We develop a continuum theory to model low energy excitations of a generic four-band time reversal invariant electronic system with boundaries. We propose a variational energy functional for the wavefunctions which allows us to derive natural boundary conditions valid for such systems. Our formulation is particularly suited for developing a continuum theory of the protected edge/surface excitations of topological insulators both in two and three dimensions. By a detailed comparison of our analytical formulation with tight binding calculations of ribbons of topological insulators modelled by the Bernevig-Hughes-Zhang (BHZ) Hamiltonian, we show that the continuum theory with a natural boundary condition provides an appropriate description of the low energy physics.

We address how the nature of linearly dispersing edge states of two-dimensional (2D) topological insulators evolves with increasing electron-electron correlation engendered by a Hubbard-like on-site repulsion U in finite ribbons of two models of topological band insulators. Using an inhomogeneous cluster slave-rotor mean-field method developed here, we show that electronic correlations drive the topologically nontrivial phase into a Mott insulating phase via two different routes. In a synchronous transition, the entire ribbon attains a Mott insulating state at one critical U that depends weakly on the width of the ribbon. In the second, asynchronous route, Mott localization first occurs on the edge layers at a smaller critical value of electronic interaction, which then propagates into the bulk as U is further increased until all layers of the ribbon become Mott localized. We show that the kind of Mott transition that takes place is determined by certain properties of the linearly dispersing edge states which characterize the topological resilience to Mott localization.

We propose a model to realize a fermionic superfluid state in an optical lattice circumventing the cooling problem. Our proposal exploits the idea of tuning the interaction in a characteristically low entropy state, a band-insulator in an optical bilayer system, to obtain a superfluid. By performing a detailed analysis of the model including fluctuations and augmented by a variational quantum Monte Carlo calculations of the ground state, we show that the superfluid state obtained has high transition temperature of the order of the hopping energy. Our system is designed to suppress other competing orders such as a charge density wave. We suggest a laboratory realization of this model via an orthogonally shaken optical lattice bilayer.

We present an analytical theory for the magnetic phase diagram for zigzag edge terminated honeycomb nanoribbons described by a Hubbard model with an interaction parameter U . We show that the edge magnetic moment varies as ln U and uncover its dependence on the width W of the ribbon. The physics of this owes its origin to the sensory organ like response of the nanoribbons, demonstrating that considerations beyond the usual Stoner-Landau theory are necessary to understand the magnetism of these systems. A first order magnetic transition from an anti-parallel orientation of the moments on opposite edges to a parallel orientation occurs upon doping with holes or electrons. The critical doping for this transition is shown to depend inversely on the width of the ribbon. Using variational Monte-Carlo calculations, we show that magnetism is robust to fluctuations. Additionally, we show that the magnetic phase diagram is generic to zigzag edge terminated nanostructures such as nanodots. Furthermore, we perform first principles modeling to show how such magnetic transitions can be realized in substituted graphene nanoribbons.PACS numbers: 73.22.Pr Interest and activity in magnetic nanostructures has been driven by their possible application in nanoelectronic/spintronic devices, with graphene based systems grabbing a significant fraction of the attention. Magnetism at the zigzag terminated edges of graphene has been studied by first principles calculations [11,12], and by a simplified effective Hubbard model [13][14][15][16][17][18][19] described by a hopping parameter t and a site-local repulsion U . Ref. 15 showed that the magnetism in graphene is robust to "shape disorder" of the nanostructure, while ref. 17 studied finite width graphene nanoribbons including the effects of doping. There are also studies of defect induced magnetism [20,21] and of magnetism of other nanostructures [14,22]. There are encouraging recent experimental signatures of magnetism [23][24][25], along with suggestions [26] that extraneous effects such as reconstruction would render the magnetism fragile.The origin of magnetic moment in zigzag edge terminated honeycomb nanostructures has been attributed to the edge states[2, 27, 28] -localized electronic states which have most weight at the edges and die exponentially in the bulk. [29][30][31] These states are of topological origin [32] and have been experimentally observed using scanning tunneling microscopy. [33,34] Magnetism at the edges is attributed to the Stoner mechanism(see, e.g., [35]) and is best discussed in terms of a Landau theory.[35] The ground state energy of the system is expressed aswhere M is the magnetic order parameter, a, b > 0 are positive constants that depend on the microscopics, and U c is a critical value of the on-site repulsion. For U < U c , the energy is minimum when M = 0, i. e., system is non-magnetic. At U = U c , there is a quantum phase transition to the magnetic state, and for U > U c one finds |M | ∼ √ U − U c . Stoner theory [36], based on linear response...