Session F29: Novel Topological Phases: Theory

Chiral topological superconductor and half-integer conductance plateau from~quantum anomalous hall plateau transition

We propose to realize a two-dimensional chiral topological superconducting (TSC) state from the quantum anomalous hall plateau transition in a magnetic topological insulator thin LM through the proximity E ECT to a conventional s-wave superconductor. This state has a full pairing gap in the bulk and a single chiral majorana mode at the edge. The optimal condition for realizing such chiral tsc is to have inequivalent superconducting pairing amplitudes on top and bottom surfaces~of the doped magnetic topological insulator. We further propose several transport experiments to~detect the chiral TSC. One unique signature is that the conductance will be quantized into a half-integer plateau at the coercive eld in this hybrid system. in particular, with the point contact~formed by a superconducting junction, the conductance oscillates between E2=2H AND E2=H with the~frequency determined by the voltage across the junction. we close by discussing the feasibility of~these experimental proposals.   

Bona fide interaction-driven topological phase transition in correlated SPT states

It is expected the interplay between non-trivial band topology and strong electron correlation will lead to very rich physics. Thus a controlled study of the competition between topology and correlation is of great interest. Here, employing large-scale quantum Monte Carlo simulations, we provide a concrete example of the Kane-Mele-Hubbard model on an AA stacking bilayer honeycomb lattice with inter-layer antiferromagnetic interaction. Our simulation identified several different phases: a quantum spin-Hall insulator (QSH), a xy-plane antiferromagnetic Mott insulator (xy-AFM) and an inter-layer dimer-singlet insulator (dimer-singlet). Most importantly, a bona fide topological phase transition between the QSH and the dimer-singlet insulators, purely driven by the inter-layer antiferromagnetic interaction is found. At the transition, the spin and charge gap of the system close while the single-particle excitations remain gapped, which means that this transition has no mean field analogue and it can be viewed as a transition between bosonic SPT states. At one special point, this transition is described by a (2+1)d O(4) nonlinear sigma model with exact SO(4) symmetry, and a topological term at theta=p. Relevance of this work towards more general interacting SPT states is discussed.   

Topological Phases on Non-orientable Surfaces: Twisting by Parity Symmetry

We discuss (2+1)D topological phases on non-orientable spatial surfaces, such as M\"obius strip, real projective plane and Klein bottle, etc., which are obtained by twisting the parent topological phases by their underlying pairty symmetries through introducing parity defects. We construct the ground states on arbitrary non-orientable closed manifolds and calculate the ground state degeneracy. Such degeneracy is shown to be robust against continuous deformation of the underlying manifold. We also study the action of the mapping class group on the multiplet of ground states on the Klein bottle. The physical properties of the topological states on non-orientable surfaces are deeply related to the parity symmetric anyons which do not have a notion of orientation in their statistics. For example, the number of ground states on the projective plane equals the root of the number of distinguishable parity symmetric anyons, while the ground state degeneracy on the Klein bottle equals the total number of parity symmetric anyons; In deforming the Klein bottle, the Dehn twist encodes the topological spins whereas the Y-homeomorphism tells the particle-hole relation of the parity symmetric anyons.   

Analysis of the Kane-Mele-Kondo lattice at finite temperatures.

Recently, correlation effects on topological insulators are extensively studied because the interplay of topological properties and electron correlations is expected to induce exotic phenomena. A promising candidate for a topological insulator in heavy-fermion systems is {\$}$\backslash $mathrm\textbraceleft SmB\textbraceright \textunderscore 6{\$} where the Kondo effects play an essential role. In this article, we study the Kane-Mele-Kondo lattice at finite temperatures. By using the dynamical mean-field theory, we obtain a temperature vs. interaction phase diagram (a Doniach phase diagram). Furthermore, we have observed an intriguing crossover behavior induced by the interplay of electron correlations and topologically nontrivial properties. In the bulk system, the spin-Hall conductivity which is proportional to the spin Chern number is zero at low temperatures while the conductivity rapidly increases with increasing temperature. Correspondingly, gapless modes are restored by temperature effects at the edge sites, which are destroyed by the Kondo effect at low temperature.   

Bulk Topological Proximity Effect

Existing proximity effects stem from systems with a local order parameter, such as a local mag- netic moment or a local superconducting pairing amplitude. Here, we demonstrate that despite lacking a local order parameter, topological phases also may give rise to a proximity effect of a distinctively inverted nature. We focus on a general construction in which a topological phase is extensively coupled to a second system, and we argue that in many cases, the inverse topological order will be induced on the second system. To support our arguments, we rigorously establish this ``bulk topological proximity effect'' for all gapped free fermion topological phases and repre- sentative integrable models of interacting topological phases. We present a terrace construction which illustrates the phenomenological consequences of this proximity effect. Finally, we discuss generalizations beyond our framework, including how intrinsic topological order may also exhibit this effect.   

Electrically tunable spin polarization of chiral edge modes in a quantum anomalous Hall insulator

In the quantum anomalous Hall effect, chiral edge modes are expected to conduct spin polarized current without dissipation and thus hold great promise for future electronics and spintronics with low energy consumption. However, spin polarization of chiral edge modes has never been established in experiments. In this work, we theoretically study spin polarization of chiral edge modes in the quantum anomalous Hall effect, based on both the effective model and more realistic tight-binding model constructed from the first principles calculations. We find that spin polarization can be manipulated by tuning either a local gate voltage or the Fermi energy. We also propose to extract spin information of chiral edge modes by contacting the quantum anomalous Hall insulator to a ferromagnetic (FM) lead. The establishment of spin polarization of chiral edge modes, as well as the manipulation and detection in a fully electrical manner, will pave the way to the applications of the quantum anomalous Hall effect in spintronics.   

Topological invariants in interacting topological insulators: Success and Breakdown

The content of this talk is twofold. In the first part, we provide a paradigm of efficient numerical evaluation scheme for topological invariants via zero-frequency single-particle Green's function in quantum Monte Carlo (QMC) simulations. Especially, we introduce a periodization process to overcome the ubiquitous finite-size effect and make use of symmetry properties of the underlying systems to reduce the computational effort. This scheme is tested to be successful on models of interacting topological insulators, where there is single-particle gap closing at the transition. In the second part, we apply the above scheme to wider classes of interacting topological insulators, in which the breakdown of constructing topological invariant via single-particle Green’s functions is presented. These systems host novel interaction-driven topological phase transitions without symmetry breaking, and hence fermionic degree of freedom is not involved at the critical point, instead, collective bosonic mode become critical.   

Of Bulk and Boundaries: Generalized Transfer Matrices for Tight-Binding Models

We construct a generalized transfer matrix corresponding to noninteracting tight-binding lattice models, which can subsequently be used to compute the bulk bands as well as the edge states. Crucially, our formalism works even in cases where the hopping matrix is non-invertible. Following Hatsugai [PRL 71, 3697 (1993)], we explicitly construct the energy Riemann surfaces associated with the band structure for a specific class of systems which includes systems like Chern insulator, Dirac semimetal and graphene. The edge states can then be interpreted as non-contractible loops, with the winding number equal to the bulk Chern number. For these systems, the transfer matrix is symplectic, and hence we also describe the windings associated with the edge states on $Sp(2,\mathbb{R})$ and interpret the corresponding winding number as a Maslov index. This work is discussed in arXiv preprint arXiv:1510.04279.   

Topological Edge States with Zero Hall Conductivity in a Dimerized Hofstadter Model

The Hofstadter model is one of the most celebrated models for the study of topological properties of matter and allows the study of the quantum Hall effect in a lattice system. Indeed, the Hofstadter Hamiltonian harbors the topological chiral edge states that are responsible for the quantized Hall conductivity. Here, we show that a lattice dimerization in the Hofstadter model opens an energy gap at half-filling. What is more, we demonstrate that even if the ensuing insulator has a Chern number equal to \textit{zero}, concomitantly a doublet of edge states appear that are pinned to specific momenta. We show that the presence of these states can be understood from the topological properties of lower dimensional cuts of the system, using a mapping of the Hofstadter Hamiltonian to a collection of one-dimensional Aubry-Andr\'e-Harper (AAH) models. A sub-set of AAH chains in this collection preserve inversion symmetry. This guarantees the presence of topologically protected doublets of end modes to which the edge states are pinned. To explicitly prove the robustness of the emerging edge states, we define and calculate the topological invariant that protects them, which turns out to be an integer invariant for inversion-symmetric AAH models.   

Field theory of symmetry protected valence bond solid states in (2+1) dimensions

With the scope of identifying possible symmetry-protected topological (SPT) states, we revisit the effective field theory description of 2d antiferromagnets in terms of nonlinear sigma models with topological Berry phase terms. We focus on ground states that can be characterized as spatially-uniform valence-bond-solid states residing on a square lattice, which implies that the spin quantum number S be an even integer. A path integral representation of wave functionals allows us to study the topological properties of the ground state in terms of a field theory defined in a space whose dimensionality is reduced by one, which leads us to an interesting incarnation of the well-known Haldane-gap argument for 1d antiferromagnets. From this, we conclude that the ground state can be an SPT state when S =2$\times$ odd integer, while for S =2 $\times$ even integer it is topologically trivial. We also discuss how our method generalizes to 1d and 3d antiferromagnets.   

First-Order Character and Observable Signatures of Topological Quantum Phase Transitions

Topological quantum phase transitions are characterized by changes in global topological invariants. These invariants classify many-body systems beyond the conventional paradigm of local order parameters describing spontaneous symmetry breaking. For noninteracting electrons, it is well understood that such transitions are continuous and always accompanied by a gap closing in the energy spectrum, given that the symmetries protecting the topological phase are maintained. Here, we demonstrate that a sufficiently strong electron-electron interaction can fundamentally change the situation: we discover a topological quantum phase transition of first-order character in the genuine thermodynamic sense that occurs without a gap closing. Our theoretical study reveals the existence of a quantum critical endpoint associated with an orbital instability on the transition line between a 2D topological insulator and a trivial band insulator. Remarkably, this phenomenon entails unambiguous signatures related to the orbital occupations that can be detected experimentally. \\ Part of the results presented in this talk have been published in Phys.~Rev.~Lett. {\bf 114}, 185701 (2015)   

Construction of non-Abelian topological insulators using non-Abelian bosonization

A way to construct 2D topological insulators and superconductors is to couple an array of wires. The advantage of this construction is that one can use bosonization. Many 2D integer and fractional topological quantum states have been proposed using Abelian bosonization. In this talk we show how to use non-Abelian bosonization to construct non-Abelian topological insulators and superconductors in 2D. With the help of conformal field theory, we construct topological states whose edge states are described by coset theories of the Wess-Zumino-Witten model. In this construction, all the interactions we use to gap the bulk are physical, i.e. tunneling of electrons and current-current interactions.   

Constructing parent Hamiltonians for SU(N) ALKT states - a diagrammatic method

Over the last decade, there has been increasing experimental interest in alkaline cold atom systems which exhibit $SU(N)$ symmmetry. Theoretical work has shown that a one-dimensional $SU(N)$ chain can have $N-1$ symmetric protected states distinguished by fractionalized boundary spins. We introduce a new method for constructing $SU(N)$ invariant Hamiltonians for Haldane phases in one dimension. Working at the AKLT point where the ground state is known exactly, we show a universal form of the Hamiltonian for any appropriate choice of physical and boundary spins. We apply our method to the case where the physical spin is in the adjoint representation and obtain a general expression for the Hamiltonian as well the Transfer Matrix for any $N$. Finally we comment on the relevance of our results to the generalized Haldane conjecture.   

Topological Nonsymmorphic Crystalline Superconductors

Topological superconductors possess a nodeless superconducting gap in the bulk and gapless zero energy modes, known as ``Majorana zero modes'', at the boundary of a finite system. In this work, we introduce a new class of topological superconductors, which are protected by nonsymmorphic crystalline symmetry and thus dubbed ``topological nonsymmorphic crystalline superconductors''. We construct an explicit Bogoliubov-de Gennes type of model for this superconducting phase in the D class and show how Majorana zero modes in this model are protected by glide symmetry. Furthermore, we generalize the classification of topological nonsymmorphic crystalline superconductors to the classes with time reversal symmetry, including the DIII and BDI classes, in two dimensions. Our theory provides a guidance to search for new topological superconducting materials with nonsymmorphic crystal structures.   

Equivalence of topological insulators and superconductors

Systems of free fermions are classified by symmetry, space dimensionality, and topological properties described by K-homology. We show that by taking a many-body/Fock space viewpoint it becomes possible to establish equivalences of topological insulators and superconductors in terms of duality transformations [1]. These mappings connect topologically inequivalent systems of fermions, jumping across entries in existent classification tables, because of the phenomenon of symmetry transmutation by which a symmetry and its dual partner have identical algebraic properties but very different physical interpretations and electromagnetic response. Since our analysis extends to interacting fermion systems we also briefly discuss some such applications. To illustrate main concepts we will present dual superconducting partners of paradigmatic models, such as the Haldane Chern insulator as well as a quantum spin Hall effect graphene model. [1] Phys. Rev. B 92, 155125 (2015).