**By Francesca Finnochiaro (IMDEA)**

We learnt from classical mechanics that often, in the absence of specific constraints, in order to create a response (for instance: acceleration) in a given direction it is easier if you apply a perturbation (for instance, a force) in that same direction. If you try to have a response in a different direction you likely have to exert a higher force in order to get the same result up to the extreme case where you cannot have any response in the perfectly perpendicular direction (a typical example is the man trying to carry a machine with wheels on a railtrack, miserably suffering when he pulls in a direction that differs from that of the railway, and eventually failing in the case he tries to pull in the perpendicular direction. Every student that has passed trough such an example must have mentally asked to the man in the picture in the books: “why?” at least once). The question is: why would we expect to generate a response in a different direction, and why could we possibly want it?

One easy answer to the first question is readily given as soon as one enters the world of electromagnetism. In this world there is a force, the magnetic force, compelling charged particles to move in a direction that is perpendicular to both the magnetic field itself and the velocity of the charged particle itself. This means that this force accelerates the electrons towars a direction that is perpendicular to their direction of motion. In general, if one applies the field for a sufficiently long time and in a sufficiently large space, a charged particle succeds in completing a circular path in the plane perpendicular to the direction of motion. Such an orbit is called the cyclotron orbit and its radius is directly proportional to the velocity and inversely to the magnetic field, whereas its frequency is linear in the field.

How can we use such a force? Well, the ways in which this force (the Lorentz force) finds applications in physics are countless. Here we want to specialize with an effect, called the Hall effect, which is based on such a mechanism. Let’s take a thin metallic bar and let’s induce a steady electrical current through it. If a perpendicular magnetic field is applied, the electrons will start to be deflected towards the perpendicular direction. If the transversal dimension of the bar is such as to impede the electrons to carry out closed orbits, they will just start to accumulate on one side of the bar, leaving the opposite side depleted of negative charges. Because of this, an electric field arises in the transverse direction and with a little bit of patience the generated field gets to a strenght such as to impede further charges to accumulate. All and all, thanks to the presence of the perpendicular magnetic field, a permanent voltage is generated transverse to the direction along which the current was injected in the first place. When a phenomenon like this happens, a Hall voltage is said to have been generated, with the world Hall simply meaning “transverse to the direction of the applied current”.

This is the classical Hall effect, which is the precursor of a genuine cascade of “modified” Hall effects that have been discovered successively thanks to the advancements in quantum mechanics and importantly intertwined with the discovery of the Berry phase and all the phenomena related with it.

First of the list is the quantum Hall effect, which is basically the quantum-mechanical version of the classical one, realized in two-dimensional systems and for really strong magnetic fields, at very low temperatures. In such a situation, the Hall conductance (that is, the voltage response along y to a current applied along x) is quantized in units of e2/h. This is simply due to the fact that quantistically the closed circular orbits traveled by the electrons can only assume quantized energies. These quantized orbits are called the Landau Levels, and the mechanism giving rise to the quantized Hall conductivity is related to how many of these orbits (that is, how many of these energy levels) are populated with electrons. The integer number appearing in the conductivity formula is the number of such levels that are occupied, and it is what is called a topological number (that is, protected by symmetry against a vaste range of possible perturbations, among which many body interactions and disorder). These occupied states are metallic and only live on the surface of the system, disappearing in the bulk.

A phenomenon which is similar to the “standard” Hall effect, dubbed the anomalous Hall effect, can arise even in the absence of a magnetic field (what explain its “anomaly”!). Namely, this effect arises in magnetic materials where a spin-orbit interaction is present. The anomalous Hall effect can be due either to extrinsic mechanisms, that is disorder in crystals, that makes scattering of the electrons with impurities asymmetric with respect to the incident direction and giving rise to a transverse (Hall!) current, or to an intrinsic mechanism that is based on the gain of an anomalous term in the electrons velocity. Furthermore, if the exchange and Rashba coupling are strong enough and the Fermi energy is placed in the middle of a gap, then the conductivity only receives contributions from a number of in-gap states that live on the surface of the system, and the Hall conductance is found once again to be quantized in units of e2/h. In this case, as we should have learnt by now, one speaks of quantum anomalous Hall effect.

A different Hall effect that is relevant for spintronic and graphene physics (!) is the spin Hall effect. In its “classical” manifestation, the spin Hall effect consists in the transverse accumulation of electrons carrying opposite spin orientations when an electric field is applied. It can only have an extrinsic nature, that is, it manifests as an asymmetric (with respect to spin orientation) scattering probability when electrons are scattered by spin-orbit- active impurities. The quantum version of this effect is the celebrated quantum spin Hall effect, first predicted in graphene by Kane and Mele, for which the intrinsic spin-orbit interaction of the carbon atoms is responsibile. This state of matter is gapped in the bulk and supports the transport of spin in metallic states that propagate at the surface.

Getting to the one-bilion-dollars question: why do we care? A quantity of answers do fit. For instance, the classical Hall effect is typically used in experiments to derive the sign and density of the carriers, or, more commonly, their mobility (out of which useful quantities can be deduced, such as the mean free path). As for the quantum Hall effect, among many possible applications, it has allowed for the modern definition of the standard for electrical resistance, based on the Hall resistance quantum given by h/e2. Also, the generation of dissipationless and very robust currents at the edges of systems supporting a quantum Hall regime finds vast applications in the field of transfer and storage of information. Finally, the spin Hall effect comes with the captivating promise of making one of the new frontiers for electronics -spintronics-, happen, that is control and transport of the spin degree of freedom.

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