Deconfined Quantum Criticality

Deconfined quantum criticality proposes a continuous transition between Néel and valence-bond-solid phases in which the critical fields are fractional spinons coupled to an emergent gauge field. The two order parameters are not assembled into one microscopic Landau multiplet; instead, Néel order is a spinon bilinear and VBS order is a monopole operator. The field theory is a candidate infrared description, not proof that a given lattice transition is asymptotically continuous.

Required background. Gapped spin liquids supplies spinons and valence bonds; gauge-matter regimes supplies Higgs and confinement; critical surfaces and crossover supplies RG stability.

Helpful background. The CP(N-1) model supplies the continuum gauge theory.

Néel–VBS operator dictionary

Write the unit Néel vector as

$$\mathbf n=z^\dagger\boldsymbol\sigma z, \qquad |z_1|^2+|z_2|^2=1, \qquad z\sim e^{i\alpha}z.$$

$z_\alpha$ is a spin-$1/2$ spinon and the phase redundancy introduces $a_\mu$. The hard constraint above defines the CP1 parametrization of a unit vector. Its soft-spin extension relaxes that constraint, integrates over the amplitude of $z$, and uses the Euclidean action

$$\mathcal L=\sum_{\alpha=1}^{2}|(\partial_\mu-ia_\mu)z_\alpha|^2 +s|z|^2+u(|z|^2)^2 +\frac{1}{4e^2}f_{\mu\nu}f_{\mu\nu} +\lambda_4(\mathcal M_4+\mathcal M_4^\dagger).$$

Repeated spacetime indices are contracted with the Euclidean metric. The microscopic gauge field is compact. On the square lattice, Berry phases make a unit monopole transform like the complex VBS order parameter and permit only quadrupled monopoles in the action. Writing the critical action as “noncompact” means $\lambda_4$ is conjectured irrelevant at the fixed point, not that compactness has disappeared Senthil et al. 2004.

For $s< s_c$, spinons condense in a gauge-fixed description, $\langle\mathbf n\rangle\ne0$, and the gauge field is Higgsed. For $s>s_c$, spinons are gapped; monopole proliferation confines them and selects VBS order. At criticality, deconfined spinons and an emergent flux symmetry can appear over the scaling regime.

Dangerously irrelevant monopoles

If $\mathcal M_4$ is irrelevant at criticality but relevant in the VBS phase, two scales occur: the spin correlation length $\xi$ and a larger VBS angle-locking length $\xi_{\rm VBS}$. Finite systems with $L$ between them can show an approximately U(1)-symmetric VBS distribution even though the thermodynamic phase has only Z4 symmetry. This is a predicted crossover, not alone evidence for SO(5) symmetry.

Duality arguments suggest an emergent SO(5) rotating the three Néel components and two VBS components in some versions Wang et al. 2017. Such symmetry would imply common scaling dimensions and joint-distribution relations. It can also emerge approximately along a walking or weak-first-order flow, so asymptotic tests remain necessary.

Scope and failure modes

Check the microscopic symmetry and Berry-phase pattern, monopole quantum numbers, number of spinon flavors, allowed anisotropies, and whether only one tuning is required. A relevant singlet, relevant monopole, runaway coupling, or bimodal thermodynamic energy distribution can replace a critical point by multicriticality or weak first order. The continuum operator map remains useful for organizing a long crossover even then.

Exercise

Why is the VBS order parameter represented by a monopole rather than by $z^\dagger z$?

Solution

$z^\dagger z$ is a symmetry-singlet density. A $2\pi$ gauge-flux insertion carries the lattice rotation and translation quantum numbers of the complex VBS pattern because of microscopic Berry phases. Thus the unit monopole $\mathcal M_1$ has the VBS transformation law, while only $\mathcal M_4$ is invariant in the square-lattice action.

References

• T. Senthil, Ashvin Vishwanath, Leon Balents, Subir Sachdev, and Matthew P. A. Fisher, “Deconfined Quantum Critical Points,” Science 303 (2004) 1490–1494, doi:10.1126/science.1091806.
• Chong Wang, Adam Nahum, Max A. Metlitski, Cenke Xu, and T. Senthil, “Deconfined Quantum Critical Points: Symmetries and Dualities,” Physical Review X 7 (2017) 031051, doi:10.1103/PhysRevX.7.031051.