Quantum magnetism and criticality

TL;DR

The paper surveys quantum magnetism in two-dimensional insulating systems, detailing phases such as Néel order, valence-bond solids, and $Z_2$ spin liquids, and develops low-energy descriptions using order-parameter fields and emergent gauge fields. It contrasts conventional Landau-Ginzburg-Wilson criticality for the Néel–dimer transition with deconfined criticality that features spinons and an emergent U(1) gauge field, linking monopole fluctuations to VBS order and highlighting numerical evidence for suppressed monopoles near the transition. The discussion extends to itinerant systems, showing how superfluid–insulator transitions at various fillings map onto gauge theories with fractionalized excitations, giving rise to exotic metallic states such as fractionalized Fermi liquids and algebraic charge liquids. Finally, it connects finite-temperature quantum criticality to universal hydrodynamics and holographic dualities, underscoring broad implications for superconductivity, metallic states, and cold-atom platforms.

Abstract

Magnetic insulators have proved to be fertile ground for studying new types of quantum many body states, and I survey recent experimental and theoretical examples. The insights and methods transfer also to novel superconducting and metallic states. Of particular interest are critical quantum states, sometimes found at quantum phase transitions, which have gapless excitations with no particle- or wave-like interpretation, and control a significant portion of the finite temperature phase diagram. Remarkably, their theory is connected to holographic descriptions of Hawking radiation from black holes.

Figures (10)

• Figure 1: Néel ground state of the $S=1/2$ antiferromagnet $H_0$ will all $J_{ij} = J$ on the (a) square and (b) triangular lattices. The spin polarization is (a) collinear and (b) coplanar.
• Figure 2: The coupled dimer antiferromagnet, described by the Hamiltonian $H_0$, with $J_{ij} = J$ on the full red lines, and $J_{ij} = J/g$ on the dashed red lines. (a) The large $g$ ground state, with each ellipse representing a singlet valence bond $(\left|\uparrow \downarrow \right\rangle - \left|\downarrow \uparrow \right\rangle)/\sqrt{2}$. (b) The $S=1$ spin triplon excitation. The pair of parallel spins on the broken valence bond hops between dimers using the $J/g$ couplings.
• Figure 3: (a) Columnar VBS state of $H_0+H_1$ with $J_{ij} = J$ on all bonds. This state is the same as in Fig. \ref{['dimer']}a, but the square lattice symmetry has been broken spontaneously. Rotations by multiples of $\pi/2$ about a lattice site yield the 4 degenerate states. (b) 4-fold degenerate plaquette VBS state with the rounded squares representing $S=0$ combination of 4 spins.
• Figure 4: Caricature of a spin liquid state. The valence bonds are entangled between different pairings of the spins, only one of which is shown. Also shown are two unpaired $S=1/2$ spinons, which can move independently in the spin liquid background.
• Figure 5: Histogram of the VBS order parameter defined in Eq. (\ref{['defPsi']}) in the numerical study anders by Sandvik of $H_0+H_1$. The circular symmetry is emergent and implies a Goldstone boson which is the emergent photon $A_\mu$.