Inhomogeneous superconducting phases in the frustrated Kondo-Heisenberg chain

We use bosonization and renormalization group methods to determine the ground-state phase diagram of a one-dimensional frustrated Kondo-Heisenberg system consisting of a one-dimensional spin-$\frac{1}{2}$ Luttinger liquid coupled by a Kondo exchange interaction $J_K$ to a frustrated quantum antiferromagnetic Heisenberg chain, with a nearest-neighbor exchange coupling $J_1$ and a next-nearest-neighbor (frustrating) exchange interaction $J_2$. We analyze the interplay of quantum frustration in the antiferromagnetic chain with the Kondo exchange coupling $J_K$ with the Luttinger liquid. We discuss the structure of the phase diagram of this system as a function of the ratios $J_K/J_1$, $J_2/J_1$, and of the parameters of the Luttinger liquid. In particular, we discuss in detail the regimes in which a pair-density-wave state may be realized and its relation with the spin correlations in the frustrated antiferromagnetic chain.

Figure 1

Schematic phase diagram for the frustrated Kondo-Heisenberg chain. Here, $J_1$ and $J_2$, respectively, represent the nearest-neighbor and next-nearest-neighbor exchange interactions and $J_K$ the Kondo exchange interaction between the the spins of the frustrated Heisenberg chain and the 1D electronic system. PDW denotes the pair-density-wave superconducting phase, FSL the fractionalized spin liquid [controlled by the Toulouse fixed point (TFP)], and D+LL the phase with a dimerized Heisenberg chain coexisting with a Luttinger liquid in the 1D electronic system. The quantum phase transition between the D+LL phase and the PDW and FSL phases is most likely first order (see text). The phase boundary between the FSL and the PDW phases is of the Kosterlitz-Thouless class, for $J_K$ weak enough, but may be first order for $J_K$ large enough. We have not found any evidence of a phase transition at $J_K\to \infty$.