Heisenberg Model

The Heisenberg model describes fixed-length quantum spins coupled by rotationally invariant exchange. On a nearest-neighbor lattice its canonical Hamiltonian is

$$H = J \sum_{\langle i,j\rangle} \mathbf s_i\cdot\mathbf s_j,$$

where the dimensionless spin operators obey $\mathbf s_i^2=s(s+1)$. With this sign convention, $J\gt0$ is antiferromagnetic and $J\lt0$ is ferromagnetic.

The formula is compact, but its behavior depends decisively on spin $s$, dimension, lattice geometry, boundary conditions, temperature, and the sign and range of exchange. It includes exact dimers and ferromagnetic one-magnon states, the Bethe-ansatz-solvable spin-$1/2$ chain, semiclassical ordered magnets, frustrated lattices, and the integer-spin Haldane regime. It is therefore both a model of localized quantum magnetism and a laboratory for symmetry, entanglement, collective modes, criticality, and emergence.

Helpful background. Spin angular momentum and addition of angular momentum supply the local algebra and coupled-spin basis.

Fixed-length spins and exchange units

Choose a graph $G=(V,E)$ with $L=|V|$ sites. Each site carries one irreducible spin-$s$ degree of freedom,

$$\mathcal H_i \simeq \mathbb C^{2s+1}.$$

The many-spin Hilbert space is

$$\mathcal H = \bigotimes_{i=1}^{L} \mathcal H_i, \qquad \dim\mathcal H = (2s+1)^L.$$

Dimensionless spin operators satisfy

$$\begin{aligned} [s_i^\alpha,s_j^\beta] &= i\delta_{ij} \epsilon_{\alpha\beta\gamma} s_i^\gamma, \\ \mathbf s_i^2 &= s(s+1), \end{aligned}$$

with $\alpha,\beta,\gamma\in\{x,y,z\}$. Operators on different sites commute.

Physical angular momentum is

$$\mathbf S_i = \hbar\mathbf s_i.$$

Accordingly, the same exchange term can be written in either of two equivalent conventions:

$$J\mathbf s_i\cdot\mathbf s_j = \frac{J}{\hbar^2} \mathbf S_i\cdot\mathbf S_j.$$

Here $J$ has units of energy. A formula written directly as $J\mathbf S_i\cdot\mathbf S_j$ instead assigns different units to its symbol $J$. Numerical comparisons require checking this convention first.

For spin $1/2$,

$$\mathbf s_i = \frac{1}{2}\boldsymbol\sigma_i, \qquad \mathbf S_i = \frac{\hbar}{2}\boldsymbol\sigma_i.$$

Hamiltonian on a Graph

The isotropic exchange Hamiltonian on a general graph is

$$H_{ \mathrm{ex} } = \sum_{(i,j)\in E} J_{ij} \mathbf s_i\cdot\mathbf s_j.$$

Each unordered bond should appear once. For a uniform nearest-neighbor model,

$$H = J \sum_{\langle i,j\rangle} \mathbf s_i\cdot\mathbf s_j.$$

A field measured in energy units adds

$$H_h = -\sum_i \mathbf h_i\cdot\mathbf s_i.$$

If the physical magnetic field is $\mathbf B_i$ and the magnetic moment convention is $\boldsymbol\mu_i=g\mu_{\mathrm B}\mathbf s_i$, then $\mathbf h_i=g\mu_{\mathrm B}\mathbf B_i$. The electron magnetic moment may introduce an additional sign depending on whether the spin or magnetic-moment direction is being described.

Common extensions include anisotropic exchange,

$$H_{XYZ} = \sum_{\langle i,j\rangle} \left( J_xs_i^xs_j^x + J_ys_i^ys_j^y + J_zs_i^zs_j^z \right),$$

single-ion anisotropy $D\sum_i(s_i^z)^2$ for $s\ge1$, antisymmetric Dzyaloshinskii–Moriya exchange, longer-range couplings, and multi-spin terms. These are related models. They can change the symmetry, spectrum, ordering, and even the phase classification, so results for the isotropic model should not be transferred to them silently.

What the Sign of Exchange Means

For one bond, define its total spin

$$\mathbf s_{ij} = \mathbf s_i+\mathbf s_j.$$

The identity

$$\mathbf s_i\cdot\mathbf s_j = \frac{1}{2} \left( \mathbf s_{ij}^2 - \mathbf s_i^2 - \mathbf s_j^2 \right)$$

diagonalizes the exchange interaction in sectors of bond spin $\ell=0,1,\ldots,2s$. The corresponding bond energy is

$$E_\ell = \frac{J}{2} \left[ \ell(\ell+1) - 2s(s+1) \right].$$

Thus:

• $J\gt0$ favors the smallest available bond spin and is called antiferromagnetic;
• $J\lt0$ favors the largest bond spin and is called ferromagnetic.

These names describe the energetic tendency of each exchange bond. They do not by themselves prove long-range order. Quantum fluctuations, frustration, dimensionality, and temperature determine whether an ordered phase actually forms.

An isotropic bond couples fixed-length spins through $J\mathbf s_i\cdot\mathbf s_j$. For two spin-$1/2$ sites, $J\lt0$ places the triplet below the singlet, whereas $J\gt0$ places the singlet below the triplet. The crossing at $J=0$ makes the sign convention explicit.

Exact Spin-One-Half Dimer

The two-site spin-$1/2$ model is the basic exchange benchmark:

$$H_{ 2 } = J\mathbf s_1\cdot\mathbf s_2.$$

The singlet is

$$\lvert S\rangle = \frac{ \lvert\uparrow\downarrow\rangle - \lvert\downarrow\uparrow\rangle }{\sqrt2},$$

and the triplet is

$$\begin{aligned} \lvert T_+\rangle &= \lvert\uparrow\uparrow\rangle, \\ \lvert T_0\rangle &= \frac{ \lvert\uparrow\downarrow\rangle + \lvert\downarrow\uparrow\rangle }{\sqrt2}, \\ \lvert T_-\rangle &= \lvert\downarrow\downarrow\rangle. \end{aligned}$$

Their energies are

$$E_S = -\frac{3J}{4}, \qquad E_T = \frac{J}{4}.$$

The signed singlet–triplet separation $E_T-E_S$ is $J$. The positive excitation gap above the ground multiplet is $|J|$ for $J\ne0$. Antiferromagnetic exchange selects the entangled singlet; ferromagnetic exchange selects the three-dimensional triplet multiplet.

The exchange operator also has useful projector forms:

$$\begin{aligned} P_S &= \frac{1}{4} - \mathbf s_1\cdot\mathbf s_2, \\ P_T &= \frac{3}{4} + \mathbf s_1\cdot\mathbf s_2. \end{aligned}$$

For spin $1/2$, the swap operator is

$$P_{12} = 2\mathbf s_1\cdot\mathbf s_2 + \frac{1}{2}.$$

This identity links exchange, permutation symmetry, and singlet–triplet splitting. It is special to two spin-$1/2$ degrees of freedom in this simple form.

Thermal dimer

At inverse temperature $\beta$, the dimer partition function is

$$Z_2 = e^{3\beta J/4} + 3e^{-\beta J/4}.$$

The factor of three is the triplet degeneracy. The exact exchange correlation is

$$\left\langle \mathbf s_1\cdot\mathbf s_2 \right\rangle_\beta = -\frac{3}{4} \frac{ e^{\beta J}-1 }{ e^{\beta J}+3 }.$$

It approaches zero at high temperature. At low temperature it approaches $-3/4$ for $J\gt0$ and $1/4$ for $J\lt0$. This exactly solvable system is a useful check on thermal-state normalization, degeneracy factors, energy derivatives, and sign conventions.

Global Spin-Rotation Symmetry

Define total spin

$$\mathbf s_{ \mathrm{tot} } = \sum_i\mathbf s_i.$$

Every isotropic bond is a scalar under simultaneous rotations of all spins. Therefore

$$[H,s_{ \mathrm{tot} }^\alpha] = 0, \qquad \alpha\in\{x,y,z\},$$

and

$$[H,\mathbf s_{ \mathrm{tot} }^2] = 0.$$

The model has global $SU(2)$ spin-rotation symmetry. Its energy eigenstates can be organized into total-spin multiplets labeled by $S_{\mathrm{tot}}$ and $m$. Within an exact multiplet the energy is independent of $m$.

This global symmetry does not mean that each local spin is conserved. In general,

$$[H,\mathbf s_i] \ne 0,$$

because exchange transfers angular momentum between sites while preserving the total.

Additional symmetries depend on the graph and couplings:

• a uniform chain or Bravais lattice may have translations and point-group symmetries;
• real isotropic exchange without a field is invariant under time reversal;
• spatial inversion may exchange sites or bonds;
• a uniform field leaves rotations about the field axis and conserves the corresponding total magnetization;
• XXZ anisotropy usually retains a $U(1)$ subgroup, while generic XYZ anisotropy retains less.

Symmetry sectors reduce numerical cost and provide strict checks on spectra and matrix elements.

Chains and Boundary Conditions

For a one-dimensional chain of $L$ sites,

$$H_{ \mathrm{chain} } = J \sum_i \mathbf s_i\cdot\mathbf s_{i+1}.$$

The range of the sum must be stated:

$$\begin{array}{ll} i=1,\ldots,L-1 &\text{open boundary conditions}, \\ i=1,\ldots,L, \quad \mathbf s_{L+1}=\mathbf s_1 &\text{periodic boundary conditions}. \end{array}$$

The open chain has $L-1$ bonds; a periodic chain with $L\ge3$ has $L$ distinct bonds. A two-site dimer contains one bond unless a doubled coupling is explicitly intended. For an antiferromagnetic periodic chain, odd $L$ prevents a perfectly alternating pattern and introduces geometric frustration. These finite-size details can alter degeneracies, momentum quantum numbers, edge states, and gaps.

Ferromagnetic Ground States

Let $J\lt0$ on every bond of a connected nearest-neighbor graph. The fully polarized state

$$\lvert F\rangle = \lvert s,s,\ldots,s\rangle$$

has maximal spin on every bond and is an exact ground state. The linear span of its global rotations is the maximal-total-spin multiplet

$$S_{ \mathrm{tot} } = Ls,$$

with $2Ls+1$ states. On a periodic uniform chain its energy is

$$E_F = JLs^2.$$

For open boundaries, replace $L$ by $L-1$ in the bond count.

The polarized product state is exact because every ferromagnetic bond can minimize its energy simultaneously. This is very different from an antiferromagnet, where minimizing one bond through a singlet generally conflicts with sharing each spin among several bonds.

Exact one-magnon dispersion

On a periodic chain, create one localized spin deviation with

$$\lvert j\rangle = \frac{1}{\sqrt{2s}} s_j^- \lvert F\rangle.$$

Within the one-deviation subspace,

$$\left( H-E_F \right) \lvert j\rangle = Js \left( \lvert j-1\rangle + \lvert j+1\rangle - 2\lvert j\rangle \right).$$

Momentum states

$$\lvert k\rangle = \frac{1}{\sqrt L} \sum_j e^{ikaj} \lvert j\rangle$$

are therefore exact eigenstates with excitation energy

$$\varepsilon(k) = 2\lvert J\rvert s \left[ 1-\cos(ka) \right].$$

Near $k=0$,

$$\varepsilon(k) \simeq \lvert J\rvert s (ka)^2.$$

The $k=0$ state is proportional to $s_{\mathrm{tot}}^-\lvert F\rangle$ and lies in the same ground multiplet. The quadratic long-wavelength dispersion is characteristic of an isotropic ferromagnet. On a finite periodic chain, the smallest nonzero momentum is $2\pi/(La)$, so the first dispersing magnon energy scales as $L^{-2}$.

Why the Antiferromagnet Is More Quantum

Write one bond as

$$\mathbf s_i\cdot\mathbf s_j = s_i^zs_j^z + \frac{1}{2} \left( s_i^+s_j^- + s_i^-s_j^+ \right).$$

On a bipartite lattice, the classical Néel product state alternates the largest positive and negative $s^z$ values. Its exchange expectation is

$$\left\langle \mathbf s_i\cdot\mathbf s_j \right\rangle_{ \mathrm{N\acute eel} } = -s^2$$

on each antiparallel bond. It is generally not an eigenstate, because the transverse terms exchange neighboring spin projections.

For a dimer the exact antiferromagnetic bond energy is

$$-Js(s+1),$$

which is lower than the product-state expectation $-Js^2$. The additional lowering is a direct signature of quantum fluctuations and, for the spin-$1/2$ dimer, entanglement.

On an extended lattice, one spin cannot form an independent singlet with every neighbor. The ground state must compromise among overlapping bonds. This competition underlies resonating valence-bond descriptions, reduced ordered moments, fractionalized excitations in selected systems, and the difficulty of frustrated antiferromagnets.

Exact Spin-One-Half Antiferromagnetic Chain

For $s=1/2$, $J\gt0$, periodic boundaries, and nearest-neighbor exchange, the one-dimensional model is integrable by Bethe ansatz. In a sector with $M$ down spins, a coordinate ansatz has the schematic form

$$\psi(x_1,\ldots,x_M) = \sum_P A(P) \exp\!\left( i\sum_{a=1}^{M} k_{P_a}x_a \right),$$

for ordered positions $x_1\lt\cdots\lt x_M$. Two-body scattering relates the amplitudes $A(P)$, and periodicity quantizes the rapidities. The factorized scattering structure is special: generic perturbations destroy Bethe-ansatz integrability even though translation and spin symmetry may remain.

In the thermodynamic limit, the exact ground-state energy per site is

$$\frac{E_0}{L} = J \left( \frac{1}{4} - \ln2 \right) \simeq -0.443147J.$$

The ground state is a total-spin singlet for even $L$, not a Néel product state. It has strong antiferromagnetic correlations but no nonzero staggered magnetization in the translation-invariant one-dimensional ground state.

The low-energy spectrum is gapless. Its elementary fractional excitations are spinons carrying spin $1/2$. A local spin probe has integer spin overall and, with periodic boundaries, accesses multi-spinon continua rather than an isolated single-spinon pole. The exact lower edge of the two-spinon continuum is

$$\varepsilon_{ \mathrm L }(q) = \frac{\pi J}{2} \left\lvert \sin(qa) \right\rvert.$$

This result gives a linear low-energy scale near the gapless wavevectors. Calling every feature a magnon would miss the continuum and the fractional quantum numbers of the one-dimensional antiferromagnet.

The Heisenberg chain gives further exact results and finite-size comparisons. The spin-half chain compares its normalization with anisotropic relatives.

Integer and Half-Integer Chains

Haldane’s semiclassical mapping of the antiferromagnetic spin-$s$ chain leads to an $O(3)$ nonlinear sigma model with topological angle

$$\theta = 2\pi s.$$

The mapping distinguishes integer and half-integer spin:

• integer-spin nearest-neighbor chains are expected to have a nonzero bulk gap and short-range spin correlations;
• with one half-integer spin per unit cell, short-range interactions, spin-rotation symmetry and unbroken one-site translation, a unique gapped ground state is excluded; the uniform nearest-neighbor isotropic antiferromagnetic chain is gapless.

This statement is a low-energy classification, not the claim that every Hamiltonian with integer or half-integer onsite spin behaves identically. Dimerization, frustration, anisotropy, enlarged unit cells, long-range interactions, or symmetry breaking can change the outcome.

For the isotropic spin-1 chain, high-precision numerical work gives the Haldane gap

$$\Delta_{ \mathrm H } \simeq 0.4105J.$$

An open spin-1 chain also supports effective spin-$1/2$ edge degrees of freedom whose splitting becomes exponentially small with length. The symmetry-protected interpretation and full phase diagram belong with many-body phases and quantum matter; the key point here is that changing the onsite spin can qualitatively change the infrared physics even when the Hamiltonian has the same algebraic form.

Spin-Wave Theory Preview

Spin-wave theory represents deviations from an ordered reference state by bosons. Magnons explains their quantization, probe matrix elements and interactions. For a spin locally aligned along $+z$, the exact Holstein–Primakoff representation is

$$\begin{aligned} s_i^z &= s-a_i^\dagger a_i, \\ s_i^+ &= \sqrt{ 2s-a_i^\dagger a_i } \,a_i, \\ s_i^- &= a_i^\dagger \sqrt{ 2s-a_i^\dagger a_i }. \end{aligned}$$

The exact representation acts on the physical local subspace $0\le n_i\le2s$; unrestricted bosonic states are not additional spin states. Expanding the square roots is controlled when occupation fluctuations as well as the mean are small on the scale $2s$. Keeping the leading quadratic terms gives linear spin-wave theory. Finite-spin corrections can matter even in a one-deviation sector when extra anisotropy is present, as the exercise below shows.

For the ferromagnetic chain, the one-boson result reproduces the exact one-magnon dispersion. Interactions between magnons enter at higher boson number and higher order in the expansion.

For a nearest-neighbor bipartite antiferromagnet, one first rotates the local axes on one sublattice. On a hypercubic lattice, linear spin-wave theory gives

$$\varepsilon_{\mathbf k} = zJs \sqrt{ 1-\gamma_{\mathbf k}^{2} },$$

where $z$ is the coordination number and

$$\gamma_{\mathbf k} = \frac{1}{z} \sum_{\boldsymbol\delta} e^{i\mathbf k\cdot\boldsymbol\delta}$$

averages over nearest-neighbor vectors. The formula is most reliable for large $s$, low spin-wave density, and phases with stable long-range or strong semiclassical order.

In one dimension it predicts a gapless antiferromagnetic mode, but for the spin-$1/2$ chain it gives neither the exact velocity nor the correct spinon-continuum interpretation. Agreement on gaplessness does not make the quasiparticle content exact.

Dimensionality and Temperature

The Mermin–Wagner theorem places a precise limit on thermal ordering. For regular one- or two-dimensional lattices of finite spins with bounded finite-range exchange, continuous spin-rotation symmetry prevents ferromagnetic or antiferromagnetic long-range order at any nonzero temperature.

The conditions matter:

• the theorem concerns nonzero temperature, not automatically the ground state;
• it assumes the relevant continuous symmetry is unbroken by the Hamiltonian;
• sufficiently long-range interactions, anisotropy, or interlayer coupling can change the conclusion;
• it forbids the stated long-range order, not all correlations, crossovers, or topological phenomena.

For example, the two-dimensional square-lattice antiferromagnet can have Néel order at zero temperature while having only a large but finite correlation length at every $T\gt0$. A weak interlayer coupling or easy-axis anisotropy can then support a finite-temperature transition in a real quasi-two-dimensional material. Finite-temperature transitions explains the thermodynamic criteria; universality relates a critical point to dimension, symmetry and interaction range.

Finite systems never exhibit spontaneous symmetry breaking in exactly the same way as the infinite system. A finite-volume ground state can be a symmetry eigenstate while low-lying states, correlation functions, and order-of-limits behavior reveal the approach to an ordered thermodynamic phase. Spontaneous Symmetry Breaking develops the Anderson tower, oriented wave packets, and source-selected limit.

Frustration and Exchange Geometry

For nonuniform exchange,

$$H = \sum_{i<j} J_{ij} \mathbf s_i\cdot\mathbf s_j,$$

the signs and graph geometry determine whether preferred bond correlations can coexist.

An antiferromagnetic triangle is the simplest geometric example. No collinear assignment can make all three neighboring pairs antiparallel. Quantum mechanically, the three bonds share spins and cannot all be singlets. Frustration can enlarge ground-state degeneracy, suppress conventional order, favor noncollinear correlations, or stabilize highly entangled phases.

Competing exchanges can frustrate even a bipartite lattice. In the square-lattice $J_1$–$J_2$ model, antiferromagnetic nearest- and next-nearest-neighbor couplings prefer incompatible patterns. Disorder can add another source of competition.

Frustration is not synonymous with a spin liquid. A frustrated model may order, dimerize, freeze, remain critical, or realize a spin-liquid regime depending on parameters and dimension. Establishing the phase requires observables and scaling, not geometry alone.

Microscopic Origins and Model Boundaries

The Heisenberg model usually describes localized moments after charge and orbital excitations have been integrated out or otherwise frozen. Several microscopic mechanisms can generate exchange, including direct orbital exchange, superexchange through virtual charge fluctuations, and indirect exchange mediated by itinerant carriers.

For the single-band repulsive Hubbard model at half filling, with uniform real nearest-neighbor hopping on a bipartite lattice and $U\gg\lvert t\rvert$, eliminating virtual doublon–hole states gives

$$H_{ \mathrm{eff} } = J_{ \mathrm H } \sum_{\langle i,j\rangle} \left( \mathbf s_i\cdot\mathbf s_j - \frac{1}{4} \right) + O\!\left( \frac{t^4}{U^3} \right),$$

with

$$J_{ \mathrm H } = \frac{4t^2}{U} \gt 0.$$

The constant can be dropped at fixed half filling. The derivation and its validity conditions are explained in Effective Hamiltonians in Many-Body Systems.

The displayed fourth-order remainder uses the stated hopping geometry. Odd loops with flux can generate third-order chiral terms, so this error estimate cannot be transferred to an arbitrary graph. The leading exchange scale is $4|t_{ij}|^2/U$ for complex spin-independent bond hopping. The simple formula $4t^2/U$ is not a universal exchange law. Multiple orbitals, Hund coupling, ligand paths, spin–orbit coupling, charge-transfer energies, and higher-order virtual processes can change the magnitude, sign, range, and tensor structure of exchange.

The Heisenberg model also omits mobile charge. Away from half filling, projected hopping survives in the t–J model at first order and a spin-only Hamiltonian is generally insufficient. Likewise, strongly itinerant magnets may not admit a fixed local-spin description over the scales of interest.

The Kondo Model Preview retains both a localized spin and an itinerant conduction bath. Its central issue is screening by exchange, not a graph containing only fixed spins.

Core Observables

Magnetization

The total magnetization operator in spin units is

$$\mathbf M = \sum_i\mathbf s_i.$$

On a bipartite lattice, a staggered magnetization is

$$\mathbf M_{ \mathrm{st} } = \sum_i \eta_i\mathbf s_i,$$

where $\eta_i=+1$ on one sublattice and $-1$ on the other. A nonzero finite-system value of $\langle\mathbf M_{\mathrm{st}}\rangle$ requires a symmetry-breaking state or field; the rotationally invariant diagnostic is often $\langle\mathbf M_{\mathrm{st}}^2\rangle$ and its scaling.

Spin correlations

Equal-time correlations are

$$C^{\alpha\beta}_{ij} = \left\langle s_i^\alpha s_j^\beta \right\rangle.$$

In an $SU(2)$-invariant state,

$$\left\langle s_i^\alpha s_j^\beta \right\rangle = \frac{\delta_{\alpha\beta}}{3} \left\langle \mathbf s_i\cdot\mathbf s_j \right\rangle.$$

This relation is a useful symmetry check. It need not hold in a symmetry-broken state, in a field, or with anisotropic exchange.

Static structure factor

For positions $\mathbf r_i$, one common normalization is

$$S(\mathbf q) = \frac{1}{L} \sum_{i,j} e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \left\langle \mathbf s_i\cdot\mathbf s_j \right\rangle.$$

A peak near $\mathbf q=0$ indicates ferromagnetic correlations. On a one-dimensional bipartite chain, antiferromagnetic correlations peak near $q=\pi/a$. Peak height and width must be scaled with system size before inferring long-range order; Long-Range Order explains the extensive-versus-subextensive test.

Dynamic structure factor

A frequency-resolved convention is

$$S^{\alpha\beta}(\mathbf q,\omega) ={} \frac{1}{2\pi L} \int_{-\infty}^{\infty} dt\,e^{i\omega t} \times \sum_{i,j} e^{-i\mathbf q\cdot(\mathbf r_i-\mathbf r_j)} \left\langle s_i^\alpha(t)s_j^\beta(0) \right\rangle.$$

The normalization, Fourier sign, and thermal convention vary across fields. The dynamic structure factor distinguishes sharp magnons from continua and connects directly to scattering probes after including form factors and experimental units.

Structure factors explains the conventions, detailed balance and neutron polarization factors. A peak broadened by finite resolution is not automatically a finite quasiparticle lifetime.

Susceptibility and gap

If a uniform field enters as $-hM^z$, the equilibrium isothermal susceptibility is the derivative of $\langle M^z\rangle$ with respect to $h$. Since $[H,M^z]=0$, it is

$$\chi = \beta \left[ \left\langle (M^z)^2 \right\rangle - \left\langle M^z \right\rangle^2 \right].$$

Divide by $L$ for susceptibility per site. Multiplication by $(g\mu_{\mathrm B})^2$ converts to the response of the physical magnetic moment when $h=g\mu_{\mathrm B}B$.

For a finite system, the spectral gap is

$$\Delta_L = E_1(L)-E_0(L).$$

State whether repeated ground-state energies are included in this ordering and which symmetry sectors are compared. A degenerate multiplet, an ordered-state tower or open-chain edge states can make this difference vanish even when a bulk excitation is gapped. Diagnose the bulk using excitations above the relevant ground-state manifold and compare boundary conditions. Conversely, the ferromagnet has both an exactly degenerate ground multiplet and dispersing magnons whose nonzero energies vanish as $L^{-2}$.

Exact and Approximate Solution Status

The model’s simple definition does not imply generic solvability.

Exact structures include:

• arbitrary two-spin dimers by angular-momentum addition;
• fully polarized ferromagnetic ground states on broad classes of graphs;
• the one-magnon ferromagnetic sector;
• the spin-$1/2$ nearest-neighbor chain and selected anisotropic relatives by Bethe ansatz;
• small clusters by direct diagonalization.

Common controlled or systematically improvable approaches include:

• spin-wave and $1/s$ expansions in semiclassical ordered regimes;
• high- and low-temperature expansions;
• perturbation theory around dimers, Ising limits, or strong fields;
• matrix-product-state and density-matrix-renormalization methods in one dimension;
• quantum Monte Carlo for suitable unfrustrated models without a sign obstruction;
• exact diagonalization and Krylov methods for finite clusters;
• tensor-network, series, and linked-cluster methods in selected higher-dimensional settings.

No method is uniformly best. Frustration can create a Monte Carlo sign problem; two-dimensional entanglement growth limits tensor networks; spin-wave theory can fail without stable order; and exact diagonalization is exponentially size limited.

Exercises

Néel state is not an eigenstate

For a spin-$1/2$ bond, act with $\mathbf s_1\cdot\mathbf s_2$ on $\lvert\uparrow\downarrow\rangle$. Explain why an alternating product state is not an eigenstate of the isotropic antiferromagnetic chain.

Solution

Using

$$\mathbf s_1\cdot\mathbf s_2 = s_1^zs_2^z + \frac{1}{2} \left( s_1^+s_2^- + s_1^-s_2^+ \right),$$

gives

$$\mathbf s_1\cdot\mathbf s_2 \lvert\uparrow\downarrow\rangle = -\frac{1}{4} \lvert\uparrow\downarrow\rangle + \frac{1}{2} \lvert\downarrow\uparrow\rangle.$$

The second term is a distinct basis state, so $\lvert\uparrow\downarrow\rangle$ is not an eigenvector. Every antiparallel bond in an alternating chain has such a transverse fluctuation. The product state has the correct classical pattern but not the exact quantum correlations.

Antiferromagnetic triangle

Three spin-$1/2$ sites form a triangle with

$$H = J \left( \mathbf s_1\cdot\mathbf s_2 + \mathbf s_2\cdot\mathbf s_3 + \mathbf s_3\cdot\mathbf s_1 \right), \qquad J\gt0.$$

Find the energies for total spin $S_{\mathrm{tot}}=1/2$ and $3/2$. Why can no state make all three bonds singlets?

Solution

Use

$$\mathbf s_{\mathrm{tot}}^2 = \sum_{i=1}^{3} \mathbf s_i^2 + 2 \sum_{i<j} \mathbf s_i\cdot\mathbf s_j.$$

Since each site has $\mathbf s_i^2=3/4$,

$$H = \frac{J}{2} \left[ \mathbf s_{\mathrm{tot}}^2 - \frac{9}{4} \right].$$

Therefore

$$E_{1/2} = -\frac{3J}{4}, \qquad E_{3/2} = \frac{3J}{4}.$$

A singlet bond uses both of its spins in a total-spin-zero state. One of those spins cannot simultaneously form an independent singlet with the third spin. A direct energy argument is stronger: three independently minimized bonds would give $-9J/4$, but the exact minimum is only $-3J/4$. The ground space contains two spin-$1/2$ multiplets, hence four states; the $S=3/2$ quartet has four states. The absence of a total singlet alone would not rule out one singlet bond plus a spectator spin.

What Mermin–Wagner does and does not say

Assess each claim for a short-range isotropic Heisenberg model:

1. A two-dimensional system cannot have Néel order at $T=0$.
2. A two-dimensional system cannot have Néel long-range order at $T\gt0$.
3. Weak easy-axis anisotropy leaves the theorem’s assumptions unchanged.
4. A finite sample cannot display strong antiferromagnetic correlations.

Solution

1. False. The theorem addresses nonzero temperature; a two-dimensional ground state may order.
2. True under the theorem’s finite-range and isotropic continuous-symmetry assumptions.
3. False. Easy-axis anisotropy explicitly reduces the continuous symmetry and can permit a finite-temperature transition.
4. False. The theorem forbids thermodynamic long-range order, not a large finite correlation length, strong short-range correlations, or finite-size crossover behavior.

Single-ion anisotropy at finite spin

Add $-K(s^z)^2$ to one site polarized along $+z$. What is the exact energy change when $m=s$ becomes $m=s-1$? Compare it with the leading large-$s$ spin-wave value and explain the spin-$1/2$ case.

Solution

The difference is $-K(s-1)^2+Ks^2=K(2s-1)$. Since $s^z=s-n$ and $n^2=n+a^{\dagger2}a^2$,

$$-K(s^z)^2=-Ks^2+K(2s-1)n-Ka^{\dagger2}a^2.$$

The quartic term vanishes in the one-deviation sector. The often-used $2Ks$ is the leading large-$s$ value, with an order-one correction $-K$. For $s=1/2$, $(s^z)^2=1/4$ is constant throughout the physical local space, so this anisotropy produces no splitting.

References

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• J. des Cloizeaux and J. J. Pearson, “Spin-Wave Spectrum of the Antiferromagnetic Linear Chain”, Physical Review 128, 2131–2135 (1962).
• E. Fradkin, Field Theories of Condensed Matter Physics, 2nd ed., Cambridge University Press (2013).
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2004).
• F. D. M. Haldane, “Nonlinear Field Theory of Large-Spin Heisenberg Antiferromagnets”, Physical Review Letters 50, 1153–1156 (1983).
• W. Heisenberg, “Zur Theorie des Ferromagnetismus”, Zeitschrift für Physik 49, 619–636 (1928).
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• N. D. Mermin and H. Wagner, “Absence of Ferromagnetism or Antiferromagnetism in One- or Two-Dimensional Isotropic Heisenberg Models”, Physical Review Letters 17, 1133–1136 (1966), with erratum at p. 1307.
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