Algebraic spin liquid as the mother of many competing orders

We study the properties of a class of two-dimensional interacting critical states—dubbed algebraic spin liquids—that can arise in two-dimensional quantum magnets. A particular example that we focus on is the staggered flux spin liquid, which plays a key role in some theories of underdoped cuprate superconductors. We show that the low-energy theory of such states has much higher symmetry than the underlying microscopic spin system. This symmetry has remarkable consequences, leading in particular to the unification of a number of seemingly unrelated competing orders. The correlations of these orders—including, in the staggered flux state, the Néel vector, and the order parameter for the columnar and box valence-bond solid states—all exhibit the same slow power-law decay. Implications for experiments in the pseudogap regime of the cuprates and for numerical calculations on model systems are discussed.

Figure 1

Cartoon pictures of some of the slowly varying competing orders within the staggered flux spin liquid state. These are the Néel state (A) and the columnar (B) and box (C) valence bond solids. The shaded regions denote those groups of spins that are most strongly combined into local singlets. Note that in the sF state these orders fluctuate in both space and time. These pictures describe the character of some of the important slowly varying fluctuations, but should not be viewed as snapshots of the physics at the lattice scale, which may be quite complicated.

Figure 2

Schematic temperature-chemical potential phase diagram of an underdoped cuprate superconductor. For $\mu <{\mu }_c$ the holes are gapped and the ground state is a spin liquid Mott insulator, while when $\mu >{\mu }_c$ a finite density of holes has entered the system and the ground state is a $d$-wave superconductor. The regime labeled “QC” is controlled by the Mott quantum critical point separating these two phases, while the FS regime is best thought of in terms of fluctuating $d$-wave superconductivity. The dashed line represents the finite-temperature behavior of a real underdoped material (with a superconducting ground state) at fixed doping.

Figure 3

Representation of an arbitrary diagram contributing to the two-point function of a fermion bilinear. The two dark circles represent the bilinear’s matrix structure and are positioned at the points where the bilinear is inserted into the diagram. The shaded region is built from the ingredients of the large-$N_f$ perturbation theory described in Sec. 2B—fermion and photon lines, and the gauge vertex. The pairs $(\alpha ,i)$ denote the $\mathrm{SU}\left(N_f\right)$ flavor ($\alpha =1,\dots ,N_f$) and Dirac spin ($i=1,2$) indices of the fermions.