Pyrochlore Photons: The U(1) Spin Liquid in a S=1/2 Three-Dimensional Frustrated Magnet

TL;DR

This work demonstrates that a pair of three-dimensional frustrated spin models on pyrochlore and related lattices host a stable U(1) spin liquid with an emergent gauge structure, a gapless photon, and gapped spinons and monopoles. By deriving an exact soluble RK point and mapping the low-energy sector to Gaussian quantum electrodynamics, the authors establish a Coulomb phase with 1/r interactions, topological order, and characteristic power-law correlations. They further analyze corrections and dualities to connect microscopic models to their effective field theory, and confirm predictions via exact ground-state wavefunctions and Monte Carlo methods. The results provide a concrete, tractable route to realizing and diagnosing U(1) fractionalization in three dimensions, with tangible experimental signatures such as a T^3 contribution to the specific heat from the emergent photon.

Abstract

We study the S=1/2 Heisenberg antiferromagnet on the pyrochlore lattice in the limit of strong easy-axis exchange anisotropy. We find, using only standard techniques of degenerate perturbation theory, that the model has a U(1) gauge symmetry generated by certain local rotations about the z-axis in spin space. Upon addition of an extra local interaction in this and a related model with spins on a three-dimensional network of corner-sharing octahedra, we can write down the exact ground state wavefunction with no further approximations. Using the properties of the soluble point we show that these models enter the U(1) spin liquid phase, a novel fractionalized spin liquid with an emergent U(1) gauge structure. This phase supports gapped S^z = 1/2 spinons carrying the U(1) ``electric'' gauge charge, a gapped topological point defect or ``magnetic'' monopole, and a gapless ``photon,'' which in spin language is a gapless, linearly dispersing S^z = 0 collective mode. There are power-law spin correlations with a nontrivial angular dependence, as well as novel U(1) topological order. This state is stable to ALL zero-temperature perturbations and exists over a finite extent of the phase diagram. Using a convenient lattice version of electric-magnetic duality, we develop the effective description of the U(1) spin liquid and the adjacent soluble point in terms of Gaussian quantum electrodynamics and calculate a few of the universal properties. The resulting picture is confirmed by our numerical analysis of the soluble point wavefunction. Finally, we briefly discuss the prospects for understanding this physics in a wider range of models and for making contact with experiments.

Figures (14)

• Figure 1: Phase diagram for both models. The parameter $V/J_{ring}$ is the relative strength of the Rokhsar-Kivelson potential and the XY ring exchange that obtains in the easy-axis limit of the Heisenberg model. The soluble point is located at $V/J_{ring} = 1$, which is a special deconfined point of the adjacent $U(1)$ spin liquid. Just to the right of the soluble point the models go into an Ising ordered state. Sufficiently far to the left we expect Ising order, while at intermediate values of $V/J_{ring}$ states with broken translation symmetry but no magnetic order are also possible. Immediately to the left of the soluble point, the $U(1)$ spin liquid exists over a finite (but unknown) extent of the phase diagram.
• Figure 2: The pyrochlore lattice (left), and one up-pointing tetrahedron (right). One sublattice of tetrahedra is shaded, and the other transparent. The thickened bonds show the location of a pyrochlore hexagon. Each such hexagon is a member of one of four orientations of kagomé lattice planes. The numbering of sites in the up-pointing tetrahedron on the right is the convention used in the text. For $i=0,1,2$, the fcc Bravais lattice vector ${\bf a}_i$ points in the direction given by looking from site $3$ to site $i$.
• Figure 3: Depiction of the processes contributing to the third-order degenerate perturbation theory for the easy-axis pyrochlore Heisenberg antiferromagnet. Processes (A) and (B) give only trivial constant shifts of the energy. Process (C) leads to an XY ring exchange term acting on hexagonal plaquettes.
• Figure 4: A small piece of the diamond lattice. The links form hexagonal loops corresponding to the pyrochlore hexagons. These are the shortest possible closed paths on the diamond lattice. The hexagon with three thickened bonds depicts the dimer positions on a flippable hexagon. The alternating full and empty bonds correspond to alternating up and down spins.
• Figure 5: Illustration that the links of the cubic lattice are equivalent to the sites of a lattice of corner-sharing octahedra.