Relating the topology of Dirac Hamiltonians to quantum geometry: When the quantum metric dictates Chern numbers and winding numbers

Quantum geometry has emerged as a central and ubiquitous concept in quantum sciences, with direct consequences on quantum metrology and many-body quantum physics. In this context, two fundamental geometric quantities are known to play complementary roles:~the Fubini-Study metric, which introduces a notion of distance between quantum states defined over a parameter space, and the Berry curvature associated with Berry-phase effects and topological band structures. In fact, recent studies have revealed direct relations between these two important quantities, suggesting that topological properties can, in special cases, be deduced from the quantum metric. In this work, we establish general and exact relations between the quantum metric and the topological invariants of generic Dirac Hamiltonians. In particular, we demonstrate that topological indices (Chern numbers or winding numbers) are bounded by the quantum volume determined by the quantum metric. Our theoretical framework, which builds on the Clifford algebra of Dirac matrices, is applicable to topological insulators and semimetals of arbitrary spatial dimensions, with or without chiral symmetry. This work clarifies the role of the Fubini-Study metric in topological states of matter, suggesting unexplored topological responses and metrological applications in a broad class of quantum-engineered systems.

Time-periodic corner states from Floquet higher-order topology

The recent discoveries of higher-order topological insulators (HOTIs) have shifted the paradigm of topological materials, previously limited to topological states at boundaries of materials, to include topological states at boundaries of boundaries, such as corners. So far, all HOTI realisations have been based on static systems described by time-invariant Hamiltonians, without considering the time-variant situation. There is growing interest in Floquet systems, in which time-periodic driving can induce unconventional phenomena such as Floquet topological phases and time crystals. Recent theories have attempted to combine Floquet engineering and HOTIs, but there has been no experimental realisation so far. Here we report on the experimental demonstration of a two-dimensional (2D) Floquet HOTI in a three-dimensional (3D) acoustic lattice, with modulation along a spatial axis serving as an effective time-dependent drive. Acoustic measurements reveal Floquet corner states with double the period of the underlying drive; these oscillations are robust, like time crystal modes, except that the robustness arises from topological protection. This shows that space-time dynamics can induce anomalous higher-order topological phases unique to Floquet systems.

Topological properties of Sb(111) surface: A first-principles study

First-principles calculations based on density functional theory were performed to systematically study the electronic properties of the thin film of antimony in (111) orientation. Considering the spinorbit interaction, for stoichiometric surface, the topological states keep robust for six-bilayer case, and can be recovered in the three-bilayer film, which are guaranteed by time-reversal symmetry and inverse symmetry. For reduced surface doped by non-magnetic Bi or magnetic Mn atom, localized three-fold symmetric features can be identified. Moreover, band structures show that the non-trivial topological states stand for non-magnetic substitutional Bi atom, while can be eliminated by adsorbed or substitutional magnetic Mn atom.

Studying Quantum Materials with Scanning SQUID Microscopy

Electronic correlations give rise to fascinating macroscopic phenomena such as superconductivity, magnetism, and topological phases of matter. Although these phenomena manifest themselves macroscopically, fully understanding the underlying microscopic mechanisms often requires probing on multiple length scales. Spatial modulations on the mesoscopic scale are especially challenging to probe, owing to the limited range of suitable experimental techniques. Here, we review recent progress in scanning superconducting quantum interference device (SQUID) microscopy. We demonstrate how scanning SQUID combines unmatched magnetic field sensitivity and highly versatile designs, by surveying discoveries in unconventional superconductivity, exotic magnetism, topological states, and more. Finally, we discuss how SQUID microscopy can be further developed to answer the increasing demand for imaging new quantum materials. Expected final online publication date for the Annual Review of Condensed Matter Physics, Volume 13 is March 2022. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.