Competition between Neel, Haldane nematic, plaquette valence bond solid, and $(\pi ,\pi )$ valence bond solid phases in SU(N) analogs of $S=1$ square-lattice antiferromagnets

Abstract

We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) methods to study the phases and transitions displayed by a class of sign-free designer Hamiltonians for SU($N$) analogs of spin $S=1$ quantum antiferromagnets on the square lattice. The SU($N$) spins are generators of the single-row two-column representation (complex conjugate of single-row two-column representation) on $A$ ($B$) sublattices of the square lattice, and the Hamiltonian is designed to explore the competition between the nearest neighbour antiferromagnetic exchange couplings $J$ and four-spin interactions $Q$ that favor a plaquette-ordered valence bond solid (p-VBS) ground state. We find that this state is indeed established at large $Q/J$ for all $N>3$. For $3<N\le 9$, the ground state exhibits a direct first order quantum phase transition from a small-$Q/J$ Néel ordered antiferromagnetic state to this large-$Q/J$ p-VBS state. The ground state at $Q/J=0$ for $N\ge 10$ has been previously reported to be a valence bond nematic state, dubbed the Haldane nematic in recent literature. For small nonzero $Q/J$ and $N\ge 10$, we additionally find an unusual intermediate state in which the bond energy has a Bragg peak at wavevector $(\pi ,\pi )$ with no accompanying Bragg peaks at wavevectors $(\pi ,0)$ and $(0,\pi )$. This $(\pi ,\pi )$ state also appears to be metastable at $Q/J=0$, as evidenced by low temperature histograms of the Haldane nematic and $(\pi ,\pi )$ order parameters. Deep in the p-VBS phase of the ground state phase diagram, we find the temperature-driven melting of the p-VBS order is in the Ashkin-Teller universality class. In this regime, we identify an interesting signature of Ashkin-Teller criticality in bond correlations at wavevector $(\pi ,\pi )$; this is in addition to the expected critical fluctuations of the conventional p-VBS order parameter at wavevectors $(\pi ,0)$ and $(0,\pi )$.

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