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As we know, the classical Hall effect arises when we apply an external magnetic field
perpendicular to the direction of the current density. One can define the Hall coefficient
RH = Ey/ (jxBz), where Ey is the induced electric field. For a single band the Hall number depends only on the sign and density of carriers. That is why, at low temperatures, it
can be used to determine the number of electrons or holes in electron and hole-like pockets
respectively. However, in systems with multiple bands or near Fermi-surface topological transitions, the Hall coefficient starts to deviate from its exclusive counting nature. This anomalous
behavior can be attributed to singularities in the density of states, called Van Hove singularities [1, 2]. Most materials show logarithmic-type Van Hove singularities, which correspond to
a logarithmic divergence of the density of states at the Lifshitz transition. In our work, we are
interested in higher order Van Hove singularities, which arise from more complicated Lifshitz
transitions. Such singularities can be observed, for example, in Sr3Ru2O7 which exhibits a more
complicated singularity when an external magnetic field is applied [3, 4]. Another example is
highly overdoped graphene and twisted bilayer graphene [5–8].
In the first section of this thesis (Chapters 1-2), we extend the well-known Chambers’
formula [9] to the case of a time-dependent electric field as well as a band with non-zero
Berry Curvature. To this end, we study the solution of the Boltzmann equation to first order
in the magnetic field B and to quadratic order in the electric field E2. We note that the
equations of motion used in this work are valid to leading order in the electric and magnetic
fields. We also neglect Zeeman splitting and spin orbit coupling which would result in more
complicated expressions. We then apply our generalized Chambers’ formula to study three
different systems. The first is a simple rectangular lattice [10], the second is a simple model
for highly doped graphene and the last is the Haldane model as an example of a system with
non-zero Berry Curvature. In all of the above cases, we study the effect of the high order Van
Hove singularities to the Hall coefficient of the system.
In the second section of the work (Chapter 3), we study the Reactive Hall response [11]
(T→0, ω→0) for the case of a rectangular lattice, to which we have added a spin-orbit
coupling in the form of Rashba interaction. To do this, we consider a magnetic field in the ydirection modulated by a one component wave vector q. Furthermore, we study the “screening”
(or slow) response, by first taking the ω→0 limit and then the q→0 limit. Through our
analysis, we analytically derive an equation that connects the Hall constant RH with the Drude
weight D of the system. We then apply our formula to study the behavior of the Hall constant
of our system for different values of the hopping elements. This gives us a qualitative image of
the sign change of the carriers close to a metal-insulator transition. Once again, we observe that
closely to the Van Hove singularity points, the behavior of the Hall constant rapidly changes.
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Hall effect in 2D spin-orbit and topological materials with Van Hove singularities
