Model Encyclopedia

The Model Encyclopedia catalogs reusable idealizations of quantum systems. A model card identifies the degrees of freedom, Hilbert space, dynamical law, parameters, observables, solvability, validity regime, and the canonical pages where its physics is derived.

A model is more than a Hamiltonian formula. The same matrix can describe a spin, two localized orbitals, or two atomic levels, while the same named model can change when its dimension, boundary conditions, interactions, statistics, or approximation scheme changes. The card’s job is to expose those choices before a formula is reused.

Find a Model by Question

Question	Start here
Which model introduces continuous spectra and wave packets?	Free Particle
Which bound model introduces ladder operators and zero-point energy?	Harmonic Oscillator
Which model organizes Coulomb spectra and orbital structure?	Hydrogen Atom
Which ideal model shows magnetic quantization?	Landau-Level System
Which finite model captures mixing, avoided crossings, and coherent transfer?	Two-Level System
Which lattice models introduce quantum magnetism and criticality?	Ising Chain, Heisenberg Chain, or XY Model
Which models contrast bosonic and fermionic statistics?	Ideal Bose Gas and Ideal Fermi Gas
Which models combine hopping with local interactions?	Hubbard or Bose–Hubbard
Which model introduces pairing and quasiparticles?	BCS Model
Which models couple matter to one quantized mode?	Rabi, Jaynes–Cummings, or Dicke
Which models isolate error-correction structure?	Stabilizer Circuit or Surface Code
Which one-qubit models describe standard noise?	Depolarizing Channel or Amplitude-Damping Channel
Which band models introduce topology or Dirac cones?	SSH, Kitaev Chain, or Graphene Dirac Model

For a compact operator expression, use the Operator and Hamiltonian Library. For a single equation with assumptions and units, use the Formula Compendium. A model card answers the broader question: what physical or computational problem has been defined?

What Defines a Quantum Model

A useful specification can be summarized schematically as

$$\mathcal M = \left( \mathcal H, \mathcal G, \boldsymbol\lambda, \mathcal O, \mathcal I, \mathcal R \right),$$

where:

• $\mathcal H$ is the Hilbert space, including tensor factors, particle statistics, geometry, and boundary conditions;
• $\mathcal G$ is the dynamical law, such as a Hamiltonian, time-dependent propagator, quantum channel, or master-equation generator;
• $\boldsymbol\lambda$ is the set of physical parameters and controls;
• $\mathcal O$ is the set of observables or operational outputs of interest;
• $\mathcal I$ specifies initial states, ensembles, temperature, filling, or preparation procedures;
• $\mathcal R$ states the regime, approximation hierarchy, and interpretation.

Not every card needs this notation, but every mature card must communicate the corresponding information. A Hamiltonian without its Hilbert space is incomplete, and a Hilbert space plus Hamiltonian may still be insufficient for an open, driven, thermal, or measurement-based model.

Model Card Contract

Each mature model card should make the following items findable without reconstructing them from prose.

Field	What the card must state
One-sentence description	The model’s physical setup and distinctive idealization
Degrees of freedom	Particles, sites, modes, spins, fields, or logical qubits
Hilbert space	Configuration space, tensor product or Fock space, statistics, and boundary data
Dynamical law	Hamiltonian, channel, circuit rules, or master equation
Parameters	Meanings, units, signs, allowed ranges, and control status
Solvability	Exactly which quantity is exact, approximate, numerical, or unknown
Spectrum and states	Relevant quantum numbers, sectors, degeneracies, and limits
Key observables	Quantities that diagnose motion, response, correlations, phases, or errors
What it teaches	The reusable conceptual or methodological lesson
Validity	Idealizations, truncations, omitted interactions, and failure conditions
Canonical home	One page or volume owning the full explanation
Resources	Formula cards, worked examples, notebooks, benchmarks, and references
Variants	Nearby models and the term that changes between them

A short card can satisfy this contract by linking outward. It should not reproduce a canonical derivation merely to appear self-contained.

Solvability Labels

Solvability is a statement about a specified model and specified outputs, not a prestige label.

Label	Meaning
Exactly solvable	A stated spectrum, eigenbasis, propagator, or observable has a closed analytic construction under declared assumptions
Integrable	The model has enough conserved structure for a specialized exact method in a specified dimension, geometry, and sector
Quadratic or Gaussian	A canonical transformation reduces the dynamics to independent modes
Perturbative	Results are organized in a controlled small parameter and retained order
Asymptotic or semiclassical	Accuracy is tied to a scale limit rather than convergence at fixed parameters
Mean-field	Interactions are replaced by a self-consistent effective field; fluctuations are omitted or added separately
Numerically tractable	A method and convergence test are known for a stated size, geometry, sign structure, or tensor-network regime
Phenomenological	Parameters encode observed behavior without a complete microscopic derivation
Benchmark model	The model is selected because independent analytic or numerical checks are available

Several labels can apply at once. A quadratic model may be exactly solvable; an interacting model may be integrable only in one dimension; a mean-field Hamiltonian may be diagonalized exactly after an uncontrolled approximation.

The phrase exact diagonalization names a finite-matrix numerical method. It does not mean that the underlying many-body family has a closed-form solution as system size grows.

Before accepting an exact claim, ask:

1. Which Hamiltonian or dynamical map is meant?
2. Which dimension and boundary conditions are assumed?
3. Which parameter regime or symmetry sector is included?
4. Which quantity is obtained exactly?
5. Is the statement finite-size, thermodynamic-limit, ground-state, equilibrium, or dynamical?

Exactly Solvable and Canonical Models

Exactly Solvable Models collects controlled laboratories for spectra, eigenstates, propagation, and scattering.

Model	Defining structure	What is solved	Canonical use
Free Particle	Kinetic energy on a stated geometry	Fourier-diagonal spectrum and propagation	Continuous spectra, packets, normalization
Harmonic Oscillator	Positive quadratic Hamiltonian	Complete number-state spectrum and dynamics	Ladder operators, Gaussian states, normal modes
Hydrogen Atom	Nonrelativistic Coulomb two-body reduction	Bound spectrum and separated eigenstates	Central potentials, angular structure, degeneracy
Landau-Level System	Ideal charged particle in a uniform magnetic field	Landau levels and degeneracy structure	Gauge choice, magnetic quantization, edge of QH physics
Pöschl–Teller Potential	Special hyperbolic well or barrier family	Parameter-dependent bound or scattering data	Shape invariance and reflectionless cases

Exactness is conditional. Boundaries alter the free spectrum, anharmonicity alters oscillator spacing, relativistic and radiative corrections alter hydrogen, disorder lifts ideal Landau degeneracy, and only specific Pöschl–Teller parameterizations possess the advertised closed forms.

Spin and Finite-Dimensional Models

Spin Models ranges from one effective two-state degree of freedom to interacting chains.

Model	Degrees of freedom	Main structure	Solution character
Two-Level System	One $\mathbb C^2$ sector	General Hermitian $2\times2$ dynamics	Exactly diagonalizable when static
Ising Chain	Local spins on a chain	Ising exchange with specified field	Classical or quantum; integrability is version-dependent
Heisenberg Chain	Local spins with exchange	Rotationally symmetric or anisotropic couplings	Bethe-ansatz integrable in important one-dimensional cases
XY Model	Planar spin chain	Anisotropic $x$-$y$ exchange and field	Standard one-dimensional forms map to quadratic fermions

For spin chains, state the spin representation, lattice, interaction range, anisotropy, field direction, operator normalization, coupling sign, and boundary conditions. The label Heisenberg model alone does not select one Hamiltonian.

Many-Body Models

Many-Body Models identifies statistics, ensembles, interactions, phases, and solution methods. The Many-Body Model Encyclopedia owns the more detailed convention-complete dossiers.

Model	Defining structure	Main observables	Solution character
Ideal Fermi Gas	Noninteracting fermionic modes	Occupations, Fermi energy, equation of state	Mode-factorized; geometry and ensemble still matter
Ideal Bose Gas	Noninteracting bosonic modes	Occupations, condensate fraction, thermodynamics	Mode-factorized; condensation depends on dimension and limit
Hubbard Model	Fermion hopping plus onsite interaction	Correlations, magnetism, charge response	General problem numerical; exact in special limits and cases
Bose–Hubbard Model	Boson hopping plus onsite interaction	Number fluctuations, coherence, Mott and superfluid diagnostics	Approximate or numerical in general
BCS Model	Reduced pairing interaction	Gap, quasiparticle spectrum, condensate response	Standard treatment is a controlled or phenomenological mean-field reduction

Particle number, chemical potential, temperature, filling, dimensionality, lattice geometry, and thermodynamic limit are model data. A phase diagram cannot be transported between conventions without translating them.

Light–Matter Models

Light–Matter Models compares few-mode quantum-optical Hamiltonians.

Model	Matter and field content	Terms retained	Solution character
Rabi Model	One two-level system and one bosonic mode	Rotating and counter-rotating couplings	Analytic structure plus truncation or specialized methods
Jaynes–Cummings Model	One two-level system and one bosonic mode	Rotating-wave excitation exchange	Exactly block diagonal under its assumptions
Dicke Model	Many two-level systems and one collective mode	Convention-dependent collective coupling	Symmetry reduction, approximations, and numerical treatment

Specify whether the field is classical or quantized, how $g$ is normalized, whether the rotating-wave approximation is made, how many emitters and modes are retained, and whether drive or loss is present. The Jaynes–Cummings model is not the full dipole-coupling Hamiltonian.

Quantum Information Models

Quantum Information Models includes operational circuit, code, and channel models as well as Hamiltonian systems.

Model	Mathematical object	Main use	Essential scope
Stabilizer Circuit	Clifford operations, Pauli preparations and measurements	Efficient algebraic tracking and code structure	Not every efficiently simulable circuit is a stabilizer circuit
Surface Code	Local stabilizer code with boundaries and syndrome extraction	Logical encoding and fault-tolerance studies	Performance depends on noise model, circuit, decoder, and geometry
Depolarizing Channel	One-qubit completely positive trace-preserving map	Isotropic noise benchmark	Parameter conventions differ across sources
Amplitude-Damping Channel	Nonunital completely positive trace-preserving map	Energy-relaxation benchmark	Preferred basis and time parameter must be stated

A quantum channel is a dynamical map on density operators, not a Hamiltonian. A code model requires physical-qubit layout, checks, logical operators, syndrome protocol, and decoder before a threshold or resource claim becomes meaningful.

Condensed-Matter and Topological Models

Condensed-Matter Models collects lattice and low-energy band models.

Model	Defining structure	Main lesson	Validity boundary
Tight-Binding Chain	Orbitals with hopping on a one-dimensional lattice	Band formation and boundary-sensitive modes	Truncated orbital and hopping basis
Graphene Dirac Model	Two-sublattice low-energy expansion near valleys	Dirac cones, chirality, and valley structure	Valid near the expansion points and within retained terms
SSH Model	Alternating nearest-neighbor hopping	Chiral symmetry, winding, and boundary states	Topological statement assumes a gap and symmetry
Kitaev Chain	Spinless p-wave superconducting chain	Bogoliubov form and Majorana boundary modes	Mean-field, symmetry, gap, and boundary assumptions

For every band model, record lattice vectors, unit cell, orbital ordering, Brillouin-zone convention, filling, boundary termination, interactions, disorder, and protecting symmetries. A band Hamiltonian is not a complete microscopic material description.

Compare Models Systematically

When deciding whether two formulas describe the same model, compare the following in order.

1. Degrees of freedom: particle coordinates, sites, modes, spins, internal levels, or logical qubits.
2. State space: first quantization, Fock space, tensor-product spins, code subspace, or operator space.
3. Statistics and constraints: bosonic, fermionic, distinguishable, gauge constrained, fixed parity, or hard-core.
4. Geometry: continuum or lattice, dimensionality, topology, and boundary conditions.
5. Dynamics: closed Hamiltonian, driven Hamiltonian, channel, circuit, or master equation.
6. Parameters: dimensions, signs, normalization, control protocol, and scale hierarchy.
7. Symmetries: exact, approximate, explicitly broken, or emergent.
8. Observables: spectrum, scattering, correlations, response, order parameters, logical errors, or trajectories.
9. State or ensemble: initial preparation, particle number, filling, temperature, or steady-state condition.
10. Solution claim: quantity, method, approximation order, finite-size regime, and convergence evidence.

Two Hamiltonians related by a unitary basis change may define the same abstract closed-system dynamics while their physical observables use different dictionaries. Conversely, identical-looking formulas on different domains or tensor factors can define different models.

Canonical Ownership

Need	Canonical layer
Compact operator expression and domain	Operator and Hamiltonian Library
Physical setup and model taxonomy	Model card in this encyclopedia
Full explanation or derivation	Subject volume linked by the card
Reusable equation with assumptions	Formula Compendium
Numerical implementation and convergence	Software, Notebooks, and Benchmarks
Convention-complete many-body dossier	Many-Body Model Encyclopedia
Experimental realization	Application or experiment page in the relevant volume

This division enforces one canonical home. The model card states what is being modeled and where to continue; it does not become a second textbook chapter.

Reliability Checklist

Before importing a result from one model into another, verify:

• the Hilbert space and operator domain agree;
• the same boundary conditions and geometry are used;
• all parameters have compatible units, signs, and normalizations;
• the same particle statistics, spin content, and tensor ordering apply;
• the model is closed, driven, open, thermal, or measurement-based in the same sense;
• the cited solution concerns the desired observable and parameter regime;
• finite-size and thermodynamic-limit statements are not mixed;
• an effective truncation remains separated from omitted states;
• an exact statement has not been inherited from an approximate Hamiltonian without qualification;
• numerical claims include convergence against basis size, lattice size, time step, bond dimension, sampling error, or another relevant control.

Common Mistakes

• Treating a model name as a complete mathematical specification.
• Assuming one standard convention exists for every named Hamiltonian.
• Calling a physical system exactly solvable because an idealized limit is solvable.
• Presenting a mean-field diagonalization as an exact solution of the interacting model.
• Using exact diagonalization as a synonym for analytic exact solvability.
• Generalizing an integrability statement across dimensions, boundary conditions, anisotropies, or perturbations.
• Comparing phase diagrams without translating coupling and filling conventions.
• Confusing bare model parameters with directly measured observables.
• Treating a low-energy effective model as valid across the entire spectrum or Brillouin zone.
• Calling a noise channel a Hamiltonian.
• Quoting a code threshold without its noise model and decoder.
• Duplicating a canonical derivation inside several model cards.

Exercises

1. Same matrix, different model

The Hamiltonian

$$\hat H=\epsilon\sigma_z-J\sigma_x$$

can describe a spin-$1/2$ in a magnetic field or a particle restricted to two localized sites. List model data that distinguish the two realizations even though their spectra agree.

Solution

Both have eigenvalues

$$E_\pm=\pm\sqrt{\epsilon^2+J^2}.$$

For a physical spin, the basis may be eigenstates of $S_z$, the Pauli matrices represent spin components, and $\epsilon$ and $J$ are components of a magnetic or effective field multiplied by the magnetic moment convention. Position is not represented by $\sigma_z$ unless an additional encoding is declared.

For two localized sites, the basis states identify left and right orbitals, $\epsilon$ is a site-energy bias, $J$ is a tunneling amplitude, and $\sigma_z$ is proportional to the population imbalance. Spatial position, current, and coupling to electric fields use the site dictionary.

The abstract unitary dynamics is equivalent after the parameter map, but preparations, observables, environmental couplings, and the validity of the two-state truncation differ. The model card must state those data.

2. Audit an exactness claim

Critique the sentence: “The Heisenberg model is exactly solvable.” Rewrite it as a defensible reference claim.

Solution

The sentence leaves the lattice, dimension, spin, interaction range, anisotropy, field, boundary conditions, and target quantity unspecified. Generic Heisenberg models are not covered by one exact solution.

A defensible version is: “The one-dimensional nearest-neighbor spin-$1/2$ isotropic Heisenberg chain is Bethe-ansatz integrable under standard boundary choices; obtaining a particular finite-size or thermodynamic observable still requires the corresponding Bethe equations and limit.” Related XXZ chains also have integrable regimes, but adding generic longer-range couplings, disorder, or fields can destroy that structure.

This revision identifies the model family, method, and scope without implying that every geometry and observable has a simple closed form.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, 2 vols., Wiley, 1977.
• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press, 2010.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press, 2011.
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press, 1997.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
• H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, 2002.