We first present a simple proof that for any bipartite lattice at half filling the RKKY interaction is antiferromagnetic between impurities on opposite (i.e., A and B) sublattices and is ferromagnetic between impurities on the same sublattices. This result is valid on all length scales. We then focus on the honeycomb lattice and examine the theorem in the long distance limit by performing the low energy calculation using Dirac electrons. To find the universal (cutoff free) result we perform the calculation in smooth cutoff schemes, as we show that the calculation based on a sharp cutoff leads to wrong results. We also find the long distance behavior of the RKKY interaction between "plaquette" impurities in both coherent and incoherent regimes.

The study of natural images and how our brain processes them has been an area of intense research in neuroscience, psychology, and computer science. We introduced a unique approach to studying natural images by decomposing images into a hierarchy of layers at different logarithmic intensity scales and mapping them to a quasi-2D magnet. The layers were in different phases: "cold" and ordered at large-intensity scales, "hot" and disordered at small-intensity scales, and going through a second-order phase transition at intermediate scales. There was a single "critical" layer in the hierarchy that exhibited long-range correlation similar to that found in the 2D Ising model of ferromagnetism at the critical temperature. We also determined the interactions between layers mapped from natural images and found mutual inhibition that generated locally "frustrated" antiferromagnetic states. Almost all information in natural images was concentrated in a few layers near the phase transition, which has biological implications and also points to the hierarchical origin of scale invariance in natural images.critical point | vision | generative models O ur visual system evolved to survive in nature with scenes of mountains, rivers, trees, and other animals (1). The neural representations of visual inputs are related to their statistical structure (1-3). Structures in nature come in a hierarchy of sizes that cannot be separated, a signature of scale invariance, which also occurs near a critical point in many physical systems. The classic example of a critical point is a uniaxial ferromagnetic system going through a second-order phase transition in a zero magnetic field by increasing temperature. At the critical point, the system loses its magnetization due to thermal fluctuations. There are large regions ("islands") that are magnetized in one direction but are surrounded by large regions ("seas") that are magnetized in the opposite direction. The seas themselves are embedded in bigger islands, ad infinitum. The total magnetization is zero, but the correlation length diverges, which is visualized by growth of the sizes of seas and islands with the system size. At the critical point, the system is free of a length scale because fluctuations occur at scales of all lengths. The infinite correlation length is thus intricately linked with scale invariance. The scale invariance in natural images was first characterized by the 1/f 2 spatial power spectrum of pixel intensities (2). Here, we study scaling properties of natural images at a deeper level by finding a hierarchy of statistical structures, in which the scale invariance emerges near a second-order phase transition.Images are preprocessed in the retina by a complex network with ∼55 distinct cell types in mammals (5). The cerebral cortex receives a spatiotemporal stream of spikes that contain all the information in the visual inputs that has been coded by the retina. Understanding the hierarchies of statistical structures in natural images is essential for better understanding how ...

We study the Kondo-Heisenberg model on the honeycomb lattice at half-filling. Due to the vanishing of the density of states at the fermi level, the Kondo insulator disappears at a finite Kondo coupling even in the absence of the Heisenberg exchange. We adopt a large-N formulation of this model and use the renormalization group machinery to study systematically the second order phase transition of the Kondo insulator (KI) to the algebraic spin liquid (ASL). We note that neither phase breaks any physical symmetry, so that the transition is not described by the standard Ginzburg-Landau-Wilson critical point. We find a stable Lorentz-invariant fixed point that controls this second order phase transition. We calculate the exponent ν of the diverging length scale near the transition. The quasi-particle weight of the conduction electron vanishes at this KI-ASL fixed point, indicating non-Fermi liquid behavior. The algebraic decay exponent of the staggered spin correlation is calculated at the fixed point and in the ASL phase. We find a jump in this exponent at the transition point.

Data sets with high dimensionality such as natural images, speech, and text have been analyzed with methods from condensed matter physics. Here we compare recent approaches taken to relate the scale invariance of natural images to critical phenomena. We also examine the method of studying high-dimensional data through specific heat curves by applying the analysis to noncritical systems: 1D samples taken from natural images and 2D binary pink noise. Through these examples, we concluded that due to small sample sizes, specific heat is not a reliable measure for gauging whether high-dimensional data are critical. We argue that identifying order parameters and universality classes is a more reliable way to identify criticality in high-dimensional data.