Abstract
Spin ice is a frustrated magnetic system that at low temperatures exhibits a Coulomb phase, a classical spin liquid with topological order and deconfined excitations. This work establishes the presence of a Coulomb phase with coexisting ferromagnetic order in a microscopic model of classical spin ice subject to uniaxial lattice distortion. General theoretical arguments are presented for the presence of such a phase, and its existence is confirmed using Monte Carlo results. This example is used to illustrate generic properties of spin liquids with magnetic order, including deconfinement of monopoles, signatures in the neutron-scattering structure factor, and critical behavior at phase transitions. An analogous phase, a superfluid with spontaneously broken particle-hole symmetry, is demonstrated in a model of hard-core lattice bosons, related to spin ice through the quantum-classical correspondence.
Highlights
Spin liquids are phases where magnetic degrees of freedom exhibit strong local correlations, despite the persistence of large fluctuations,[1] of either quantum mechanical or thermal origin
A precise definition of a quantum spin liquid (QSL) can be phrased in terms of long-range entanglement,[1] while the Coulomb phase,[9] the classical spin liquid (CSL) that is of primary interest here, can be defined through deconfinement of fractionalized “monopole” excitations.[8]
We present Monte Carlo (MC) results that confirm both of these statements, and illustrate the generic properties of ordered spin liquids, including the structure factor for elastic neutron scattering
Summary
Spin liquids are phases where magnetic degrees of freedom exhibit strong local correlations, despite the persistence of large fluctuations,[1] of either quantum mechanical or thermal origin. A precise definition of a quantum spin liquid (QSL) can be phrased in terms of long-range entanglement,[1] while the Coulomb phase,[9] the classical spin liquid (CSL) that is of primary interest here, can be defined through deconfinement of fractionalized “monopole” excitations.[8] Experimental evidence exists for a Coulomb phase in the spin-ice compounds, which can be treated as classical at relevant temperatures.[10] These definitions provide positive characterizations for QSL and CSL phases, and make clear the possibility of a magnetically ordered spin liquid, in which spin-liquid phenomena coexist with conventional symmetry-breaking order. While the numerical results are consistent with this prediction, larger system sizes would be required for a definitive confirmation of the universality class This phase transition provides another interesting example of the diversity of critical phenomena that exists in the neighborhood of spin-liquid phases. In the Appendix, the classical–quantum mapping developed in Ref. 14 is applied to this system, and the resulting quantum model is related to a problem of hard-core quantum bosons studied by Rokhsar and Kotliar.[15]
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