Finite-Temperature Methods

A small thermal system may need only a direct Gibbs trace. An equilibrium propagator may be easier to calculate in imaginary time, while a real-frequency spectrum may require a spectral representation and careful reconstruction from limited data. A quench starting from a thermal state instead needs real-time evolution.

This chapter explains how to choose among these methods and translate their conventions. The Finite-Temperature QM Overview introduces their common structure.

Required background. Enter through Quantum Statistical Mechanics able to declare the ensemble, trace domain, Gibbs operator, partition function, held-fixed controls, and whether the relevant generator is $H$ or $H-\mu N$.

Helpful background. Correlation Functions and Linear Response supplies real-time correlator and response conventions. Field operators and Fock space, Fourier series, complex analysis, Euclidean path integrals, and numerical inverse problems are branch-specific preparation rather than universal prerequisites.

Imaginary time and the thermal trace

Write a grand-canonical state as $\rho=e^{-\beta K}/\Xi$, where $K=H-\mu N$. If imaginary time $\tau$ has units of inverse energy, define $A(\tau)=e^{\tau K}Ae^{-\tau K}$ and use the interval $0\leq\tau<\beta$. An alternative time-valued coordinate has interval $\beta\hbar$ and exponent $\tau K/\hbar$; the two conventions must not be mixed.

Thermal cyclicity, combined with the chosen time ordering, leads to periodic even-operator correlators and antiperiodic odd-fermion propagators. In the inverse-energy convention their Fourier grids have energy units:

$$\nu_n^{B}=\frac{2\pi n}{\beta},\qquad \nu_n^{F}=\frac{(2n+1)\pi}{\beta}.$$

Divide by $\hbar$ to obtain angular frequencies. A density operator made from two fermion fields is even and normally uses the bosonic grid. Constituent statistics alone therefore does not identify the grid.

These statements require a thermal trace compatible with the operator algebra. For a trace restricted to one particle-number sector, a single creation or annihilation field leaves that sector; the usual one-function antiperiodicity cannot simply be assumed. Number-conserving observables do not have that particular obstruction. The thermal Green-function and KMS pages explain the conditions.

Also distinguish equilibrium weighting from laboratory dynamics. Evolution with $K$ moves the chemical potential into the frequency convention; evolving with $H$ instead leads to the corresponding charge-dependent thermal twist. Applying both shifts without derivation counts the chemical potential twice. Imaginary-time propagation is nonunitary and does not describe the physical history of a quench.

Choose a thermal method

The KMS and path-integral branches can be studied without completing every Matsubara calculation.

1. Establish the common structure. Begin with Finite-Temperature QM Overview. It connects the Gibbs generator, compact imaginary time, thermal ordering, frequency grids, spectral data, path integrals, and real-time boundaries with links to the detailed treatments.
2. Learn the semigroup and boundary distinction. Imaginary Time separates nonunitary imaginary-time evolution, open projection kernels, and the closed thermal trace. Imaginary time is not laboratory time.
3. Enter the compact-frequency branch. Matsubara Formalism Preview develops graded ordering and Fourier analysis on the thermal circle. Bosonic and Fermionic Matsubara Frequencies is the convention and arithmetic reference for the two grids.
4. Build a thermal propagator only after declaring its channel. Thermal Green Functions treats imaginary-time ordering, equal-time jumps, free benchmarks, and interacting two-point functions. It is not automatically a causal response function.
5. Connect representations before continuing data. Spectral Representation identifies the common analytic object linking Euclidean, Matsubara, spectral, and retarded forms, including static bosonic contributions. Analytic Continuation then distinguishes exact boundary-value identities from ill-conditioned inference using finite uncertain data.
6. Choose the coordinate path branch for traced configuration histories. Path Integrals for Statistical Mechanics develops the closed thermal trace, time slicing, permutation closure, ring-polymer picture, and regulator checks.
7. Choose the Fock-space path branch only with its algebra prepared. Coherent-State Path Integrals Preview adds bosonic coherent states, fermionic Grassmann variables, normal-symbol and ordering cautions, and the connection to field theory.
8. Use KMS as the deeper equilibrium test. After the overview and time-dependent correlations, KMS Condition Preview explains thermal cyclicity, analyticity, detailed balance, and equilibrium beyond finite trace-class density operators. Stationarity alone is weaker.
9. Switch formalisms for an initial-value question. Real-Time Thermal Dynamics Preview starts from a thermal state but organizes quenches, drives, greater and lesser functions, and closed time paths. Analytic continuation of equilibrium data is not nonequilibrium evolution.

Example: three questions at finite temperature

For a finite-temperature fermion model, the mean particle number is a direct grand-canonical trace and needs no Matsubara machinery. Its equilibrium addition and removal spectrum can be organized through an odd, antiperiodic thermal Green function, the fermionic frequency grid, and a spectral representation; if only noisy imaginary-axis data are available, continuation becomes a regularized inverse problem with limited resolution. A sudden interaction quench from the same thermal state is instead an initial-value problem requiring real-time contour evolution. The initial density operator is shared, but the computed object and warranted inference differ.

Continue with equilibrium or dynamics

Quantum Statistical Mechanics supplies the ensemble and thermodynamic derivatives. Correlations and Response defines the real-time observables that thermal methods help calculate. Quantum Monte Carlo connects thermal representations to numerical sampling.

For a drive or quench, continue to Nonequilibrium Dynamics. For collective thermal singularities, see Finite-Temperature Phase Transitions. The finite-temperature field-theory bridge develops the next conceptual steps.

Common pitfalls

“Imaginary time is physical time with $t=-i\tau$.” It is a nonunitary equilibrium representation or projection coordinate, not laboratory evolution.

“Fermions always use fermionic frequencies.” Grid parity belongs to the operator channel. An even density or spin correlator on a fermionic system normally uses bosonic frequencies.

“Chemical potential can be inserted twice for safety.” Evolving with $H-\mu N$ and also applying the equivalent thermal twist double-counts $\mu$ unless a deliberately different convention is derived.

“A thermal Green function is a retarded response.” Imaginary-time ordering and causal commutator support define different objects connected only through the stated spectral and analytic structure.

“Replace $i\omega_n$ by $\omega+i0^+$ in the data.” The symbolic rule presupposes a known analytic function. Reconstructing it from finite noisy values is ill conditioned and requires priors, covariance, validation, and honest resolution limits.

“A smooth continued peak has a measured lifetime.” Regularization and plotting broadening create widths. Intrinsic linewidth requires stability against resolution, size, method, and sum-rule checks.

“The continuum path integral defines its own measure.” It is shorthand for a regulated finite-slice construction whose ordering, Jacobian, normalization, boundary data, and limiting procedure matter.

“Stationary means KMS equilibrium.” Stationarity lacks the thermal analyticity and cyclicity needed for detailed balance and equilibrium fluctuation relations.

“Imaginary-time methods solve a quench.” A driven initial-value problem needs real-time evolution; equilibrium continuation does not create its history.

Exercises

Exercise 1: Route four thermal problems

Choose the minimum route for (a) the exact free energy of a finite spin system, (b) a fermion addition spectrum inferred from quantum Monte Carlo data, (c) a coordinate-space thermal path integral, and (d) a sudden quench from a Gibbs state.

Solution

For (a), remain in Quantum Statistical Mechanics and evaluate the direct trace. For (b), use Overview → Imaginary Time → Matsubara Formalism and Frequencies → Thermal Green Functions → Spectral Representation → Analytic Continuation, with covariance and resolution tests. For (c), use Overview → Imaginary Time → Path Integrals for Statistical Mechanics and keep the trace boundary condition and time-slice regulator explicit. For (d), use Overview → KMS to define the initial equilibrium state, then Real-Time Thermal Dynamics; imaginary-time continuation alone cannot evolve the quench.

Exercise 2: Repair a convention chain

Repair this proposal: “For grand-canonical fermions, evolve with $H-\mu N$, add the chemical-potential twist again, impose periodicity because the measured density is made of fermions, and read the width of a continued peak as its lifetime.”

Solution

Choose either $H-\mu N$ evolution or the equivalent $H$-evolution twist and carry that convention consistently. The density is parity even, so its thermal correlator normally has bosonic periodicity and a bosonic Matsubara grid. A continued width is only a regularization-dependent inference until covariance, resolution, finite-size, sum-rule, and model tests establish that an intrinsic linewidth is actually resolved.

References

• A. Altland and B. Simons, Condensed Matter Field Theory, 2nd ed., Cambridge University Press (2010).
• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015).
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003).
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000).
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, Westview Press (1998).