Origin of artificial electrodynamics in three-dimensional bosonic models

Several simple models of strongly correlated bosons on three-dimensional lattices have been shown to possess exotic fractionalized Mott insulating phases with a gapless “photon” excitation. In this paper we show how to view the physics of this “Coulomb” state in terms of the excitations of proximate superfluid. We argue for the presence of ordered vortex cores with a broken discrete symmetry in the nearby superfluid phase and that proliferating these degenerate but distinct vortices with equal amplitudes produces the Coulomb phase. This provides a simple physical description of the origin of the exotic excitations of the Coulomb state. The physical picture is formalized by means of a dual description of three-dimensional bosonic systems in terms of fluctuating quantum mechanical vortex loops. Such a dual formulation is extensively developed. It is shown how the Coulomb phase (as well as various other familiar phases) of three-dimensional bosonic systems may be described in this vortex loop theory. For bosons at half-filling and the closely related system of spin-$1∕2$ quantum magnets on a cubic lattice, fractionalized phases as well as bond- or “box”-ordered states are possible. The latter are analyzed by an extension of techniques previously developed in two spatial dimensions. The relation between these “confining” phases with broken translational symmetry and the fractionalized Coulomb phase is exposed.

Figure 1

View of the cross section of a vortex line in a three-dimensional superfluid of bosons at half-filling on a cubic lattice. Two checkerboard ordering patterns—where the boson density is higher on one of the two sublattices—are possible inside the vortex core.

Figure 2

Vortex lines with different core orders are indicated as $A$ and $B$. Left panel: a domain wall associated with the order in the core. If the superfluid is disordered by proliferating both vortex species simultaneously while keeping such domain walls gapped, the Coulomb phase results. The gapped domain walls survive as the monopoles of the Coulomb phase. Right panel: a “roton” loop is formed by bringing together a vortex with one ordered core and an antivortex with the other core. These particular roton loops correspond to loops of emergent magnetic flux which are strongly fluctuating in the Coulomb phase. The artificial photon is associated with fluctuations of these loops.

Figure 3

Explicit model that realizes the Coulomb phase. Boson islands are located on the link-centered sites of a simple cubic lattice. The islands can be also viewed as forming a network of corner-sharing octahedra, and two octahedron units are shown. In the Hamiltonian, Eq. (1), we stipulate a term $U{N}_r^2$ which prefers charge neutrality of each octahedron.

Figure 4

Schematic phase diagram of the boson model, Eq. (1), exhibiting the stable phases discussed in the text; note that we do not know the details for intermediate-coupling strengths and whether some other states may intervene.

Figure 5

Schematic phase diagram of the compact $U\left(1\right)$ gauge theory with two matter fields, Eq. (5). The dashed line in the superfluid phase represents the conjectured vortex core transition from a unique vortex to two physically distinct vortices (there is no transition in the superfluid ground state across this line).

Figure 6

Description of the monopole Berry phases for bosons at half-filling on a simple cubic lattice in terms of a static gauge potential seen by the monopoles hopping on the dual lattice. (a) ${\mathfrak{h}}_{\rho \sigma }^0$ gives a flux of $\pm \pi ∕3$ through each spatial placket of the dual lattice with the flux oriented from the $A$ to $B$ sublattice of the direct lattice. (b) Our gauge choice for $e^{i{X}_{\rho }^0}$, which realizes the fluxes $e^{i{\mathfrak{h}}_{\rho \sigma }^0}$ (Eq. (46)).

Figure 7

(a) Schematic picture of the columnar valence bond solid (VBS) state obtained when $v_8>0$. We have an increased monopole density (and also energy density) on the shaded $z$-even planes, which is interpreted as having valence bonds of the original spin model preferentially crossing these planes. Arrows along the placket edges indicate monopole currents. (b) 3D box VBS state is obtained when $v_8<0$. For clarity, only the planes with increased monopole density are shown ($x,y,z$ all even), and the dimers resonate around the cubes centered where these planes meet.

Figure 8

Quantum Ising model that realizes $P^{*}$ phase in two dimensions. Spins are located on the link-centered sites of a square lattice. Each shaded diamond indicates the $U$ term in the Hamiltonian, Eq. (D1).