Transport and criticality in topological systems and spin models

Abstract

This thesis presents work done on transport in topological insulators and graphene-based systems, and quantum criticality in one- and two-dimensional spin models. In particular we study the following: transport on surfaces of three-dimensional topological insulators in the presence of time-independent and time-dependent barriers, Majorana modes in a one-dimensional topological insulator in proximity with a $s$-wave superconductor, the phase diagram of the Hubbard model on a triangular lattice periodically driven by an in-plane electric field, quantum criticality of a Ising model with three-spin interactions and a transverse field, the origin of spin-orbit coupling in a graphene-WSe$_2$ heterostructure, and a prediction of edge states in trilayer graphene. In the first chapter, we give a brief introduction to the concepts relevant to the rest of the thesis such as topological insulators, superconductivity, Floquet theory for studying periodically driven Hamiltonians, graphene and the spin-orbit coupling terms, quantum phase transitions, and the transverse field Ising model. In the second chapter, we consider a thin-film topological insulator (TI) in which the top and the bottom surfaces are separated by a small distance. The hybridisation between the states on the top and bottom surfaces of this system is characterized by a coupling strength $\lambda$. We study the various features of transport when a potential or magnetic barrier is applied on one of the surfaces. We find that the conductance $G$ of this system oscillates with the barrier strength with the period of oscillations varying with the coupling strength $\lambda$. This gives us an indirect way of estimating the extent of hybridisation in such thin films by looking at the conductance. The period of these oscillations changes from $2\pi$ to $\pi$ as $\lambda$ increases from zero to a value close to the energy of the incident electrons. Next we study the effects of a magnetic barrier, and we find that the conductance reaches a non-zero and $\lambda$-dependent value as the barrier strength is increased. This is in sharp contrast to the behavior of the conductance of a single TI surface where it approaches zero with increasing magnetic barrier strength. We also find oscillations in the case of a magnetic barrier for large barrier widths. The period of these oscillations depends on $\lambda$. In the third chapter, we consider a similar magnetic barrier whose strength is periodically driven in time. We explore the behaviour of the conductance as a function of the driving parameters. Such a barrier can be realised by shining linearly polarised light over a region of width $L$ on the surface of a TI. We find that the conductance of this system exhibits a number of interesting features like prominent peaks and dips as the parameters of the system are varied. This also paves the way to have an optical (electromagnetic) control over the electrical current in such junctions where we can go from a high-conductance regime to a low-conductance regime or vice versa by tuning the amplitude and frequency of the light. We also see that this system can act as a frequency detector or an optically controlled switch as a function of the incident energy of the electron. In the fourth chapter, we consider a model of a TI which is now constricted to a narrow and long strip running along the $x-$direction. We study what happens to the Majorana modes when such a system is placed in proximity to an $s$-wave superconductor. This model hosts a spin-dependent chirality and only has a right-moving spin-up and a left-moving spin-down branch. We find that this leads to a number of unusual features, such as only one zero energy Majorana mode at each end of a finite system, a single Andreev bound state at a Josephson junction instead of two states, and multiple Shapiro steps for particular frequencies of an AC driving. In the fifth chapter, we study a Hubbard model on a triangular lattice at half-filling in the limit of large interaction. At half-filling, this is known to describe a Heisenberg spin Hamiltonian with equal nearest-neighbour couplings. We study the effects of driving this system periodically with an in-plane electric field. Taking the driving to be the perturbation, we find, using Floquet perturbation theory, that the effective Hamiltonian up to third order has two-spin Heisenberg couplings with different magnitudes in the three different directions of the triangular lattice. We also get a three-spin interaction chiral term in the third order with its sign being opposite on up- and down-pointing triangles. We study the ground state phase diagram as a function of the three couplings using exact diagonalization. We find that driving leads to new phases in the system apart from the spiral phase. We have three collinear ordered phases, one coplanar ordered phase, and three disordered (spin-liquid) phases. These phases are distinguished by looking at the peaks of the static spin structure function $S(\vec{q})$ in the Brillouin zone, the ground state fidelity susceptibility, the minimum value of the correlation function $C(\vec{r})$ in real space, and the crossings of the energies of the ground state and first excited state. In the sixth chapter, we consider a one-dimensional Ising model with a three-spin interaction with a transverse field of magnitude $h$. We find that this model has duality and a second-order phase transition at the self-dual point $h=1$. We find from finite-size scaling that the correlation length exponent $\nu$ is close to $0.8$ in this model. Having a dynamical critical exponent $z=1$ and a central charge $c=1$, we find that the model displays weak universality and lies somewhere in the middle of the Ashkin-Teller line of models, with the two extreme limits of the line being the transverse field Ising and four-state Potts models. Unlike the transverse Ising model, our model is non-integrable, with the level spacing statistics being governed by the Wigner-Dyson Gaussian orthogonal ensemble. We also find that this model has a subset of zero energy states which are rather special as they are independent of the value of $h$ and have very low entanglement entropy compared to the states in the neighbourhood of the energy eigenvalues. These states are quantum many-body scars and they violate the eigenstate thermalisation hypothesis (ETH). Chapters $7.1$ and $7.2$ describe works done in collaboration with some experimental groups. In Chapter $7.1$, we study the system of graphene-WSe$_2$ heterostructure where we have a strong proximity-induced spin-orbit coupling. The quantum Shubnikov-de Haas (SdH) oscillations observed experimentally show a beating implying the presence of two closely spaced frequencies. The energy dispersion thus extracted is then studied theoretically using an effective Hamiltonian with all possible spin-orbit couplings present. The Fermi velocity of the sample is about $1.5$ times that of pristine graphene. The data fitting and perturbation calculations show that the spin-splitting energy of nearly $5$ meV comes dominantly from the valley-Zeeman and Rashba spin-orbit couplings in the system. In chapter $7.2$, we study a system of trilayer graphene under the influence of a perpendicular electric field. The non-local and local resistance measurements done in this system show a scaling relation given by $R_{NL} \sim R_{L}^{\alpha}$ with $\alpha =1$ for a range of values of the displacement field. The value of $\alpha$ is seen to be close to 1 up to temperatures around which the bulk gap closes in the system. This strongly suggests that the transport is dominated in this sample by edge modes. We study a theoretical model for trilayer graphene with displacement fields consistent with the experiments, and show that in this regime the valley Chern number is non-zero with a large value of $2.5$ for a given valley and a given spin. We also show that the system host zig-zag edge modes for the displacement fields of interest, although they are not protected from backscattering. A simple resistor circuit model that mimics the inter-valley scattering through dissipation then explains the linear relation between the non-local and local resistances. At the end, we summarise our results and discuss possible future studies in these areas of research