Mapping reservoir-enhanced superconductivity to near-long-range magnetic order in the undoped 1D Anderson- and Kondo-lattices

TL;DR

This work establishes an exact particle-hole mapping at half-filling that connects the 1D Kivelson bilayer reservoir-enhanced superconductivity setup to the undoped 1D Anderson- and Kondo-lattices, enabling a unified, non-perturbative study of these systems with quasi-exact numerics (DMRG and AFQMC). The analysis shows that, at weak interlayer coupling, the systems exhibit near-long-range superconducting and CDW (or AFM in the duals) behavior on accessible length scales due to extended-range coupling mediated by the metallic layer, but the back-action of the pairing layer induces a finite spin/charge gap in the metal, turning the effective mediation into an exponentially decaying interaction and preventing true long-range order in 1D. Consequently, the RKKY-like coupling is effectively short-ranged, consistent with Mermin-Wagner constraints in 1D, though large finite systems can display dramatic coherence and slow decays. The results bridge reservoir-enhanced superconductivity and Kondo-lattice physics, offering experimentally accessible tests in quasi-1D heavy fermion compounds, ad-atom chains, and ultracold atomic gases, and providing a foundation for exploring doped regimes and higher dimensions where new ordered phases may emerge.

Abstract

The undoped Kondo necklace in 1D is a paradigmatic and well understood model of a Kondo insulator. This work performs the first large-scale study of the 1D Anderson-lattice underlying the Kondo necklace with quasi-exact numerical methods, comparing this with the perturbative effective 1D Kondo-necklace model derived from the former. This study is based on an exact mapping of the Anderson model to one of a superconducting pairing layer connected to a metallic reservoir which is valid in arbitrary spatial dimensions, thereby linking the previously disparate areas of reservoir-enhanced superconductivity, following Kivelson's pioneering proposals, and that of periodic Kondo-systems. Our work reveals that below the length-scales on which the insulating state sets in, which can be very large, superconducting and density-density correlations are degenerate and may both appear to approach an almost ordered state, to a degree that far exceeds that of any isolated 1D pairing layer with short-range interactions. We trace these effects to the effective extended-range coupling that the metallic layer mediates within the pairing layer. These results translate directly to the appearance of near-long-range magnetic order at intermediate scales in the Kondo-systems, and explain the strong renormalization of the RKKY-coupling that we effectively observe, in terms of the back-action of the pairing layer onto the metallic layer. The effects we predict could be tested either by local probes of quasi-1D heavy fermion compounds such as CeCo$_2$Ga$_8$, in engineered chains of ad-atoms or in ultracold atomic gases.

Figures (9)

• Figure 1: (a) Schematic figure of the 1D Kivelson bilayer proposal, $\hat{H}_{\rm KBP}$ at half-filling, \ref{['model_equ']}). (b) Kondo necklace: effective model \ref{['eq:hkbpj']} resulting from \ref{['model_equ']} using Schrieffer-Wolff transformation.
• Figure 2: $C_{\rm p}(i,j)$ (left column, ), $|N_{\rm p}(i,j)|$ (left column, ), $C_{\rm m}(i,j)$ (right column, ) and $|N_{\rm m}(i,j)|$ (right column, ) with $L=200$. (a)$\&$(b)$U=4, t_{\perp}=0.55,t_{\rm m}=1$(c)$\&$(d)$U=10, t_{\perp}=3.0,t_{\rm m}=10$(e)$\&$(f)$U=10, t_{\perp}=4.0,t_{\rm m}=10.$
• Figure 3: Charge gap $\Delta _{\rm c}(L)$ in regime 1 of the (a) microscopic and (b) effective models with $L=100,t_{\rm m}=1.0$. (a)$t_{\perp}=0.5$ (circle), $t_{\perp}=0.55$ (square), $t_{\perp}=0.6$ (diamond) , $t_{\perp}=0.65$ (pentagon). (b)$J_{\perp}=0.25$ (circle), $J_{\perp}=0.3025$ (square), $J_{\perp}=0.36$ (diamond) , $J_{\perp}=0.4225$ (pentagon). (c)$\Delta _{\rm c}$ in regime 2 for the microscopic model (circle, $t_{\perp}=3.0$) and the effective model (square, $J_{\perp}=3.6$) with $L=100,t_{\rm m}=10.0$. (d) The gap $E_{1}-E_{0}$ (difference between first excited and ground state energy ) against $t_{\rm m}$ with $t_{\perp}=1.2$, $L=100$ in regime 1. Red square marks the gap from exact calculation of an isolated dimer, $t_{\rm m}=0$.
• Figure 4: Comparison of pair-pair correlation function between microscopic model (open circles) and effective model (full diamonds) in regime 1 with $L=100,t_{\rm m}=1.0$. (a)$t_{\perp}=0.5, J_{\perp}=0.25$, (b)$t_{\perp}=0.55,J_{\perp}=0.3025$, (c)$t_{\perp}=0.6,J_{\perp}=0.36$ and (d)$t_{\perp}=0.65,J_{\perp}=0.4225$.
• Figure 5: (a)$K^{-1}_{\rm p}$ vs. $t_{\perp}$ for $\hat{H}_{\rm KBP}$ in regime 1 at ${L=100}$. $t_{\rm m}=1.0$ (square), $t_{\rm m}=2.0$ (circle), $t_{\rm m}=3.0$ (diamond) and $t_{\rm m}=4.0$ (pentagon). (b)$\Delta _{\rm c}(L)$ for $\hat{H}_{\rm KBP}$ at ${t_\perp=1.2}$. $t_{\rm m}=3.0$ (diamond), $t_{\rm m}=4.0$ (pentagon), $t_{\rm m}=5.0$ (x). (c)$K^{-1}_{\rm p}$ vs. $t_{\perp}$ for $\hat{H}_{\rm KBP}$ in regime 2 at ${L=100}$. $t_{\rm m}=2.0$ (otimes), $t_{\rm m}=4.0$ (square), $t_{\rm m}=6.0$ (diamond), $t_{\rm m}=8.0$ (circle), $t_{\rm m}=10.0$ (triangle) and $t_{\rm m}=12.0$ (pentagon).