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We investigate the effects of magnetic and nonmagnetic impurities on the two-dimensional surface states of three-dimensional topological insulators (TIs). Modeling weak and strong TIs using a generic four-band Hamiltonian, which allows for a breaking of inversion and time-reversal symmetries and takes into account random local potentials as well as the Zeeman and orbital effects of external magnetic fields, we compute the local density of states, the single-particle spectral function, and the conductance for a (contacted) slab geometry by numerically exact techniques based on kernel polynomial expansion and Greens function approaches. We show that bulk disorder refills the suface-state Dirac gap induced by a homogeneous magnetic field with states, whereas orbital (Peierls-phase) disorder perserves the gap feature. The former effect is more pronounced in weak TIs than in strong TIs. At moderate randomness, disorder-induced conducting channels appear in the surface layer, promoting diffusive metallicity. Random Zeeman fields rapidly destroy any conducting surface states. Imprinting quantum dots on a TIs surface, we demonstrate that carrier transport can be easily tuned by varying the gate voltage, even to the point where quasi-bound dot states may appear.
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Our understanding of topological insulators is based on an underlying crystalline lattice where the local electronic degrees of freedom at different sites hybridize with each other in ways that produce nontrivial band topology, and the search for material systems to realize such phases have been strongly influenced by this. Here we theoretically demonstrate topological insulators in systems with a random distribution of sites in space, i. e., a random lattice. This is achieved by constructing hopping models on random lattices whose ground states possess nontrivial topological nature (characterized e. g., by Bott indices) that manifests as quantized conductances in systems with a boundary. By tuning parameters such as the density of sites (for a given range of fermion hopping), we can achieve transitions from trivial to topological phases. We discuss interesting features of these transitions. In two spatial dimensions, we show this for all five symmetry classes (A, AII, D, DIII and C) that are known to host nontrivial topology in crystalline systems. We expect similar physics to be realizable in any dimension and provide an explicit example of a $Z_2$ topological insulator on a random lattice in three spatial dimensions. Our study not only provides a deeper understanding of the topological phases of non-interacting fermions, but also suggests new directions in the pursuit of the laboratory realization of topological quantum matter.
We consider extended Hubbard models with repulsive interactions on a Honeycomb lattice and the transitions from the semi-metal phase at half-filling to Mott insulating phases. In particular, due to the frustrating nature of the second-neighbor repulsive interactions, topological Mott phases displaying the quantum Hall and the quantum spin Hall effects are found for spinless and spinful fermion models, respectively. We present the mean-field phase diagram and consider the effects of fluctuations within the random phase approximation (RPA). Functional renormalization group analysis also show that these states can be favored over the topologically trivial Mott insulating states.
The traditional concept of phase transitions has, in recent years, been widened in a number of interesting ways. The concept of a topological phase transition separating phases with a different ground state topology, rather than phases of different symmetries, has become a large widely studied field in its own right. Additionally an analogy between phase transitions, described by non-analyticities in the derivatives of the free energy, and non-analyticities which occur in dynamically evolving correlation functions has been drawn. These are called dynamical phase transitions and one is often now far from the equilibrium situation. In these short lecture notes we will give a brief overview of the history of these concepts, focusing in particular on the way in which dynamical phase transitions themselves can be used to shed light on topological phase transitions and topological phases. We will go on to focus, first, on the effect which the topologically protected edge states, which are one of the interesting consequences of topological phases, have on dynamical phase transitions. Second we will consider what happens in the experimentally relevant situations where the system begins either in a thermal state rather than the ground state, or exchanges particles with an external environment.
The modern theory of electric polarization has recently been extended to higher multipole moments, such as quadrupole and octupole moments. The higher electric multipole insulators are essentially topological crystalline phases protected by underlying crystalline symmetries. Henceforth, it is natural to ask what are the consequences of symmetry breaking in these higher multipole insulators. In this work, we investigate topological phases and the consequences of symmetry breaking in generalized electric quadrupole insulators. Explicitly, we generalize the Benalcazar-Bernevig-Hughes model by adding specific terms in order to break the crystalline and non-spatial symmetries. Our results show that chiral symmetry breaking induces an indirect gap phase which hides corner modes in bulk bands, ruining the topological quadrupole phase. We also demonstrate that quadrupole moments can remain quantized even when mirror symmetries are absent in a generalized model. Furthermore, it is shown that topological quadrupole phase is robust against a unique type of disorder presented in the system.
Topological insulators in three dimensions are characterized by a Z2-valued topological invariant, which consists of a strong index and three weak indices. In the presence of disorder, only the strong index survives. This paper studies the topological invariant in disordered three-dimensional system by viewing it as a super-cell of an infinite periodic system. As an application of this method we show that the strong index becomes non-trivial when strong enough disorder is introduced into a trivial insulator with spin-orbit coupling, realizing a strong topological Anderson insulator. We also numerically extract the gap range and determine the phase boundaries of this topological phase, which ?ts well with those obtained from self-consistent Born approximation (SCBA) and the transport calculations.
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