Question

I am reading about the Quantum Hall Effect. In a course, they wrote that How does one get a Hall effect? The key is to break time-reversal symmetry. A flowing current breaks time-reversal symmetry, while an electric field doesn’t. Hence, any system with a Hall effect must somehow break time-reversal symmetry

And I don't know what it means. About TR symmetry means that the same physical laws describe a considered system with t→−t
. Also, I know that with this transformation E→E
, which means that TR symmetry doesn't change the sign of the electric field. But why does flowing current break this symmetry?

Answer 1

The flowing current breaks time-reversal symmetry because, unlike an electric field, the associated movement of charges and the magnetic fields they create would reverse direction under time reversal, whereas the electric field would not. This breaking of symmetry is evident in the Hall effect, where the direction of current flow and the resulting magnetic force on charges lead to a measurable Hall voltage.

Step-by-step explanation:

The Quantum Hall Effect is a phenomenon that arises when you have a two-dimensional electron system at very low temperatures in a strong magnetic field. When discussing time-reversal symmetry (TR symmetry), we look at whether the laws of physics remain unchanged if we reverse the direction of time (i.e., t → -t). An electric field is invariant under time reversal, meaning it doesn't change direction if time is reversed; however, a current, which involves the movement of charges, would reverse direction if time did. This actual movement and the associated magnetic fields it generates do not stay the same when time is reversed, hence the flowing current breaks time-reversal symmetry.

In the context of the Hall effect, charge carriers (electrons or holes) flowing through a conductor in the presence of a perpendicular magnetic field experience a force due to the magnetic field. This causes a build-up of charges on one side and creates a measurable voltage across the conductor; this is the Hall emf. The presence of this induced emf indicates that TR symmetry is broken because the direction of current flow determines the direction of charge accumulation, and thus, the direction of the Hall voltage, which would reverse if time were reversed.

Therefore, the flowing current's ability to break TR symmetry is directly linked to the creation of a magnetic field, which is not invariant under time reversal. Hence, any system exhibiting the Hall effect inherently violates time-reversal symmetry.