Quantum phase diagram of the half filled Hubbard model with bond-charge interaction

Using quantum field theory and bosonization, we determine the quantum phase diagram of the one-dimensional Hubbard model with bond-charge interaction $X$ in addition to the usual Coulomb repulsion $U$ at half-filling, for small values of the interactions. We show that it is essential to take into account formally irrelevant terms of order $X$. They generate relevant terms proportional to $X^2$ in the flow of the renormalization group (RG). These terms are calculated using operator product expansions. The model shows three phases separated by a charge transition at $U=U_c$ and a spin transition at $U=U_s>U_c$. For $U<U_c$ singlet superconducting correlations dominate, while for $U>U_s$, the system is in the spin-density wave phase as in the usual Hubbard model. For intermediate values $U_c<U<U_s$, the system is in a spontaneously dimerized bond-ordered wave phase, which is absent in the ordinary Hubbard model with $X=0$. We obtain that the charge transition remains at $U_c=0$ for $X \neq 0$. Solving the RG equations for the spin sector, we provide an analytical expression for $U_s(X)$. The results, with only one adjustable parameter, are in excellent agreement with numerical ones for $X < t/2$ where $t$ is the hopping.