Quantum skyrmion Hall effect

We consider the problem of magnetic charges in $(2+1)$ dimensions for a torus geometry in real-space, subjected to an inverted Lorentz force due to an external electric field applied normal to the surface of the torus. We compute the Hall conductivity associated with transport of these charges for the case of negligible gapless excitations and global $\mathrm{U}(1)$ charge conservation symmetry, and find it is proportional to an integer-valued topological invariant $\mathcal{Q}$, corresponding to a magnetic quantum Hall effect (MQHE). We identify a lattice model realizing this physics in the absence of an external electric field. Based on this, we identify a generalization of the MQHE to be quantized transport of magnetic skyrmions, the quantum skyrmion Hall effect (QSkHE), with a $\mathrm{U}(1)$ easy-plane anisotropy of magnetic skyrmions and effective conservation of charge associated with magnetic skyrmions yielding incompressibility, provided a hierarchy of energy scales is respected. As the lattice model may be characterized both by a total Chern number and the topological invariant $\mathcal{Q}$, we furthermore outline a possible field theory for electric charges, magnetic charges, and correlations between magnetic and electric charges approximated as composite particles, on a two-torus, to handle the scenario of intermediate-strength correlations between electric and magnetic charges modeled as composite particles. We map this problem to a generalized $(4+1)$D theory of the quantum Hall effect for the composite particles.Comment: 9 pages, 6 figure

Design principles for shift current photovoltaics

While the basic principles and limitations of conventional solar cells are well understood, relatively little attention has gone toward maximizing the potential efficiency of photovoltaic devices based on shift currents. In this work, we outline simple design principles for the optimization of shift currents for frequencies near the band gap, derived from the analysis of a general effective model. The use of a novel sum rule allows us to express the band edge shift current in terms of a few model parameters and to show it depends explicitly on wavefunctions via Berry connections in addition to standard band structure. We use our approach to identify two new classes of shift current photovoltaics, ferroelectric polymer films and single-layer orthorhombic monochalcogenides such as GeS. We introduce tight-binding models for these systems, and show that they exhibit the largest shift current responsivities at the band edge reported so far. Moreover, exploring the parameter space of these models we find photoresponsivities that can exceed $100$ mA/W. Our results show how the study of the shift current via effective models allows one to improve the possible efficiency of devices based on this mechanism and better grasp their potential to compete with conventional solar cells.Comment: 10 pages, 4 figures, AC and BMF share equal contributions. Published in Nature Communication

Time-reversal invariant topological skyrmion phases

Topological phases realized in time-reversal invariant (TRI) systems are foundational to experimental study of the broader canon of topological condensed matter as they do not require exotic magnetic orders for realization. We therefore introduce topological skyrmion phases of matter realized in TRI systems as a foundational step towards experimental realization of topological skyrmion phases. A novel bulk-boundary correspondence hidden from the ten-fold way classification scheme is revealed by the presence of a non-trivial value of a $\mathbb{Z}_2$ spin skyrmion invariant. This quantized topological invariant gives a finer description of the topology in 2D TRI systems as it indicates the presence or absence of robust helical edge states for open boundary conditions, in cases where the $\mathbb{Z}_2$ invariant computed with projectors onto occupied states takes a trivial value. Physically, we show this hidden bulk-boundary correspondence derives from additional spin-momentum-locking of the helical edge states associated with the topological skyrmion phase. ARPES techniques and transport measurements can detect these signatures of topological spin-momentum-locking and helical gapless modes. Our work therefore lays the foundation for experimental study of these phases of matter

Topology in the Sierpi\'nski-Hofstadter problem

Using the Sierpi\'nski carpet and gasket, we investigate whether fractal lattices embedded in two-dimensional space can support topological phases when subjected to a homogeneous external magnetic field. To this end, we study the localization property of eigenstates, the Chern number, and the evolution of energy level statistics when disorder is introduced. Combining these theoretical tools, we identify regions in the phase diagram of both the carpet and the gasket, for which the systems exhibit properties normally associated to gapless topological phases with a mobility edge.Comment: 9 pages, 8 figure

Multiplicative Majorana zero-modes

Topological qubits composed of unpaired Majorana zero-modes are under intense experimental and theoretical scrutiny in efforts to realize practical quantum computation schemes. In this work, we show the minimum four \textit{unpaired} Majorana zero-modes required for a topological qubit according to braiding schemes and control of entanglement for gate operations are inherent to multiplicative topological phases, which realize symmetry-protected tensor products -- and maximally-entangled Bell states -- of unpaired Majorana zero-modes known as multiplicative Majorana zero-modes. We introduce multiplicative Majorana zero-modes as topologically-protected boundary states of both one and two-dimensional multiplicative topological phases, using methods reliant on multiplicative topology to construct relevant Hamiltonians from the Kitaev chain model. We furthermore characterize topology in the bulk and on the boundary with established methods while also introducing techniques to overcome challenges in characterizing multiplicative topology. In the process, we explore the potential of these multiplicative topological phases for an alternative to braiding-based topological quantum computation schemes, in which gate operations are performed through topological phase transitions.Comment: 24 pages, 23 figures, and 2 tables in main text, 11 pages in supplementary material

Multiplicative topological semimetals

Exhaustive study of topological semimetal phases of matter in equilibriated electonic systems and myriad extensions has built upon the foundations laid by earlier introduction and study of the Weyl semimetal, with broad applications in topologically-protected quantum computing, spintronics, and optical devices. We extend recent introduction of multiplicative topological phases to find previously-overlooked topological semimetal phases of electronic systems in equilibrium, with minimal symmetry-protection. We show these multiplicative topological semimetal phases exhibit rich and distinctive bulk-boundary correspondence and response signatures that greatly expand understanding of consequences of topology in condensed matter settings, such as the limits on Fermi arc connectivity and structure, and transport signatures such as the chiral anomaly. Our work therefore lays the foundation for extensive future study of multiplicative topological semimetal phases.Comment: 16 pages and 11 figures in main text, 4 pages and 1 figure in supplementary material

Defect bulk-boundary correspondence of topological skyrmion phases of matter

Unpaired Majorana zero-modes are central to topological quantum computation schemes as building blocks of topological qubits, and are therefore under intense experimental and theoretical investigation. Their generalizations to parafermions and Fibonacci anyons are also of great interest, in particular for universal quantum computation schemes. In this work, we find a different generalization of Majorana zero-modes in effectively non-interacting systems, which are zero-energy bound states that exhibit a cross structure -- two straight, perpendicular lines in the complex plane -- composed of the complex number entries of the zero-mode wavefunction on a lattice, rather than a single straight line formed by complex number entries of the wavefunction on a lattice as in the case of an unpaired Majorana zero-mode. These cross zero-modes are realized for topological skyrmion phases under certain open boundary conditions when their characteristic momentum-space spin textures trap topological defects. They therefore serve as a second type of bulk-boundary correspondence for the topological skyrmion phases. In the process of characterizing this defect bulk-boundary correspondence, we develop recipes for constructing physically-relevant model Hamiltonians for topological skyrmion phases, efficient methods for computing the skyrmion number, and introduce three-dimensional topological skyrmion phases into the literature.Comment: 16 pages and 13 figures in main text, 15 pages and 1 figure in the supplementary material

Topological quantum criticality from multiplicative topological phases

Symmetry-protected topological phases (SPTs) characterized by short-range entanglement include many states essential to understanding of topological condensed matter physics, and the extension to gapless SPTs provides essential understanding of their consequences. In this work, we identify a fundamental connection between gapless SPTs and recently-introduced multiplicative topological phases, demonstrating that multiplicative topological phases are an intuitive and general approach to realizing concrete models for gapless SPTs. In particular, they are naturally well-suited to realizing higher-dimensional, stable, and intrinsic gapless SPTs through combination of canonical topological insulator and semimetal models with critical gapless models in symmetry-protected tensor product constructions, opening avenues to far broader and deeper investigation of topology via short-range entanglement

Higher-Order Topological Insulators

Three-dimensional topological (crystalline) insulators are materials with an insulating bulk, but conducting surface states which are topologically protected by time-reversal (or spatial) symmetries. Here, we extend the notion of three-dimensional topological insulators to systems that host no gapless surface states, but exhibit topologically protected gapless hinge states. Their topological character is protected by spatio-temporal symmetries, of which we present two cases: (1) Chiral higher-order topological insulators protected by the combination of time-reversal and a four-fold rotation symmetry. Their hinge states are chiral modes and the bulk topology is $\mathbb{Z}_2$-classified. (2) Helical higher-order topological insulators protected by time-reversal and mirror symmetries. Their hinge states come in Kramers pairs and the bulk topology is $\mathbb{Z}$-classified. We provide the topological invariants for both cases. Furthermore we show that SnTe as well as surface-modified Bi$_2$TeI, BiSe, and BiTe are helical higher-order topological insulators and propose a realistic experimental setup to detect the hinge states.Comment: 8 pages (4 figures) and 16 pages supplemental material (7 figures