Quantum Spin Ice with Frustrated Transverse Exchange: From a π-Flux Phase to a Nematic Quantum Spin Liquid

Spectroscopic Signatures of Fractionalization in Octupolar Quantum Spin Ice

Recent investigations on the dipolar-octupolar compounds Ce_{2}Zr_{2}O_{7} and Ce_{2}Sn_{2}O_{7} suggest that they may stabilize so-called π-flux octupolar quantum spin ice (π-O-QSI), a novel three-dimensional quantum spin liquid hosting emergent photons. Confirmation of such an exotic phase would require the prediction of a distinctive signature and its subsequent experimental observation. So far, however, theoretical predictions for any such sharp smoking-gun signatures are lacking. In this Letter, we thoroughly investigate O-QSI using an extension of gauge mean-field theory. This framework produces a phase diagram consistent with previous work and an energy-integrated neutron scattering signal with intensity-modulated rod motifs, as reported in experiments and numerical studies. We predict that the dynamical spin structure factor of π-O-QSI is characterized by a broad continuum with three distinctive peaks as a consequence of the two mostly flat spinon bands. These three peaks should be measurable by high-resolution inelastic neutron scattering. Such spectroscopic signatures would be clear evidence for the realization of π-flux quantum spin ice.

Abundance of Hard-Hexagon Crystals in the Quantum Pyrochlore Antiferromagnet

We propose a simple family of valence-bond crystals as potential ground states of the S=1/2 and S=1 Heisenberg antiferromagnet on the pyrochlore lattice. Exponentially numerous in the linear size of the system, these can be visualized as hard-hexagon coverings, with each hexagon representing a resonating valence-bond ring. This ensemble spontaneously breaks rotation, inversion, and translation symmetries. A simple, yet accurate, variational wave function allows a precise determination of the energy, confirmed by the density matrix renormalization group and numerical linked cluster expansion, and extended by an analysis of excited states. The identification of the origin of the stability indicates applicability to a broad class of frustrated lattices, which we demonstrate for the checkerboard and ruby lattices. Our work suggests a perspective on such quantum magnets, in which unfrustrated motifs are effectively uncoupled by the frustration of their interactions.

L-based numerical linked cluster expansion for square lattice models

We introduce a numerical linked cluster expansion for square-lattice models whose building block is an L-shape cluster. For the spin-1/2 models studied in this work, we find that this expansion exhibits a similar or better convergence of the bare sums than that of the (larger) square-shaped clusters and can be used with resummation techniques (like the site- and bond-based expansions) to obtain results at even lower temperatures. We compare the performance of weak- and strong-embedding versions of this expansion in various spin-1/2 models and show that the strong-embedding version is preferable because of its convergence properties and lower computational cost. Finally, we show that the expansion based on the L-shape cluster can be naturally used to study properties of lattice models that smoothly connect the square and triangular lattice geometries.

Uncovering Footprints of Dipolar-Octupolar Quantum Spin Ice from Neutron Scattering Signatures

Recent experiments on Ce_{2}Zr_{2}O_{7} suggest that this material may host a novel form of quantum spin ice, a three-dimensional quantum spin liquid with an emergent photon. The Ce^{3+} local moments on the pyrochlore lattice are described by pseudospin-1/2 degrees of freedom, whose components transform as dipolar and octupolar moments under symmetry operations. In principle, there exist four possible quantum spin ice regimes, depending on whether the Ising component is in the dipolar or octupolar channel, and two possible flux configurations of the emergent gauge field. In this Letter, using exact diagonalization and molecular dynamics, we investigate the equal-time and dynamical spin structure factors in all four quantum spin ice regimes using quantum and classical calculations. Contrasting the distinct signatures of quantum and classical results for the four possible quantum spin ice regimes and elucidating the role of quantum fluctuations, we show that the quantum structure factor computed for the π-flux octupolar quantum spin ice regime is most compatible with the neutron scattering results on Ce_{2}Zr_{2}O_{7}.

Possible Inversion Symmetry Breaking in the S=1/2 Pyrochlore Heisenberg Magnet

We address the ground-state properties of the long-standing and much-studied three-dimensional quantum spin liquid candidate, the S=1/2 pyrochlore Heisenberg antiferromagnet. By using SU(2) density-matrix renormalization group (DMRG), we are able to access cluster sizes of up to 128 spins. Our most striking finding is a robust spontaneous inversion symmetry breaking, reflected in an energy density difference between the two sublattices of tetrahedra, familiar as a starting point of earlier perturbative treatments. We also determine the ground-state energy, E_{0}/N_{sites}=-0.490(6)J, by combining extrapolations of DMRG with those of a numerical linked cluster expansion. These findings suggest a scenario in which a finite-temperature spin liquid regime gives way to a symmetry-broken state at low temperatures.

A quantum liquid of magnetic octupoles on the pyrochlore lattice

Spin liquids are highly correlated yet disordered states formed by the entanglement of magnetic dipoles¹. Theories define such states using gauge fields and deconfined quasiparticle excitations that emerge from a local constraint governing the ground state of a frustrated magnet. For example, the '2-in-2-out' ice rule for dipole moments on a tetrahedron can lead to a quantum spin ice^2-4 in rare-earth pyrochlores. However, f-electron ions often carry multipole degrees of freedom of higher rank than dipoles, leading to intriguing behaviours and 'hidden' orders^5-6. Here we show that the correlated ground state of a Ce³⁺-based pyrochlore, Ce₂Sn₂O₇, is a quantum liquid of magnetic octupoles. Our neutron scattering results are consistent with a fluid-like state where degrees of freedom have a more complex magnetization density than that of magnetic dipoles. The nature and strength of the octupole-octupole couplings, together with the existence of a continuum of excitations attributed to spinons, provides further evidence for a quantum ice of octupoles governed by a '2-plus-2-minus' rule^7-8. Our work identifies Ce₂Sn₂O₇ as a unique example of frustrated multipoles forming a 'hidden' topological order, thus generalizing observations on quantum spin liquids to multipolar phases that can support novel types of emergent fields and excitations.

Rank-2 U(1) Spin Liquid on the Breathing Pyrochlore Lattice

Higher-rank generalizations of electrodynamics have recently attracted considerable attention because of their ability to host "fracton" excitations, with connections to both fracton topological order and gravity. However, the search for higher-rank gauge theories in experiment has been greatly hindered by the lack of materially relevant microscopic models. Here we show how a spin liquid described by rank-2 U(1) gauge theory can arise in a magnet on the breathing pyrochlore lattice. We identify Yb-based breathing pyrochlores as candidate systems, and make explicit predictions for how the rank-2 U(1) spin liquid would manifest itself in experiment.

Order out of a Coulomb Phase and Higgs Transition: Frustrated Transverse Interactions of Nd_{2}Zr_{2}O_{7}

The pyrochlore material Nd_{2}Zr_{2}O_{7} with an "all-in-all-out" (AIAO) magnetic order shows novel quantum moment fragmentation with gapped flat dynamical spin ice modes. The parametrized spin Hamiltonian with a dominant frustrated ferromagnetic transverse term reveals a proximity to a U(1) spin liquid. Here we study the magnetic excitations of Nd_{2}Zr_{2}O_{7} above the ordering temperature (T_{N}) using high-energy-resolution inelastic neutron scattering. We find strong spin ice correlations at zero energy with the disappearance of gapped magnon excitations of the AIAO order. It seems that the gap to the dynamical spin ice closes above T_{N} and the system enters a quantum spin ice state competing with and suppressing the AIAO order. Classical Monte Carlo simulations, molecular dynamics, and quantum boson calculations support the existence of a Coulombic phase above T_{N}. Our findings relate the magnetic ordering of Nd_{2}Zr_{2}O_{7} with the Higgs mechanism and provide explanations for several previously reported experimental features.

Magnetic Order with Fractionalized Excitations in Pyrochlore Magnets with Strong Spin-Orbit Coupling

A recent inelastic neutron scattering experiment on the pyrochlore magnet Yb₂Ti₂O₇ uncovers an unusual scattering continuum in the spin excitation spectrum despite the splayed ferromagnetic order in the ground state. While there exist well defined spin wave excitations at high magnetic fields, the one magnon modes and the two magnon continuum start to strongly overlap upon decreasing the field, and eventually they become the scattering continuum at zero field. Motivated by these observations, we investigate the possible emergence of a magnetically ordered ground state with fractionalized excitations in the spin model with the exchange parameters determined from two previous experiments. Using the fermionic parton mean field theory, we show that the magnetically ordered state with fractionalized excitations can arise as a stable mean field ground state in the presence of sufficiently strong quantum fluctuations. The spin excitation spectrum in such a ground state is computed and shown to have the scattering continuum. Upon increasing the field, the fractionalized magnetically ordered state is suppressed, and is eventually replaced by the conventional magnetically ordered phase at high fields, which is consistent with the experimental data. We discuss further implications of these results to the experiments and possible improvements on the theoretical analysis.