Ricci-flat graphs with girth at least five

A graph is called Ricci-flat if its Ricci-curvatures vanish on all edges. Here we use the definition of Ricci-cruvature on graphs given in [Lin-Lu-Yau, Tohoku Math., 2011], which is a variation of [Ollivier, J. Funct. Math., 2009]. In this paper, we classified all Ricci-flat connected graphs with girth at least five: they are the infinite path, cycle $C_n$ ($n\geq 6$), the dodecahedral graph, the Petersen graph, and the half-dodecahedral graph. We also construct many Ricci-flat graphs with girth 3 or 4 by using the root systems of simple Lie algebras.Comment: 14 pages, 15 figure

Universal transport near a quantum critical Mott transition in two dimensions

We discuss the universal transport signatures near a zero-temperature continuous Mott transition between a Fermi liquid (FL) and a quantum spin liquid in two spatial dimensions. The correlation-driven transition occurs at fixed filling and involves fractionalization of the electron: upon entering the spin liquid, a Fermi surface of neutral spinons coupled to an internal gauge field emerges. We present a controlled calculation of the value of the zero temperature universal resistivity jump predicted to occur at the transition. More generally, the behavior of the universal scaling function that collapses the temperature and pressure dependent resistivity is derived, and is shown to bear a strong imprint of the emergent gauge fluctuations. We further predict a universal jump of the thermal conductivity across the Mott transition, which derives from the breaking of conformal invariance by the damped gauge field, and leads to a violation of the Wiedemann-Franz law in the quantum critical region. A connection to organic salts is made, where such a transition might occur. Finally, we present some transport results for the pure rotor O(N) CFT.Comment: 27 pages, 16 figure

Bending crystals: Emergence of fractal dislocation structures

We provide a minimal continuum model for mesoscale plasticity, explaining the cellular dislocation structures observed in deformed crystals. Our dislocation density tensor evolves from random, smooth initial conditions to form self-similar structures strikingly similar to those seen experimentally - reproducing both the fractal morphologies and some features of the scaling of cell sizes and misorientations analyzed experimentally. Our model provides a framework for understanding emergent dislocation structures on the mesoscale, a bridge across a computationally demanding mesoscale gap in the multiscale modeling program, and a new example of self-similar structure formation in non-equilibrium systems.Comment: 4 pages, 4 figures, 5 movies (They can be found at http://www.lassp.cornell.edu/sethna/Plasticity/SelfSimilarity.html .) In press at Phys. Rev. Let

Slip Failure of Embankment on Soft Marine Clay

The results of an investigation carried out at a site before and after the slip failure of an embankment on soft marine clay are presented. An evaluation of the possible causes for the slip failure and the remedial measures taken after the failure are also included. It appears that overstressing of the ground might have led to the movement of the soft soil. The compressible soils were subjected to an increase in stress without a comparable increase in foundation shear strength. This increase in horizontal stress resulted in the horizontal displacement of the soil as evident by the presence of the tension cracks in the fill just before slip failure occurred