Interplay of spin and charge order in the electron-doped cuprates

We study magnetic and charge order in the electron-doped high-$T_c$ cuprates based on the one-band Hubbard model with onsite $U$ and nearest-neighbor $V$ interactions. To investigate the interplay between the orders, we employ the Kotliar-Ruckenstein slave-boson method and analyze fluctuations descending from an antiferromagnetic parent state. Our analysis reveals incommensurate charge order whose ordering vector matches the doping dependence of resonant inelastic x-ray scattering (RIXS) measurements in ${\mathrm{Nd}}_{2-x}{\mathrm{Ce}}_x{\mathrm{CuO}}_4$ (NCCO). From our calculations of paramagnon dispersion as well as dynamical charge and spin structure factors, we reproduce the characteristic features of the RIXS signal.

Figure 1

Phase diagram in the $U\text{−}V$ plane at four fixed values of density $n$ as inferred from the stability analysis of the mean-field phases PM and AFM. The emerging charge orders are phase-separated (PS), incommensurate (iCO), or commensurate, i.e., checkerboard charge-order (CCO) or ferrimagnetism (FIM).

Figure 2

Phase diagram of emerging CO as a function of electron-doping $x$ and the critical interaction ratio $V_{cr}/U_{cr}$. Different orders are visualized by idealized sketches, depicting how doped charges arrange in a real-space picture that combines magnetic ($\uparrow ,\downarrow$) and charge (circle size) order.

Figure 3

Incommensurate charge order (iCO) wavevector inferred from SB fluctuations in comparison to RXS data in NCCO adopted from Ref. [9]. Our result for ${\mathit{Q}}_{\text{iCO}}$ follows the diameter of the Fermi surface (FS) that relates to the doping $x$ via Luttinger's theorem (dashed line).

Figure 4

Paramagnon dispersion on the high-symmetry path, which appears as bright line in the color-coded plot of $\text{Im}{\chi }_s^t(\mathit{q},\omega )$ at $(U,V)=(5.5,0.55)$, along with RIXS data in NCCO (circles with error bars) adopted from Ref. [30]. The absolute energy scale in our model is fitted to $t=0.52\mathrm{eV}$. The dashed line displays linear spin-wave theory with the velocity $c_s=1.24\mathrm{eV}\AA$ adopted from Ref. [30].

Figure 5

Charge ($S_c$) and longitudinal ($S_s^l$) spin-structure factor for $x=0.108$ and $(U,V)\approx (U_{cr},V_{cr})$, i.e., (8.7,0) and (11.5,1.15) as function of $\mathit{q}=(q_x,0)$ for the energy cutoffs ${\mathrm{\Omega }}_{\text{min}}=0.01, {\mathrm{\Omega }}_{\text{max}}=2$. The inset shows RIXS data in NCCO adopted from Ref. [11] with the same doping and comparable energy cutoffs.