SQUID or Superconducting QUantum Interference Device can be used to sense magnetic field produced even by one single electron. A scanning SQUID-on-tip (SOT) allows to image magnetic field produced by a single electron on a scale of tens-of-nanometer. This tool is frequently used to probe magnetism originating from different sources, for example, topological and non-topological currents, cyclotron orbits due to Landau quantization, electron flow thorugh channels etc. The SOT can also act as a local thermal sensor to probe heat dissipated in nanometer scale. More details can be found in 1, 2, 3.

Scanning tunneling spectroscopy/ microscopy (STS/M) is used to study the electronic properties of surfaces at the atomic scale. STS measures the tunneling current between a sharp tip and a sample, allowing the determination of the local density of states (LDOS) of the sample, which provides information about the electronic structure and energy levels of the sample. In case of a superconducting sample STS can be used to study superconducting vortices. More details at 1.

Electron wavepacket in a topological material (with non-zero Chern number) can have a self-rotation magnetic moment, as well as, a centre of mass drift current resulting in Chern magnetism. It is possible to image locally these two magnetic moments using a SQUID-on-tip (SOT). In presence of non-zero magnetic field, electrons in a 2-dimensional system (like graphene heterostructures) form Landau levels. An SOT can be used also to image this orbital magnetism. Distinguishing between these different sources of magnetism can help solve puzzles like anomalous hall effect, ferromagnetism etc. in twisted graphene systems. More details at 1.

In the mixed state of a type-II superconductor magnetic flux lines thread through the superconductor in forms of quantized flux. This quantum of flux is surrounded by a circulating supercurrent and is called a vortex, which has a normal metallic core. In 3-dimension the vortices look like tubes of flux, while in 2-dimension the vortices can be thought of particles in 2-d. The vortices repel each other and in general form a hexagonal lattice. This vortex lattice is an ideal platform to study statistical mechanics problems, such as lattice melting. One such example is the famous BKTHNY two-step melting, in which an exotic hexatic fluid phase is realized in case of a very soft vortex lattice. More details can be found at 1, 2, 3, 4.

We have used transport and STM studies to deal with a well-studied yet extremely important topic: superconductor to insulator transition (SIT) and its nature. There have been a lot of studies in the topic which ultimately asks which path the SIT follows -- is it bosonic where phase fluctuation drives the superconducting state into an aggregate of phase-decoherent Cooper pairs, or is it fermionic where Coulomb interaction affects the amplitude of the superconducting order parameter? Understanding the nature of SIT even holds the key to understanding aspects of high-Tc superconductors, like pseudogap phase and/or bad-metallic phase. However there is no single consensus as to which path the SIT would take for a given material and needs an individual inspection for each of them. More details can be found at 1, 2, 3.

Due to inter-vortex repulsion, vortices in superconductor form a triangular array, also known as the Abrikosov lattice, characterised by a length-scale namely, vortex lattice constant. If the superconductor has a periodic array of pinning centers, it introduces another length-scale in the picture, lattice constant of the pinning array. If these two length-scales match (commensurate state), the vortex lattice is strongly pinned and becomes incompressible, mimicking a Mott insulator. On the other hand in the incommensurate state, the vortex lattice stays compressible. The incompressible Mott-like vortex state can undergo a transition into a metal-like state where the vortices are freed from the periodic pinning array. This Mott to metal-like transition is due to a collective behaviour which requires much less energy than the pinning energy of the underlying pinning array. More details can be found at 1.