Quantum Criticality and Deconfinement in Phase Transitions Between Valence Bond Solids

TL;DR

This paper demonstrates direct second-order transitions between two valence bond solid phases in spin-1/2 quantum antiferromagnets, realized generically on a bilayer honeycomb lattice. The critical theory is a Gaussian fixed line with dynamical exponent $z=2$, described by a deconfined $U(1)$ gauge field (a gapless photon) and deconfined spin-1/2 spinons at criticality, with monopoles and a marginal quartic term providing dangerously irrelevant perturbations. A rich nearby phase structure emerges as a tilted height-field develops an incomplete devil's staircase of commensurate VBS states, and the Rokhsar–Kivelson point is identified as a special, highly fine-tuned multicritical point. The results connect deconfined criticality concepts to dimer-model frameworks and offer predictions for bilayer realizations, finite-temperature behavior, and potential extensions to single-layer and higher-spin systems.

Abstract

We consider spin-half quantum antiferromagnets in two spatial dimensions in the quantum limit, where the spins are in a valence bond solid (VBS) phase. The transitions between two such VBS phases is studied. In some cases, an interesting second order transition controlled by a fixed line with varying critical exponents if found. A specific example is provided by an antiferromagnetically coupled bilayer system on the honeycomb lattice where a continuous quantum phase transition can generically exist between two VBS phases. Furthermore, these critical points are deconfined, in the sense that gapped spin-1/2 spinon excitations emerge right at the transition. The low energy physics of this critical point (upto marginally irrelevant interactions) contains just a free quadratically dispersing `photon'. The phase structure on one side of this continuous transition is very intricate consisting of a series of infinitely closely spaced further transitions in a `devil's staircase' form. Analogies with previous examples of deconfined quantum criticality are emphasized. Closely related transitions in single layer systems are explored. These are second order only at some multicritical points. The solvable Rokshar-Kivelson point of quantum dimer models of single layer systems is found to correspond to a very special non-generic multicritical point

Figures (2)

• Figure 1: Caricature of VBS phases on the bilayer honeycomb lattice. (a) The zero tilt state, with singlet bonds (thick lines) on the interlayer rungs. Note, this state does not break any lattice symmetry. (b) One of six possible maximally tilted (staggered) phases.
• Figure 2: Schematic depiction of the phase diagram of VBSs, the vertical axis in the plots is roughly the the parameter $\rho$ and the staggered state has the maximum tilt. The generic phase diagrams expected for the bilayer honeycomb lattice are shown in (a) and (b) (and also of the single layer honeycomb lattice after tuning one parameter, the cubic term, to zero). (a) The continuous transition (shown with the dashed line) is from a zero tilt phase to a region where the tilt (${\bf Q}$) exhibits a a devil's staircase structure, contours of equal tilt (thin solid lines) are shown. The critical line ends in a multicritical point $M'$ beyond which the transition is first order (solid line). The horizontal axis here may be thought of as the coefficient of the quartic term. (b) An alternate scenario, there is again a continuous transition to a region with the devil's staircase structure for the tilt. Here however, there is a multicritical point $M$ that is adjacent to the staggered state which could control a zero tilt to staggered state transition. The horizontal axis here may be thought of as the energy cost of the maximally tilted (staggered) state. (c) The RK point for the single layer square/honeycomb lattice - an infinite number of parameters need to be tuned to access this plane. Exact degeneracy of the different winding number sector ground states implies that states with arbitrary tilt lie infinitesimally close to the RK point as shown.