Exact Realization of SO s 5 d Symmetry in Extended Hubbard Models

m S. C. Zhang [1,2] recently conjectured that highTc cuprate compounds possess an approximate SO(5) s metry. His theory aims to explain the proximity o superconducting (SC) and antiferromagnetic (AF) phas in the phase diagram, and to account for the low-ener excitations as approximate SO(5) Goldstone mode Antiferromagnetism and superconductivity are unified one grand order parameter field smx , my, mz, ReC, Im Cd, behaving as a five-component vector, where the fi three elements are Cartesian components of the stagg magnetization andC is a spin-singlet SC order paramete (Here “ReC” ; 12 sC 1 C yd, etc.) In this picture, small symmetry-breaking terms drive the system in a “supersp flop” between antiferromagnetism and superconductivi just as, in a magnet with approximate SO(3) symmetr competing spin-space anisotropies and external field c drive a “spin-flop” transition between magnetic orde along thez axis and in thexy plane [1]. The SO(5) theory, while positing an intimate relation ship between SC and AF order, does not imply that t pairing mechanism is AF fluctuations [3,4]. Rather, quantifies the notion (also relevant to superfluid 3He) that there need not be a sharp difference between interacti mediated by magnetic and “charge” (number) fluctuation I pass over Zhang’s specific mechanism (whereby the s tem accommodates doping by switching from the AF sta to a symmetry-related SC state which has a different p ticle number), for SO(5) symmetry can be valid even another sort of perturbation is found responsible for th symmetry breaking and the AF-SC transition. The 41 meV mode observed in spin-flip neutron scatte ing on YBa2Cu3O7 [5] is interpreted as a Goldstone mod of SO(5) with a gap due to the symmetry-violating term analogous to the anisotropy gap in a spin-wave branch the unixial magnet [1]. These excitations are created “ p̂y” operators [6] [SO(5) generators that mix magnet and SC components] [7]. They are charged bosons w the quantum numbers of “preformed” Cooper pairs, a presumably carry the current in the “normal” metal [1]; has been speculated [9] that this explains the linear te perature dependence of the normal-state resistivity. To the extent that SO(5)-violating terms are small (a in Zhang’s phase diagram “ A” [1]), relations between AF