Model-to-Volume Cross-Link Index

The same model can answer different questions: what its Hamiltonian means, how its spectrum is calculated, or how it describes atoms or a material. Use this index to move between a precise model specification, a derivation, a numerical example, and an application.

Helpful background. The Model Encyclopedia compares the major families. If you start from a measured property rather than a model name, begin with Choosing a Model for Quantum Matter.

Many-body models by physical question

The first link in each row specifies the model, including its Hilbert space and boundary conventions. The second column suggests what to study next.

Gases and collective motion

Model	Derivations and examples
Ideal Bose gas	Thermodynamics and condensation; formula sheet and numerical gas integral
Ideal Fermi gas	Fermi surfaces and thermodynamics; formula sheet and numerical gas integral
Harmonic chain	Phonon normal modes and the chain’s explicit dynamical matrix
Lieb–Liniger gas	Low-dimensional quantum gases and contact bosons as a field theory
Tonks–Girardeau gas	The impenetrable limit and its Bose–Fermi map; Luttinger liquid for long-wavelength correlations

Spins and lattice particles

Model	Derivations and examples
Spin-1/2 chain	Spin Hamiltonians and symmetry sectors
Transverse-field Ising chain	Fermion solution and critical behavior; finite-ring spectrum and gap scaling
Heisenberg chain	Exchange physics and a four-site spin ring
XXZ chain	Anisotropy, phases, and Luttinger behavior; XXX denotes its isotropic exchange limit
Tight-binding chain	Bands, Fourier diagonalization, and boundary twists
Hubbard dimer	Exact two-site spectrum and its numerical example
Hubbard chain	Generic Hubbard physics and a four-site calculation
Bose–Hubbard dimer	Lattice-boson physics and a fixed-number two-site example
Bose–Hubbard chain	One-dimensional limits and finite-size scaling

Pairing and impurity physics

Model	Derivations and examples
Reduced BCS model	Richardson’s exact finite-level pairing solution; compare BCS mean field and its gap-equation example
Anderson impurity	Charge fluctuations, spectral peaks, and regimes; the single-impurity Anderson model (SIAM) differs from Anderson localization
Kondo model	Screening and scaling and the Anderson-to-Kondo reduction; also called an $s$–$d$ exchange model in this setting
Sachdev–Ye–Kitaev model	Large-$N$ methods and operator growth; specify Majorana or complex fermions and the interaction/disorder convention

For a short Hamiltonian lookup, use Common Many-Body Hamiltonians or Common Spin Hamiltonians. For calculations, start with Computational Many-Body QM and the runnable examples in Reproducible Notebooks.

From a model to an experiment or material

Question	Suggested route
How do trapped atoms realize a Bose or Fermi gas?	Bose–Einstein condensates and degenerate Fermi gases
How do lattice depth, confinement, and preparation affect a Hubbard realization?	Optical lattices and ultracold-atom quantum simulation
Which spin model describes a magnetic material?	Magnetism and Spin Systems
How does a pairing model connect to Meissner response, vortices, or superconducting materials?	Superfluidity and Superconductivity and London Theory
How is a model encoded in an information-processing simulator?	Quantum Information Roadmap
How do continuum fields and renormalization emerge?	Bridge to QFT

A useful comparison keeps both the model assumptions and the experimental observable visible. For example, matching a Hubbard hopping amplitude does not establish that a single-band model captures all optical transitions in a material; omitted orbitals may contribute at the probe energy.

Controlled limits connecting models

Noninteracting lattice particles. At $U=0$, a Hubbard Hamiltonian becomes quadratic. Fermions fill its one-particle levels; bosonic equilibrium also requires the chemical potential below the lowest one-particle energy. The shared one-body problem is tight binding.

Hubbard to Heisenberg. At exactly one fermion per site, $U\gg |t|$, and energies well below the charge gap, virtual double occupation produces $J=4|t|^2/U$ at leading order. The effective Hamiltonian derivation includes the accompanying density term. With mobile holes, projected hopping survives: a pure spin Hamiltonian is no longer the complete low-energy model.

Anderson to Kondo. An impurity local moment can be isolated when empty and doubly occupied impurity states cost much more than the low-energy scales. Schrieffer–Wolff projection then gives spin exchange and potential scattering. The projected Kondo model cannot reproduce the impurity’s high-energy charge-transfer peaks.

Lieb–Liniger to Tonks–Girardeau. For repulsive one-dimensional contact bosons, $\gamma=mg_{\mathrm{1D}}/(\hbar^2n)\to+\infty$ gives impenetrability. The Bose–Fermi map relates energies and density observables, with compatible boundary conditions. Off-diagonal bosonic correlations still differ from those of free fermions.

Hard-core bosons to spins. Projecting a repulsive Bose–Hubbard model onto $n_i=0,1$ maps hopping to $XY$ exchange, while density interactions generate longitudinal spin coupling. At finite $U$, virtual double occupation gives corrections. See Bose–Hubbard and XXZ.

Ising spins to quadratic fermions. The nearest-neighbor transverse-field chain maps to free fermions by Jordan–Wigner transformation. Periodic spin boundaries give parity-dependent fermion boundaries. A generic longitudinal field destroys that free-fermion solution.

Harmonic chain to a phonon field. Long wavelengths compared with the lattice spacing give an elastic continuum with a fixed mass density and elastic modulus. The approximation loses Brillouin-zone details; phonons connect the two descriptions.

Finite pairing to bulk mean field. The reduced pairing Hamiltonian conserves number and has Richardson solutions. Under the usual extensive coupling scaling and bulk limit, the root distribution connects to BCS gap and number equations. An unprojected BCS state is not generally an exact finite-system eigenstate.

Exercises

Charge peaks and a low-energy model

An Anderson impurity has a well-formed local moment. Can its projected Kondo Hamiltonian predict the energies of the empty-impurity and doubly occupied charge peaks?

Solution

No. Those charge states have been eliminated in constructing the low-energy Hilbert space. The Kondo model can describe the remaining spin response and screening in its regime of validity, but charge-transfer spectra require the Anderson model or a description retaining the charge states.

Doping a strong-coupling Hubbard model

Why does $U\gg|t|$ alone not justify replacing a doped Hubbard model by a Heisenberg Hamiltonian?

Solution

Large $U$ suppresses double occupation but does not remove empty sites. A fermion can hop into a neighboring hole without paying $U$, so projected hopping of order $t$ remains alongside exchange of order $t^2/U$. A spin-only description loses that charge motion. At exactly one particle per site, the projected hopping vanishes and the leading dynamics is exchange.

References

• M. Girardeau, “Relationship between Systems of Impenetrable Bosons and Fermions in One Dimension,” Journal of Mathematical Physics 1, 516–523 (1960), doi:10.1063/1.1703687.
• E. Lieb, T. Schultz, and D. Mattis, “Two Soluble Models of an Antiferromagnetic Chain,” Annals of Physics 16, 407–466 (1961), doi:10.1016/0003-4916(61)90115-4.
• E. H. Lieb and W. Liniger, “Exact Analysis of an Interacting Bose Gas. I. The General Solution and the Ground State,” Physical Review 130, 1605–1616 (1963), doi:10.1103/PhysRev.130.1605.
• R. W. Richardson, “A Restricted Class of Exact Eigenstates of the Pairing-Force Hamiltonian,” Physics Letters 3, 277–279 (1963), doi:10.1016/0031-9163(63)90259-2.
• J. R. Schrieffer and P. A. Wolff, “Relation between the Anderson and Kondo Hamiltonians,” Physical Review 149, 491–492 (1966), doi:10.1103/PhysRev.149.491.