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Quantum Mechanics

Many-Body Model Families

A many-body model turns a physical question into a specified set of degrees of freedom, a Hamiltonian, and a state. This chapter compares eighteen recurring models, from ideal gases and spin chains to pairing and impurity problems. Start with the question you want to answer: a ground-state energy, a thermal property, a correlation function, and a time-dependent response can require different methods even for the same Hamiltonian.

Helpful background. Symbols and Conventions fixes indices, units, and normalizations; Many-Body Hilbert Spaces explains sectors and local spaces; Lattice Models introduces hopping and exchange.

What specifies a many-body model?

Section titled “What specifies a many-body model?”

Consider nearest-neighbor hopping,

Ht=−t∑⟨i,j⟩(ci†cj+cj†ci).H_t=-t\sum_{\langle i,j\rangle}(c_i^\dagger c_j+c_j^\dagger c_i).

Each undirected bond is counted once and tt is real. The formula still leaves essential choices open. Are the cic_i bosonic or fermionic operators? Is the geometry a chain or a two-dimensional lattice? Are the ends open or joined? How many particles are present? With those choices fixed, the same expression can describe a single-particle band, a free Fermi sea, or a lattice Bose gas.

For equilibrium calculations, distinguish the energy operator HH from the grand-canonical generator K=H−μNK=H-\mu N. At fixed particle number, occupations of different modes are constrained by their sum. In an ideal grand-canonical gas, mode probabilities factorize because both the energy and number operators are sums over independent modes. These are different statistical questions even when the one-particle spectrum is identical.

Spin normalization changes quoted couplings

Section titled “Spin normalization changes quoted couplings”

Let Sja=σja/2S_j^a=\sigma_j^a/2 denote dimensionless spin operators; the angular momentum is ℏSja\hbar S_j^a. Then

H=−Jσ∑jσjzσj+1z−hσ∑jσjx=−JS∑jSjzSj+1z−hS∑jSjx,\begin{aligned} H&=-J_\sigma\sum_j\sigma_j^z\sigma_{j+1}^z-h_\sigma\sum_j\sigma_j^x\\ &=-J_S\sum_j S_j^zS_{j+1}^z-h_S\sum_j S_j^x, \end{aligned}

with JS=4JσJ_S=4J_\sigma and hS=2hσh_S=2h_\sigma. The familiar critical ratio hσ/Jσ=1h_\sigma/J_\sigma=1 for the uniform ferromagnetic infinite Ising chain therefore becomes hS/JS=1/2h_S/J_S=1/2. Comparing the numerical ratio without converting the operators would compare different physical problems.

Gases and harmonic motion

Section titled “Gases and harmonic motion”
Model What it teaches Scope of the solution
Ideal Bose gas Bose occupation and condensation Grand-canonical mode factorization is exact; condensation depends on the spectrum and thermodynamic limit.
Ideal Fermi gas Pauli filling and the Fermi surface Occupations follow from the one-particle spectrum; low-temperature expansions need smooth spectral data near the chemical potential.
Harmonic chain Collective normal modes and phonons A stable quadratic Hamiltonian reduces to independent modes, with zero modes treated separately.
Lieb–Liniger model Repulsive contact interactions in one dimension Bethe equations determine the spectrum; correlation functions require additional work.
Tonks–Girardeau gas Impenetrable bosons A Bose–Fermi mapping relates spectra and density observables, but not all off-diagonal correlations.

For a translation-invariant unpinned harmonic chain, uniform displacement costs no potential energy. Its center-of-mass coordinate is not an ordinary positive-frequency oscillator. Fixing it is a physical constraint; retaining it requires a separate treatment. Neither choice is equivalent to silently dropping a mode.

The Lieb–Liniger coupling is commonly written

γ=mg1Dℏ2n,\gamma=\frac{m g_{1\mathrm D}}{\hbar^2 n},

where nn is line density and g1D>0g_{1\mathrm D}>0 is the contact coupling. At fixed density, γ→∞\gamma\to\infty gives the Tonks–Girardeau limit. Lieb and Liniger’s exact solution and Girardeau’s mapping explain this connection; equality of energies does not imply equality of bosonic and fermionic momentum distributions.

Spin and lattice models

Section titled “Spin and lattice models”
Model What it teaches Scope of the solution
Spin-1/2 chain Local spins, interactions, and symmetries A family of models; local dimension two does not imply exact solvability.
Transverse-field Ising chain Competition between alignment and transverse fluctuations The nearest-neighbor chain without a longitudinal field maps to quadratic fermions.
Heisenberg chain Isotropic exchange The uniform spin-1/2 chain is integrable; generic lattices and couplings are not.
XXZ chain Exchange anisotropy The standard uniform chain is integrable, with phases depending on anisotropy and magnetization.
Tight-binding chain Hopping, bands, and boundary phases The translation-invariant quadratic problem is diagonal in momentum.
Hubbard dimer Charge motion versus onsite repulsion Small fixed-number sectors can be diagonalized completely.
Hubbard chain Interacting lattice fermions The uniform one-dimensional model has a Bethe-ansatz solution.
Bose–Hubbard dimer Boson tunneling and interaction The fixed-NN sector has dimension N+1N+1.
Bose–Hubbard chain Coherence and localization of lattice bosons Several special limits are solvable; the generic soft-core chain is not integrable.

An exact change of variables need not make a problem free. The Jordan–Wigner transformation remains an exact operator mapping when added interactions make the transformed Hamiltonian nonquadratic. For a ring, it also relates the fermionic boundary term to the parity sector. Do not impose periodic fermion boundaries independently of that sector.

At half filling and U≫∣t∣U\gg |t|, a repulsive Hubbard system has a low-energy spin description with leading nearest-neighbor exchange Jeff=4∣t∣2/UJ_{\mathrm{eff}}=4|t|^2/U. Virtual charge fluctuations generate this interaction. The approximation describes energies well below the charge-excitation scale; it cannot reproduce the Hubbard model’s full charge dynamics. See Effective Hamiltonians for the projection and its corrections.

Pairing, impurities, and random interactions

Section titled “Pairing, impurities, and random interactions”
Model What it teaches Scope of the solution
Reduced BCS model Pair scattering among time-reversed levels Richardson’s finite-level solution differs from the symmetry-breaking BCS mean-field approximation.
Anderson impurity model An interacting level hybridized with a bath The U=0U=0 limit is quadratic; interactions introduce charge and spin correlations.
Kondo model Exchange between an impurity spin and a bath Weak-coupling scaling and low-temperature screening describe different regimes.
SYK model Random all-to-all interactions The disorder-averaged large-NN limit has a controlled saddle; a finite realization remains a separate problem.

The Anderson model reduces to a Kondo spin model when empty and doubly occupied impurity configurations are costly compared with the energies being studied. This removes impurity charge states, so it is a low-energy approximation rather than an identity. The Anderson-to-Kondo derivation specifies the charge gaps, exchange convention, and accompanying potential scattering.

What does “solved” mean?

Section titled “What does “solved” mean?”

A finite matrix can be diagonalized to numerical precision without solving its infinite-lattice counterpart. Conversely, an integrable model may provide exact energy equations while leaving a difficult calculation of local dynamical correlations. Always attach the method to the particular quantity it determines.

Method What to check before using its result
Finite diagonalization System size, sector, energy zero, and numerical residuals.
Quadratic-mode solution Stability, zero modes, boundary conditions, and the transformation of observables.
Bethe ansatz The integrable Hamiltonian and boundaries, completeness or state restrictions, and the observable being computed.
Controlled expansion Its small parameter and the size of corrections for the chosen observable.
Numerical extrapolation Separate basis or algorithm convergence at fixed size from the limit of increasing physical system size.

For a first numerical example, compare the Hubbard or Bose–Hubbard dimer with Benchmark Problems. For the derivation behind a model, use its linked article. For a quick comparison of operator and coupling conventions, use Common Many-Body Hamiltonians.

Exercises

Section titled “Exercises”

1. Count a bosonic dimer sector. Two bosonic modes contain exactly NN particles. Find the dimension of the Hilbert space. Does choosing a local occupation cutoff smaller than NN preserve that space?

Solution

The states are ∣n,N−n⟩\lvert n,N-n\rangle with n=0,1,…,Nn=0,1,\ldots,N, giving N+1N+1 states. Number conservation already supplies a finite space. A local cutoff below NN excludes legitimate states such as ∣N,0⟩\lvert N,0\rangle; it is an additional approximation, not a necessary way to make this problem finite.

2. Interpret an energy claim. A nearest-neighbor tight-binding calculation reports a ground-state energy −2t-2t, with t>0t>0. Explain one setting where this is correct and why it is not a general many-particle answer.

Solution

For one particle on a uniform periodic one-dimensional chain, with the usual bond convention and at least three sites, ϵ(k)=−2tcos⁡(ka)\epsilon(k)=-2t\cos(ka). The allowed k=0k=0 state has energy −2t-2t. Several noninteracting bosons can occupy that state, giving −2Nt-2Nt at fixed number NN. Spinless fermions instead fill distinct momentum states, so their ground-state energy is the sum of the NN lowest one-particle energies. Open ends or a boundary twist can also change the minimum at finite size.

3. Separate two convergence questions. An energy no longer changes visibly when a tensor-network bond dimension is increased at fixed chain length. Has the thermodynamic ground-state energy been found?

Solution

No. This checks sensitivity to one approximation at that chain length. One must also assess optimization or truncation errors and compare increasing lengths with specified boundaries. A small energy change alone need not bound the error in correlations, gaps, or entanglement.

References

Section titled “References”