Magnetic structure factor in cuprate superconductors: Evidence for charged meron vortices

In this paper we demonstrate that a ground state consisting of a quantum liquid of merons and antimerons can quantitatively describe the position and orientation of the incommensurate peaks observed in elastic neutron scattering on cuprates, with essentially no free or adjustable parameters. At small dopings, the relevant length scale is the transverse size of a meron-antimeron pair, defined by the doping concentration. This results in a displacement of the incommensurate peaks from $(\pi ,\pi )$ which is proportional to doping. At higher dopings, the meron core size becomes relevant, leading to a saturation of the displacement. The formation of stripe-like phases at $\delta =1∕8$ and the suppression of the superconductivity is also recaptured in this model.

Figure 1

Spin distribution ${\vec{S}}_n$ for an isolated meron vortex on a $10\times 10$ lattice. The meron is centered on the plaquette marked by M. The spins of either magnetic sublattice rotate by $2\pi$ on any closed path that surrounds the vortex core. This self-consistent configuration was obtained within the static Hartree-Fock approximation for the spin-flux Hamiltonian (see Ref. 19).

Figure 2

Spin distribution $S_n$ for a chain with a domain wall centered at $x_o=0.5$ and with a domain-wall radius $\rho =1$ (upper panel). For convenience, ${\tilde{S}}_n\left(Q\right)=\exp \left(iQna\right)S_n$ is shown in the lower panel. By symmetry, $F_m\left(Q\right)={\sum }_n{\tilde{S}}_n\left(Q\right)=0$.

Figure 3

Spin distribution $S_n$ for a cyclic chain with 32 sites and four domain walls with $\rho =1$ centered at $x_o=4.5$, 12.5, 20.5, and 28.5 (upper panel). ${\tilde{S}}_n\left(Q\right)=\exp \left(iQna\right)S_n$ is shown in the middle panel. By symmetry, $F_m\left(Q\right)={\sum }_n{\tilde{S}}_n\left(Q\right)=0$. In the lower panel we show $\mathsf{Re}{\tilde{S}}_n\left(q_d\right)=\mathsf{Re}\left[\exp \left(i{q}_{d}na\right)S_n\right]$. Clearly, $F_m\left(q_d\right)={\sum }_n{\tilde{S}}_n\left(q_d\right)\ne 0$.

Figure 4

Absolute value of the magnetic structure factor for the chain shown in Fig. 3, as a function of $qa$. $\mid F_m\left(q\right)\mid$ has two main peaks at $q_d=Q\pm \pi ∕d$, where $d=8a$ is the average distance between consecutive domain walls. Smaller satellite peaks at higher order harmonics are also visible.

Figure 5

Schematic representation of a 1D chain with several charged domain walls. Each domain wall mediates the transition from one AFM ground state to the other one and has a hole trapped in its core. A 2D plane with parallel stripes can be thought of as being obtained from the corresponding 1D chain through translation along the stripe backbones. The doping holes are trapped in the stripe backbone.

Figure 6

Schematic representation of the 1D charged domain wall. For simplicity, we only show the orientation of the spins on one magnetic sublattice (the spins on the other sublattice are locally antiparallel). If we perform a $\pm 2\pi$ rotation of this structure about an axis centered in its core, we obtain a 2D spin vortex or antivortex, with the doping hole trapped in the vortex core. We call such objects merons or antimerons.

Figure 7

Spin distribution for a tightly bound meron-antimeron pair. The meron (M) and the antimeron (A) are localized on neighboring plaquettes. The total winding number of the pair is zero. The magnetic AFM order is disturbed only in the neighborhood of the vortices. This self-consistent configuration was obtained within the static Hartree-Fock approximation for the spin-flux Hamiltonian (see Ref. 19).

Figure 8

Upper panel: spin distribution $\vec{S}\left(\vec{r}\right)$ for a random collection of 15 merons and 15 antimerons on a $20\times 20$ lattice. The merons are centered on plaquettes marked by dark squares, while the antimerons are centered on plaquettes marked by light squares. Lower panel: same distribution as in the upper panel, but with cyclic boundary conditions imposed. This leads to the appearance of an extra meron [centered at (1.5, 20.5)] and antimeron [centered at (5.5, 20.5)].

Figure 9

The average structure factor for 1000 random configurations corresponding to the average dopings $\delta =0.02$ (upper) and $\delta =0.04$ (lower panel). The wave vectors are measured in units of $2\pi ∕a$. The four incommensurate peaks are aligned along the diagonals, in agreement with experimental observations for such extremely underdoped samples.

Figure 10

The average structure factor of 1000 random configurations corresponding to an average doping $\delta =0.100\pm 0.003$ (also, see Fig. 11). The wave vectors are measured in units of $2\pi ∕a$. The structure factor is suppressed at the AFM wave-vector $Q=(\pi ∕a,\pi ∕a)$ but exhibits four displaced peaks.

Figure 11

Histogram showing the number of configurations corresponding to a given number $N_0$ of pairs of vortices and antivortices in the set of 1000 random configurations whose average structure factor is shown in Fig. 10.

Figure 12

Average magnetic structure factor along two directions in the Brillouin zone indicated in the inset, for dopings $\delta =0.05$, 0.10, 0.15. The wave vector is measured in units of $2\pi ∕a$. The positions of the experimentally measured peaks are indicated on each graph (short lines), and are found to be in excellent agreement with those predicted in our model.

Figure 13

Same as in Fig. 12, but for different cuts in the Brillouin zone, as indicated in the insets.

Figure 14

The collinear shift ${\delta }_q$, in units of $2\pi ∕a$, as a function of the doping $\delta$. The full circles indicate the results obtained for liquids of merons and antimerons. The empty circles are experimental results for Sr-doped samples, and the squares are experimental results for Nd-doped samples, from Ref. 5.

Figure 15

The structure factors for the two meron-antimeron pairs shown in Figs. 16a, 16b, respectively. The wave vector is measured in units of $2\pi ∕a$. In both cases, a pair of satellite peaks appears, with an orientation perpendicular to that of the meron-antimeron axis in real space.

Figure 16

A diagonal (top) and a horizontal (bottom) meron-antimeron pair is depicted. The corresponding magnetic structure factors are shown in Figs. 16a, 16b.

Figure 17

The collinear shift $\delta q$, in units of $2\pi ∕a$, as a function of core size $\rho$ (measured in lattice constant units $a$). The full circles, full squares, full diamonds, empty squares and empty circles correspond to $\delta =0.05$, 0.08, 0.10, 0.15, and 0.17, respectively. While for $\rho \to 0$ we have $\delta q\approx \delta$, for increasing $\rho$ the shift $\delta q$ decreases, and the effect is more pronounced for larger concentrations.

Figure 18

Self-consistent spin distribution for the configuration of lowest energy found after adding a 3% anisotropy in the hopping integrals, at $\delta =1∕8$, for $U∕t=5$ in the spin-flux phase (from Ref. 19). The merons and antimerons arrange on horizontal lines, leading to a structure similar to that suggested by Tranquada in Ref. 28.

Figure 19

Stripe configuration suggested by Tranquada in Ref. 28, in order to explain results obtained through neutron diffraction for charge and spin ordering in ${\mathrm{La}}_{1.48}{\mathrm{Nd}}_{0.4}{\mathrm{Sr}}_{0.12}\mathrm{Cu}{\mathrm{O}}_4$.