Skip to main content

Research Repository

Advanced Search

Emergent SO(5) symmetry at the columnar ordering transition in the classical cubic dimer model

Sreejith, G.J.; Powell, Stephen; Nahum, Adam


G.J. Sreejith

Adam Nahum


The classical cubic-lattice dimer model undergoes an unconventional transition between a columnar crystal and a dimer liquid, in the same universality class as the deconfined quantum critical point in spin-1/2 anti-ferromagnets but with very different microscopic physics and microscopic symmetries. Using Monte Carlo simulations, we show that this transition has emergent SO(5) symmetry relating quantities characterizing the two phases. While the low-temperature phase has a conventional order parameter, the defining property of the Coulomb liquid on the high-temperature side is deconfinement of monomers, and so SO(5) relates fundamentally different types of objects. Studying linear system sizes up to = 96, we find that this symmetry applies with an excellent precision, consistently improving with system size over this range. It is remarkable that SO(5) emerges in a system as basic as the cubic dimer model, with only simple discrete degrees of freedom. Our results are important evidence for the generality of the SO(5) symmetry that has been proposed for the NCCP 1 field theory. We describe an interpretation for these results in terms of a consistent hypothesis for the renormalization-group flow structure, allowing for the possibility that SO(5) may ultimately be a near-symmetry rather than exact. The classical dimer model on the cubic lattice illustrates three key mechanisms in three-dimensional (3D) critical phenomena. Two of these are the appearance of artificial gauge fields, and unconventional phase transitions where topologi-cal effects play a key role. The third, which we demonstrate here, is the emergence in the infrared (IR) of unusual non-abelian symmetries that would be impossible at a conventional Wilson–Fisher-like critical point. The close-packed dimer model has a power-law correlated 'Coulomb' phase [1, 2], governed by an emergent U(1) gauge field whose conserved flux arises from a 'magnetic field' defined in terms of dimers. A remarkable phase transition [3] separates this liquid from a 'columnar' phase, illustrated in Fig. 1(a), in which the dimers form a crystal, spontaneously breaking lattice symmetries. Despite being entirely classical, this transition is not described by Ginzburg–Landau theory, but is instead a Higgs transition of the U(1) gauge theory [4– 6]. The effective field theory is the noncompact CP 1 model (NCCP 1), in which the gauge field couples to a two-component bosonic matter field that condenses at the transition. NCCP 1 is also the effective field theory for the 'decon-fined' Néel–valence-bond solid (VBS) phase transition [7, 8] in 2+1D quantum antiferromagnets [9–18] and a related lattice loop model [19]. This raises the possibility that the dimer model exhibits a surprising emergent symmetry: Simulations of the loop model show SO(5) symmetry emerging at large scales [20]—either exactly or to an extremely good approximation. Earlier work on topological sigma models for decon-fined critical points [21, 22] revealed that SO(5) is a consistent possibility in the IR, despite the fact that it cannot be made manifest in the gauge theory [38]. The Néel–VBS transition involves a three-component antiferromagnetic order parameter and a two-component VBS order parameter; SO(5) allows all five components to be rotated into each other. This symmetry can be understood through a set of dualities for NCCP 1 and related theories [23]. Here we use Monte Carlo simulations to demonstrate emergent SO(5) at the dimer ordering transition. This large symmetry is particularly striking in a discrete classical model with no internal symmetries at all, only spatial symmetries together with a local constraint that is equivalent to U(1) symmetry in a dual representation. SO(5) furthermore unifies operators of conceptually distinct types, rotating the crystal order parameter—a conventional observable in terms of dimers— into 'monopole' operators that insert or remove monomers, and cannot be measured in the ensemble of dimer configurations. Together these yield a five-component SO(5) super-spin. The emergent symmetry group is therefore identical to that of the Néel–VBS transition. But it should be noted that the microscopic symmetries of the latter—roughly speaking , SO(3) × (lattice symmetries)—are very different from the (lattice symmetries) × U(1) in the dimer model. Previously, SO(5) has been demonstrated directly only in a single lattice model [20], and is also supported by level de-generacies in the JQ model [24], both realizations of the Néel– VBS transition. Its presence in the dimer model is particularly important because the IR behaviour of NCCP 1 is subtle and remains controversial [11, 12, 15–19, 23, 25–27]. The simplest explanation for SO(5) would be flow to a fixed point where FIG. 1. Dimer model phases and interactions. (a) Columnar phase (one of six symmetry-related ground states). (b) Disordered configuration , typical of high-temperature Coulomb phase. (c) Pairs of nearest-neighbor parallel dimers (back face of cube) contribute energy − 2. (d) Four parallel dimers around a cube contribute 4 .


Sreejith, G., Powell, S., & Nahum, A. (2019). Emergent SO(5) symmetry at the columnar ordering transition in the classical cubic dimer model. Physical Review Letters, 122(8),

Journal Article Type Article
Acceptance Date Jan 31, 2019
Online Publication Date Feb 27, 2019
Publication Date Mar 1, 2019
Deposit Date Feb 1, 2019
Publicly Available Date Feb 4, 2019
Journal Physical Review Letters
Print ISSN 0031-9007
Electronic ISSN 1079-7114
Publisher American Physical Society
Peer Reviewed Peer Reviewed
Volume 122
Issue 8
Article Number 080601
Keywords General Physics and Astronomy
Public URL
Publisher URL
Additional Information Emergent SO(5) Symmetry at the Columnar Ordering Transition in the Classical Cubic Dimer Model, G.?J. Sreejith, Stephen Powell, and Adam Nahum, Phys. Rev. Lett. 122, 080601.


You might also like

Downloadable Citations