Valence-bond solids, vestigial order, and emergent SO(5) symmetry in a two-dimensional quantum magnet

Abstract

We introduce a quantum spin-1/2 model with many-body correlated Heisenberg-type interactions on the two-dimensional square lattice, designed so that the system can host a fourfold degenerate plaquette valencebond solid (PVBS) ground state that spontaneously breaks Z 4 symmetry. The system is sign-problem free and amenable to large-scale quantum Monte Carlo simulations, thus allowing us to carry out a detailed study of the quantum phase transition between the standard Néel antiferromagnetic (AFM) and PVBS states. We find a first-order transition, in contrast to previously studied continuous transitions from the AFM phase into a columnar valence-bond solid (CVBS) phase. The theory of deconfined quantum criticality predicts generic continuous AFM-CVBS and AFM-PVBS transitions, and, in one version of the theory, the two critical order parameters transform under SO(5) symmetry. Emergent SO(5) symmetry has indeed been observed in studies of the AFM-CVBS transition, and here we show that the first-order AFM-PVBS transition also exhibits SO(5) symmetry at the transition point. Such unexpected symmetry of the coexistence state, which implies a lack of energy barriers between the coexisting phases, has recently been observed at other first-order transitions, but the case presented here is the first example with SO(5) symmetry. The extended symmetry may indicate that the transition is connected to a deconfined critical point. We also discuss the first-order transition in the context of a recent proposal of spinons with fracton properties in the PVBS state, concluding that the fracton scenario is unlikely. Furthermore, we discover a novel type of eightfold degenerate VBS phase, arising when the PVBS state breaks a remaining Z 2 symmetry. This second phase transition, which is continuous, implies that the PVBS phase can be regarded as an intermediate "vestigial" phase, a concept recently introduced to describe multistage phase transitions involving a continuous symmetry. Here we construct a six-dimensional order parameter and also introduce a general graph-theoretic approach to describe the two-stage discrete symmetry breaking. We discuss different ways of breaking the symmetries in one or two stages at zero and finite temperatures. In the latter case, we observe fluctuation-induced first-order transitions, which are hallmarks of vestigial phase transitions. We also mention possible connections of the AFM-PVBS transition to the SO(5) theory of high-T c superconductivity.

Quantum Criticality and Spin Liquid Phase in the Shastry-Sutherland model

“…In some variants of the J-Q model, clearly first-order transitions were observed [5,60,61]. The "checker-board" J-Q (CBJQ) model [5] (and a closely related loop model [62]) has a Z 2 breaking PS phase like that in the SS model.…”

Fractional spin excitations in the infinite-layer cuprate CaCuO$_2$

“…It is therefore natural to correlate the high-energy continuum, which we interpret as made of fractionalized magnetic excitations, to the strong J c which in this material cannot be regarded as a small perturbation. The presence of spinons in the spectrum of the Heisenberg AF with strong multi-spin couplings (of which the ring exchange is an example) has been predicted some years ago and is now well established [20][21][22]32]. Interestingly, a recent exact diagonalization study [57] performed on the EHM Hamiltonian has shown that, at J c /J ≈ 1 the excitation spectrum at ( 1 /2, 0) breaks into a continuum of states very close in energy, similarly to what has been calculated for Neél-RVB and Neél-VBS transitions [20,[28][29][30][31].…”

“…square lattices have been predicted in several Heisenberglike Hamiltonians [11,19,20,22,[28][29][30]32] spinon pairs have been often invoked to explain the anomalous highenergy dynamics of cuprates [10] and isostructural iridates [12]. However, an undisputed experimental evidence of their existence in 2D is still missing.…”

“…1) among first (J 1 ), second (J 2 ) and third (J 3 ) nearest neighbours, plus another multi-spin term that couples four spins across a Cu 4 O 8 plaquettes, usually called ring exchange [6,7,27]. It has been put forward that the partial deconfinement of spinons occurs in the presence of frustrating next-nearest neighbor [28][29][30][31] or multi-spin couplings [20][21][22]32]. In particular, many studies have established that J c is by far the most important term [6,7,15].…”

Building models of topological quantum criticality from pivot Hamiltonians

“…We argue that, remarkably, these two regimes are separated by a deconfined quantum critical point (DQCP), which we will support through numerics and field theory arguments. In fact, its universality is that of the SO(5) DQCP, which has been studied before [2,[60][61][62][63][64][65][66][67][68][69][70], e.g., as a direct transition between an SO(3) Néel state (the analogue of the blue shaded region in Fig. 2) and a 6), perturbed by a trivial paramagnet (horizontal direction) and a same-sublattice Ising coupling (vertical direction).…”