Transition between critical antiferromagnetic phases in the $J_1$-$J_2$ spin chain

Abstract

The $J_1$-$J_2$ spin chain is one of the canonical models of quantum magnetism, and has long been known to host a critical antiferromagnetic phase with power-law decay of spin correlations. We show in this Letter that there are, in fact, \textit{two} distinct critical antiferromagnetic phases, where the roles of the local dimer field and its dual field are interchanged: the `Affleck-Haldane' phase near the Heisenberg point $J_2 = 0$, where the dimer field that parametrises local singlet order is gapless and part of a joint $O(4)$ Néel-singlet order parameter; and the `Zirnbauer' phase which appears at sufficiently large ferromagnetic $J_2$, where the dimer field is gapped out and its \textit{dual} field -- the instanton density of the $O(3)$ Néel field -- is critical instead. The phases are so-named because each realises one of the competing pictures for how the $O(3)$ non-linear sigma model with a topological theta term renormalises to the $\mathfrak{\hat{su}}(2)_1$ Wess-Zumino-Witten model. We support these predictions with density matrix renormalisation group calculations.

Figures (3)

• Figure 1: (a) Sketch of the ${J_1{\text{--}}J_2}$ spin chain, showing the nearest-neighbour $J_1$ bonds (black, thick) and next-nearest-neighbour $J_2$ bonds (grey, thin). (b) Sketch of the phase diagram for $J_1 > 0$. The critical Affleck-Haldane phase around $J_2 = 0$ is described by the $O(4)$ NLSM with a Wess-Zumino term ($O(4)_{\mathrm{WZ}}$), where the $O(4)$ field comprises both local Néel and dimer order (cf. Eqs. \ref{['eq:Neel_order']} & \ref{['eq:singlet_order']}). At $J_2^* \approx -1.3$ the model transitions to the critical Zirnbauer phase: the dimer field acquires a mass and decouples from the joint-order parameter, and its dual field -- the instanton density of the Néel field -- is gapless instead. The critical part of both phases is the same conformal field theory in the infrared ($\hat{\mathfrak{su}}(2)_1$) but the microscopic nature of the critical field changes and can be detected in the dimer correlation functions, cf. Fig. \ref{['fig:DMRG']}.
• Figure 2: Flow diagram from the RG equations \ref{['eq:m2_flow_equation']} & \ref{['eq:g_flow_equation']} of the decoupled action \ref{['eq:Potts_phase_action']} (no numbers are shown because the choice of ultraviolet cutoff $\Lambda$ is arbitrary). Arrows indicate flow to the infrared. For sufficiently large bare values of $M^2$ we remain in the Zirnbauer phase where the dimer field is gapped (orange trajectories). As the bare value of $M^2$ is lowered, however, it will eventually flow to zero and the joint order parameter will recouple in the infrared (blue trajectories), leading us to the Affleck-Haldane phase (though the flow equations should not be trusted at small $M^2$). All the flows are away from the unstable $c = 2$ fixed point at $g = M^2 = 0$ (which is the direct sum of $O(3)_\pi$ with a free massless boson) in accordance with Zamolodchikov's $c$-theorem zamolodchikov1986irreversibility; though this unstable fixed point cannot be accessed by the spin chain as there is only one tuning parameter $J_2$.
• Figure 3: Transition between the Affleck-Haldane and Zirnbauer phases in the $J_1$--$J_2$ chain. (a), (b) Dimer correlations for $J_2 = -1$ and $-2$, respectively; dotted lines are the fits to Eq. \ref{['eq:dimer_corr_fit']}. We observe a region of exponential decay in the latter, but not the former. (c) Ratio $B/A$ (cf. Eq. \ref{['eq:dimer_corr_fit']}) of the exponential and power-law terms in the dimer correlators \ref{['eq:dimer_corrs']}; we find the exponential-decay term appears suddenly around $J_2^*\approx -1.3$. Error bars are obtained from the covariance matrix by Gaussian error propagation.