Quantum Many-Body Theory and Statistical Mechanics Reference

Use these reference pages when you need a definition, a formula, or a translation between conventions. Each entry links to a fuller explanation when you need to understand the result or check its assumptions. The tables below group the sixteen pages by the questions they answer.

Required background. Symbols and Conventions fixes indices, normalizations, and units. Site Conventions explains the notation used throughout the site.

Symbols, ensembles, and ideal gases

Your question	Where to look	Check before substituting
What does a symbol or Fourier transform mean?	Symbols and Conventions	Units, signs, and factors of $2\pi$ and $\hbar$.
Which partition function or derivative applies?	Ensemble Formula Sheet	Fixed particle number versus fixed chemical potential, and which variables the derivative holds constant.
How do Bose, Fermi, and classical occupations compare?	Quantum Gas Formula Sheet	Spectrum, degeneracy, and whether a dilute approximation applies.
What sets Fermi energy and degeneracy pressure?	Fermi Gas Formula Sheet	Dimension and the definition of number density.
When does a Bose condensate occur?	Bose Gas Formula Sheet	Low-energy density of states and the thermodynamic limit.

A formula for mean occupation is not a statement that each mode has a definite particle number. Likewise, agreement of mean thermodynamic quantities between ensembles does not imply identical number fluctuations. Return to Quantum Statistics or Quantum Statistical Mechanics when that distinction affects the question.

Hamiltonians, operators, and model choices

Your question	Where to look	Check before substituting
What is the standard form of a gas, hopping, pairing, or impurity Hamiltonian?	Common Many-Body Hamiltonians	Particle statistics, sector, boundaries, and whether the expression is $H$ or $H-\mu N$.
How are spin exchange and field terms normalized?	Common Spin Hamiltonians	Spin versus Pauli operators and bond counting.
Which operator identity can simplify this expression?	Operator Identities	Commutator versus anticommutator, ordering, and operator domains.
Where can I learn about a named model?	Model-to-Volume Cross-Link Index	Whether you need a derivation, a numerical example, or an application.

For a more leisurely comparison, read the Model Encyclopedia. A name such as “Hubbard model” still needs a geometry, filling, boundary conditions, and coupling convention before it specifies a calculation.

Correlations, response, and specialist terms

Your question	Where to look	Check before substituting
Which correlation function is being used?	Correlation Function Definitions	Operator order, connected subtraction, state, and transform.
How does a weak source change an observable?	Linear Response Formula Sheet	Source sign, contact terms, and the order of static or uniform limits.
Which imaginary-time frequencies occur?	Matsubara Frequency Table	Operator parity and whether the frequency variable has energy or inverse-time units.
What do critical exponents describe?	Critical Exponent Glossary	Observable, spatial dimension, and scaling assumptions.
Which entanglement quantity fits this state?	Many-Body Entanglement Glossary	Pure versus mixed state, subsystem, and logarithm base.

The Matsubara grid follows the operator’s thermal boundary condition, not merely the name of the particles. For example, an even density operator made from fermion fields has a bosonic grid. See Finite-Temperature Methods for the state and trace assumptions behind these statements.

Why two static responses can differ

Here is a useful warning with a concrete calculation. Let a finite equilibrium system have Hamiltonian $H$, and perturb it by $-fB$, where $[H,B]=0$. The isolated retarded response of $B$ to that source vanishes: $B(t)=B$, so the commutator $[B(t),B]$ is zero. An isolated system initially diagonal in their common basis keeps the same populations.

Now allow the system to equilibrate at a fixed temperature after the field changes. In the Gibbs state,

$$\left.\frac{\partial\langle B\rangle_f}{\partial f}\right|_{f=0} =\beta\bigl(\langle B^2\rangle-\langle B\rangle^2\bigr), \qquad \beta=\frac{1}{k_{\mathrm B}T}.$$

This can be nonzero because equilibration changes the populations. Thus a thermodynamic derivative and an isolated zero-frequency commutator response can answer different questions. For spatially varying sources, the order of momentum, frequency, and large-system limits also matters. The response chapter develops those distinctions.

Practice with particles and modes

Use Identical-Particle Exercises to practice exchange symmetry, normalization, and spin. Continue with Fock-Space Exercises for occupations, ladder factors, fermionic signs, and changes of mode basis. Both collections include solutions so you can compare the reasoning, not just a final expression.

Exercises

1. A thermal two-level response. In the example above, take $H=0$ and $B=\sigma_z$. Compute the equilibrium mean of $B$ for a source $-fB$, and compare its zero-field slope with the isolated response.

Solution

The energies of the $B=+1$ and $B=-1$ states are $-f$ and $+f$, respectively. Their Boltzmann weights give

$$\langle B\rangle_f=\frac{e^{\beta f}-e^{-\beta f}}{e^{\beta f}+e^{-\beta f}} =\tanh(\beta f).$$

The slope at zero field is $\beta$, equal to $\beta\operatorname{Var}(B)$ in the unpolarized state. An isolated initially unpolarized state stays unpolarized under this commuting Hamiltonian, giving zero response. The difference is the ability to redistribute populations.

2. Check a spin coefficient. A reference writes $J\,\boldsymbol\sigma_1\cdot\boldsymbol\sigma_2$. Your calculation uses dimensionless $\mathbf S=\boldsymbol\sigma/2$. What coefficient should multiply $\mathbf S_1\cdot\mathbf S_2$?

Solution

The coefficient is $4J$, since each Pauli operator contributes a factor of two. A term linear in spin would acquire only one factor of two. This is why an exchange-to-field ratio must be translated term by term.

References

• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015), for ensembles, correlations, and model conventions.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover Publications (2003; reprint of the 1971 edition), for thermal correlation functions and linear response.
• S. Sachdev, Quantum Phase Transitions, 2nd ed., Cambridge University Press (2011), for critical phenomena and their limiting procedures.