Electronic properties of disordered valence-bond stripes in cuprate superconductors

We calculate single-particle properties of short-range-ordered stripe states using Monte Carlo simulations of collective charge-density wave (CDW) order parameters coupled to fermions on a $2d$ square lattice. For superconducting bond-centered stripes with a $d$-wave form factor, we find a valence-bond “glass” which coexists with low-energy quasiparticles featuring interference phenomena, in agreement with recent scanning tunneling microscopy (STM) measurements on underdoped ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ and ${\text{Ca}}_{2-x}{\text{Na}}_x{\text{CuO}}_2{\text{Cl}}_2$. Together with earlier work, our calculations provide a link between CDW signatures seen in STM and those in magnetic neutron scattering.

Figure 1

(Color online) Local densities for ordered CDW states, calculated from ${\mathcal{S}}_c+{\mathcal{S}}_{c\psi }$ for constant $\psi$. The figures show both site and bond densities, $⟨1-c_{i\sigma }^{†}{c}_{i\sigma }⟩$ and $⟨c_{i\sigma }^{†}{c}_{i+\Delta ,\sigma }+\text{H}\text{.c}\text{.}⟩$ (Ref. 40) for $8^2$ Cu sites; black squares are centers of ${\text{CuO}}_2$ plaquettes. (a) Stripes with ${\kappa }_1=0$, ${\kappa }_2=-{\kappa }_3=0.03$, $-{\kappa }_4={\kappa }_5=0.007$ ($d$-wave). (b) Stripes with ${\kappa }_1=0.022$, ${\kappa }_{2\dots 5}=0$ ($s$-wave). (c) Checkerboard [$d$-wave as in (a)]. (d) Checkerboard [$s$-wave as in (b)].

Figure 2

(Color online) Local observables for pinned short-range-ordered stripes [$\kappa$ parameters as in Fig. 1a, correlation length $\xi \approx 10$], showing ${20}^2$ Cu sites of the ${64}^2$ sample. (a) Order parameter field $Q_x+Q_y$. (b) Local density (including bonds) as in Fig. 1. (c) LDOS at $-0.3\text{eV}$. (d) $Z\left(\vec{r},E\right)$ at 0.3 eV. The arrows mark the rows or columns along which the LDOS is shown in Fig. 5.

Figure 3

(Color online) Fourier-transformed local density [from sites and bonds (Ref. 40)] for short-range-ordered stripes, obtained from averaging 40 MC configurations on a ${64}^2$ lattice. Left: ${\kappa }_1=0$, ${\kappa }_2=-{\kappa }_3=0.03$, $-{\kappa }_4={\kappa }_5=0.007$ ($d$-wave). Right: ${\kappa }_1=0.022$, ${\kappa }_{2\dots 5}=0$ ($s$-wave). The dark/light (red/green) arrows indicate the peaks at $\left(\pi /2,0\right)$ and $\left(3\pi /2,0\right)$. (The strong peaks at the zone boundary arise from artificially setting the density to zero in the plaquette centers.)

Figure 4

(Color online) LDOS $\rho \left(\vec{r},E\right)$ at negative (left) and positive (middle) bias, together with $Z\left(\vec{r},E\right)$ (right), for pinned short-range-ordered stripes with additional impurities on a ${64}^2$ lattice, at energies 50, 24, and 8 meV (from top to bottom). $\kappa$ parameters are as in Fig. 1a; for details see text.

Figure 5

(Color online) LDOS spectra corresponding to the data in Fig. 2, along the two paths indicated by arrows in Figs. 2a, 2b. Spectra have been broadened by 7 meV and shifted for clarity.