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arXiv:2001.10045 · Physical Review Research

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

Jun Takahashi, Anders W. Sandvik

Published Sep 22, 2020• 40 citations
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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.

DOI
10.1103/physrevresearch.2.033459

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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.
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We study the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free S=1S=1S=1 square-lattice model. This is the same kind of transition that for S=1/2S=1/2S=1/2 has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice S=1/2S=1/2S=1/2 N\'eel-VBS transition have found consistency with the deconfined critical point scenario with no direct evidence for first-order behavior. In contrast to the S=1/2S=1/2S=1/2 case, in our quantum Monte Carlo simulations for the S=1S=1S=1 model, we present unambiguous evidence for a direct conventional first-order quantum phase transition. Classic signs of a first-order transition demonstrating coexistence including double-peaked histograms and switching behavior are observed. The sharp contrast from the S=1/2S=1/2S=1/2 case is remarkable; we hypothesize that this is a striking demonstration of the role of the size of the quantum spin in the phase diagram of two-dimensional lattice models.
We resolve the nature of the quantum phase transition between antiferromagnetic and valence-bond solid ground states in square-lattice spin-1/2 magnets - the putative deconfined quantum-critical point. Using quantum Monte Carlo simulations of J-Q models, in which Heisenberg exchange J competes with multi-spin interactions Qn formed by n singlet projectors, we demonstrate first-order discontinuities increasing with n. For n=2 and n=3, the first-order signatures are very weak and we extract universal scaling dimensions of a proximate multicritical point. Their values coincide closely with known bounds for an SO(5) symmetric conformal field theory (CFT) containing a relevant SO(5) singlet operator that destabilizes the generic deconfined critical point. We characterize the emergent SO(5) symmetry by its scaling dimensions of the leading irrelevant perturbations. Using the J-Q2-Q6 model, we tune the transition from weak to stronger first-order and find scaling of the coexisting order parameters with a large exponent beta ~ 0.85, in agreement with the relevant scaling dimensions of the multicritical point. The exponent that had previously been identified with the correlation length does not have a corresponding CFT level, and we explain it as pseudo-critical length scale of bubbles of one phase inside the other that are compressed in size by an SO(5)-violating irrelevant repulsive interaction between the two order parameters. The bubble scale reflects a hierarchy of excitation gaps generated by the SO(5) perturbation in the neighborhood of the first-order transition, akin to a Higgs mechanism. We argue that the multicritical point is the tip of a gapless spin liquid phase in frustrated Heisenberg models into which the J-Q models can be deformed, but the spin liquid, and perhaps the multicritical point itself, exist outside the regime amenable to sign-free simulations.
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We utilize density matrix renormalization group and fully augmented matrix product states (FAMPS) methods to investigate the valence bond solid (VBS) phase in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice Heisenberg model. To differentiate between the columnar valence bond solid and plaquette valence bond solid (PVBS) phases, we introduce an anisotropy ${\mathrm{\ensuremath{\Delta}}}_{y}$ in the nearest-neighboring coupling in the yyy direction, aiming at detecting the possible spontaneous rotational symmetry breaking in the VBS phase. In the calculations, we push the bond dimension to as large as D=25000D=25\phantom{\rule{0.16em}{0ex}}000D=25000 in FAMPS, simulating systems at a maximum size of $14\ifmmode\times\else\texttimes\fi{}14$. With a careful extrapolation of the truncation errors and appropriate finite-size scaling, followed by finite ${\mathrm{\ensuremath{\Delta}}}_{y}$ scaling analysis of the VBS dimer order parameters, we identify the VBS phase as a PVBS type, meaning there is no spontaneous rotational symmetry breaking in the VBS phase. This paper not only supports the presence of PVBS order in the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ square lattice Heisenberg model, but also highlights the capabilities of FAMPS in the study of two-dimensional quantum many-body systems.
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