We present a simple prescription to flatten isolated Bloch bands with a nonzero Chern number. We first show that approximate flattening of bands with a nonzero Chern number is possible by tuning ratios of nearest-neighbor and next-nearest-neighbor hoppings in the Haldane model and, similarly, in the chiral-π-flux square lattice model. Then we show that perfect flattening can be attained with further range hoppings that decrease exponentially with distance. Finally, we add interactions to the model and present exact diagonalization results for a small system at 1/3 filling that support (i) the existence of a spectral gap, (ii) that the ground state is a topological state, and (iii) that the Hall conductance is quantized.

Electron fractionalization is intimately related to topology. In one-dimensional systems, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall fluids, where time-reversal symmetry is broken by a large external magnetic field. Recently, there has been a tremendous effort in the search for examples of fractionalization in two-dimensional systems with time-reversal symmetry. In this letter, we show that fractionally charged topological excitations exist on graphenelike structures, where quasiparticles are described by two flavors of Dirac fermions and time-reversal symmetry is respected. The topological zero-modes are mathematically similar to fractional vortices in p-wave superconductors. They correspond to a twist in the phase in the mass of the Dirac fermions, akin to cosmic strings in particle physics. In low dimensional systems, the excitation spectrum sometimes contains quasiparticles with fractionalized quantum numbers. A famous example of fractionalization was obtained in one-dimension (1D) by Jackiw and Rebbi [1] and by Su, Schrieffer and Hegger [2]. They showed the existence of charge e/2 states, with polyacetelene as a physical realization of such phenomena. In these systems, a charge density wave develops and the ground state is two-fold degenerate. The fractionalized states correspond to mid-gap or zero-mode solutions that are sustained at the domain wall (a soliton) interpolating between the two-degenerate vacua.The fractional quantum Hall effect provides an example of fractionalization in two-dimensions (2D). Not only do the Laughlin quasiparticles have fractional charge [3], but they also have fractional (anyon) statistics [4,5]. Time-reversal symmetry (TRS) is broken due to the strong magnetic field, leaving as an outstanding problem the search for systems where fractionalization is realized without the breaking of TRS. The motivation for such a quest stems from speculations that fractionalization may play a role in the mechanism for high-temperature superconductivity [6,7,8]. Progress has been made on finding model systems, such as dimer models, in which monomers defects act as fractionalized (and deconfined, in the case of the triangular lattice) excitations.In this letter, we present a mechanism to fractionalize the electron in graphenelike systems that leaves TRS unbroken. The excitation spectrum of honeycomb lattices, which have been known theoretically for a few decades to be described by Dirac fermions [9,10], is now the subject of many recent studies since single and few atomic-layer graphite samples have been realized experimentally [11]. Quantum number fractionalization is intimately related to topology and here we find that a twist or a vortex in an order parameter for a mass gap gives rise to a single mid-gap state at zero energy. Such twist in the mass of the Dirac fermions in graphenelike structure is the analogous in 2+1 space-time dimensions of a cosmic string in 3+1-dimensions [12].The zero-...

We classify all possible 36 gap-opening instabilities in graphene-like structures in two dimensions, i.e., masses of Dirac Hamiltonian when the spin, valley, and superconducting channels are included. These 36 order parameters break up into 56 possible quintuplets of masses that add in quadrature, and hence do not compete and thus can coexist. There is additionally a 6th competing mass, the one added by Haldane to obtain the quantum Hall effect in graphene without magnetic fields, that breaks time-reversal symmetry and competes with all other masses in any of the quintuplets. Topological defects in these 5-dimensional order parameters can generically bind excitations with fractionalized quantum numbers. The problem simplifies greatly if we consider spin-rotation invariant systems without superconductivity. In such simplified systems, the possible masses are only 4 and correspond to the Kekulé dimerization pattern, the staggered chemical potential, and the Haldane mass. Vortices in the Kekulé pattern are topological defects that have Abelian fractional statistics in the presence of the Haldane term. We calculate the statistical angle by integrating out the massive fermions and constructing the effective field theory for the system. Finally, we discuss how one can have generically non-Landau-Ginzburg-type transitions, with direct transitions between phases characterized by distinct order parameters.I.

Using a Kac-Moody current algebra with U (1/1) × U (1/1) graded symmetry, we describe a class of (possibly disordered) critical points in two spatial dimensions. The critical points are labelled by the triplets (l, m, k j ), where l is an odd integer, m is an integer, and k j is real. For most such critical points, we show that there are infinite hierarchies of relevant operators with negative scaling dimensions. To interpret this result, we show that the line of critical points (1, 1, k j > 0) is realized by a field theory of massless Dirac fermions in the presence of U (N ) vector gauge-like static impurities. Along the disordered critical line (1, 1, k j > 0), we find an infinite hierarchy of relevant operators with negative scaling dimensions {∆ q |q ∈ IN}, which are related to the disorder average over the q-th moment of the single-particle Green function. Those relevant operators can be induced by non-Gaussian moments of the probability distribution of a mass-like static disorder. 71.55Jv,11.10Gh

We calculate nonperturbatively the multifractal scaling exponents of the critical wave function for two dimensional Dirac fermions in the presence of a random magnetic field. We do so by arguing that the multifractal scaling exponents can be expressed in terms of the free energy of random directed polymers on a Cayley tree. We find a weak-strong disorder transition for the multifractal scaling exponents of the wave function that is parallel to the freezing or glassy transition of the random polymer model. [S0031-9007(96)01600-6] PACS numbers: 02.50.Fz, 05.40.+ j, 64.60.Ak Starting with the work of Wegner, it has been proposed that at the localization transition the critical wave functions are multifractal [1]. Early predictions of multifractality in the theory of localization have relied on renormalization group arguments applied to various nonlinear s models. Although the validity of such perturbative approaches has been questioned [2], a consensus has developed on the multifractal nature of wave functions at the localization transition on the basis of numerical simulations [3].In this Letter, we will approach multifractality at the localization transition from the following perspective. We consider Dirac fermions moving in a plane and with a static random magnetic field normal to the plane. The pure model can be derived by taking the continuum limit of various tight-binding Hamiltonians. Examples are the Chalker-Coddington network model, d-wave superconductors, and degenerate semiconductors [4], for which, typically, the density of states for the pure system has a V -shaped singularity at the Fermi energy. General considerations on the conductivity tensor [5] and calculations of the inverse participation ratio [6] predict that the random magnetic field localizes wave functions with energies close to the Fermi energy, whereas exactly at the Fermi energy the (critical) wave functions remain extended. Thus the model describes a metal-insulator transition in two dimensions.A very useful property of this model is that the wave function at the Fermi energy can be calculated exactly for any realization of the random magnetic field b͑x͒ = 2 F͑x͒, and is given by c͑x͒ exp͓2F͑x͔͒ [7]. It has been suggested [8] that the multifractal properties of the critical wave functions c are closely related to those of the so-called prelocalized states of a two-dimensional metallic cavity [9]. We will view the Dirac fermion probability density~jcj 2 as a random surface with a distribution controlled by the disorder. We will explore the connection between spatial averages of powers of the density and the partition function for random directed polymers. From this point of view, we will be able to explore nonperturbatively the multifractal nature of the critical wave functions. We will assume that the random magnetic field b͑x͒ = 2 F͑x͒ is Gaussian distributed through P͓F͔~exp͕2Here the dimensionless variance g measures the disorder strength. Thus the total flux through the plane is constant in our model, although it may fluctuate l...

The multifractal scaling exponents are calculated for the critical wave function of a twodimensional Dirac fermion in the presence of a random magnetic field. It is shown that the problem of calculating the multifractal spectrum maps into the thermodynamics of a static particle in a random potential. The multifractal exponents are simply given in terms of thermodynamic functions, such as free energy and entropy, which are argued to be self-averaging in the thermodynamic limit. These thermodynamic functions are shown to coincide exactly with those of a Generalized Random Energy Model, in agreement with previous results obtained using Gaussian field theories in an ultrametric space.

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models.

A weakly disordered quasi-one-dimensional tight-binding hopping model with N rows is considered. The probability distribution of the Landauer conductance is calculated exactly in the middle of the band, where´ 0, and it is shown that a delocalization transition at this energy takes place if and only if N is odd. This even-odd effect is explained by level repulsion of the transmission eigenvalues.[S0031-9007(98)06686-1] PACS numbers: 71.10. Fd, 11.30.Rd, 72.15.Rn The existence of delocalization transitions in a disordered one-dimensional system is surprising, as it goes against the general wisdom that disordered systems in less than two dimensions are localized [1]. Nevertheless a delocalization transition in one dimension goes back to Dyson's work on models for a glass in 1953 [2,3]. Dyson's one-dimensional glass is related to a large variety of disordered systems: a one-dimensional tightbinding model with nearest-neighbor random hopping [3], a two-dimensional asymmetric random bond Ising model [4] which is equivalent to the one-dimensional random quantum Ising chain [5], one-dimensional random bond quantum XY models [6] and more generally random XYZ spin-1͞2 Heisenberg models [7], and narrow gap semiconductors [8]. These models are of current interest in view of their rich physics: new universality classes, logarithmic scaling, the existence of strong fluctuations calling for a distinction between average and typical properties. They might also be useful laboratories to address the problem of disorder induced quantum phase transitions in higher dimensions such as the plateau transition between insulating Hall states in the quantum Hall effect [9,10].The one-dimensional nearest-neighbor random hopping model is described by the Hamiltonianwhere the operators c y n and c n are creation and annihilation operators for spinless fermions, respectively, and the hopping parameter t n t 1 dt n consists of a nonrandom part t and a fluctuating part dt n . The fundamental symmetry of the Hamiltonian (1) that distinguishes it from one-dimensional systems with on-site disorder is the presence of a sublattice (or chiral) symmetry: particles can hop only from even-to odd-numbered sites. The energy´ 0 is special since it corresponds to a logarithmically diverging mean density of states [2]. Furthermore, there are several independent correlation lengths that diverge for´! 0 [7] indicating that the energy´ 0 represents a (disorder induced) quantum critical point [4,6,7,9]. In particular, at´ 0 the conductance exhibits large fluctuations superimposed on an algebraically decaying mean value [10]. By contrast, for nonzero energy the system described by Eq. (1) is noncriticial resulting in standard localized behavior: A typical sample is well characterized by ͗log g͘, which is proportional to L and has relatively small sample-tosample fluctuations.A different type of delocalization in one-dimensional disordered systems was considered recently by Hatano and Nelson [11], who considered a chain with onsite disorder and an imaginary ve...