N=1 Supersymmetric Spin-Charge Separation in effective gauge theories of planar magnetic superconductors

We present a N=1 Supersymmetric extension of a spin-charge separated effective $SU(2)\times U_S(1)$ `particle-hole' gauge theory of excitations about the nodes of the gap of a d-wave planar magnetic superconductor. The supersymmetry is achieved without introducing extra degrees of freedom, as compared to the non-supersymmetric models. The only exception, the introduction of gaugino fieds, finds a natural physical interpretation as describing interlayer coupling in the statistical model. The low-energy continuum theory is described by a relativistic (2+1)-dimensional supersymmetric $CP^1$ $\sigma$-model with Gross-Neveu-Thirring-type four-fermion interactions. We emphasize the crucial r\^ole of the $CP^1$ constraint in inducing a non-trivial dynamical mass generation for fermions (and thus superconductivity), in a way compatible with manifest N=1 supersymmetry. We also give a preliminary discussion of non-perturbative effects. We argue that supersymmetry suppresses the dangerous for superconductivity instanton contributions to the mass of the perturbatively massless gauge boson of the unbroken U(1) subgroup of SU(2). Finally, we point out the possibility of applying these ideas to effective gauge models of spin-charge separation in one-space dimensional superconducting chains of holons, which, for example, have recently been claimed to be important in the stripe phase of underdoped cuprates.