9.2 Heisenberg model

Heisenberg Model Basics

The Heisenberg model describes quantum mechanical interactions between magnetic moments (spins) arranged on a lattice. It's one of the most important models in statistical mechanics for understanding how cooperative spin behavior gives rise to magnetism, phase transitions, and exotic quantum states. Where the Ising model restricts spins to a single axis, the Heisenberg model treats spins as full three-component vectors, capturing the richer physics of real magnetic materials.

Spin Interactions

The interactions between spins in the Heisenberg model originate from quantum mechanical exchange interactions. These arise because of the overlap between electronic wavefunctions combined with the Pauli exclusion principle, not from classical dipole-dipole forces.

• Ferromagnetic exchange favors parallel spin alignment (spins pointing the same direction)
• Antiferromagnetic exchange favors antiparallel alignment (spins pointing opposite directions)
• Interaction strength decreases with distance, so most treatments only include nearest-neighbor pairs
• Both direct exchange (wavefunction overlap between neighboring atoms) and superexchange (mediated through a non-magnetic ion, common in oxides) contribute

Hamiltonian Formulation

The Heisenberg Hamiltonian expresses the total energy of the spin system:

$H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j$

Here $\mathbf{S}_i$ and $\mathbf{S}_j$ are quantum spin operators at lattice sites $i$ and $j$. The angle brackets $\langle i,j \rangle$ mean the sum runs over nearest-neighbor pairs only (each pair counted once). The dot product $\mathbf{S}_i \cdot \mathbf{S}_j = S_i^x S_j^x + S_i^y S_j^y + S_i^z S_j^z$ reflects the full vector nature of the spins, coupling all three components equally in the isotropic case.

Exchange Coupling Constant

The parameter $J$ controls everything about the magnetic character of the model:

• $J > 0$: ferromagnetic coupling (parallel alignment lowers energy)
• $J < 0$: antiferromagnetic coupling (antiparallel alignment lowers energy)
• Its magnitude sets the energy scale, which in turn determines the critical temperature
• $J$ depends on material-specific properties like atomic orbital overlap and interatomic distance
• It can be extracted experimentally (e.g., from inelastic neutron scattering data) or computed from first-principles electronic structure calculations

Types of Heisenberg Models

Different physical systems correspond to different dimensionalities and symmetries. The dimension of the lattice has profound consequences for whether the system can develop long-range magnetic order.

Isotropic vs. Anisotropic

The isotropic Heisenberg model treats all spin components equally, preserving full rotational symmetry in spin space (SU(2) symmetry). In real materials, anisotropy often breaks this symmetry:

• Crystal field effects and spin-orbit coupling introduce preferred directions for spin alignment
• Anisotropy determines the easy axis (or easy plane) of magnetization and affects domain wall structure
• Anisotropic models interpolate between the isotropic Heisenberg limit and the Ising limit, producing richer phase diagrams

One-Dimensional Chain

Spins arranged in a linear chain with nearest-neighbor interactions.

• Certain cases are exactly solvable using the Bethe ansatz (1931)
• The Mermin-Wagner theorem forbids long-range magnetic order at any finite temperature for the isotropic model in 1D (and 2D), because thermal fluctuations of the continuous order parameter destroy order
• Quantum effects are strongest here: the spin-1/2 antiferromagnetic chain has gapless excitations (spinons), while integer-spin chains have a gap (Haldane conjecture, confirmed numerically and experimentally)
• Realized in quasi-1D materials like KCuF₃ and Sr₂CuO₃

Two-Dimensional Lattice

Spins on a planar lattice (square, triangular, honeycomb, kagome).

• The Mermin-Wagner theorem still forbids finite-temperature long-range order for the isotropic model, but quasi-long-range order and Kosterlitz-Thouless-type transitions can occur
• Quantum fluctuations are strong and can stabilize exotic ground states, especially on frustrated geometries like the triangular or kagome lattice
• Directly relevant to layered magnetic materials and the parent compounds of high-temperature cuprate superconductors (e.g., La₂CuO₄, where the Cu²⁺ spins form a 2D square-lattice antiferromagnet)

Three-Dimensional Lattice

The most realistic setting for bulk magnets.

• Long-range magnetic order is stable below a well-defined critical temperature
• Supports conventional second-order phase transitions with 3D Heisenberg universality class critical exponents
• Lattice structures include simple cubic, BCC, FCC, and others
• Describes common ferromagnets (Fe, Co, Ni) and antiferromagnets (MnO, NiO)

Quantum Mechanical Aspects

The Heisenberg model is intrinsically quantum mechanical. The spin operators don't commute, which leads to quantum fluctuations that have no classical analog and can qualitatively change the physics.

Spin Operators

Spin operators $\mathbf{S} = (S^x, S^y, S^z)$ represent the intrinsic angular momentum at each site.

• For spin-1/2 systems, the components are proportional to the Pauli matrices: $S^\alpha = \frac{\hbar}{2}\sigma^\alpha$
• Raising and lowering operators $S^+ = S^x + iS^y$ and $S^- = S^x - iS^y$ flip spin states up or down by one unit
• $S^z$ eigenvalues range from $-S$ to $+S$ in integer steps, giving $2S+1$ states per site
• The total Hilbert space dimension grows as $(2S+1)^N$ for $N$ sites, which is why exact solutions are so hard

Commutation Relations

The spin operators obey the SU(2) algebra:

$[S^x, S^y] = i\hbar S^z$

with cyclic permutations for the other pairs. These commutation relations mean you cannot simultaneously know all three components of a spin with arbitrary precision (a spin uncertainty principle). This is the root cause of quantum fluctuations: even in the ground state, spins are not perfectly aligned but fluctuate around their average direction.

Ground State Properties

The ground state depends strongly on lattice geometry and the sign of $J$:

• For a ferromagnet ($J > 0$), the ground state is the fully aligned state $|\uparrow\uparrow\uparrow\cdots\rangle$, which is an exact eigenstate
• For an antiferromagnet, the classical Néel state (alternating up/down) is not an exact eigenstate due to quantum fluctuations. The true ground state is an entangled superposition with reduced sublattice magnetization
• On frustrated lattices, quantum fluctuations can prevent any ordering, potentially producing quantum spin liquids with long-range entanglement but no broken symmetry
• Computational methods for finding ground states include variational approaches, exact diagonalization (small systems), and DMRG (1D and quasi-1D systems)

Phase Transitions

The Heisenberg model exhibits continuous (second-order) phase transitions between magnetically ordered and disordered (paramagnetic) states. The critical behavior falls into the Heisenberg universality class, distinct from the Ising universality class.

Critical Temperature

The critical temperature $T_c$ is where long-range order vanishes:

• For ferromagnets, this is the Curie temperature; for antiferromagnets, the Néel temperature
• It reflects the competition between exchange energy ($\sim J$), which favors order, and thermal energy ($\sim k_B T$), which favors disorder
• $T_c$ scales with $J$ and increases with coordination number (number of nearest neighbors)
• Mean-field theory overestimates $T_c$ because it neglects fluctuations; more accurate values come from Monte Carlo simulations or series expansions

Spontaneous Magnetization

Below $T_c$, a ferromagnet develops a nonzero spontaneous magnetization $M$ even without an external field. This is the order parameter for the transition.

Near the critical point, it follows a power law:

$M \sim (T_c - T)^{\beta}$

The critical exponent $\beta$ depends on dimensionality and universality class. For the 3D Heisenberg model, $\beta \approx 0.365$, compared to $\beta \approx 0.326$ for 3D Ising. The magnetization vanishes continuously at $T_c$, characteristic of a second-order transition.

Correlation Functions

The spin-spin correlation function $\langle \mathbf{S}_i \cdot \mathbf{S}_j \rangle$ measures how correlated spins are as a function of separation:

• Above $T_c$: correlations decay exponentially with distance, $\sim e^{-r/\xi}$, where $\xi$ is the correlation length
• At $T_c$: correlations decay as a power law, $\sim r^{-(d-2+\eta)}$, signaling scale invariance. The exponent $\eta \approx 0.037$ for the 3D Heisenberg model
• Below $T_c$: long-range order means correlations persist to infinite distance
• The correlation length $\xi$ diverges at $T_c$ as $\xi \sim |T - T_c|^{-\nu}$
• These functions connect directly to experiment through neutron scattering cross-sections

Approximation Methods

Exact solutions of the Heisenberg model are rare (the 1D spin-1/2 chain via Bethe ansatz being a notable exception). In higher dimensions, you need approximation methods, each with its own regime of validity.

Mean-Field Theory

The simplest analytical approach. Each spin is treated as if it interacts with an effective average field produced by all its neighbors, rather than with the actual fluctuating spins.

1. Replace $\mathbf{S}_j$ with its thermal average $\langle \mathbf{S}_j \rangle = m\hat{z}$ (assuming ordering along $z$)
2. The Hamiltonian becomes a sum of single-spin problems in an effective field $h_{\text{eff}} = zJm$, where $z$ is the coordination number
3. Solve self-consistently: compute $m$ from the Brillouin function and require it to match the assumed value

Mean-field theory gives qualitatively correct phase diagrams and predicts $T_c^{MF} = \frac{2}{3}zJS(S+1)/k_B$ for spin-$S$. However, it overestimates $T_c$ and gives classical (mean-field) critical exponents ($\beta = 1/2$, $\nu = 1/2$) that don't match experiment in 2D or 3D. It becomes exact in the limit of infinite dimensions or infinite-range interactions.

Spin-Wave Theory

Describes the low-energy excitations above an ordered ground state. These excitations are magnons, quantized spin waves.

1. Express spin operators in terms of bosonic operators using the Holstein-Primakoff transformation
2. Expand to leading order (linear spin-wave theory), yielding a quadratic bosonic Hamiltonian
3. Diagonalize via Fourier transform to get the magnon dispersion relation $\omega(\mathbf{k})$

For a ferromagnet, the dispersion is quadratic at long wavelengths: $\omega \propto k^2$. For an antiferromagnet, it's linear: $\omega \propto k$. Spin-wave theory accurately predicts low-temperature thermodynamics, including Bloch's $T^{3/2}$ law for the magnetization reduction in 3D ferromagnets. It breaks down near $T_c$ where fluctuations become large.

Renormalization Group

The RG approach is the most powerful framework for understanding critical behavior.

• Systematically integrates out short-wavelength fluctuations, producing an effective theory at longer length scales
• Fixed points of the RG flow correspond to universality classes
• Allows calculation of critical exponents via the $\epsilon$-expansion ($\epsilon = 4 - d$) or other schemes
• Explains why different systems share the same critical exponents: they flow to the same fixed point
• Applies to both classical thermal transitions and quantum phase transitions (where imaginary time acts as an extra dimension)

Applications in Materials

Ferromagnets

Ferromagnets have $J > 0$, so spins align parallel below the Curie temperature, producing a net magnetization.

• Classic examples: Fe ($T_c \approx 1043$ K), Co ($T_c \approx 1388$ K), Ni ($T_c \approx 627$ K)
• The Heisenberg model captures their spin-wave spectra, temperature-dependent magnetization, and critical behavior
• Technological applications include permanent magnets, data storage media, electric motors, and transformers

Antiferromagnets

Antiferromagnets have $J < 0$, producing antiparallel spin alignment on two sublattices with zero net magnetization.

• The ordering temperature is called the Néel temperature $T_N$
• Examples: Cr ($T_N \approx 311$ K), MnO ($T_N \approx 118$ K), NiO ($T_N \approx 525$ K)
• Antiferromagnetic order is detected through neutron diffraction (magnetic Bragg peaks at the antiferromagnetic wavevector)
• Growing interest for spintronics applications because antiferromagnets are insensitive to external magnetic fields and have ultrafast dynamics

Frustrated Systems

Frustration occurs when the lattice geometry prevents all pairwise interactions from being satisfied simultaneously. The classic example: three antiferromagnetically coupled spins on a triangle can't all be antiparallel to each other.

• Frustrated lattices include the triangular, kagome, and pyrochlore geometries
• Frustration suppresses conventional order and can stabilize exotic ground states like quantum spin liquids (no symmetry breaking, long-range entanglement, fractionalized excitations)
• Candidate spin liquid materials include herbertsmithite (ZnCu₃(OH)₆Cl₂, kagome lattice) and certain pyrochlore compounds
• Spin glasses arise when frustration combines with disorder

Numerical Simulations

When analytical methods fall short, numerical techniques provide quantitative results. The main challenge is the exponential growth of the quantum Hilbert space with system size.

Monte Carlo Methods

For the classical Heisenberg model (treating spins as classical vectors), Monte Carlo sampling is the workhorse method:

1. Start from a random spin configuration
2. Propose a local update (e.g., rotate a single spin)
3. Accept or reject based on the Metropolis criterion: accept if energy decreases, otherwise accept with probability $e^{-\Delta E / k_B T}$
4. Repeat to generate a Markov chain of configurations sampling the Boltzmann distribution

Cluster algorithms (Wolff, Swendsen-Wang) flip correlated groups of spins at once, dramatically reducing critical slowing down near $T_c$. Monte Carlo gives accurate estimates of $T_c$, critical exponents, and thermodynamic quantities for large systems.

Quantum Monte Carlo

Extends Monte Carlo to the quantum Heisenberg model by mapping the quantum partition function onto a classical problem in one higher dimension.

• World-line and stochastic series expansion (SSE) are common formulations
• Provides unbiased results for ground-state and finite-temperature properties
• The sign problem is the major limitation: for frustrated systems or fermionic models, the statistical weights can become negative, causing exponential growth in statistical errors. This restricts QMC to certain classes of Hamiltonians (e.g., unfrustrated bipartite lattices)

Exact Diagonalization

Directly constructs and diagonalizes the Hamiltonian matrix in the full Hilbert space.

• Gives the complete energy spectrum and all eigenstates, so you can compute any observable exactly
• Symmetries (translation, rotation, spin inversion) reduce the matrix size through block diagonalization
• Practical limit: roughly 36-44 spin-1/2 sites with current hardware, depending on symmetries
• The exponential scaling ($(2S+1)^N$) is a hard wall; for 40 spin-1/2 sites, the Hilbert space has $\sim 10^{12}$ states
• Particularly valuable for studying ground-state entanglement, spectral gaps, and benchmarking approximate methods

Extensions and Variations

XXZ Model

The XXZ model introduces anisotropy between the in-plane and out-of-plane spin components:

$H = -J \sum_{\langle i,j \rangle} \left( S_i^x S_j^x + S_i^y S_j^y + \Delta S_i^z S_j^z \right)$

The anisotropy parameter $\Delta$ interpolates between different limits:

• $\Delta = 0$: XY model (only in-plane coupling)
• $\Delta = 1$: isotropic Heisenberg model
• $\Delta \to \infty$: Ising model (only $z$-component coupling)

The 1D spin-1/2 XXZ chain is exactly solvable by Bethe ansatz and exhibits a quantum phase transition from a gapless XY phase ($|\Delta| < 1$) to a gapped Néel phase ($\Delta > 1$). The XXZ model describes materials with easy-plane or easy-axis magnetic anisotropy.

Ising Model Comparison

The Ising model is the extreme anisotropic limit of the Heisenberg model, retaining only the $S^z_i S^z_j$ interaction. Spins take discrete values (up or down) rather than being continuous vectors.

• Exactly solved in 1D (no phase transition) and 2D on a square lattice (Onsager, 1944; $T_c = \frac{2J}{k_B \ln(1+\sqrt{2})}$)
• Has a phase transition for $d \geq 2$, unlike the isotropic Heisenberg model which requires $d \geq 3$ for finite-temperature order (by Mermin-Wagner)
• The discrete symmetry ($\mathbb{Z}_2$) of the Ising model versus the continuous symmetry (SU(2)) of the Heisenberg model is the fundamental reason for this difference

Heisenberg-Kitaev Model

Combines isotropic Heisenberg exchange with bond-dependent Kitaev interactions, where different spin components couple on different bond directions:

$H = -J_H \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j - J_K \sum_{\langle i,j \rangle_\gamma} S_i^\gamma S_j^\gamma$

Here $\gamma = x, y, z$ depends on the bond direction. This model is relevant for materials with strong spin-orbit coupling on honeycomb lattices, such as $\alpha$-RuCl₃ and certain iridates (Na₂IrO₃, Li₂IrO₃). The pure Kitaev limit is exactly solvable and hosts a quantum spin liquid ground state with Majorana fermion excitations, making it a candidate platform for topological quantum computation.

Experimental Realizations

Magnetic Insulators

Materials with localized magnetic moments and negligible itinerant electrons are the most direct realizations of the Heisenberg model.

• Transition metal oxides (MnO, NiO, La₂CuO₄) and fluorides (KCuF₃, K₂CuF₄) provide examples across 1D, 2D, and 3D
• Neutron scattering is the primary probe: it directly measures the spin-wave dispersion and correlation functions
• Other techniques include magnetic susceptibility, specific heat, NMR/ESR, and muon spin rotation
• Chemical substitution, pressure, and applied fields allow tuning of exchange parameters

Ultracold Atoms

Neutral atoms trapped in optical lattices (periodic potentials created by interfering laser beams) can simulate the Heisenberg model.

• At half-filling in a Mott insulator regime, superexchange interactions between neighboring sites reproduce the Heisenberg Hamiltonian
• The lattice geometry, dimensionality, and interaction strength are all tunable by adjusting laser parameters
• Quantum gas microscopy enables site-resolved imaging of individual spins, giving direct access to correlation functions
• Temperatures in the nanokelvin range are needed, and reaching the relevant energy scales ($\sim J/k_B$) remains an experimental challenge

Quantum Simulators

Engineered quantum platforms can implement Heisenberg-type interactions in highly controlled settings:

• Trapped ions: spin-spin interactions mediated by shared motional modes, with tunable range and strength
• Superconducting qubits: programmable couplings on various lattice geometries
• Nitrogen-vacancy centers in diamond: arrays of interacting spin defects
• These platforms access regimes difficult to reach in solid-state materials, including non-equilibrium dynamics, quantum quenches, and real-time evolution
• They serve both as tools for studying fundamental many-body physics and as testbeds for quantum computing architectures