Dynamical spin structure factors of quantum spin nematic states

Dynamical spin structure factors of quantum spin nematic states are calculated in a spin-$\frac{1}{2}$ square-lattice $J_1$-$J_2$ model with ferromagnetic $J_1$ and competing antiferromagnetic $J_2$ interactions. To this end, we use a fermion representation, generalizing it to $N$ flavors. We begin with a spin-triplet pairing state of fermion fields, called $Z_2$ planar state, which is a stable saddle-point solution in the large-$N$ limit in a finite parameter range where the couplings $J_1$ and $J_2$ compete strongly [R. Shindou and T. Momoi, Phys. Rev. B 80, 064410 (2009)]. Using a large-$N$ expansion, we take into account fluctuations around this saddle point up to corrections of order $1/N$. The dynamical spin structure factors thus obtained signify the existence of gapless $q$-linear director-wave (spin-wave) modes at $\mathit{q}=(0,0)$ and gapped “gauge-field”-like collective modes at $\mathit{q}=(\pi ,\pi )$, whose spectral weight vanishes as a linear and quadratic function of the momentum, respectively. The low-energy collective modes contain fluctuations of nematic-director, spin, and gauge degrees of freedom. Associated with the gapless $q$-linear modes, we evaluate the temperature dependence of the nuclear spin relaxation rate $1/T_1$ in the low-temperature regime as $1/T_1\propto T^{2d-1}$, where $d$ is the effective spatial dimension.

Figure 1

Square-lattice $J_1$-$J_2$ frustrated ferromagnetic model, in which the nearest-neighbor exchange $J_1$ is ferromagnetic and the next-nearest-neighbor exchange $J_2$ is antiferromagnetic.

Figure 2

Mean-field phase diagram of the $S=\frac{1}{2}$ square-lattice $J_1$-$J_2$ model in the large-$N$ limit. In the strong $J_2$ regime, the mean-field ground state consists of only two decoupled spin-singlet pairing states defined on $J_2$ bonds, which are often called $\pi$-flux states (Refs. 3, 4, and 27). When ferromagnetic $J_1$ increases, the mean-field ground state acquires finite spin-triplet pairing amplitudes on the ferromagnetic bonds, which connect the decoupled $\pi$-flux states (Ref. 9). These states are called $Z_2$ planar state and $\text{U}\left(1\right)$ planar state, depending on the amplitude of singlet pairings, and both of them are characterized as spin nematic states. In the strong $J_1$ regime, the mean-field ground state contains only spin-triplet pairings, which corresponds to a fully polarized ferromagnetic state (FM state) (Ref. 23). Energetics of the mean-field solutions conclude that $J_{c,1}=1.325$, $J_{c,2}=1.0448$, and $J_{c,3}=1.02$.

Figure 3

Configurations of (a) $d$ vectors for spin-triplet pairings and (b) directors corresponding to the quadrupolar moments in the $Z_2$ planar state on the square lattice.

Figure 4

(a) Dispersion relation of spinon band ${\xi }_{\mathit{k}}$ in the $Z_2$ planar state at $J_2=1.1J_1$. (b) Contour plot of ${\xi }_{\mathit{k}}$, showing eight minima at $\mathit{k}=(\pm \frac{\pi }{2},0),(0,\pm \frac{\pi }{2}),(\pm \frac{\pi }{2},\pi ),(\pi ,\pm \frac{\pi }{2})$, and eight maxima at $\mathit{k}=(0,0),(\pm \frac{\pi }{2},\pm \frac{\pi }{2}),(0,\pi ),(\pi ,0),(\pi ,\pi )$.

Figure 5

Contour plots of (a) Im${\chi }_{zz}^{\left(0\right)}(\mathit{q},\epsilon )$ and (b) Im${\chi }_{+-}^{\left(0\right)}(\mathit{q},\epsilon )$ at $J_2=1.1J_1$. Both of them consist of broad continuum spectra, which have finite weight only for $\epsilon \ge {\omega }_c\simeq 0.3$. (c) Path of the momentum $\mathit{q}$ in (a) and (b), which runs from $(0,0)$ to $(\pi ,\pi )$, to $(\pi ,0)$, and back to $(0,0)$. (d) Feynman diagram for Im${\chi }_{\mu \mu }^{\left(0\right)}(\mathit{q},i{\epsilon }_m)$. Solid lines denote the fermion single-particle Green's functions and dashed lines denote the external magnetic fields.

Figure 6

Static structure factors in the mean-field approximation: (a) $C_{zz}^{\left(0\right)}(\mathit{q},\tau =0)$ and (b) $C_{+-}^{\left(0\right)}(\mathit{q},\tau =0)$ at $J_2=1.1J_1$.

Figure 7

Diagrams of interactions in Eq. (48) containing the renormalized vertex parts (a) ${\overline{\mathcal{S}}}^{(0,3)}$, (b) ${\overline{\mathcal{S}}}^{(1,1)}$, and (c) ${\overline{\mathcal{S}}}^{(2,1)}$.

Figure 8

$1/N$ contributions to correlation functions, where dotted lines denote external fields, wavy lines are the RPA propagators, and solid lines are single-particle Green's functions. Diagram (a) contributes to ${\chi }_{\mu \mu ,\mathrm{II}}^{aa}(q,i{\epsilon }_n)$, while diagrams (b)–(d) to ${\chi }_{\mu \mu ,\mathrm{I}}^{aa}(q,i{\epsilon }_n)$. They contribute to (a) spin-wave term, (b), (d) Hartree-Fock (HF) term with renormalized single-particle Green's functions, and (c) HF term with a vertex correction.

Figure 9

Excitation energy spectrum in the dynamical structure factor $\mathrm{Im}{\chi }_{zz}(\mathit{q},\epsilon )$ in the $Z_2$ planar ground state at $J_2/J_1=1.1$. (a) Spectral wight of the collective modes. (b) Momentum-energy dispersion relation, showing the characteristic collective modes (${\mathit{\varphi }}_1$, ${\mathit{\phi }}_1$, ${\mathit{\phi }}_2$, ${\mathit{\rho }}_1$) given in the text. The momentum runs along three high-symmetric $q$ points [see Fig. 5c]. The gray zones denote Stoner continuum. (c) Momentum-energy dispersion relation for the lowest collective modes in the first quadrant of the Brillouin zone.

Figure 10

Excitation energy spectrum in the dynamical structure factor $\mathrm{Im}{\chi }_{+-}(\mathit{q},\epsilon )$ in the $Z_2$ planar ground state at $J_2/J_1=1.1$. (a) Spectral wight of the collective modes. (b) Momentum-energy dispersion relations, showing the collective modes (${\mathit{\varphi }}_2$, ${\mathit{\varphi }}_3$, ${\mathit{\phi }}_3$, ${\mathit{\phi }}_4$) given in the text. The gray zones denote Stoner continuum. (c) Momentum-energy dispersion relation for the lowest collective modes in the first quadrant of the Brillouin zone.

Figure 11

(a) Coefficients of eigenmodes ${\mathit{\phi }}_1$, ${\mathit{\phi }}_2$, and ${\mathit{\phi }}_3$ [Eqs. (73, 74, 75)] as a function of $J_2/J_1$, where the units are taken as ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2=1$ and ${\epsilon }^2+{\zeta }^2=1$. At the critical point $J_2/J_1=J_{c,2}$, $\alpha$, $\gamma$, and $\epsilon$ are reduced to zero, while $\beta$, $\delta$, and $\zeta$, respectively, converge to $D/\sqrt{D^2+{\chi }^2}$, $\chi /\sqrt{D^2+{\chi }^2}$, and 1. (b) Mass of eigenmodes as a function of $J_2/J_1$. Negative energy gaps indicate the presence of instabilities. A: Mass of ${\mathit{\phi }}_3$ and ${\mathit{\phi }}_4$ modes at $\mathit{q}=(\pi ,\pi )$ in the $Z_2$ planar phase and mass of ${\mathit{e}}_2^2$ and ${\mathit{e}}_2^1$ modes at $\mathit{q}=(\pi ,\pi )$ in the $\text{U}\left(1\right)$ planar phase. B: Mass of ${\mathit{\phi }}_1$ and ${\mathit{\phi }}_2$ modes at $\mathit{q}=(\pi ,\pi )$. C: Mass of ${\mathit{e}}_8^3$ mode at $\mathit{q}=(\pi ,0)$ and mass of ${\mathit{e}}_7^3$ mode at $\mathit{q}=(0,\pi )$. D: Mass of ${\mathit{e}}_{12}^3$ mode at $\mathit{q}=(\pi ,\pi )$.

Figure 12

Dynamical structure factor $\mathrm{Im}{\chi }_{zz}(\mathit{q},\epsilon )$ at the critical point $J_2/J_1=J_{c,2}=1.0448$ between the $Z_2$ planar and $\text{U}\left(1\right)$ planar phases. The gray zones denote Stoner continuum. (a) Spectral wight of the collective modes. (b) Momentum-energy dispersion relations for the collective modes, showing that the lowest excitations at the $(\pi ,\pi )$ point become gapless.

Figure 13

Photonlike momentum-energy dispersion in $\mathrm{Im}{\chi }_{zz}(\mathit{q},\epsilon )$ at the $(\pi ,\pi )$ point in the $\text{U}\left(1\right)$ planar phase at $J_2/J_1=1.025<J_{c,2}$.

Figure 14

(a) Raman scattering process. (b) Each vertex part consists of two internal lines (wavy lines; gapless director-wave modes) and one external line (dashed line; magnetic field).