Ordering and criticality in an underscreened Kondo chain

Motivated by the nickel valence controversy in the perovskite nickelate $R$NiO${}_3$, we consider a one-dimensional underscreened Kondo chain consisting of alternating spin-1 (“nickel”) and electron (“oxygen”) sites, which in addition to the usual electron hopping and spin-spin interaction between the $S=1$ spin and the electron also contains a spin-mediated electron hopping term. Using the density-matrix renormalization group (DMRG), we obtained the zero-temperature phase diagram of the model, as well as various correlation functions in each phase. Importantly, for a certain range of parameters the model exhibits a quasi-long-range spiral (QS) order. To understand the DMRG results, we construct a mean-field theory based on a Schwinger fermion decomposition of the $S=1$ spins, from which we argue that the QS phase corresponds to a phase in proximity to the spin Bose metal state proposed by Sheng, Motrunich, and Fisher [Phys. Rev. B 79, 205112 (2009)]. Notably, we find no evidence for a phase with the symmetry of “nickel”-centered charge order, which has been argued to arise due to site-selective Kondo screening of half the $S=1$ spins, and suggest that order of this type occurs only due to an additional energy gain from spontaneous lattice distortions, not present in this model.

Figure 1

The nickelate chain and the $t$-$J$-$J^{\prime }$ Hamiltonian (bottom panel), as derived from a Hubbard-typed model (top panel).

Figure 2

Phase diagram obtained from the semiclassical computation, where AF stands for a antiferromagnetic state in which wave vector $q=\pi$, F stands for a ferromagnetic phase in which wave vector $q=0$, and S stands for a spiral phase in which $0<q<\pi$. Note that all three phases meet at the point $J=J^{\prime }=t$.

Figure 3

DMRG block used in the calculations.

Figure 4

Phase diagram from the DMRG calculation. Here F stands for the ferromagnetic phase, QAF stands for the quasi-long-range antiferromagnetic phase, CD stands for the charge-density ordered phase, and QS stands for the quasi-long-range spiral phase. The dashed line within the CD phase represents the boundary in which the peak in $\langle S_k^z{S}_{-k}^z\rangle$ changes from $k=\pi$ to $k=\pi \pm \delta$.

Figure 5

(a) Nickel spin-spin correlation $\langle S_k^z{S}_{-k}^z\rangle$, (b) electron spin-spin correlation $\langle s_k^z{s}_{-k}^z\rangle$, and (c) electron density-density correlation $\langle \delta n_k\delta n_{-k}\rangle$ for (orange, filled circular symbols) $J/t=0.4$ and $J^{\prime }/t=1.6$, (green, empty square symbols) $J/t=0.8$ and $J^{\prime }/t=4.0$, (blue, empty diamond symbols) $J/t=1.6$ and $J^{\prime }/t=4.0$, and (purple, Greek cross symbols) $J/t=2.4$ and $J^{\prime }/t=4.0$. The insets of panels (a) and (b) show the details of the respective correlation for $J/t=2.4$ and $J^{\prime }/t=4.0$ in the range $0.5<k/\pi <1.5$.

Figure 6

The nearest-neighbor nickel-oxygen spin-spin correlation $\langle {\mathbf{s}}_{i\pm 1/2}·{\mathbf{S}}_i\rangle$ for the characteristic points in parameter space used in Table , with system size $N=48$. The two correlations are combined and the index $i$ now labels the bond center of the correlation. The positions of the nickel and the oxygen sites are indicated by the labels above the curves. The inset shows finite-size scaling of the oscillation amplitude in $\langle {\mathbf{s}}_{i\pm 1/2}·{\mathbf{S}}_i\rangle$ for $J=0.4$ and $J^{\prime }=1.6$ using a second-order polynomial fit.

Figure 7

The development of charge-density wave order parameter $\delta n_{\mathrm{CD}}=\langle n_{N/2}\rangle -\langle n_{N/2+1}\rangle$ as $J^{\prime }/t$ increases while $J/t=0.8$ is fixed. The inset (a) shows $\langle n_i\rangle$ as function of unit-cell index $i$ for (black, filled square symbols) $J/t=0.8$ and $J^{\prime }/t=2.4$ and (blue, filled diamond symbols) $J/t=0.8$ and $J^{\prime }/t=4.0$, with system size $N=96$. The inset (b) shows examples of finite-size scaling of $\delta n_{\mathrm{CD}}$ at different $J^{\prime }$ using second-order polynomial functions.

Figure 8

Locations of the singularity (resp. peak) in the ferromagnetic and QS phases (resp. CD phase) as function of $J^{\prime }/t$ when $J/t=2.0$ is fixed.

Figure 9

Finite-size scaling of the charge gap for the characteristic points in parameter space used in Table .

Figure 10

The lower branch of the von Neumann entanglement entropies, obtained from the lower envelope of the full entropy shown in the inset, as a function of the conformally transformed length (Ref. 17) ${\ell }^{\prime }=(L/\pi )\sin (\pi \ell /L)$ of the subsystem A for the characteristic points in parameter space in Table ($L$ is the system size). The solid lines are linear fits using $S_A=\frac{c}{6}\ln {\ell }^{\prime }$, which determines the central charge $c$.

Figure 11

Dependence of $r/t$ on $(J+J^{\prime })/t$ in the class of mean-field Ansätze defined by $Q_a=Q_a^{\prime }=r{[1,0]}^T$.

Figure 12

Typical mean-field spectra obtained from $H^{\mathrm{MF}}$ for the class of mean-field Ansätze defined by $Q_a=Q_a^{\prime }=r{[1,0]}^T$. Note that the degenerate flat bands at zero energy are not plotted.

Figure 13

Schematic of the Kondo underscreening, in which the electrons screen out only half of the nickel spin on each nickel site, leaving behind an unscreened spin-1/2.

Figure 14

Classical phase diagram of the effective spin-1/2 model Eq. (8) truncated at (a) ${\mathcal{J}}_2$ and (b) ${\mathcal{J}}_8$. Here filled gray triangular symbols indicate the ferromagnetic phase, filled orange circular symbols indicate the antiferromagnetic phase, and unfilled teal square symbols indicate the incommensurate spiral phase. The lower left corners of the phase diagrams are excluded because the gap in $H^{\mathrm{MF}}$ is too small to accurately calculate ${\mathcal{J}}_{ij}$.

Figure 15

Classical spiral angle $\theta$ of the effective spin-1/2 model Eq. (8) as function of $J^{\prime }$ along the line cut (a) $J^{\prime }=4J$ and (b) $J^{\prime }=2J$. The red, solid (blue, dashed) curve with filled circle (cross) symbols corresponds to truncation at ${\mathcal{J}}_2$ (${\mathcal{J}}_8$).

Figure 16

Quantum phase diagram of the effective spin-1/2 model Eq. (8), with ${\mathcal{J}}_{ij}$ computed from Eq. (12) and truncated at ${\mathcal{J}}_2$. Here filled gray triangular symbols indicate the ferromagnetic phase, filled orange circular symbols indicate the quasi-long-range antiferromagnetic phase, unfilled green square symbols indicate the dimer phase with spin-spin correlation peaked at $k=\pi$, and unfilled violet diamond symbols indicate the dimer phase with spin-spin correlation peaked at an incommensurate wave vector. The lower left corner of the phase diagram is excluded because the gap in $H^{\mathrm{MF}}$ is too small to accurately calculate ${\mathcal{J}}_{ij}$.

Figure 17

Illustration of the two inequivalent oxygen sites when the effective spin-1/2 model is in the dimer phase.

Figure 18

Results from energy minimization for the general mean-field Ansätze. Here unfilled blue circle (red cross) indicates states for which $({\sum }_a|\langle f_{ia\alpha }^{†}{c}_{i+1/2,\alpha }\rangle -\langle f_{ia\alpha }^{†}{c}_{i-1/2,\alpha }\rangle {{|}^2)}^{1/2}$ $\le$ ($>$) $0.05$.

Figure 19

Band structures obtained from $H^{\mathrm{MF}}$ with the additional $f$-hopping term $t^{\prime }$, for $r/t=r^{\prime }/t=1.812$ and various values of $t^{\prime }$ and $\theta$. The colors encode the eigenvector composition of the bands, with red $\sim$$c_k$, green $\sim$$f_{1k}$, and blue $\sim$$f_{2k}$ [the detailed color scheme is shown in (d)]. Panel (e) also defines the convention for the bosonization treatment.