Deconfined Quantum Critical Points: Symmetries and Dualities

The deconfined quantum critical point (QCP), separating the Néel and valence bond solid phases in a 2D antiferromagnet, was proposed as an example of $(2+1)\mathrm{D}$ criticality fundamentally different from standard Landau-Ginzburg-Wilson-Fisher criticality. In this work, we present multiple equivalent descriptions of deconfined QCPs, and use these to address the possibility of enlarged emergent symmetries in the low-energy limit. The easy-plane deconfined QCP, besides its previously discussed self-duality, is dual to $N_f=2$ fermionic quantum electrodynamics, which has its own self-duality and hence may have an $\mathrm{O}(4)\times Z_2^T$ symmetry. We propose several dualities for the deconfined QCP with SU(2) spin symmetry which together make natural the emergence of a previously suggested SO(5) symmetry rotating the Néel and valence bond solid orders. These emergent symmetries are implemented anomalously. The associated infrared theories can also be viewed as surface descriptions of $(3+1)\mathrm{D}$ topological paramagnets, giving further insight into the dualities. We describe a number of numerical tests of these dualities. We also discuss the possibility of “pseudocritical” behavior for deconfined critical points, and the meaning of the dualities and emergent symmetries in such a scenario.

Popular Summary

Quantum matter, formed by many interacting electrons and often extremely challenging to understand, can sometimes be described by seemingly very different theories. These theories are said to be “dual” to each other, and a hard question in one theory may become a simple one in another. Here, we use this idea of “duality” to understand a class of exotic quantum magnets, known as deconfined quantum critical points (DQCPs). We show that they are connected to the problem of electrons interacting with electromagnetic fields in two (instead of three) dimensions, a connection that unveils some previously hidden properties of these theories.

More precisely, we show that the quantum field theories that describe DQCPs—a class of exotic quantum phase transitions—are dual to quantum electrodynamics in two spatial dimensions (and some of its variants). This duality reveals some deep structures of both theories that were previously hidden. In particular, a large symmetry, which is absent microscopically in both theories, is predicted to emerge at the critical point. Our approach explains a previous observation, seen in a numerical analysis, of a large emergent symmetry in a class of DQCPs. We also generate many other predictions for both DQCP and quantum electrodynamics for future numerical tests.

We expect our work to inspire future developments on both deconfined quantum criticality and quantum electrodynamics, especially on the numerical side.

Figure 1

Mean-field phase diagram for the mass term $m_1|z_1{|}^2+m_2|z_2{|}^2$ in the easy-plane ${\mathrm{NCCP}}^1$ model. The same phase diagram is obtained from the QED theory with mass term $m_1{\overline{\psi }}_1{\psi }_1-m_2{\overline{\psi }}_2{\psi }_2$. The upper panel is realized in the context of the quantum magnet, where the $\mathrm{U}(1{)}_{B_2}$ symmetry is only emergent. The lower panel is realized in the context of the integer quantum Hall transition of bosons, where the two superfluid phases correspond to superfluids of the first or second layer, i.e., up or down components of the pseudospin. (Going beyond mean field will move the phase boundaries away from the axes.)

Figure 2

Phase diagram near the SO(5)-invariant fixed point with perturbation of the form ${\lambda }_1(X_{11}+X_{22})+{\lambda }_2{X}_{55}$.

Figure 3

Phase diagram near the SO(5)-invariant fixed point with perturbation of the form $-{\tilde{\lambda }}_1(|z_1{|}^2+|z_2{|}^2)+{\tilde{\lambda }}_2[(|z_1{|}^2-|z_2{|}^2{)}^2-\frac{1}{3}(|z_1{|}^2+|z_2{|}^2{)}^2]$. This is the natural perturbation to consider in the context of deconfined criticality in quantum magnets. The emergent SO(5) symmetry requires that the slope of the lower transition line is twice that of the upper transition line.

Figure 4

Phase diagram near the SO(5)-invariant fixed point with perturbation of the form $m_{\varphi }{\varphi }^2+m_{\psi }({\overline{\psi }}_1{\psi }_1-{\overline{\psi }}_2{\psi }_2)$. This is the natural perturbation to consider in the context of ${\mathrm{QED}}_3$-Gross-Neveu theory. The emergent SO(5) symmetry predicts that the slope of the transition lines is related to the relative amplitude of the correlation functions of the two operators, Eq. (73)