Magnetic excitations of stripes and checkerboards in the cuprates

We discuss the magnetic excitations of well-ordered stripe and checkerboard phases, including the high energy magnetic excitations of recent interest and possible connections to the “resonance peak” in cuprate superconductors. Using a suitably parametrized Heisenberg model and spin wave theory, we study a variety of magnetically ordered configurations, including vertical and diagonal site- and bond-centered stripes and simple checkerboards. We calculate the expected neutron scattering intensities as a function of energy and momentum. At zero frequency, the satellite peaks of even square-wave stripes are suppressed by as much as a factor of $34$ below the intensity of the main incommensurate peaks. We further find that at low energy, spin wave cones may not always be resolvable experimentally. Rather, the intensity as a function of position around the cone depends strongly on the coupling across the stripe domain walls. At intermediate energy, we find a saddle point at $(\pi ,\pi )$ for a range of couplings, and discuss its possible connection to the “resonance peak” observed in neutron scattering experiments on cuprate superconductors. At high energy, various structures are possible as a function of coupling strength and configuration, including a high energy square-shaped continuum originally attributed to the quantum excitations of spin ladders. On the other hand, we find that simple checkerboard patterns are inconsistent with experimental results from neutron scattering.

Figure 1

(Color online) Stripe configurations studied in this paper. The coupling between nearest neighbor spins is $J_a$, and the coupling between spins across a domain wall is $J_b$, as indicated in the figures. (a) VS4: Vertical site-centered stripes with spacing $d=4$. (b) VB4: Vertical bond-centered stripes with spacing $d=4$. (c) DS3: Diagonal site-centered stripes with spacing $d=3$. (d) DB2: Diagonal bond-centered stripes with spacing $d=2$.

Figure 2

(Color online) Checkerboard patterns. The coupling between nearest neighbor spins is $J_a$, and the coupling between spins across a domain wall is $J_b$, as indicated in the figures. (a) CS3: Checkerboard sited-centered pattern with spacing $d=3$. (b) CS4: Checkerboard site-centered pattern with spacing $d=4$. (c) CB2: Checkerboard bond-centered pattern with spacing $d=2$. (d) CB3: Checkerboard bond-centered pattern with spacing $d=3$.

Figure 3

(Color online) VS4 at $J_b=0.4J_a$: Constant energy cuts with windows $0.2Ja$ for twinned vertical, site-centered stripes of spacing $d=4$ at $J_b=0.4J_a$. The energy $E$ is in units of $J_{a}S$.

Figure 4

(Color online) VS4 at $J_b=J_a$: Constant energy cuts with windows $0.2Ja$ for twinned vertical, site-centered stripes of spacing $d=4$ at $J_b=J_a$. The energy $E$ is in units of $J_{a}S$.

Figure 5

(Color online) VB4 at $J_b=-0.4J_a$: Constant energy cuts with windows $0.2Ja$ for twinned vertical, bond-centered stripes of spacing $d=4$ at $J_b=-0.4J_a$. The energy $E$ is in units of $J_{a}S$.

Figure 6

(Color online) VB4 at $J_b=-1.5J_a$: Constant energy cuts with windows $0.2Ja$ for twinned vertical, bond-centered stripes of spacing $d=4$ at $J_b=-1.5J_a$. The energy $E$ is in units of $J_{a}S$.

Figure 7

Intensity ratio between the “inner” and “outer” branches of the spin-wave cones as a function of coupling ratio $\mid J_b∕J_a\mid$ for vertical stripes. The ratio is calculated at constant low energy $E=0.4J_{a}S$. The solid points are our numerical calculations of the intensity ratio. The solid line is a fit to the results, as explained in the text. (a) VS4: Vertical, site-centered stripes of spacing $d=4$. (b) VB4: Vertical, bond-centered stripes of spacing $d=4$.

Figure 8

(Color online) DB2 at $J_b=-0.1J_a$: Constant energy cuts with windows of $0.2Ja$ for diagonal, bond-centered stripes of spacing $d=2$ with $J_b=-0.1J_a$. The left column is untwinned and the right one is twinned. The energy $E$ is in units of $J_{a}S$.

Figure 9

(Color online) DS3 at $J_b=0.1J_a$: Constant energy cuts with windows of $0.2Ja$ for diagonal, site-centered stripes of spacing $d=3$ at $J_b=0.1J_a$. The left column is untwinned and the right one is twinned. The energy $E$ is in units of $J_{a}S$.

Figure 10

(Color online) DS3 at $J_b=0.5J_a$: Constant energy cuts with windows of $0.2Ja$ for diagonal, site-centered stripes of spacing $d=3$ at $J_b=0.5J_a$. Each panel is twinned. The energy $E$ is in units of $J_{a}S$.

Figure 11

Intensity ratio between the “inner” and “outer” branches of the spin-wave cones as a function of coupling ratio $\mid J_b∕J_a\mid$ for diagonal stripes. The solid points are our numerical calculations of the intensity ratio. The solid line is a fit to the results, as explained in the text. (a) DB2: Intensity ratio for diagonal bond-centered stripes of spacing $d=2$ at $E=0.4J_{a}S$. (b) DS3: Intensity ratio for diagonal site-centered stripes of spacing $d=3$ at $E=J_{a}S$.

Figure 12

(Color online) CS3 at $J_b=0.1J_a$: Dispersion and intensities for a site-centered checkerboard pattern with stripe spacing $d=3$ at $J_b=0.1J_a$. The upper pannel is along $(k_x,\pi )$ direction; the lower pannel corresponds to the diagonal direction $(k_x,k_x)$. The energy $E$ is in units of $J_{a}S$.

Figure 13

CS3 at $J_b=0.1J_a$: Constant energy cuts for site-centered checkerboard pattern with stripe spacing $d=3$ at $J_b=0.1J_a$. The energy $E$ is in units of $J_{a}S$.