Doping dependence of spin excitations in the stripe phase of high-$T_c$ superconductors

Based on the time-dependent Gutzwiller approximation for the extended Hubbard model, we calculate the energy and momentum dependence of spin excitations for striped ground states. Our starting point correctly reproduces the observed doping dependence of the incommensurability in La-based cuprates and the dispersion of magnetic modes in the insulating parent compound. This allows us to make quantitative predictions for the doping evolution of the dispersion of magnetic modes in the stripe phase including the energy ${\omega }_0$ and intensity of the resonance peak as well as the velocity $c$ of the spin-wave-like Goldstone mode. In the underdoped regime $n_h<1∕8$, we find a weak linear dependence of ${\omega }_0$ on doping, whereas the resonance energy significantly shifts to higher values when the charge concentration in the stripes starts to deviate from half-filling for $n_h>1∕8$. The velocity $c$ is nonmonotonous with a minimum at $1∕8$ in coincidence with a well known anomaly in $T_c$. Our calculations are in good agreement with available experimental data. We also compare our results with analogous computations based on linear spin-wave theory.

Figure 1

Unit cell used for the $d=4$ stripe computation. The length of the spin arrows corresponds to the LSWT computations. (For the GA computations, the spin magnitude is modulated as shown in Fig. 4.) We also show the interactions used in the LSWT computation: a uniform value of the AFM interaction $J$ across all the bonds except for the ferromagnetic bonds where the sign is opposite to ensure stability and the modulus is $J_F=0.2J$.

Figure 2

(Color online) The central polygon is the first magnetic Brillouin zone for $d=4$ stripes. Thick lines enclose the extended Brillouin zone. The dots are a set of magnetic reciprocal superlattice vectors that define a set of higher magnetic Brillouin zones with total volume equal to the EBZ volume. Magnetic modes and single-particle states can be classified by the momentum in the central magnetic Brillouin zone and a band index. The magnetic reciprocal superlattice vectors closest to $(\pm \pi ,\pm \pi )$ are referred to as $Q^s$.

Figure 3

Energy and wave-vector dependence of magnetic excitations in the half-filled system as obtained from the $\mathrm{GA}+\mathrm{RPA}$ approach for $U∕t=7.5$, $t^{\prime }∕t=-0.2$, and $t=342\mathrm{meV}$. Squares and triangles correspond to data points from INS experiments at $T=10\mathrm{K}$ on ${\mathrm{La}}_2{\mathrm{CuO}}_4$ by Coldea et al. (Ref. 72).

Figure 4

Profile of the charge (top panels), staggered magnetic order (middle panels), and effective interaction $V_{ii}$ (lowest panels) for $d=4$ and 10 bond-centered stripes in the GA. The dashed line in the lowest panels indicates the “bare” spin interaction of the $\mathrm{HF}+\mathrm{RPA}$ $V_{ii}^{\mathrm{HF}}=-U∕t=-7.5$.

Figure 5

(Color online) Dispersion of magnetic excitations along the cut $(\pi ,0)\to (\pi ,\pi )$ and $(\pi ,\pi )\to (0,\pi )$ for BC stripes and dopings $n_h⩽1∕8$ and stripe separation $d⩾4$. The dispersion is obtained from five contours of $\omega {\chi }_q^{″}\left(\omega \right)$ (which is dimensionless) at regular intervals between 0 and 7.

Figure 6

(Color online) Dispersion of magnetic excitations along the cut $(\pi ,0)\to (\pi ,\pi )$ and $(\pi ,\pi )\to (0,\pi )$ for BC and SC stripes and $n_h=0.13$ and stripe separation is $d=4$. The SC data have been shifted by $10\mathrm{meV}$ for clarity. We also show the fit of the acoustic branch with Eq. (13). The dispersion is obtained from four contours of $\omega {\chi }_q^{″}\left(\omega \right)$ at regular intervals between 0 and 7.

Figure 7

(Color online) Dispersion of magnetic excitations along the cut $(\pi ,0)\to (\pi ,\pi )$ and $(\pi ,\pi )\to (0,\pi )$ for BC stripes at $d=4$ and different dopings. The dispersion is obtained from five contours of $\omega {\chi }_q^{″}\left(\omega \right)$ at regular intervals between 0 and 7.

Figure 8

(Color online) Dispersion of magnetic excitations along the cut $(\pi ,0)\to (\pi ,\pi )$ and $(\pi ,\pi )\to (0,\pi )$ for BC stripes oriented along the $y$ direction within LSWT for $d=8$ (top) and $d=4$ (bottom). The radius of the points is proportional to the intensity times the energy of the excitation. Modes with negligible weight are plotted with a small dot. We used a uniform value of the AFM interaction $J$ across all the bonds except for the ferromagnetic bonds where the sign is opposite to ensure stability and the modulus is $J_F=0.2J$ as shown in Fig. 1.

Figure 9

(Color online) Left panel: Dispersion of magnetic excitations along the cut $(\pi ,\pi )\to (\pi ∕4,\pi )$ for BC stripes for dopings $n_h=0.1$ (short dashed line), 0.125 (full line), and 0.16 (long dashed line). We include the experimental data from Ref. 11 for $n_h=0.125$ (triangles) and from Ref. 13 for $n_h=0.16$ (squares). Right panel: velocity of the collective mode $c$ divided the lattice spacing $a$. We report the theoretical result (squares) and the experimental result (triangles) both obtained by fitting Eq. (13) to the data. The diamond at $n_h=0$ is the spin-wave velocity in the insulator.

Figure 10

Contour plot of ${\chi }_q^{″}\left(\omega \right)$ as a function of wave vector for $\omega =45.6\mathrm{meV}$ and $n_h=0.125$. The plot range for the axes is the same as in Fig. 1(c) of Ref. 19. Wave vectors are in units of $2\pi ∕a$.

Figure 11

(Color online) Dispersion of magnetic excitations along the cut $(\pi ,0)\to (\pi ,\pi )$ and $(\pi ,\pi )\to (0,\pi )$ for BC stripes at $d=4$ and $n_h=0.125$. We have averaged the data over the two possible orientations of the stripe, vertical and horizontal. The dispersion is obtained from five contours of $\omega {\chi }_q^{″}\left(\omega \right)$ at regular intervals between 0 and 10. We also show the experimental data from Ref. 11 (courtesy of J. Tranquada). The latter horizontal “error” bars indicate the half-width of the peaks, whereas vertical “error” bars indicate an energy interval of data integration.

Figure 12

Contour plots of ${\chi }_q^{″}\left(\omega \right)$ as a function of wave vector at selected energies for doping $n_h=0.08$. The underlying GA solution is a $d=6$ BC stripe array. The wave-vector coordinate system has been rotated by ${45}^{\circ }$ in such a way that the magnetic Brillouin zone of an antiferromagnet occupies all the window of the plot with ${\mathbf{Q}}_{\mathrm{AF}}=(\pi ,\pi )$ at the center. Wave vectors are in units of $2\pi ∕\left[\sqrt{(}2\right)a]$.

Figure 13

Contour plots of ${\chi }_q^{″}\left(\omega \right)$ as a function of wave vector at selected energies for doping $n_h=0.16$. The underlying GA solution is a $d=6$ BC stripe array. The wave-vector units are as in Fig. 12.

Figure 14

(Color online) Doping dependence of the resonance frequency and intensity for BC (left panels) and SC (right panels) stripes within the $\mathrm{GA}+\mathrm{RPA}$. The circles for $n_h>1∕8$ correspond to $d=4$ solutions, whereas the triangle corresponds to the $d=3$ solution. The full diamond symbol is the experimental result from Ref. 11. The full squares in (a) are the LSWT result using $J_F=0.2J$ and $J=100\mathrm{meV}$.

Figure 15

(a) A schematic cut perpendicular to the stripe through the Néel-type spin structure of $d=4$ SC stripes. (b) Resulting spin structure upon adding transverse spin components $S_{\perp }\sim \cos \left(\mathbf{QR}\right)$ with $\mathbf{Q}=(\pi ,\pi )$.

Figure 16

Density of magnetic excitations $S\left(\omega \right)$ (see text) for three doping levels $n_h=0.083$, 0.1, 0.125, and 0.132. Theoretical curves include a factor $Z_d{\eta }_{\perp }=0.25$. The vertical bars indicate the energies of the saddle point at ${\mathbf{Q}}_{\mathrm{AF}}$ and the roton-type minimum in the dispersion along the stripes. Experimental data for $n_h=0.125$ are from Ref. 11.