Magnetism arises from spin — quantum mechanical angular momentum with no classical analog. From ferromagnetism to frustrated magnets to quantum spin liquids, magnetic materials display a rich variety of quantum phases driven by exchange interactions.
Distinguish ferromagnetic, antiferromagnetic, and frustrated geometries and explain how frustration produces spin liquids.
Apply Larmor precession to describe NMR signal generation and spatial encoding in MRI.
MG.1 Paramagnetism and Diamagnetism
In an external field H, a free spin-J atom acquires energy −g_J μ_B m_J H. The partition function Z = Σ e^(g_J μ_B m_J H/(k_BT)) gives the magnetization:
At high T (x ≪ 1): M ≈ N g_J² μ_B² J(J+1) H/(3k_BT) — Curie law: χ = C/T. At low T (x ≫ 1): M → N g_J μ_B J (saturation).
Diamagnetism: all materials are weakly diamagnetic. The orbital response of closed shells to H gives χ_dia = −Ze²⟨r²⟩/(6m_e c²) per atom (Langevin). Negative susceptibility: induced magnetization opposes applied field. Superconductors are perfect diamagnets (Meissner effect): χ = −1.
MG.2 Exchange Interaction and the Heisenberg Model
Magnetism in solids comes from the exchange interaction — a purely quantum mechanical effect arising from the overlap of wavefunctions and the Pauli exclusion principle. For two electrons on adjacent atoms:
J > 0: ferromagnetic exchange (parallel spins lower energy — triplet state preferred). J < 0: antiferromagnetic exchange (antiparallel spins — singlet preferred). Exchange integral: J = ⟨↑↓|1/r₁₂|↑↓⟩ − ⟨↑↓|1/r₁₂|↓↑⟩ (direct vs exchange term).
The full Heisenberg modelon a lattice: H = −J Σ_(ij) S_i · S_j − g μ_B H Σᵢ S_i^z. For J > 0: ferromagnet. For J < 0: antiferromagnet. The Ising model (1D solvable): H = −J Σ σᵢ σᵢ₊₁.
MG.3 Ferromagnetism — Mean-Field Theory
In mean-field theory, replace S_j · S_i → ⟨S_j⟩ · S_i = m S_i^z (Weiss molecular field approximation). Each spin sees an effective field H_eff = H + λm where λ = 2zJ/(g μ_B)² is the Weiss constant (z = coordination number).
Solve for J:J=3kBTC/(2zS(S+1))=3×1.38×10−23×1043/(2×8×1×2)=4.31×10−20/32=1.35×10−21J=0.0084 eV.
Compare:Thisistheeffectiveexchangeperbond.Theactual3dexchangeinFeis100meV—10×larger.Mean−fieldoverestimatesTCbyincludingallfluctuationsasiftheyallpointinthemean−fielddirection.ActualquantumMonteCarlogivescorrectTCwithpropertreatment of fluctuations.
Spin waves:The low-T excitations of a ferromagnet are magnons — collective spin-wave modes with dispersion \omega_quadratic,unlikephononswhicharelinear).Magnoncontributiontoheatcapacity:CV∝T^(3/2) (Bloch T^(3/2) law).
MG.4 Antiferromagnetism and Frustration
For J < 0 on a bipartite lattice (two sublattices A and B), the ground state has spins on A pointing up and B pointing down — antiferromagnetism. Néel temperature T_N = 2z|J|S(S+1)/(3k_B) (same formula as T_C). Neutron scattering (magnetic Bragg peaks) detects the antiferromagnetic order.
Geometric frustration: on a triangular or kagome lattice, you cannot simultaneously satisfy all antiferromagnetic bonds. For three spins on a triangle: two can be antiparallel but the third cannot be antiparallel to both. This frustration prevents conventional magnetic order and can give rise to:
Spin liquid: highly entangled state with no long-range order even at T = 0. Ground state is a superposition of many configurations (resonating valence bonds, proposed by Anderson 1973 for high-T_c). Characterized by fractional excitations (spinons) and topological order. Candidate materials: herbertsmithite (ZnCu₃(OH)₆Cl₂), RVB models of cuprate superconductors.
MG.5 Spin Dynamics and NMR
A spin in a magnetic field H precesses (Larmor precession):
For protons: γ = 2.675×10⁸ rad/(s·T) → Larmor frequency ν_L = γH/(2π). At H = 3 T (clinical MRI): ν_L = 127 MHz (RF range).
NMR (Nuclear Magnetic Resonance): RF pulse at ν_L tips the magnetization. Relaxation times T₁ (longitudinal, spin-lattice) and T₂ (transverse, spin-spin) encode local chemical environment. MRI adds a gradient field to encode spatial position in the Larmor frequency: frequency encodes location. Fourier transform of FID (free induction decay) → image. Nobel Prizes: Bloch & Purcell 1952, Lauterbur & Mansfield 2003.
Definition MG.1 — Common Traps
Magnetization is material response: it is not the same quantity as applied H or total B.
Exchange is quantum mechanical: ferromagnetism is not caused by classical dipole alignment alone.
Domains reduce magnetic energy: a demagnetized sample can still have ordered domains.
Curie temperature marks collective order: above it, thermal disorder destroys spontaneous magnetization.
ExplainthephysicsofMRI.Howdoesspatialencodingwork?WhydoesSNRscaleasH02?What is achievable resolution at 3 T?
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Intermediate
4.Describe the Kitaev honeycomb model and its exact solution via Majorana fermions. Why is this a quantum spin liquid and what physical materials realize it?