Instabilities near the onset of spin density wave order in metals

We discuss the low energy theory of two-dimensional metals near the onset of spin density wave order. It is well known that such a metal has a superconducting instability induced by the formation of spin-singlet pairs of electrons, with the pairing amplitude changing sign between regions of the Fermi surface connected by the spin density wave ordering wavevector. Here we review recent arguments that there is an additional instability which is nearly as strong: towards the onset of a modulated bond order which is locally an Ising-nematic order. This new instability is a consequence of an emergent "pseudospin" symmetry of the low energy theory---the symmetry maps the sign-changing pairing amplitude to the bond order parameter.Comment: 14 pages, 9 figures; contribution to the special issue of the New Journal of Physics on "Fermiology of Cuprates", edited by Mike Norman and Cyril Prous

Solving the Tower of Hanoi with Random Moves

We prove the exact formulae for the expected number of moves to solve several variants of the Tower of Hanoi puzzle with 3 pegs and n disks, when each move is chosen uniformly randomly from the set of all valid moves. We further present an alternative proof for one of the formulae that couples a theorem about expected commute times of random walks on graphs with the delta-to-wye transformation used in the analysis of three-phase AC systems for electrical power distribution