Kinetic obstruction to pairing in the doped Kitaev-Heisenberg ladder

Abstract

We investigate the hole-doped Kitaev-Heisenberg ($t$-$J$-$K$) model on a two-leg ladder geometry using the density-matrix renormalization group (DMRG). We first consider the behavior of the antiferromagnetic Kitaev (AFK) spin-liquid phase as a function of hopping strength $t$ and doping level. This reveals intriguing pairing tendencies only for $\frac{t}{K} \lesssim 0.65$, consistent with prior results on three-leg ladders, and firmly supports the emerging picture that the physics of doped Kitaev spin liquids strongly depends on the kinetic energy of the doped holes. Analysis of one- and two-hole doping uncovers close links between the spatial profiles of the plaquette operator and the charge density. We construct a doping-dependent phase diagram for antiferromagnetic Heisenberg interactions and intermediate hopping $t=1$. Upon doping, the rung-singlet region develops dominant superconducting correlations. Charge-density-wave correlations dominate at weak doping near the transition to the stripy phase. Spin-density wave-like behavior is found in the AFK and ferromagnetic Kitaev limits, and in the stripy phase.

Figures (14)

• Figure 1: Kitaev–Heisenberg model and doping-dependent phase diagram.(a) Schematic of the two-leg honeycomb ladder. The three inequivalent bonds, $x$, $y$, and $z$, are shown as solid green, blue, and red lines, respectively. Dashed red lines indicate $z$-type bonds connecting sites across the periodic boundary of the ladder. (b)– (e) Pictorial representations of the (b) zigzag (ZZ), (c) stripy (ST), (d) ferromagnetic (FM), and (e) rung-singlet (RS) magnetic phases occurring at half-filling. Ellipses in (e) indicate singlet states formed by the two enclosed spins. (f) Doping-dependent phase diagram obtained by parametrizing the interactions as $J=\cos\phi$ and $K=\sin\phi$ over the interval $-\pi/2 \le \phi \le \pi/2$, with fixed intermediate hopping $t=1$. The phase diagram at half-filling is shown at the bottom, below the hatched area. Upon finite doping (above the hatched area), the rung-singlet phase evolves into a superconducting (SC) phase, whereas the ferromagnetic Kitaev (FK) phase evolves into a commensurate spin-density-wave (SDW) order SDW$_1$ and the stripy phase evolves into an incommensurate SDW$_2$. A narrow range dominated by charge-density wave (CDW) correlations is also found near the ST-RS transition. At low doping levels in the AFK limit a potential disorder phase (DS) is identified. At higher doping levels it evolves into an incommensurate SDW$_3$ phase.
• Figure 2: Hole-pair formation.(a) Binding energy $\Delta E$ for different system sizes $N$, obtained using DMRG for $-\pi/2 \le \phi \le \pi/2$ with fixed hopping $t=1$, where the couplings are parametrized as $J=\cos\phi$ and $K=\sin\phi$. At this intermediate hopping strength, pair formation is found only in the rung singlet phase. (b) Binding energy $\Delta E$ as a function of hopping $t$ in the AFK limit ($K=1$, $J=0$) for different system sizes $N$. (c) Binding energy $\Delta E$ as a function of $t$ in the FK limit ($K=-1$, $J=0$) for different $N$. In both the AFK and FK regimes, the vertical dashed cyan line marks the hopping strength at which the binding energy becomes non-negative, occurring at $t^c_\mathrm{AFK} \approx 0.65K$ and $t^c_\mathrm{FK} \approx 0.1|K|$, respectively.
• Figure 3: Average plaquette operator and spin correlations as a function of hopping. (a) Average plaquette operator $P_{\mathrm{avg}}=\frac{1}{N_p}\sum_i \langle W_p(l) \rangle$ at the Kitaev points $\phi=\pm \pi/2$ as a function of the hopping amplitude $t$ for one ($n_h=1$) and two ($n_h=2$) doped holes. At half-filling $W_p$ is a good quantum number and the ground state has $P_\mathrm{avg}=1$. Deviations from this value thus capture deviations from the QSL state. Dotted lines indicate reference values of $P_{\mathrm{avg}}$ for $n$ broken plaquettes. The top line corresponds to $n=1$, with descending lines representing $n = 2, 3, 4$, respectively. (b,c) Spin–spin correlation function $S(r)$ for two doped holes at different values of $t$ in the (b) AFK and (c) FK regions. These numerical calculations were performed for a fixed system size $N=2\times63$.
• Figure 4: Spatial correspondence between the local plaquette operator and charge-density profile in the two-hole doped system (AFK regime).(a) Spatial profiles of the local plaquette expectation value $\langle W_p(l) \rangle$ in the AFK limit, plotted for various hopping amplitudes $t$. (b) Corresponding plaquette-averaged charge density $n_{p}(l)$ in the two-hole doped system, displayed for increasing hopping strength $t$. Notably, $\langle W_p(l) \rangle$ and $n_{p}(l)$ exhibit strong spatial congruence: a single minimum in the range of $t\leq 0.6K$ where pair formation occurs, and two minima for $t\geq 0.8K$ where doped holes repel instead of forming bound pairs. This correlation underscores that the dopants locally disturb the spin liquid background in the plaquettes around where the holes are delocalized. Data are shown for a fixed system size of $N = 2 \times 63$.
• Figure 5: Comparison of averaged correlation functions in the four-hole doped AFK regime with various hopping amplitudes.(a) Charge–charge correlations $C(r)$, (b) spin–spin correlations $S(r)$, (c) singlet pair–pair $P^{z}_{S}(r)$, and (d) triplet pair–pair $P^{z}_{T}(r)$ correlations vs. distance $r$, plotted for representative values of the hopping amplitude $t$. The pair-pair correlations are strongly suppressed for $t\geq 0.8K$, consistent with the absence of pair formation. This is accompanied by enhanced spin-spin correlations, signaling a transition into a SDW phase. Data correspond to a fixed system size of $N = 2 \times 63$.