Monopole Holes in a Partially Ordered Spin Liquid

Abstract

If spin liquids have been famously defined by what they are not, i.e. ordered, the past years have seen the frontier between order and spin liquid starting to fade, with a growing number of materials whose low-temperature physics cannot be explained without co-existence of (partial) magnetic order and spin fluctuations. Here we study an example of such co-existence in the presence of magnetic dipolar interactions, related to spin ice, where the order is long range and the fluctuations support a Coulomb gauge field. Topological defects are effectively coupled via energetic and entropic Coulomb interactions, the latter one being stronger than for the spin-ice ground state. Depending on whether these defects break the divergence-free condition of the Coulomb gauge field or the long-range order, they are respectively categorized as monopoles -as in spin ice -or monopole holes, in analogy with electron holes in semiconductors. The long-range order plays the role of a fully-occupied valence band, while the Coulomb spin liquid can be seen as an empty conducting band. These results are discussed in the context of other lattices and models which support a similar co-existence of Coulomb gauge field and long-range order. We conclude this work by explaining how dipolar interactions lift the spin liquid degeneracy at very low energy scale by maximizing the number of flippable plaquettes, in light of the equivalent quantum dimer model.The possibility to recast the collective behavior of electrons and atoms as elegant emergent phenomena is probably one of the most fascinating aspect of condensed matter. The emergence of quasi-particles does not only allow for a deeper understanding of the problem at hand but has also often led to surprising connections across physics, as recently illustrated by photon-like magnetic excitations in quantum spin ice 1-4 . Such approach takes an enhanced flavor when the particle has not yet been observed at high energy, or may not exist. This is for example the case for Majorana fermions observed in nanowires coupled to superconductors 5 , and for magnetic monopoles and their underlying Coulomb gauge field 6 .Coulomb gauge theories have emerged from a variety of discrete models, on the kagome lattice 7 , in fully-packed loop models 8 , itinerant-electron systems at partial-filling 9-11 and chemically disordered materials such as CsNiCrF 6 12 . We refer the interested reader to the excellent review by Chris Henley on this topic 13 . But its most famous experimental realization has probably been observed in the classical spin liquid ground state of spin ice materials Dy 2 Ti 2 O 7 and Ho 2 Ti 2 O 7 , where topological excitations take the form of magnetic monopoles effectively interacting via long-range Coulomb interactions.Interestingly, a Coulomb phase is not incompatible with partial magnetic order. For example, the stability of a Coulomb ferromagnet has been discussed in the context of the generalized Quantum Spin Ice model 14,15 and observed over a finite temperature window in a classical spin...

Out-of-equilibrium dynamics and extended textures of topological defects in spin ice

“…As a result, charge relaxation becomes extremely slow, while the magnetization decays rather quickly. The system finally forms the co-called fragmented Coulomb spin liquid (FCSL) [37][38][39] . In the FCSL charges are long-range ordered but the spin texture remains disordered, extensively degenerate and described by a Coulomb gauge theory.…”

“…The FCSL has been predicted theoretically 40,41 and observed experimentally [42][43][44] in nano-lithographic artificial kagome ice whose geometry prevents the existence of a charge-free Coulomb phase 45 . But in three dimensions, it has so far only been partially stabilized at equilibrium in the spin-ice model with dipolar interactions 46 , or requires four-body interactions 39 or the suppression of double charges 37,38 . The nonequilibrium magnetic-field quench proposed here provides a promising tool to realize this state and demonstrates the possibility of engineering a macroscopic state via nonequilibrium techniques 47 .…”

“…As a result, the initial charge order imposed by the field quench at t = 0 persists while the magnetization quickly vanishes. The resulting metastable phase is known as the fragmented Coulomb spin liquid (FCSL) [37][38][39] , where long-range (charge) order co-exists with a (Coulomb) spin liquid. The FCSL is a very elusive phase, which makes the present field-quench protocol a rare opportunity to realize it in a realistic model.…”

“…12 after the extinction of the total magnetization, it comes from the fact that, as opposed to the proliferation of vacuum tetrahedra observed for −0.25 J, the mechanism along the [111] direction is mediated by the propagation of pairs of local excitations: vacuum tetrahedra and double charges 39 . Both excitations can fragment into two single charges: one which is fixed and respects the initial charge order, remnant of the field quench, and the second one which can move along the [111] direction.…”

“…The presence of a crossover instead of a phase transition is confirmed by Bethe-lattice calculations, but could not be completely ruled out by simulations for 0.24 < J < 0.25. On a side note, the small negative value of M Q.Q for J = 0 (nearest-neighbor spin-ice model) is a consequence of the entropic Coulomb interactions between topological defects in spin ice 39,49,56 . …”