Exploring Quantum Spin Liquids: Magnetic Frustration & Unique Excitations

Fun with Quantum Spin Liquids Ruoyu Yin Jan 22, 2018

Contents • · Concept of Magnetic frustration • · Spin liquids • (classical) spin ice • resonating valence bond • · Experimental identification of QSL • Anomaly Raman scattering • inelastic neutron scattering

Magnetic Frustration Frustration: the presence of competing forces that cannot be simultaneously satisfied. As a consequence, a plenitude of ground states may result at zero temperature. Classical examples: 1 spin triangle 2 water ice

Example of Frustration: spin triangle A triangular lattice with near-neighbor spins coupling antiferromagnetically. What is the ground state of such a system? 6-fold degeneracy

Another example： water ice

Another example： water ice For N water molecules, number of hydrogen atoms is 2N. There are 2 different positions for every single atom, and thus the total condition number is . In addition, for every 16 conditions, 6 of that keep original water molecule structure. Upper bound of the ground states: Correspondingly the configurational entropy

Correspondence: spin ice Kagome lattice Spins reside at 4 corners of an tetrahedral lattice, with 2 spins pointing inward and 2 outward.

Analogies to electromagnetism Vector field Spin direction 2 in 2 out Divergence free ‘artificial’ magnetic field: Spins are fluctuating, the magnetic field also fluctuates. However, because the magnetic lines do not start or end, the fluctuations include long loops of flux. spin-spin correlation（by Young-blood）

Quantum spin liquids Spin liquid: the constituent spins are highly correlated but still fluctuate strongly down to a temperature of absolute zero. , no enough thermal energy to support the fluctuation. So, in low temperature, spins in classical spin liquid materials are ordered.

Valence Bond Solid Each spin is part of a valence bond. The ground state of a single VB is spin-0 and non-magnetic. The full state is the product of all VBs, and two nearest spins are highly entangled.

Resonating Valence Bond To build a quantum spin liquid, we should introduce VBs of different direction and theyshould long range. This state does not break the lattice symmetry and spins are long-distancecorrelated. The problem is whether this is a ground state.

An important nature of QSLs: fractional excitations Electron-like (spin and charge) Excitations in most phase of matter Magnon-like (Spin and charge neutral) An example of magnon: Excitations in QSL: spinons, spin and charge neutral Not associated with real-space spins or charge density

Lattices that host QSL Square Lattice Triangular Lattice Kagome Lattice (Mott insulator) Kitaev Lattice ()

Experimental identification of QSLs 1Specific heat measurements: give information about low energy DOS 2 Inelastic neutron scattering: gives information about the nature of excitations and correlations

Experimental identification of QSLs 3 optical measurements: Raman spectra reveal anomaly phonon behavior possibly induced by spin-phonon coupling 4 Resonant inelastic X-ray scattering: similar to INS, but providing information concerning high-energy excitations instead of high-resolution. Frustration parameter a good indication of QSL:

, host of Kitaev QSL Heisenberg-Kitaev model : Kitaev coupling strength : Heisenberg coupling strength for single crystal for powder PRL 114, 147201(2015) DOI: 10.1103/PhysRevLett.114.147201

, host of Kitaev QSL A broad continuum shows up at low temperature(<100K), which is not easily explained by conventional two-magnon scattering or structural disordering, but is consistent with the theoretical prediction for the Kitaev spin liquid. PRL 114, 147201(2015) DOI: 10.1103/PhysRevLett.114.147201

, host of Kitaev QSL Temperature dependence of the magnet scattering PRL 114, 147201(2015) DOI: 10.1103/PhysRevLett.114.147201

, host of Kitaev QSL Temperature dependence of QES Quasielastic Scattering Herbertsmithite PRL 114, 147201(2015) DOI: 10.1103/PhysRevLett.114.147201

, host of Kitaev QSL Fitting: Fano form PRL 114, 147201(2015) DOI: 10.1103/PhysRevLett.114.147201

, host of Kitaev QSL Results from Raman scattering of powder : 1. Continuum that cannot be explained by conventional magnon scattering. 2. This continuum corresponds well with theoretically predicted Kitaev spin liquid. 3. Anomaly temperature dependence of phonon self-energy. 4. The Kitaev coupling strength (by theory). PRL 114, 147201(2015) DOI: 10.1103/PhysRevLett.114.147201

Inelastic Neutron Scattering measurement of NS of single crystal NS of powder Specific heat DOI: 10.1038/NMAT4604

Inelastic Neutron Scattering measurement of Collective magnetic modes measured with INS using at different Temperature Below Above DOI: 10.1038/NMAT4604

Inelastic Neutron Scattering measurement of Spin wave theory calculation for ZZ order DOI: 10.1038/NMAT4604

Inelastic Neutron Scattering measurement of Results from a Kitaev QSL response function DOI: 10.1038/NMAT4604

Inelastic Neutron Scattering measurement of Results from INS measurement: 1. Kitaev QSL model corresponds well with experimental data, which strongly indicatesthat is proximate to QSL phase. 2. The feature survives above , indicating that the mode is not directly connected to the existence of long-range magnetic order. This is likely to beMajorana fermion excitation. 3. Kitaev coupling strength derived from INS measurement is 5.5meV, while that fromRaman scattering is 8meV. DOI: 10.1038/NMAT4604

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Exploring Quantum Spin Liquids: Magnetic Frustration & Unique Excitations