Topological Phase Boundary Research Articles

Massive Dirac Fermion Behavior in a Low Bandgap Graphene Nanoribbon Near a Topological Phase Boundary.

Graphene nanoribbons (GNRs) have attracted much interest due to their largely modifiable electronic properties. Manifestation of these properties requires atomically precise GNRs which can be achieved through a bottom-up synthesis approach. This has recently been applied to the synthesis of width-modulated GNRs hosting topological electronic quantum phases, with valence electronic properties that are well captured by the Su-Schrieffer-Heeger (SSH) model describing a 1D chain of interacting dimers. Here, ultralow bandgap GNRs with charge carriers behaving as massive Dirac fermions can be realized when their valence electrons represent an SSH chain close to the topological phase boundary, i.e., when the intra- and interdimer coupling become approximately equal. Such a system has been achieved via on-surface synthesis based on readily available pyrene-based precursors and the resulting GNRs are characterized by scanning probe methods. The pyrene-based GNRs (pGNRs) can be processed under ambient conditions and incorporated as the active material in a field effect transistor. A quasi-metallic transport behavior is observed at room temperature, whereas at low temperature, the pGNRs behave as quantum dots showing single-electron tunneling and Coulomb blockade. This study may enable the realization of devices based on carbon nanomaterials with exotic quantum properties.

Unraveling the topology of ZrTe5 by changing temperature

We study the effects of temperature on the topological nature of ZrTe$_5$, which sits near the phase boundary between strong and weak topological insulating orders. Using first-principles calculations, we show that the band gap and the topological indices of ZrTe$_5$ are extremely sensitive to thermal expansion, and that the electron-phonon interaction accounts for more than a third of the total temperature-dependent effects in ZrTe$_5$. We find that the temperature dependence of the band gap has an opposite sign in the strong and weak topological insulator phases. Based on this insight, we propose a robust and unambiguous method to determine the topological nature of any given sample of ZrTe$_5$: if the band gap decreases with temperature it is a strong topological insulator, and if it increases with temperature it is a weak topological insulator. An analogous strategy is expected to be generally applicable to other materials and to become particularly important in the vicinity of topological phase boundaries where other methods provide ambiguous results.

Local engineering of topological phase in monolayer MoS2

Monolayer transition metal dichalcogenides (TMDCs) with the 1T′ structure are a new class of large-gap two-dimensional (2D) topological insulators, hosting topologically protected conduction channels on the edges. However, the 1T′ phase is metastable compared to the 2H phase for most of 2D TMDCs, among which the 1T′ phase is least favored in monolayer MoS2. Here we report a clean and controllable technique to locally induce nanometer-sized 1T′ phase in monolayer 2H-MoS2 via a weak Argon-plasma treatment, resulting in topological phase boundaries of high density. We found that the stabilization of 1T′ phase arises from the concerted effects of S vacancies and the tensile strain. Scanning tunneling spectroscopy (STS) clearly reveals a spin-orbit band gap (~60 meV) and topologically protected in-gap states residing at the 1T′-2H phase boundary, which are corroborated by density-functional theory (DFT) calculations. The strategy developed in this work can be generalized to a large variety of TMDCs materials, with potentials to realize scalable electronics and spintronics with low dissipation.

Braiding and gapped boundaries in fracton topological phases

We study gapped boundaries of Abelian type-I fracton systems in three spatial dimensions. Using the X-cube model as our motivating example, we give a conjecture, with partial proof, of the conditions for a boundary to be gapped. In order to state our conjecture, we use a precise definition of fracton braiding and show that bulk braiding of fractons has several features that make it \textit{insufficient} to classify gapped boundaries. Most notable among these is that bulk braiding is sensitive to geometry and is "nonreciprocal," that is, braiding an excitation $a$ around $b$ need not yield the same phase as braiding $b$ around $a$. Instead, we define fractonic "boundary braiding," which resolves these difficulties in the presence of a boundary. We then conjecture that a boundary of an Abelian fracton system is gapped if and only if a "boundary Lagrangian subgroup" of excitations is condensed at the boundary, this is a generalization of the condition for a gapped boundary in two spatial dimensions, but it relies on boundary braiding instead of bulk braiding. We also discuss the distinctness of gapped boundaries and transitions between different topological orders on gapped boundaries.

Interacting Hofstadter Interface.

Two-dimensional topological insulators possess conducting edge states at their boundary while being insulating in the bulk. We investigate the edge state emergent at a smooth topological phase boundary of interacting fermions within a full real-space analysis of the time-reversal invariant Hofstadter-Hubbard model. We characterize the localization of the edge state and the topological phase boundary by means of the local compressibility, the spectral density, a generalized local spin Chern marker as well as the Hall response and find good agreement between all these quantities. Computing the edge state spectra at the interface we observe robustness of the edge state against fermionic two-body interactions and conclude that interactions only shift its position. Hence the bulk-boundary correspondence for the interacting system is confirmed. Since experimental probing of edge states remains a challenge in ultracold atom setups, we propose the detection of the local compressibility by measuring correlations with a quantum gas microscope.

Higher winding number in a nonunitary photonic quantum walk

Topological matter exhibits exotic properties yet phases characterized by large topological invariants are difficult to implement, despite rapid experimental progress. A promising route toward higher topological invariants is via engineered Floquet systems, particularly in photonics, where flexible control holds the potential of extending the study of conventional topological matter to novel regimes. Here we implement a one-dimensional photonic quantum walk to explore large winding numbers. By introducing partial measurements and hence loss into the system, we detect winding numbers of three and four in multi-step non-unitary quantum walks, which agree well with theoretical predictions. Moreover, by probing statistical moments of the walker, we identify locations of topological phase transitions in the system, and reveal the breaking of pseudo-unitary near topological phase boundaries. As the winding numbers are associated with non-unitary time evolution, our investigation enriches understanding of topological phenomena in non-unitary settings.

Tunable magnetic topological insulating phases in monolayer CrI3

We present results indicating that Chern insulator states can be achieved in the recently synthesized pristine chromium triiodide (${\mathrm{CrI}}_{3}$) by either electron or hole doping. Our first-principles density-functional-theory calculations confirmed that monolayer ${\mathrm{CrI}}_{3}$ show nontrivial Chern number $C$ in both the valence and conduction bands. By introducing on-site Coulomb interaction or epitaxial strain, the doped ${\mathrm{CrI}}_{3}$ exhibit a series of topological quantum phase transitions between multiple Chern insulator phases as well as semimetal-to-insulator transitions. We show that the covalency of Cr $d$-I $p$ bands controlled by the on-site Coulomb interaction $U$ and strain is one of the key ingredients determining the topological phase and semimetal-insulator phase boundary.

Universal quantum criticality in static and Floquet-Majorana chains

The topological phase transitions in static and periodically driven Kitaev chains are investigated by means of a renormalization group (RG) approach. These transitions, across which the numbers of static or Floquet Majorana edge modes change, are accompanied by divergences of the relevant Berry connections. These divergences at certain high symmetry points in momentum space form the basis of the RG approach, through which topological phase boundaries are identified as a function of system parameters. We also introduce several aspects to characterize the quantum criticality of the topological phase transitions in both static and Floquet systems: a correlation function that measures the overlap of Majorana-Wannier functions, the decay length of the Majorana edge mode and a scaling law relating the critical exponents. These indicate a common universal critical behavior for topological phase transitions, in both static and periodically driven chains. For the latter, the RG flows additionally display intriguing features related to gap closures at non-high symmetry points due to momentarily frozen dynamics.

Topological phase and chiral edge states of bilayer graphene with staggered sublattice potentials and Hubbard interaction

Gated heterostructures containing bilayer graphene with staggered sublattice potentials are investigated by tight binding model with Rashba spin-orbital coupling and Hubbard interaction. The topological phase diagrams depend on the combinations of substrates and the Hubbard interaction. The presence of the staggered sublattice potential favor the topological phase transition with small Rashba spin-orbital coupling strength. The presence of the Hubbard interaction modified the topological phase boundaries, increasing the minimal spin-orbital coupling strength for topological phase transition. A phase space of topological semi-metal with indirect band gap is identified in the non-interacting systems. For the bilayer graphene with different staggered sublattice potentials in the two layers, the conditions for the zigzag nanoribbons to host edge polarized chiral edge states are discussed. The conditions require moderate or vanishing Rashba spin-orbital coupling strength, as well as proper range of the gate voltage. The conditions for the systems with and without the Hubbard interaction are compared. The edge polarization can be controlled by the gate voltage.

Correction to: Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

There were two errors in the original publication. First, the term B K in Eq. (2.20) was not well-defined in the case of non-normal subgroups K.

Robustness of Majorana edge modes and topological order: Exact results for the symmetric interacting Kitaev chain with disorder

We investigate the robustness of Majorana edge modes under disorder and interactions. We exploit a recently found mapping of the interacting Kitaev chain in the symmetric region ($\mu = 0$, $t = \Delta$) to free fermions. Extending the exact solution to the disordered case allows us to calculate analytically the topological phase boundary for all interaction and disorder strengths, which has been thought to be only accessible numerically. We discover a regime in which moderate disorder in the interaction matrix elements enhances topological order well into the strongly interacting regime $U > t$. We also derive the explicit form of the many-body Majorana edge wave function revealing how it is dressed by many-particle fluctuations from interactions. The qualitative features of our analytical results are valid beyond the fine-tuned integrable point as expected from the robustness of topological order and as corroborated here by an exact diagonalization study of small systems.

Defects between gapped boundaries in two-dimensional topological phases of matter

Defects between gapped boundaries provide a possible physical realization of projective non-abelian braid statistics. A notable example is the projective Majorana/parafermion braid statistics of boundary defects in fractional quantum Hall/topological insulator and superconductor heterostructures. In this paper, we develop general theories to analyze the topological properties and projective braiding of boundary defects of topological phases of matter in two spatial dimensions. We present commuting Hamiltonians to realize defects between gapped boundaries in any $(2+1)D$ untwisted Dijkgraaf-Witten theory, and use these to describe their topological properties such as their quantum dimension. By modeling the algebraic structure of boundary defects through multi-fusion categories, we establish a bulk-edge correspondence between certain boundary defects and symmetry defects in the bulk. Even though it is not clear how to physically braid the defects, this correspondence elucidates the projective braid statistics for many classes of boundary defects, both amongst themselves and with bulk anyons. Specifically, three such classes of importance to condensed matter physics/topological quantum computation are studied in detail: (1) A boundary defect version of Majorana and parafermion zero modes, (2) a similar version of genons in bilayer theories, and (3) boundary defects in $\mathfrak{D}(S_3)$.

Phase vortices of the quenched Haldane model

Using the recently developed Bloch-state tomography technique, the quasimomentum $\mathbf{k}$-dependent Bloch states ${[sin\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right),\phantom{\rule{0.28em}{0ex}}\ensuremath{-}cos\left({\ensuremath{\theta}}_{\mathbf{k}}/2\right){e}^{i{\ensuremath{\phi}}_{\mathbf{k}}}]}^{T}$ of a two-band tight-binding model with two sublattices can be mapped out. We show that if we prepare the initial Bloch state as the lower-band eigenstate of a topologically trivial Haldane Hamiltonian ${H}_{i}$ and then quench the Haldane Hamiltonian to ${H}_{f}$, the time-dependent azimuthal phase ${\ensuremath{\phi}}_{\mathbf{k}}(t)$ supports two types of vortices. The first type of vortices is static, with the corresponding Bloch vectors pointing to the north pole (${\ensuremath{\theta}}_{\mathbf{k}}=0$). The second type of vortices is dynamical, with the corresponding Bloch vectors pointing to the south pole (${\ensuremath{\theta}}_{\mathbf{k}}=\ensuremath{\pi}$). In the $({k}_{x},{k}_{y},t)$ space, the linking number between the trajectories of these two types of vortices exactly equals the Chern number of the lower band of ${H}_{f}$, which provides an alternative method to directly map out the topological phase boundaries of the Haldane model.

Topological superradiant state in Fermi gases with cavity induced spin–orbit coupling

Coherently driven atomic gases inside optical cavities hold great promise for generating rich dynamics and exotic states of matter. It was shown recently that an exotic topological superradiant state exists in a two-component degenerate Fermi gas coupled to a cavity, where local order parameters coexist with global topological invariants. In this work, we characterize in detail various properties of this exotic state, focusing on the feedback interactions between the atoms and the cavity field. In particular, we demonstrate that cavity-induced interband coupling plays a crucial role in inducing the topological phase transition between the conventional and topological superradiant states. We analyze the interesting signatures in the cavity field left by the closing and reopening of the atomic bulk gap across the topological phase boundary and discuss the robustness of the topological superradiant state by investigating the steady-state phase diagram under various conditions. Furthermore, we consider the interaction effect and discuss the interplay between the pairing order in atomic ensembles and the superradiance of the cavity mode. Our work provides many valuable insights into the unique cavity--atom hybrid system under study and is helpful for future experimental exploration of the topological superradiant state.

Hamiltonian and Algebraic Theories of Gapped Boundaries in Topological Phases of Matter

We present an exactly solvable lattice Hamiltonian to realize gapped boundaries of Kitaev's quantum double models for Dijkgraaf-Witten theories. We classify the elementary excitations on the boundary, and systematically describe the bulk-to-boundary condensation procedure. We also present the parallel algebraic/categorical structure of gapped boundaries.

Gapped boundaries, group cohomology and fault-tolerant logical gates

This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the d-dimensional quantum double model by using d-cocycles functions (d≥2). We point out that the system supports m-dimensional excitations (m<d), which we shall call fluctuating charges, that are superpositions of point-like electric charges characterized by m-dimensional bosonic SPT wavefunctions. There exist gapped boundaries where electric charges or magnetic fluxes may not condense by themselves, but may condense only when accompanied by fluctuating charges. Magnetic fluxes and codimension-2 fluctuating charges exhibit non-trivial multi-excitation braiding statistics, involving more than two excitations. The statistical angle can be computed by taking slant products of underlying cocycle functions sequentially. We find that excitations that may condense into a gapped boundary can be characterized by trivial multi-excitation braiding statistics, generalizing the notion of the Lagrangian subgroup. As an application, we construct fault-tolerantly implementable logical gates for the d-dimensional quantum double model by using d-cocycle functions. Namely, corresponding logical gates belong to the dth level of the Clifford hierarchy, but are outside of the (d−1)th level, if cocycle functions have non-trivial sequences of slant products.

Bose-Einstein condensate in an optical lattice with Raman-assisted two-dimensional spin-orbit coupling

In a recent experiment by Wu {\textit et al.} (arXiv:1511.08170), a Raman-assisted two-dimensional spin-orbit coupling has been realized for a Bose-Einstein condensate in an optical lattice potential. In light of this exciting progress, we study in detail key properties of the system. As the Raman lasers inevitably couple atoms to high-lying bands, the behaviors of the system in both the single- and many-particle sectors are significantly affected. In particular, the high-band effects enhance the plane-wave phase and lead to the emergence of "roton" gaps at low Zeeman fields. Furthermore, we identify high-band-induced topological phase boundaries in both the single-particle and the quasi-particle spectra. We then derive an effective two-band model, which captures the high-band physics in the experimentally relevant regime. Our results not only offer valuable insights into the novel two-dimensional lattice spin-orbit coupling, but also provide a systematic formalism to model high-band effects in lattice systems with Raman-assisted spin-orbit couplings.

Topological phase boundary in a generalized Kitaev model**Project supported by the National Basic Research Program of China (Grant No. 2012CB921704).

We study the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a one-dimensional generalized Kitaev model. In the noninteracting case, we define a topological number and calculate exactly the phase diagram of the system. With addition of the next-nearest-neighbor hopping, the change of phase boundary between the topological and trivial regions can be described by an effective shift of the chemical potential. In the interacting case, we obtain the entanglement spectrum, the degeneracies of which correspond to the topological edge modes, by using the infinite time-evolving block decimation method. The results show that the interactions change the phase boundary as adding an effective chemical potential which can be explained by the change of the average number of particles.

Effects of the electrostatic environment on the Majorana nanowire devices

One of the promising platforms for creating Majorana bound states is a hybrid nanostructure consisting of a semiconducting nanowire covered by a superconductor. We analyze the previously disregarded role of electrostatic interaction in these devices. Our main result is that Coulomb interaction causes the chemical potential to respond to an applied magnetic field, while spin–orbit interaction and screening by the superconducting lead suppress this response. Consequently, the electrostatic environment influences two properties of Majorana devices: the shape of the topological phase boundary and the oscillations of the Majorana splitting energy. We demonstrate that both properties show a non-universal behavior, and depend on the details of the electrostatic environment. We show that when the wire only contains a single electron mode, the experimentally accessible inverse self-capacitance of this mode fully captures the interplay between electrostatics and Zeeman field. This offers a way to compare theoretical predictions with experiments.

Helical edge states and topological phase transitions in phononic systems using bi-layered lattices

We propose a framework to realize helical edge states in phononic systems using two identical lattices with interlayer couplings between them. A methodology is presented to systematically transform a quantum mechanical lattice which exhibits edge states to a phononic lattice, thereby developing a family of lattices with edge states. Parameter spaces with topological phase boundaries in the vicinity of the transformed system are illustrated to demonstrate the robustness to mechanical imperfections. A potential realization in terms of fundamental mechanical building blocks is presented for the hexagonal and Lieb lattices. The lattices are composed of passive components and the building blocks are a set of disks and linear springs. Furthermore, by varying the spring stiffness, topological phase transitions are observed, illustrating the potential for tunability of our lattices.