Microscopic nature of $4a_0\times4a_0$ plaquettes in stripe LDOS and $2a_0$ shift

Abstract

Scanning tunneling microscopy (STM) serves as a powerful pictorial tool for visualizing the local density of states (LDOS) of an individual stripe, which strongly intertwines with superconductivity in the underdoped cuprates. The exotic LDOS map patterns thus appear as the key to uncovering the mystery of the underlying microscopic mechanisms. With the quantum color string model framework, we reveal that the microscopic origin of the ubiquitous $4a_0\times4a_0$ plaquettes is closely related to spinon singlet pairs. Moreover, by comparing our data with LDOS of cuprates, we identify an effect of particle-hole symmetry breaking (PHSB): a $2a_0$ shift, which is confirmed in a longer stripe ($L=18$).Our work offers a fresh wavefunction-based perspective for interpreting STM signals in experiments and may advance the microscopic comprehension of high-$T_c$ cuprates.

Figures (7)

• Figure 1: (a) The schematic illustrations for a half-filled QCS (green lines) with $L=10$. The endpoints of QCSs are pinned by external fields (green pins in (a), (b2), and (c2)). Spinon singlets (orange ellipses) and the most relevant processes defined in Eq. \ref{['Eq1']} (enclosed in black dashed boxes) are highlighted. The columns from left to right in (b) and (c): DOS vs. energy $\omega$, LDOS maps at specific DOS peaks ($\bullet$), and the microscopic mechanisms for the three-bar (b3) and two-bar (c3) patterns.
• Figure 2: (a) The largest-weight basis in $\ket{g^{N+1}}$. Inset: the deviation in spinon density distribution $\Delta \rho_\textbf{r}(i_x,\,i_y)$. (b) LDOS vs energy $\omega$ along the straight cut line indicated in the inset (same as Fig. \ref{['fig1']}(b2)).
• Figure 3: (a,b) Stripy patterns in the LDOS map observed in hole-doped Ca$_2$CuO$_2$Cl$_2$ (doping $p=0.03$) with an energy bias of $\pm50$mV, as presented in Figs. 2(d,e) of Ref. yayuZRsinglet. The DMRG-calculated LDOS map at $L=14$ is shown for (c) $\ket{g^{N+1}}$ below the Fermi surface and for (d) $\ket{g^{N-1}}$ above the Fermi surface. The yellow boxes highlight the $4a_0\times4a_0$ structure, accompanied by the same red reference line. (e) The LDOS map of a longer stripe ($L=18$), displayed below (left-lower) and above (right-upper) the Fermi surface. (f) The LDOS along a green arrow in (e) for $\ket{g^{N+1}}$ (blue line) and $\ket{g^{N-1}}$ (red line), with black lines indicating Cu atom positions.
• Figure 4: (a) The ladder pattern in the LDOS map observed in hole-doped Ca$_2$CuO$_2$Cl$_2$ (doping $p=0.03$) with an energy bias of $\pm400$mV, as presented in Fig. 2(g) of Ref. yayuZRsinglet. For a half-filled QCS with $L=10$, (b) the DOS vs. $\omega$, and the LDOS maps for (d,f) the first peak and (c,f) another higher peak above the Fermi surface are plotted. In (b), the first peak ($\bullet$) and the second peak ($\blacklozenge$) are highlighted. We set $\Gamma^z_\text{max}=6a_0$ for (b,c,d), while $\Gamma^z_\text{max}=4a_0$ for (e,f).
• Figure S1: (a,c) Left: The LDOS map for $\ket{g^{N+1}}$. Right: The LDOS distribution for $\ket{g^{N+1}}$ (blue line) and $\ket{g^{N-1}}$ (red line) along the stripe $y$ mirror axis (blue dashed line in the left panel). (b,d) The DOS vs. energy $\omega$, where the state $\ket{g^{N+1}}$ ($\ket{g^{N-1}}$) indicated by $\bullet$ ($\blacktriangle$). For (a,b), all QCSM parameters are identical to those in Fig. \ref{['fig1']}(b) of the main text. In contrast, for (c,d), we only turn off $J_{xy}$, setting it to $J_{xy}=0$.