arXiv subjects

Alexander C. Tyner

Publications and source records attributed to Alexander C. Tyner.

27 records · Page 2Linked to original sources

arXiv2024

Three-dimensional $\mathbb{Z}$ topological insulators without reflection symmetry

In recent decades, the Altland-Zirnabuer (AZ) table has proven incredibly powerful in delineating constraints for topological classification of a given band-insulator based on dimension and (nonspatial) symmetry class, and has also been expanded by considering additional crystalline symmetries. Nevertheless, realizing a three-dimensional (3D), time-reversal symmetric (class AII) topological insulator (TI) in the absence of reflection symmetries, with a classification beyond the $\mathbb{Z}_{2}$ paradigm remains an open problem. In this work we present a general procedure for constructing such systems within the framework of projected topological branes (PTBs). In particular, a 3D projected brane from a "parent" four-dimensional topological insulator exhibits a $\mathbb{Z}$ topological classification, corroborated through its response to the inserted bulk monopole loop. More generally, PTBs have been demonstrated to be an effective route to performing dimensional reduction and embedding the topology of a $(d+1)$-dimensional "parent" Hamiltonian in $d$ dimensions, yielding lower-dimensional topological phases beyond the AZ classification without additional symmetries. Our findings should be relevant for the metamaterial platforms, such as photonic and phonic crystals, topolectric circuits, and designer systems.

cond-mat.mes-hall↗

arXiv2023

Solitons and real-space screening of bulk topology of quantum materials

Recent years have seen multiple high-throughput studies reveal an immense number of topological materials through use of symmetry indicators. Despite this success, three-dimensional topological insulators (TI) admitting a band-gap larger than Bi$_{2}$Se$_{3}$ and two-dimensional TIs admitting a band gap larger than $β$-bismuthene, two of the originally proposed TIs, remain extremely rare. Simultaneously, a significant effort has been made to understand and identify topological phases ``invisible" to symmetry indicators. Such phases offer a unique opportunity to expand the search for a large band-gap TI, however their identification requires sophisticated probes of bulk topology. Magnetic flux tubes or vortices have emerged as one such probe in two-dimensions when inserted into the bulk. In this work, we develop an automated workflow to perform vortex insertion and apply it to a current database of high-quality, experimentally realized, two-dimensional insulators. The results reveal multiple novel two-dimensional topological insulators supporting large bands gaps, including the 1H-MX$_{2}$ (M=Mo,W) and (X=S,Se,Te) family of transition metal dichalcogenides. Our work has broad implications for current theoretical and experimental efforts to employ these materials in superconducting and Moire systems.

cond-mat.mtrl-sci↗

arXiv2023

Fundamentals of crystalline Hopf insulators

Three-dimensional, crystalline Hopf insulators are generic members of unitary Wigner-Dyson class, which can break all global discrete symmetries and point group symmetries. In the absence of first Chern number for any two-dimensional plane of Brillouin zone, the Hopf invariant $N_H \in \mathbb{Z}$. But in the presence of Chern number $N_H \in \mathbb{Z}_{2q}$, where $q$ is the greatest common divisor of Chern numbers for $xy$, $yz$, and $xz$ planes of Brillouin zone. How does $N_H$ affect topological quantization of isotropic, magneto-electric coefficient? We answer this question with calculations of Witten effect for a test, magnetic monopole. Furthermore, we construct $N$-band tight-binding models of Hopf insulators and demonstrate their topological stability against spectral flattening.

cond-mat.mes-hall↗

arXiv2022

Part II: Witten effect and $\mathbb{Z}$-classification of axion angle $θ=n π$

The non-trivial third homotopy class of three-dimensional topological insulators leads to quantized, magneto-electric coefficient or axion angle $θ= n π$, with $n \in \mathbb{Z}$. In Part I, we developed tools for computing $n$ from a staggered symmetry-indicator $κ_{AF,j}$ and Wilson loops of non-Abelian, Berry connection in momentum-space, which clearly distinguished between magneto-electrically trivial ($n=0$), and non-trivial ($n=2s$) topological crystalline insulators. In this work, we perform $\mathbb{Z}$-classification of real-space, topological response or $θ$ by carrying out thought experiments with magnetic, Dirac monopoles. We demonstrate this for non-magnetic and magnetic topological insulators by computing induced electric charge on monopoles or Witten effect. We show that both first- and higher- order topological insulators can exhibit quantized, magneto-electric response, irrespective of the presence of gapless surface-states, and corner-states. Special attention is paid to the response of octupolar higher-order topological insulator, which was originally predicted to be magneto-electrically trivial. The important roles of fermion zero-modes, $\mathcal{CP}$, and flavor symmetries are critically addressed. Our work outlines a unified theoretical framework for addressing dc topological response and topological quantum phase transitions, which cannot be reliably predicted by symmetry-based classification scheme.

cond-mat.mes-hall↗

arXiv2022

Non-Abelian Stokes theorem and quantized Berry flux

Band topology of anomalous quantum Hall insulators can be precisely addressed by computing Chern numbers of constituent non-degenerate bands that describe quantized, Abelian Berry flux through two-dimensional Brillouin zone. Can Chern numbers be defined for $SU(2)$ Berry connection of two-fold degenerate bands of materials preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries or combined $\mathcal{PT}$ symmetry, without detailed knowledge of underlying basis? We affirmatively answer this question by employing a non-Abelian generalization of Stokes' theorem and describe a manifestly gauge-invariant method for computing magnitudes of quantized $SU(2)$ Berry flux (spin-Chern number) from eigenvalues of Wilson loops. The power of this method is elucidated by performing $\mathbb{N}$-classification of \emph{ab initio} band structures of three-dimensional, Dirac materials. Our work outlines a unified framework for addressing first-order and higher-order topology of insulators and semimetals, without relying on detailed symmetry data.

cond-mat.mtrl-sci↗

arXiv2022

Spin-charge separation and quantum spin Hall effect of $β$-bismuthene

Field theory arguments suggest the possibility of $\mathbb{Z}$-classification of quantum spin Hall effect with magnetic flux tubes, that cause separation of spin and charge degrees of freedom, and pumping of spin or Kramers pair. However, the \emph{proof of principle} demonstration of spin-charge separation is yet to be accomplished for realistic, \emph{ab initio} band structures of spin-orbit-coupled materials, lacking spin-conservation law. In this work, we perform thought experiments with magnetic flux tubes on $β$-bismuthene to demonstrate spin-charge separation, and quantized pumping of spin for three insulating states that can be accessed by tuning filling fractions. With a combined analysis of momentum-space topology and real-space response, we identify important role of topologically non-trivial bands, supporting even integer winding numbers, which cannot be inferred from symmetry-based indicators. Our work sets a new standard for prediction of two-dimensional, quantum spin-Hall materials, based on precise bulk invariant and universal topological response.

cond-mat.mes-hall↗

arXiv2022

Part I: Staggered index and 3D winding number of Kramers-degenerate bands

For three-dimensional (3D) crystalline insulators, preserving space-inversion ($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries, the third homotopy class of two-fold, Kramers-degenerate bands is described by a 3D winding number $n_{3,j} \in \mathbb{Z}$, where $j$ is the band index. It governs space group symmetry-protected, instanton or tunneling configurations of $SU(2)$ Berry connection, and the quantization of magneto-electric coefficient $θ_j = n_{3,j} π$. We show that $|n_{3,j}|$ for realistic, \emph{ab initio} band structures can be identified from a staggered symmetry-indicator $κ_{AF,j} \in \mathbb{Z}$ and the gauge-invariant spectrum of $SU(2)$ Wilson loops. The procedure is elucidated for $4$-band and $8$-band tight-binding models and \emph{ab initio} band structure of Bi, which is a $\mathbb{Z}_2$-trivial, higher-order, topological crystalline insulator. When the tunneling is protected by $C_{nh}$ and $D_{nh}$ point groups, the proposed method can also identify the signed winding number $n_{3,j}$. Our analysis distinguishes between magneto-electrically trivial ($θ=0$) and non-trivial ($θ=2 s π$, with $s \neq 0$) topological crystalline insulators. In Part II, we demonstrate $\mathbb{Z}$-classification of $θ$ by computing induced electric charge (Witten effect) on magnetic Dirac monopoles.

cond-mat.mtrl-sci↗

arXiv2022

Topology of three-dimensional Dirac semimetals and generalized quantum spin Hall systems without gapless edge modes

Usually the quantum spin Hall states are expected to possess gapless, helical edge modes. Are there clean, non-interacting, quantum spin Hall states without gapless, edge modes? We show the generic, $n$-fold-symmetric, momentum planes of three-dimensional, stable Dirac semi-metals, which are orthogonal to the direction of nodal separation are examples of such generalized quantum spin Hall systems. We demonstrate that the planes lying between two Dirac points and the celebrated Bernevig-Zhang-Hughes model support identical quantized, non-Abelian Berry flux of magnitude $2 π$. Consequently, both systems exhibit spin-charge separation in response to electromagnetic, $π$-flux vortex. The Dirac points are identified as the unit-strength, monopoles of $SO(5)$ Berry connection, describing topological quantum phase transitions between generalized, quantum spin Hall and trivial insulators. Our work identifies precise bulk invariant and quantized response of Dirac semimetals and shows that many two-dimensional higher-order topological insulators can be understood as generalized quantum spin Hall systems, possessing gapped edge states.

cond-mat.mes-hall↗

arXiv2022

Mixed-order topology of Benalcazar-Bernevig-Hughes models

Benalcazar-Bernevig-Hughes (BBH) models, defined on $D$-dimensional simple cubic lattice, are paradigmatic toy models for studying $D$-th order topology and corner-localized, mid-gap states. Under periodic boundary conditions, the Wilson loops of non-Abelian Berry connection of BBH models along all high-symmetry axes have been argued to exhibit gapped spectra, which predict gapped surface-states under open boundary conditions. In this work, we identify 1D, 2D, and 3D topological invariants for characterizing higher order topological insulators. Further, we demonstrate the existence of cubic-symmetry-protected, gapless spectra of Wilson loops and surface-states along the body diagonal directions of the Brillouin zone of BBH models. We show the gapless surface-states are described by $2^{D-1}$-component, massless Dirac fermions. Thus, BBH models can exhibit the signatures of first and $D$-th order topological insulators, depending on the details of externally imposed boundary conditions.

cond-mat.mes-hall↗