Theorists have discovered topological insulators that are insulating in their interior and on their surfaces but have conducting channels at corners or along edges.

(d - 2)-dimensional edge states of rotation symmetry protected topological states

The combination of spin-orbit coupling with interactions results in many exotic phases of matter. In this Letter, we investigate the superconducting pairing instability of the two-dimensional extended Hubbard model with both Rashba and Dresselhaus spin-orbit coupling within the mean-field level at both zero and finite temperature. We find that both first- and second-order time-reversal symmetry breaking topological gapped phases can be achieved under appropriate parameters and temperature regimes due to the presence of a favored even-parity s+id-wave pairing even in the absence of an external magnetic field or intrinsic magnetism. This results in two branches of chiral Majorana edge states on each edge or a single zero-energy Majorana corner state at each corner of the sample. Interestingly, we also find that not only does tuning the doping level lead to a direct topological phase transition between these two distinct topological gapped phases, but also using the temperature as a highly controllable and reversible tuning knob leads to different direct temperature-driven topological phase transitions between gapped and gapless topological superconducting phases. Our findings suggest new possibilities in interacting spin-orbit coupled systems by unifying both first- and higher-order topological superconductors in a simple but realistic microscopic model.

First- and Second-Order Topological Superconductivity and Temperature-Driven Topological Phase Transitions in the Extended Hubbard Model with Spin-Orbit Coupling.

Majorana bound states often occur at the end of a 1D topological superconductor. Validated by a new bulk invariant and an intuitive edge argument, we show the emergence of one Majorana Kramers pair at each corner of a square-shaped 2D topological insulator proximitized by an s_{±}-wave (e.g., Fe-based) superconductor. We obtain a phase diagram that addresses the relaxation of crystal symmetry and edge orientation. We propose two experimental realizations in candidate materials. Our scheme offers a higher-order and higher-temperature route for exploring non-Abelian quasiparticles.

High-Temperature Majorana Corner States

The band topology of a superconductor is known to have profound impact on the existence of Majorana zero modes in vortices. As iron-based superconductors with band inversion and ${s}_{\ifmmode\pm\else\textpm\fi{}}$-wave pairing can give rise to time-reversal invariant second-order topological superconductivity, manifested by the presence of helical Majorana hinge states in three dimensions, we are motivated to investigate the interplay between the second-order topology and the vortex lines in both weak- and strong-Zeeman-field regimes. In the weak-Zeeman-field regime, we find that vortex lines far away from the hinges are topologically nontrivial in the weakly doped regime, regardless of whether the second-order topology is present or not. However, when the superconductor falls into the second-order topological phase and a topological vortex line is moved close to the helical Majorana hinge states, we find that their hybridization will trivialize the vortex line and transfer robust Majorana zero modes to the hinges. Furthermore, when the Zeeman field is large enough, we find that the helical Majorana hinge states are changed into chiral Majorana hinge modes and thus a chiral second-order topological superconducting phase is realized. In this regime, the vortex lines are always topologically trivial, no matter how far away they are from the chiral Majorana hinge modes. By incorporating a realistic assumption of inhomogeneous superconductivity, our findings can explain the recent experimental observation of the peculiar coexistence and evolution of topologically nontrivial and trivial vortex lines in iron-based superconductors.

Vortex-line topology in iron-based superconductors with and without second-order topology

Topology, with its abstract mathematical constructs, often manifests itself in physics and has a pivotal role in our understanding of natural phenomena. Notably, the discovery of topological phases in condensed-matter systems has changed the modern conception of phases of matter. The global nature of topological ordering, however, makes direct experimental probing an outstanding challenge. Present experimental tools are mainly indirect and, as a result, are inadequate for studying the topology of physical systems at a fundamental level. Here we employ the exquisite control afforded by state-of-the-art superconducting quantum circuits to investigate topological properties of various quantum systems. The essence of our approach is to infer geometric curvature by measuring the deflection of quantum trajectories in the curved space of the Hamiltonian. Topological properties are then revealed by integrating the curvature over closed surfaces, a quantum analogue of the Gauss–Bonnet theorem. We benchmark our technique by investigating basic topological concepts of the historically important Haldane model after mapping the momentum space of this condensed-matter model to the parameter space of a single-qubit Hamiltonian. In addition to constructing the topological phase diagram, we are able to visualize the microscopic spin texture of the associated states and their evolution across a topological phase transition. Going beyond non-interacting systems, we demonstrate the power of our method by studying topology in an interacting quantum system. This required a new qubit architecture that allows for simultaneous control over every term in a two-qubit Hamiltonian. By exploring the parameter space of this Hamiltonian, we discover the emergence of an interaction-induced topological phase. Our work establishes a powerful, generalizable experimental platform to study topological phenomena in quantum systems.

http://absimage.aps.org/image/MAR15/MWS_MAR15-2014-008774.pdf

Observation of topological transitions in interacting quantum circuits

Higher-order topological phases give rise to new bulk and boundary physics, as well as new classes of topological phase transitions. While the realization of higher-order topological phases has been confirmed in many platforms by detecting the existence of gapless boundary modes, a direct determination of the higher-order topology and related topological phase transitions through the bulk in experiments has still been lacking. To bridge the gap, in this work we carry out the simulation of a two-dimensional second-order topological phase in a superconducting qubit. Owing to the great flexibility and controllability of the quantum simulator, we observe the realization of higher-order topology directly through the measurement of the pseudo-spin texture in momentum space of the bulk for the first time, in sharp contrast to previous experiments based on the detection of gapless boundary modes in real space. Also through the measurement of the evolution of pseudo-spin texture with parameters, we further observe novel topological phase transitions from the second-order topological phase to the trivial phase, as well as to the first-order topological phase with nonzero Chern number. Our work sheds new light on the study of higher-order topological phases and topological phase transitions.

Simulation of higher-order topological phases and related topological phase Transitions in a Superconducting Qubit

Quantum simulation can be implemented in pure digital or analog ways, each with their pros and cons. By taking advantage of the universality of a digital route and the efficiency of analog simulation, hybrid digital–analog approaches can enrich the possibilities for quantum simulation. We use a hybrid approach to experimentally perform a quantum simulation of phase-controlled dynamics resulting from a closed-contour interaction (CCI) within certain multi-level systems in superconducting quantum circuits. Due to symmetry constraints, such systems cannot host an inherent CCI. Nevertheless, by assembling analog modules corresponding to their natural evolutions and specially designed digital modules constructed from standard quantum logic gates, we can bypass such constraints and realize an effective CCI in these systems. Based on this realization, we demonstrate a variety of related and interesting phenomena, including phase-controlled chiral dynamics, separation of chiral enantiomers, and a mechanism to generate entangled states based on CCI.

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Dapeng Yu

Papers

Simulation of higher-order topological phases and related topological phase Transitions in a Superconducting Qubit

Experimental realization of phase-controlled dynamics with hybrid digital–analog approach