbit coupling.

Weyl semimetals are three-dimensional (3D) gapless topological phases with Weyl cones in the bulk band. According to lattice theory, Weyl cones must come in pairs, with the minimum number of cones being two. A semimetal with only two Weyl cones is an ideal Weyl semimetal (IWSM). Here we report the experimental realization of an IWSM band by engineering 3D spin-orbit coupling for ultracold atoms. The topological Weyl points are clearly measured via the virtual slicing imaging technique in equilibrium and are further resolved in the quench dynamics. The realization of an IWSM band opens an avenue to investigate various exotic phenomena that are difficult to access in solids.

Topological and symmetry-enriched random quantum critical points

We study how symmetry can enrich strong-randomness quantum critical points and phases, and lead to robust topological edge modes coexisting with critical bulk fluctuations. These are the disordered analogues of gapless topological phases. Using real-space and density matrix renormalization group approaches, we analyze the boundary and bulk critical behavior of such symmetry-enriched random quantum spin chains. We uncover a new class of symmetry-enriched infinite randomness fixed points: while local bulk properties are indistinguishable from conventional random singlet phases, nonlocal observables and boundary critical behavior are controlled by a different renormalization group fixed point. We also illustrate how such new quantum critical points emerge naturally in Floquet systems.

Higher-order topological Anderson insulators

We study disorder effects in a two-dimensional system with chiral symmetry and find that disorder can induce a quadrupole topological insulating phase (a higher-order topological phase with quadrupole moments) from a topologically trivial phase. Their topological properties manifest in a topological invariant defined based on effective boundary Hamiltonians, the quadrupole moment, and zero-energy corner modes. We find gapped and gapless topological phases and a Griffiths regime. In the gapless topological phase, all the states are localized, while in the Griffiths regime, the states at zero energy become multifractal. We further apply the self-consistent Born approximation to show that the induced topological phase arises from disorder renormalized masses. We finally introduce a practical experimental scheme with topoelectrical circuits where the predicted topological phenomena can be observed by impedance measurements. Our work opens the door to studying higher-order topological Anderson insulators and their localization properties.