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Recently, there has been a considerable interest in various novel two-dimensional (2D) materials, such as graphene, topological insulators, etc. These materials host a plethora of exotic phenomena owing to their unconventional electronic structure. Physics of these 2D materials is understood fairly well, so a natural generalization is to assemble these materials into three-dimensional (3D) stacks. In this thesis, we study a number of multilayer systems, where the interlayer interaction plays a salient role.

We commence with studying graphene multilayers coupled via interlayer tunneling amplitude. We calculate the energy spectrum of the system in magnetic field B parallel to the layers. The parallel magnetic field leads to a relative gauge shift of the momentum spaces of the individual 2D layers. When the interlayer tunneling is introduced, we find the Landau levels. We observe two qualitatively distinct domains in the Landau spectrum and analyze them using semiclassical arguments. Then, we include electric field E perpendicular to the layers, and analyze the spectrum in the crossed-field geometry. If the fields are in resonance E = vB , where v is the velocity of carriers in graphene, the wave-functions delocalize in the direction along the field E. We compare this prediction to a tunneling spectroscopy study of a graphite mesa in the parallel magnetic field. Indeed, the tunneling spectrum displays a peak, which grows linearly with the applied magnetic field B, and is, thus, consistent with our theoretical analysis. Then, we move on to a discussion of Z2 topological insulators within the Shock-ley model. We generalize the one dimensional (1D) Shockley model by replacing atomic sites of the original model by the 2D Rashba spin-orbit layers. We analyze surface states of a topological insulator using a construction of vortex lines in the 3D momentum space. We also study a topological insulator in a thin film geometry, where the opposite surface states are coupled by the tunneling amplitude. We cal-culate the tunneling current between the opposite surfaces and a spin polarization of the current as a function of the in-plane magnetic field. We conclude with studying a novel chiral order in cuprates. We construct a helical interlayer pattern of loop-currents. The interlayer magnetic coupling and magnetoelectric effect lead to optical gyrotropy.