N=1 Supersymmetric Spin-Charge Separation in Effective Gauge Theories of Planar Magnetic Superconductors

Abstract

We present a N = 1 Supersymmetric extension of a spin-charge separated effective SU (2) × 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 σ-model with Gross-Neveu-Thirring-type four-fermion interactions. We emphasize the crucial rôle 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.

Nodal liquids in extendedt−Jmodels and dynamical supersymmetry

“…Below we shall only outline the main results. Some technical details on the formalism are given in [13] and reviewed in Appendix B. Since it has been argued that U S (1) is responsible for dynamical mass generation (and superconductivity) in the model of [3] we shall ignore the non-Abelian SU(2) interactions, keeping only the Abelian However since the latter argument is not rigorous, it would be desirable to supersymmetrise the full group in order to check the phenomenon of dynamical mass generation.…”

“…As discussed in detail in [13,23], and reviewed briefly in Appendix B, the conditions for N = 1 supersymmetric extensions of a CP 1 σ model is that the constraint is of the standard CP 1 form (41), supplemented by attractive four-fermion interactions of the Gross-Neveu type (52), whose coupling is related to the coupling constant of the kinetic z-magnon terms of the σ-model in a way such as to guarantee the balance between bosonic and fermionic degrees of freedom Specifically, in terms of component fields, the pertinent lagrangian reads:…”

“…Some comments are now in order: First, it is quite important to remark that in the model of [3], where the antiferromagnetic structure of the theory is encoded in a colour (non-Abelian) degree of freedom of the spin-charge separated composite electron operator (1) on a single lattice geometry, there is a matching between the bosonic (z spinon fields) and fermionic (Ψ holon fields) physical degrees of freedom, as required by supersymmetry, without the need for duplicating them by introducing "unphysical" degrees of freedom [13].…”

“…Since the supersymmetric version of the CP σ-model, of interest to us here, contains -as we discuss in Appendix B -both Gross-Neveu and Thirring (ψγ µ ψ) 2 interactions in it component form [13], we turn next our attention to a brief review of a renormalization group study of such mixed models.…”

FromN=1toN=2supersymmetries in 2+1 dimensions

“…Finally, this work might be used in the context of effective models for high-temperature (planar) superconductivity [14], where the initial N = 1 supermultiplets are built out of composites of spinons and holons in the spin-charge separation framework.…”