Nematic Order in Square Lattice Frustrated Ferromagnets

We present a new scenario for the breakdown of ferromagnetic order in a two-dimensional quantum magnet with competing ferromagnetic and antiferromagnetic interactions. In this, dynamical effects lead to the formation of two-magnon bound states, which undergo Bose-Einstein condensation, giving rise to bond-centered nematic order. This scenario is explored in some detail for an extended Heisenberg model on a square lattice. In particular, we present numerical evidence confirming the existence of a state with $d$-wave nematic correlations but no long-range magnetic order, lying between the saturated ferromagnetic and collinear antiferromagnetic phases of the ferromagnetic $J_1\mathrm{\text{−}}{J}_2$ model. We argue by continuity of spectra that this phase is also present in a model with 4-spin cyclic exchange.

Figure 1

Phase diagram for the spin-$1/2$ $J_1\mathrm{\text{−}}{J}_2\mathrm{\text{−}}K$ Heisenberg model on a square lattice, in the parameter range where $J_1<0$ is FM and both $J_2\ge 0$ and $K\ge 0$ are AF. Solid lines on the $J_2/|J_1|$ and $K/|J_1|$ axes denote known Néel phases. A single nematic phase interpolates between the FM occurring for large FM $J_1$ and these two, different, AF ordered states.

Figure 2

Schematic phase diagram of the FM $J_1\mathrm{\text{−}}{J}_2\mathrm{\text{−}}K$ extended Heisenberg model in applied field, showing how the two-magnon instability crosses over from 1st to 2nd order in applied field and is eventually succeeded by a conventional one-magnon instability. The inset shows representative spectra for 1st and 2nd order transitions into $d$-wave nematic state (high-spin states only).

Figure 3

Spectrum of extended Heisenberg model for $N=36$ spin cluster with $J_1=-1$, $J_2=0.4$, $K=0$, showing the marked alternation in energies of the lowest lying states in even and odd spin sectors. The inset shows the double peak structure of the heat capacity $c_V(T)$ for clusters of $N=16$ and $N=20$ spins and the same set of parameters.

Figure 4

Finite-size scaling of the energy gaps of extended Heisenberg model with $J_1=-1$, $J_2=0.4$, $K=0$, for $A_1$ and $B_1$ symmetry states (lower) and $E$ ones (upper) in the even and odd spin sectors, respectively.

Figure 5

Bond-nematic correlations of an extended Heisenberg model with $J_1=-1$, $J_2=0.4$, $K=0$, for a cluster of $N=32$ spins, showing the $d$-wave character of the nematic phase. Correlations are measured relative to the reference bond 1–2.