Spin modulations in the Rashba-Hubbard chain -- a tensor network study

Abstract

Uniform spin-orbit coupling in an open single-band Hubbard chain is an exactly removable \(SU(2)\) gauge field at the Hamiltonian level, but not at the level of laboratory-frame spin correlations. We study this separation using density matrix renormalization group calculations for the repulsive one-dimensional Rashba-Hubbard chain. For open boundary conditions, a site-dependent spin rotation maps the model with hopping \(t\) and Rashba spin-orbit strength \(λ\) onto the ordinary Hubbard chain with renormalized hopping \(t_λ=\sqrt{t^2+λ^2}\). Consequently, charge and energy diagnostics are affected only through the bandwidth renormalization, which is quadratic in weak \(λ/t\). Spin correlations, however, respond already at linear order because the same transformation rotates the local spin basis by the wave vector \(k_{\rm so}=2\arctan(λ/t)\). We use DMRG to verify this observable consequence across the filling diagram of finite open chains. The filling structure follows the gauge-equivalent Hubbard model, whereas the spin structure factor shows the predicted spin-orbit sidebands. A dominant Hubbard-chain magnetic wave vector \(k_0\) is transformed into components at \(k_0\pm k_{\rm so}\), folded into the open-chain Brillouin zone. At half filling, where \(k_0=π\), the two sidebands fold onto a single, in-plane, spin spiral wave with \(k=π-k_{\rm so}<π\). Away from half filling, the incommensurate Hubbard spin response splits into two distinct spin-orbit-shifted components, producing a real-space beating pattern. Our results provide a filling-resolved tensor-network benchmark for the exactly removable limit of one-dimensional spin-orbit coupling, and establish a controlled reference point for ladders, multiorbital chains, rings, proximitized wires, and higher-dimensional Hubbard systems where spin-orbit coupling can no longer be gauged away.