Effective action of the weakly doped $t\text{−}J$ model and spin-wave excitations in the spin-glass phase of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$

We derive the low-energy effective field theory of the extended $t\text{−}J$ model in the regime of light doping. The action includes a previously unknown Berry phase term, which is discussed in detail. We use this effective field theory to calculate spin-wave excitations in the disordered spin spiral state of ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x\mathrm{Cu}{\mathrm{O}}_4$ (the spin-glass phase). We predict an excitation spectrum with two distinct branches: The in-plane mode has the usual linear spin-wave dispersion and the out-of-plane mode shows nontrivial doping dependent features. We also calculate the intensities for inelastic neutron scattering in these modes.

Figure 1

(Color online) (a) Characteristic ground state configuration of a particular realization at doping $x=0.05$. The impurity pseudospins $\frac{1}{2}{\Psi }_i\vec{\sigma }{\Psi }_i$ (circles) are oriented along the $z$ axis. Full (open) circles correspond to values $-\frac{1}{2}$ $(+\frac{1}{2})$. Small arrows represent the $\vec{n}$ field. The system forms domains stretched along the $a$ direction, in which all the pseudospins are aligned in parallel. (b) Uniform spiral state [Eq. (12)] with average pitch [Eq. (13)]. Apart from the absence of disorder, a uniform spiral is very similar to the true ground state and thus a good starting point to describe excitations. The corresponding structure factors are shown in Fig. 2.

Figure 2

(Color online) (a) Neutron scattering probability $S_{\mathbf{q}}$ for doping $x=0.04$. Full circles correspond to experimental observations taken from Fig. 4 in Ref. 6, with normalized intensities. The solid line represents our simulation (Ref. 25) containing no fitting parameters. The dashed line indicates the positions of the delta function peaks obtained from a uniform spiral. (b) Incommensurability $\delta$ (half the distance between the two peaks measured in reciprocal lattice units of the tetragonal lattice) as a function of doping. Accurate Monte Carlo calculations (dots) are in agreement with experimental data (squares) taken from Refs. 3, 5, 45, see Fig. 6 of Ref. 3. The uniform spiral (dashed line) captures the essential properties of the spin-glass ground state.

Figure 3

Spectrum of in-plane spin waves excited in neutron scattering at doping $x=0.05$. The momentum is directed along the orthorhombic $b$ direction and the incommensurable vector [Eq. (13)] is $Q\approx 0.39$. There are two linear branches that start at $k_b=\pm Q$ and that are broadened due to disorder and ultimately disappear at $\mid \mathbf{k}\pm \mathbf{Q}\mid \sim \sqrt{x}$. For larger momenta, spin waves reappear with the usual dispersion of the Néel antiferromagnet. The frequency at the intersection ${\mathbf{k}}_b=0$ scales linearly with doping.

Figure 4

Polarization operator describing the scattering of a spin wave (dashed line) off a pseudospin (solid line). The corresponding pseudospin-flip vertices (circles) are given by Eq. (20).

Figure 5

Spectrum of out-of-plane spin-wave excitations at doping $x=0.05$. The momentum is directed along the orthorhombic $b$ direction. The inverse radius of the impurity wave function is $\kappa =0.4$ and the incommensurable vector [Eq. (13)] is $Q\approx 0.39$. Both the lower and upper branches are getting broad and ultimately disappear due to disorder at $\mid \mathbf{k}\pm \mathbf{Q}\mid \sim \kappa$. The upper branch has a very small spectral weight and is thus difficult to observe in neutron scattering. At larger momenta, the spin waves reappear with usual dispersion of the Néel antiferromagnet. The frequency at the top of the dome at ${\mathbf{k}}_b=0$ scales quadratically with doping.