Intertwined nematic orders in a frustrated ferromagnet

We investigate the quantum phases of the frustrated spin-$\frac{1}{2}{J}_1-J_2-J_3$ Heisenberg model on the square lattice with ferromagnetic $J_1$ and antiferromagnetic $J_2$ and $J_3$ interactions. Using the pseudofermion functional renormalization group technique, we find an intermediate paramagnetic phase located between classically ordered ferromagnetic, stripy antiferromagnetic, and incommensurate spiral phases. We observe that quantum fluctuations lead to significant shifts of the spiral pitch angles compared to the classical limit. By computing the response of the system with respect to various spin rotation and lattice symmetry-breaking perturbations, we identify a complex interplay between different nematic spin states in the paramagnetic phase. While retaining time-reversal invariance, these phases either break spin-rotation symmetry, lattice-rotation symmetry, or a combination of both. We therefore propose the $J_1-J_2-J_3$ Heisenberg model on the square lattice as a paradigmatic example where different intimately connected types of nematic orders emerge in the same model.

Figure 1

(a) Quantum phase diagram of the spin-$\frac{1}{2}{J}_1-J_2-J_3$ Heisenberg model [Eq. (1)] obtained within PFFRG. The coupling constants are normalized such that $|J_1|+J_2+J_3=1$. The phase diagram hosts a large PM domain featuring $s$-SN, $d$-SN, and LN orders. Interpolating colors indicate regions of uncertainties near the phase boundaries. The corresponding classical phase diagram is shown in the upper left. A depiction of the exchange couplings is shown in the upper right. (b)–(e) Illustrations of the real space pattern (upper row) and momentum-space resolved magnetic susceptibility profile (lower row) in units of $1/|J_1|$ for magnetism at ordering vectors $\mathbf{Q}=(0,0),(\pi ,0),(q,0)$, and $(q,q)$, evaluated at the parameters $\left(\right|J_1|,J_2,J_3)=(0.84,0.08,0.08),(0.52,0.42,0.06),(0.48,0.38,0.14),\text{and}(0.52,0.06,0.42)$, respectively.

Figure 2

Illustration of the biasing patterns. (a) and (b) The oval with $+$($-$) sign represents a perturbation proportional to $\left|{\sum }_i({\widehat{S}}_i^z{\widehat{S}}_{i+\widehat{e}}^z-{\widehat{S}}_i^x{\widehat{S}}_{i+\widehat{e}}^x-{\widehat{S}}_i^y{\widehat{S}}_{i+\widehat{e}}^y)\right|$ with a positive (negative) prefactor and the respective nearest neighbor lattice vector $\widehat{e}=\widehat{x},\widehat{y}$. (c) The perturbation is proportional to the difference in the thickness of the two diagonals, i.e., $\langle {\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_{i+\widehat{x}+\widehat{y}}\rangle -\langle {\widehat{\mathbf{S}}}_i·{\widehat{\mathbf{S}}}_{i+\widehat{x}-\widehat{y}}\rangle$.

Figure 3

Deviation $\delta \mathbf{Q}=\mathbf{Q}-{\mathbf{Q}}_{\text{cl}}$ of the ordering wave vector $\mathbf{Q}$ from its classical value ${\mathbf{Q}}_{\text{cl}}$ as a function of $J_2/\left|J_1\right|$ and $J_3/\left|J_1\right|$. The shifts in the black regions are identically zero, and the gray region denotes the PM phase. The classical boundaries are marked with white dashed lines. The maximum shifts in the $(q,q)$ and $(q,0)$ HM phases are $\approx 37\%$ and $\approx 100\%$, respectively, of the classical values.

Figure 4

Magnetic susceptibility profiles in the PM regime of the quantum phase diagram evaluated at the corresponding labeled points.

Figure 5

(a) Representative RG flows of the magnetic susceptibilities at the ordering wave vectors of the four ordered regimes of Fig. 1 and the PM regime, evaluated at the following data points $\left(\right|J_1|,J_2,J_3)$: (i) FM: $(0.90,0.00,0.10)$, (ii) $(\pi ,0)$: $(0.52,0.42,0.06)$, (iii) $(q,0)$ HM: $(0.46,0.42,0.12)$, (iv) $(q,q)$ HM: $(0.52,0.06,0.42)$, and (v) PM: $(0.66,0.26,0.08)$. The points at which the solid lines become dashed (marked by arrows) indicate an instability in the flow and express the onset of order. In the smooth flow (green curve) indicating paramagnetism, no such instability is found. (b)–(d) Representative nematic responses [Eq. (6)] inside the three PM phases of Fig. 1, evaluated at the data points (0.74,0.14,0.12) (b), (0.68,0.30,0.02) (c), and (0.12,0.52,0.36) (d). The dark purple color curve (${\mathrm{LN}}_{\mathrm{NN}}$) corresponds to a lattice nematic bias on the NN bonds.