Necessary and sufficient conditions for flat bands in MM-dimensional NN-band lattices with complex-valued nearest-neighbour hopping

We formulate the necessary and sufficient conditions for the existence of dispersionless energy eigenvalues (so-called `flat bands') and their associated compact localized eigenstates in $M$-dimensional tight-binding lattices with $N$ sites per unit cell and complex-amplitude nearest-neighbour tunneling between the lattice sites. The degrees of freedom $M$ can be traded for longer-range complex hopping in lattices with reduced dimensionality. We show the conditions explicitly for $(M = 1, N\leq4)$, $(M = 2, N = 2,3)$, and $(M = 3, N = 2,3)$, and outline their systematic construction for arbitrary $N$, $M$. If and only if the conditions are satisfied, then the system has one or more flat bands. By way of an example, we obtain new classes of flat band lattice geometries by solving the conditions for the lattice parameters in special cases.Comment: 7 pages, 4 figure

Chiral Flat Bands: Existence, Engineering and Stability

We study flat bands in bipartite tight-binding networks with discrete translational invariance. Chiral flat bands with chiral symmetry eigenenergy E = 0 and host compact localized eigenstates for finite range hopping. For a bipartite network with a majority sublattice chiral flat bands emerge. We present a simple generating principle of chiral flat band networks and as a showcase add to the previously observed cases a number of new potentially realizable chiral flat bands in various lattice dimensions. Chiral symmetry respecting network perturbations - including disorder and synthetic magnetic fields - preserve both the flatband and the modified compact localized states. Chiral flatbands are spectrally protected by gaps, and pseudogaps in the presence of disorder due to Griffiths effects

Competing Antiferromagnetic and Spin-Glass Phases in a Hollandite Structure

We introduce a simple lattice model with Ising spins to explain recent experimental results on spin freezing in a hollandite-type structure. We argue that geometrical frustration of the lattice in combination with nearest-neighbour antiferromagnetic (AFM) interactions is responsible for the appearance of a spin-glass phase in presence of disorder. We investigate this system numerically using parallel tempering. The model reproduces the magnetic behaviour of oxides with hollandite structure, such as $\alpha-\text{MnO}_2$ and presents a rich phenomenology: in absence of disorder three types of ground states are possible, depending on the relative strength of the interactions, namely AFM ordered and two different disordered, macroscopically degenerate families of ground states. Remarkably, for sets of AFM couplings having an AFM ground state in the clean system, there exists a critical value of the disorder for which the ground state is replaced by a spin-glass phase while maintaining all couplings AFM. To the best of our knowledge this is the only existing model that presents this kind of transition with short-range AFM interactions. We argue that this model could be useful to understand the relation between AFM coupling, disorder and the appearance of a spin-glass phase.Comment: 8 pages, 7 figure

Multifractality of correlated two-particle bound states in quasiperiodic chains

We consider the quasiperiodic Aubry-Andr\'e chain in the insulating regime with localised single-particle states. Adding local interaction leads to the emergence of extended correlated two-particle bound states. We analyse the nature of these states including their multifractality properties. We use a projected Green function method to compute numerically participation numbers of eigenstates and analyse their dependence on the energy and the system size. We then perform a scaling analysis. We observe multifractality of correlated extended two-particle bound states, which we confirm independently through exact diagonalisation.Comment: 7 pages, 8 figure

Localization of spin waves in disordered quantum rotors

We study the dynamics of excitations in a system of $O(N)$ quantum rotors in the presence of random fields and random anisotropies. Below the lower critical dimension $d_{\mathrm{lc}}=4$ the system exhibits a quasi-long-range order with a power-law decay of correlations. At zero temperature the spin waves are localized at the length scale $L_{\mathrm{loc}}$ beyond which the quantum tunneling is exponentially suppressed $c \sim e^{-(L/L_{\mathrm{loc}})^{2(\theta+1)}}$. At finite temperature $T$ the spin waves propagate by thermal activation over energy barriers that scales as $L^{\theta}$. Above $d_{\mathrm{lc}}$ the system undergoes an order-disorder phase transition with activated dynamics such that the relaxation time grows with the correlation length $\xi$ as $\tau \sim e^{C \xi^\theta/T}$ at finite temperature and as $\tau \sim e^{C' \xi^{2(\theta+1)}/\hbar^2}$ in the vicinity of the quantum critical point.Comment: 8 pages, 2 figures, revtex

Order Induced by Dilution in Pyrochlore XY Antiferromagnets

XY pyrochlore antiferromagnets are well-known to exhibit order-by-disorder through both quantum and thermal selection. In this paper we consider the effect of substituting non-magnetic ions onto the magnetic sites in a pyrochlore XY model with generally anisotropic exchange tuned by a single parameter $J^{\pm\pm}/J^\pm$. The physics is controlled by two points in this space of parameters $J^{\pm\pm}/J^\pm=\pm 2$ at which there are line modes in the ground state and hence an $O(L^2)$ ground state degeneracy intermediate between that of a conventional magnet and a Coulomb phase. At each of these points, single vacancies seed pairs of line defects. Two line defects carrying incompatible spin configurations from different vacancies can cross leading to an effective one-dimensional description of the resulting spin texture. In the thermodynamic limit at finite density, we find that dilution selects a state "opposite" to the state selected by thermal and quantum disorder which is understood from the single vacancy limit. The latter finding hints at the possibility that Er$_{2-x}$Y$_x$Ti$_2$O$_7$ for small $x$ exhibits a second phase transition within the thermally selected $\psi_2$ state into a $\psi_3$ state selected by the quenched disorder.Comment: 14 pages, 12 figure

Avalanches and hysteresis in frustrated superconductors and XY-spin-glasses

We study avalanches along the hysteresis loop of long-range interacting spin-glasses with continuous XY-symmetry - which serves as a toy model of granular superconductors with long-range and frustrated Josephson couplings. We identify sudden jumps in the $T=0$ configurations of the XY-phases, as an external field is increased. They are initiated by the softest mode of the inverse susceptibility matrix becoming unstable, which induces an avalanche of phase updates (or spin alignments). We analyze the statistics of these events, and study the correlation between the non-linear avalanches and the soft mode that initiates them. We find that the avalanches follow the directions of a small fraction of the softest modes of the inverse susceptibility matrix, similarly as was found in avalanches in jammed systems. In contrast to the similar Ising spin-glass (Sherrington-Kirkpatrick) studied previously, we find that avalanches are not distributed with a scale-free power law, but rather have a typical size which scales with the system size. We also observe that the Hessians of the spin-glass minima are not part of standard random matrix ensembles as the lowest eigenvector has a fractal support.Comment: 17 pages, 12 figure

Incommensurate, helical spin ground states on the Hollandite lattice

We present a model of classical Heisenberg spins on a Hollandite lattice, which has been developed to describe the magnetic properties of $\alpha$-MnO$_2$ and similar compounds. The model has nearest neighbor interacting spins, however the strength and the sign of spin-spin interactions is anisotropic and depends on the nature of the bonds. Our analysis shows that the Hollandite lattice supports four different incommensurate and helical magnetic ground states depending on the relative strengths and signs of spin-spin interactions. We show that the incommensurate helical ground states appear due to the geometrical frustration present in the model. We demonstrate that each of the four helical incommensurate magnetic phases are continuously connected to four different collinear antiferromagnetic ground states as the strength of spin-spin interaction along some bonds is increased. The present results give support to the presence of helical states that have been previously suggested experimentally for Hollandite compounds. We provide an in-depth analysis of the magnetic form factors for each helical phase and describe how it could be used to identify each of these phases in neutron diffraction experiments.Comment: 11 pages, 8 figure