Particle-hole symmetry and transport properties of the flux state in   underdoped cuprates
  Transport properties are studied for the flux state with the gauge flux $\phi$ per plaquett, which may model the underdoped cuprates, with the emphasis on the particle-hole and parity/chiral symmetries.This model is reduced to the Dirac fermions in (2+1)D with a mass gap introduced by the antiferromagnetic (AF) long range order and/or the stripe formation. Without the mass gap, the Hall constant $R_H$ and the thermopower $S$ obey two-parameter scaling laws, and show the strong temperature dependence due to the recovery of the particle-hole symmetry at high temperature. The $x$-dependences of $\sigma_{xx} (\propto \sqrt{x})$ and $\sigma_{xy}$ (independent of $x$) are in a sharp contradiction with the experiments. (Here $x$ is the hole concentration.) Therefore there is no signature of the particle-hole symmetry or the massless Dirac fermions in the underdoped cuprates even above the Neel temperature $T_N$. With the mass gap introduced by the AF order, there occurs the parity anomaly for each of the Dirac fermions. However the contributions from different valleys and spins cancel with each other to result in no spontaneous Hall effect even if the time-reversal symmetry is broken with $\phi \ne \pi$. The effects of the stripes are also studied. The diagonal and vertical (horizontal) stripes have quite different influence on the transport properties. The suppression of $R_H$ occurs at low temperature only when (i) both the AF order and the vertical (horizontal) stripe coexist, and (ii) the average over the in-plane direction is taken. Discussions on the recent experiments are given from the viewpoint of these theoretical results. 
Masaru Onoda and Naoto Nagaosa