Spin-charge deconfinement and emergent ${\mathrm{AdS}}_3$ structure from a self-consistent dressing of Fierz-complete $(1+1)$d Dirac fermions

Abstract

Building on a recent derivation of spin-charge separation in $(1+1)$d paired Dirac fermions~\cite{Haddad2024}, we develop a self-consistent dressing $\ddot{I}ˆ(x)=U(x)\ddot{I}‡(x)$ for the full Fierz-complete four-fermion model, extending that result and providing a more detailed resolution of the chiral-difermion phase structure. A key feature of this approach is that the composite connection $A_{\hat{I}}{¼}^{\mathrm{dress}}=i({\partial }_{\hat{I}}¼U)U^{-1}$ encodes obstructions to local trivialization of the Dirac operator, i.e., the degree to which the background can be absorbed into the dressing. Using this fact, we prove a trivialization theorem under which three nonperturbative constructions are unified: spin-charge separation in correlated fermion systems, half-infinite Wilson-line dressing in gauge theory, and the holonomy of flat connections. Our approach shows that the three regimes of our model (chiral, difermion, intermediate), are then tied together by an emergent $\mathfrak{sl}(2,\mathbb{R})$ gauge field that binds the spin and charge degrees of freedom. In particular, the chiral-difermion transition is a deconfinement transition for these degrees of freedom, diagnosed by closed boost-sector Wilson loops that develop an area law in the chiral phase for which we compute the associated string tension. This provides a concrete realization of the conjectured Faddeev--Niemi link between spin-charge separation and confinement. We close with a unifying geometric picture in which the order-parameter manifold takes hyperbolic form $\ddot{I}{}^2-|\hat{I}”{|}^2=\ddot{I}{ƒ}^2$, promoted to ${\mathrm{AdS}}_3\cong \mathrm{SL}(2,\mathbb{R})$ on inclusion of the charge and difermion phases. The structural matching to the kinematic stage of ${\mathrm{AdS}}_3/{\mathrm{CFT}}_2$ is identified explicitly, with the conjecture that the dressed model realizes the inverse Pohlmeyer reduction of the ${\mathrm{AdS}}_3$ sigma model.

81 pages, 3 figures