Lattice Models and Spin Systems

A lattice model replaces a physical system by local degrees of freedom and specified couplings between them. It can describe atoms in an array, localized spins, electrons in orbitals, or an effective low-energy problem. The graph, local states, filling, and boundary conditions matter as much as the name of the Hamiltonian.

This chapter introduces particle, spin, and impurity models, then explains how mappings and solvable limits connect them. Start with the Lattice Models Overview when you need help specifying a model.

Required background. Enter through Many-Body Hilbert Spaces and Operators so that local tensor factors, Fock spaces, occupation notation, body rank, and real- or momentum-space representations are not left implicit.

Helpful background. Locality in Many-Body Systems separates geometric range from interaction arity, while Thermodynamic Limit and Finite-Size Effects qualify phase and scaling claims. Spin routes use preparation from Symmetry, Angular Momentum, and Spin; particle routes use the appropriate bosonic or fermionic operator algebra.

Specify the model before solving it

For a spin-$1/2$ chain, every site has two states. For a spinful fermionic site, the local states are empty, spin up, spin down, and doubly occupied. A bosonic site has infinitely many occupations unless the physical model or a numerical cutoff restricts them. These differences change both the spectrum and the computational cost.

Next specify which sites are connected and what acts on each bond. For example, $J\mathbf s_i\cdot\mathbf s_j$ uses dimensionless spin operators $\mathbf s=\boldsymbol\sigma/2$. Replacing them by Pauli matrices without changing the coefficient multiplies the interaction by four. The sign of $J$ determines whether the two-spin singlet or triplet is favored.

Boundary conditions complete the model. An open chain has ends; a ring has a closing bond and may carry a twist or flux. Under a Jordan–Wigner map, the ring closure depends on fermion parity. Compare exact limits and small systems only after these conventions agree.

Choose a model family

The following branches share a common model language, but need not all be read in order.

1. Establish the common anatomy. Begin with Lattice Models Overview. Learn how local spaces, graph data, hopping, exchange, interactions, constraints, symmetries, and universality claims fit together.
2. Treat boundaries as cross-cutting data. Consult Boundary Conditions on Lattices before interpreting finite-ring spectra, momentum grids, stiffness, Jordan–Wigner sectors, or finite-size phase signatures, even though it appears last in catalog order.
3. Choose a particle or spin trunk. Particle models normally begin with Tight-Binding Model. For a minimal discrete-symmetry spin benchmark, enter Transverse-Field Ising Model directly. For rotationally symmetric quantum magnetism, enter Heisenberg Model and continue to XXZ Spin Chain for anisotropy and integrability.
4. Branch by particle statistics and interaction. From tight binding, continue to Hubbard Model for onsite-interacting fermions, Bose–Hubbard Model for lattice bosons, Spinless Fermion Chains for the minimal interacting one-dimensional fermion route, or Anderson Impurity Model Preview for a correlated local orbital hybridized with a bath.
5. Defer strong-coupling reductions until their controls are explicit. The t–J Model Preview follows Hubbard, Heisenberg, and Effective Hamiltonians in Many-Body Systems. It is not an early core model or an algebraic identity.
6. Route impurity physics by the retained local states. Anderson retains impurity charge fluctuations. Kondo Model Preview retains a local spin coupled to a fermionic bath. Anderson → Kondo is the mechanism-first route; readers who already understand the projected spin model may enter Kondo directly.
7. Use mappings after a physical first pass. Read Jordan–Wigner Transformation after learning at least one spin chain and the fermionic operator algebra. Then return to TFIM, XXZ, or Spinless Fermion Chains with the boundary and parity caveats attached.
8. Study solvability through examples. Read Exact Solutions Preview after one quadratic example and one interacting example. It distinguishes free-mode diagonalization, Bethe ansatz, integrability, and finite exact diagonalization without promising closed forms for every observable.

Examples: a Hubbard ring and an anisotropic spin chain

Consider the phrase “a half-filled Hubbard ring at large $U/t$.” It still omits the number of sites, spin content, hopping sign and range, boundary twist, mode ordering, fixed-$N$ and spin sector, temperature, observable, and the sequence by which a bulk statement would be inferred. Start with Lattice Models Overview and Tight Binding, specify those data on the Hubbard page, and use Boundary Conditions before comparing finite spectra. A Heisenberg description at large $U/t$ is a controlled low-energy reduction with corrections and a restricted sector—not an exact rewriting of the full Hubbard model.

Now consider an open spin-$1/2$ chain with exchange anisotropy. The local space and Pauli-versus-spin normalization must be declared before deciding between Heisenberg and XXZ. Open boundaries avoid the periodic fermion-parity closure that appears under Jordan–Wigner, but edge observables and finite gaps remain boundary sensitive. A phase claim still needs a declared large-system limit and diagnostic beyond one finite spectrum.

Continue from the model to a calculation

Use Interacting Systems and Approximation Methods to choose an approximation, Correlations and Response to describe a probe, or Computational Many-Body QM to choose a numerical method. A claim about bulk order or a transition also needs the limits developed in Phases, Order, and Criticality.

The Model Encyclopedia provides compact comparisons. Quantum Matter connects the models to materials, while AMO explains atomic and optical realizations and their measurements.

Common pitfalls

“A lattice model is necessarily a crystal.” A lattice can be literal, synthetic, effective, graph based, or a regulator. The interpretation must be stated.

“A two-local Hamiltonian is geometrically local.” Interaction arity does not determine spatial range, coordination, decay, or extensivity.

“Quadratic means one particle or classical.” A quadratic Hamiltonian can act across every many-particle sector and retain exchange statistics; it is noninteracting in the adopted modes, not automatically trivial.

“The Hamiltonian name specifies the physics.” Hubbard behavior depends on filling, sign, dimension, geometry, temperature, boundaries, and the target observable—not only $U/t$.

“Large-$U$ Hubbard is exactly Heisenberg.” The spin model is a projected low-energy description with a control regime and higher-order corrections.

“Exact diagonalization proves integrability.” Finite matrices can be diagonalized exactly without an extensive family of conserved charges or an analytic many-body solution.

“Boundary conditions are just a numerical setting.” They change allowed momenta, flux response, shell structure, edge physics, and sometimes the parity sector of a mapped Hamiltonian.

Exercises

Exercise 1: Complete and route a model

A calculation is described only as “the half-filled Hubbard model on a periodic chain.” List the missing model data needed before reproducing a spectrum, then give the minimum chapter route.

Solution

Specify chain length, site and spin-orbital basis, fermionic mode ordering, hopping range and sign, onsite-interaction convention, energy units, particle-number and spin sector, precise periodic or twisted boundary, temperature or target state, observables, finite-size sequence, and numerical or analytic method. Use Lattice Models Overview → Tight-Binding Model → Hubbard Model → Boundary Conditions. Add Effective Hamiltonians and Heisenberg only if a controlled strong-coupling reduction is required.

Exercise 2: Route an exact spin-chain calculation

Route a calculation of the finite-ring transverse-field Ising spectrum and explain why neither finite exact diagonalization nor the Jordan–Wigner map alone proves generic integrability.

Solution

Read Lattice Models Overview → Transverse-Field Ising Model, then review fermionic algebra and Jordan–Wigner Transformation, consult Boundary Conditions for parity-dependent ring closure, and use Exact Solutions Preview to classify the free-mode solution. Exact diagonalization returns exact eigenvalues of a chosen finite matrix but does not establish analytic solvability at arbitrary size. Jordan–Wigner is an exact mapping of operators under stated boundary conventions; integrability follows only when the mapped Hamiltonian has the additional solvable structure.

References

• A. Auerbach, Interacting Electrons and Quantum Magnetism, Springer (1994).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• T. Giamarchi, Quantum Physics in One Dimension, Oxford University Press (2004).
• A. C. Hewson, The Kondo Problem to Heavy Fermions, Cambridge University Press (1993).
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011).