Mapping the Phase Diagram of a Frustrated Magnet: Degeneracies, Flat Bands, and Canting Cycles on the Pyrochlore Lattice

Abstract

We map the complete classical phase diagram of the spin Hamiltonian describing pyrochlore rare-earth magnets with all symmetry-allowed nearest-neighbor bond-dependent anisotropic two-spin interactions. We provide a simple derivation of the organization of spins into tensor degrees of freedom describing the multipole moments of a tetrahedron, whose components correspond to irreducible representations (irreps) of the tetrahedral symmetry group $T_d$. By parameterizing the Hamiltonian directly in terms of the energies of the individual irreps, we perform an exhaustive search of all possible irrep degeneracies which may host stable classical spin liquids. Doing so reveals four one-parameter families of models along which three phases are degenerate, all four of which merge at the Heisenberg antiferromagnet and its dual, and we give a complete three-dimensional picture of the phase diagram showing all of the phases and their intersections. The appearance of two copies of a single irrep implies an extra degenerate locus which pierces the phase boundaries at special points, yielding two additional isolated triple points. We demonstrate that one-parameter families of Hamiltonians are characterized by a topological invariant describing how the ground state spin configuration winds when adiabatically transported around this degenerate locus, analogous to a "diabolical locus". We provide a comprehensive catalog of all flat band degeneracies in the phase diagram and discuss the mechanisms that may allow for or impede the realization of a variety of classical spin liquids described by tensor gauge fields exhibiting pinch line singularities and concomitant four-fold pinch points. Lastly, we provide a list of all cases where three irreps are degenerate above the ground state, which may lead to interesting features in the spin wave spectrum within each ordered phase.

37 pages, 14 figures, 6 tables

References in corpus (53)

• "Deconfined" quantum critical points
• Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
• Pyrochlore Photons: The U(1) Spin Liquid in a S=1/2 Three-Dimensional Frustrated Magnet
• Order by disorder and spiral spin liquid in frustrated diamond lattice antiferromagnets
• Algebraic spin liquid as the mother of many competing orders
• Experimental identification of quantum spin liquids
• Foliated Quantum Field Theory of Fracton Order
• Absent pinch points and emergent clusters: further neighbour interactions in the pyrochlore Heisenberg antiferromagnet
• The case for a U(1)${}_{\pi }$ Quantum Spin Liquid Ground State in the Dipole-Octupole Pyrochlore Ce${}_2$Zr${}_2$O${}_7$
• Stiefel Liquids: Possible Non-Lagrangian Quantum Criticality from Intertwined Orders
• Possible inversion symmetry breaking in the $S=1/2$ pyrochlore Heisenberg magnet
• Topological Route to New and Unusual Coulomb Spin Liquids
• Pseudo-fermion functional renormalization group for spin models
• Ising Like Order by Disorder In The Pyrochlore Antiferromagnet with Dzyaloshinskii-Moriya Interactions
• Fragmentation in Frustrated Magnets -- A review
• Classical spin dynamics based on SU($N$) coherent states
• Generic Spiral Spin Liquids
• Order-by-disorder near criticality in $XY$ pyrochlore magnets
• Generic nearest-neighbour kagome model: XYZ & Dzyaloshinskii-Moriya, and comparison to pyrochlore
• A fractionalised "${\mathbb{Z}}_2$" classical Heisenberg spin liquid
• Quantum effects on unconventional pinch point singularities
• Frustrated Magnetism in Fluoride and Chalcogenide Pyrochlore Lattice Materials
• Semi-classical simulation of spin-1 magnets
• Combined approach to analyze and classify families of classical spin liquids
• Low energy structure of spiral spin liquids
• Competing $U(1)$ and ${\mathbb{Z}}_2$ dipolar-octupolar quantum spin liquids on the pyrochlore lattice: application to Ce${}_2$Zr${}_2$O${}_7$
• Symmetry fractionalization in the gauge mean-field theory of quantum spin ice
• Classical and quantum phases of the pyrochlore $S=1/2$ magnet with Heisenberg and Dzyaloshinskii-Moriya interactions
• Abundance of hard-hexagon crystals in the quantum pyrochlore antiferromagnet
• Symmetric $\mathrm{U}(1)$ and ${\mathbb{Z}}_2$ spin liquids on the pyrochlore lattice
• Competing Gauge Fields and Entropically-Driven Spin Liquid to Spin Liquid Transition in non-Kramers Pyrochlores
• Competing quantum spin liquids, gauge fluctuations, and anisotropic interactions in a breathing pyrochlore lattice
• Multiversality and Unnecessary Criticality in One Dimension
• Realization of fractonic quantum phases in the breathing pyrochlore lattice
• Dynamical scaling as a signature of multiple phase competition in Yb${}_2$Ti${}_2$O${}_7$
• State selection in frustrated magnets
• Emergent photons and fractionalized excitations in a quantum spin liquid
• Order-by-disorder without quantum zero-point fluctuations in the pyrochlore Heisenberg ferromagnet with Dzyaloshinskii-Moriya interactions
• Line-Graph Approach to Spiral Spin Liquids
• The Classical Heisenberg Model on the Centred Pyrochlore Lattice
• A pseudofermion functional renormalization group study of dipolar-octupolar pyrochlore magnets
• Classification of Classical Spin Liquids: Topological Quantum Chemistry and Crystalline Symmetry
• Classical origins of Landau-incompatible transitions
• Higher-Rank Spin Liquids and Spin Nematics from Competing Orders in Pyrochlore Magnets
• Dirac spin liquid as an "unnecessary" quantum critical point on square lattice antiferromagnets
• Thermodynamics and fractal dynamics of nematic spin ice, a doubly frustrated pyrochlore Ising magnet
• 2-Form U(1) Spin Liquids: Classical Model and Quantum Aspects
• Quantum effects on pyrochlore higher-rank U(1) spin liquids: Pinch-line singularities, spin nematics, and connections to oxide materials
• Human-machine collaboration: ordering mechanism of rank-2 spin liquid on breathing pyrochlore lattice
• Charge pumps, boundary modes, and the necessity of unnecessary criticality
• Exact diagonalization for spin-1/2 spin ice pyrochlores
• Pinch-line spin liquids as layered Coulomb phases and applications to cubic models
• Phase diagram of the antiferromagnetic $J_1$-$J_2$ spin-$1$ pyrochlore Heisenberg model