Finite-Temperature QM Overview

Finite-temperature quantum mechanics is the equilibrium framework in which a Gibbs operator supplies statistical weights, imaginary time organizes operator products, and thermal correlation functions connect thermodynamics to spectra and response.

The central structural identity is

$$e^{-\beta\mathcal K} = e^{-\mathcal K(\beta\hbar)/\hbar}.$$

Here $\beta=(k_{\mathrm B}T)^{-1}$ and $\mathcal K$ is the generator appropriate to the ensemble. For a canonical ensemble,

$$\mathcal K=H,$$

whereas for a grand-canonical ensemble with a conserved particle number,

$$\mathcal K=H-\mu N, \qquad [H,N]=0.$$

The Gibbs weight therefore has the algebraic form of evolution through an imaginary-time interval of length $\beta\hbar$. Taking a trace closes that interval into a thermal circle. This observation leads to imaginary-time ordering, the Kubo–Martin–Schwinger relation, bosonic and fermionic boundary conditions, discrete Matsubara frequencies, equilibrium path integrals, and a controlled bridge to real-frequency response.

The resemblance to time evolution is powerful but must be interpreted correctly. Imaginary time is a calculational coordinate for equilibrium quantum statistics. It is not laboratory time, and evolution by $e^{-\tau\mathcal K/\hbar}$ is not unitary dynamics.

Helpful background. Thermal density operators supplies Gibbs states and trace averages; partition functions supplies thermodynamic derivatives. Basic operator evolution and Fourier series are used below.

Gibbs states and imaginary-time conventions

Assume positive temperature and a self-adjoint generator $\mathcal K$ for which $e^{-\beta\mathcal K}$ is trace class. In the thermodynamic limit, local equilibrium correlations can exist even when a global normalized Gibbs density matrix does not; the finite-volume construction must then be taken to its limit.

The trace cyclicity used below applies to operators acting within the traced Hilbert space. A fixed-$N$ canonical trace therefore supports the stated KMS argument directly for number-preserving observables. For creation and annihilation fields, we use the grand-canonical trace over Fock space. Moving a number-changing field through a fixed-$N$ projector changes the sector, so the usual single-particle thermal boundary condition cannot simply be imported into that restricted trace.

This chapter keeps $\hbar$ explicit. Imaginary time $\tau$ has dimensions of time and lies on

$$0 \leq \tau \lt \beta\hbar.$$

Many texts instead use a variable with dimensions of inverse energy and write $0\leq\tau\lt\beta$. The two conventions differ by a factor of $\hbar$. A formula should never mix them silently.

We use:

Symbol	Meaning	Units
$\beta$	inverse temperature $(k_{\mathrm B}T)^{-1}$	inverse energy
$\mathcal K$	ensemble generator	energy
$\tau$	imaginary time	time
$\omega_n$	angular Matsubara frequency	inverse time
$\nu_n=\hbar\omega_n$	Matsubara energy	energy
$\mathcal T_\tau$	imaginary-time ordering	none

For a canonical calculation, replace $\mathcal K$ by $H$. For a grand-canonical calculation, using $\mathcal K=H-\mu N$ consistently is essential: it is the source of the familiar one-particle energy

$$\xi_{\mathbf k} = \epsilon_{\mathbf k}-\mu.$$

From Gibbs Weight to Imaginary Time

Real-time evolution generated by a time-independent Hamiltonian is

$$U(t) = e^{-iHt/\hbar}.$$

Formally substituting $t=-i\tau$ gives

$$U(-i\tau) = e^{-H\tau/\hbar}.$$

At the thermal interval $\tau=\beta\hbar$,

$$U(-i\beta\hbar) = e^{-\beta H}.$$

This is the unnormalized canonical Gibbs operator. More generally,

$$\rho_{\beta} = \frac{e^{-\beta\mathcal K}}{\mathcal Z}, \qquad \mathcal Z = \operatorname{Tr}e^{-\beta\mathcal K}.$$

Spectral meaning

Let

$$\mathcal K|n\rangle = \kappa_n|n\rangle.$$

Then

$$e^{-\tau\mathcal K/\hbar}|n\rangle = e^{-\tau\kappa_n/\hbar}|n\rangle.$$

Imaginary-time evolution exponentially suppresses larger $\kappa_n$ relative to smaller $\kappa_n$. It is therefore nonunitary. In particular,

$$\left( e^{-\tau\mathcal K/\hbar} \right)^\dagger e^{-\tau\mathcal K/\hbar} = e^{-2\tau\mathcal K/\hbar},$$

which is not generally the identity.

The same suppression explains both thermal weighting and ground-state projection:

• setting $\tau=\beta\hbar$ produces Boltzmann weights;
• taking a long open interval $\tau\to\infty$ selects the lowest eigenspace with nonzero trial-state overlap, provided it is a normalizable isolated eigenspace. A lower spectral bound without a ground-state eigenvector is not enough for this projection claim.

Thermal traces and ground-state projection use the same semigroup, but their boundary conditions and physical questions differ.

Thermal averages

For an observable $A$,

$$\langle A\rangle_{\beta} = \frac{ \operatorname{Tr} \left( e^{-\beta\mathcal K}A \right) }{ \mathcal Z }.$$

For any $A$ with a well-defined expectation, the one-point average is $\sum_n p_n\langle n|A|n\rangle$ in an eigenbasis of $\mathcal K$, whether or not $A$ commutes with it. Noncommutativity becomes visible in products such as $\langle A(\tau)B\rangle$: their off-diagonal matrix elements carry transition energies, which an average of classical values would miss.

Why the Trace Creates a Thermal Circle

For a particle with coordinate basis $|q\rangle$,

$$Z = \int dq\, \langle q| e^{-\beta H} |q\rangle.$$

The same coordinate appears at both ends because of the trace. After inserting many resolutions of the identity, the matrix element becomes a sum over paths with

$$q(\beta\hbar) = q(0).$$

The imaginary-time interval is therefore compact: its endpoints are identified.

The thermal trace identifies the initial and final state. Imaginary time therefore closes into a circle of circumference $\beta\hbar$. Bosonic and fermionic fields have different seam signs and hence different Fourier grids. Their correlations constrain an equilibrium spectrum.

The circle is not an extra spatial dimension in the original quantum system. It is a representation of the thermal trace. In field theory the same construction becomes a Euclidean spacetime with compact imaginary time, but additional issues such as renormalization, gauge symmetry, and Lorentzian reconstruction belong to the field-theory treatment.

Imaginary-Time Operators

Define

$$A(\tau) = e^{\tau\mathcal K/\hbar} A e^{-\tau\mathcal K/\hbar}.$$

Differentiation gives the imaginary-time Heisenberg equation

$$\hbar \frac{\partial A(\tau)}{\partial\tau} = \left[ \mathcal K,A(\tau) \right].$$

Compare this with real time:

$$i\hbar \frac{\partial A(t)}{\partial t} = \left[ A(t),H \right].$$

The missing factor of $i$ is responsible for exponential rather than oscillatory behavior.

For two operators, define the unordered imaginary-time correlator

$$C_{AB}(\tau) = \left\langle A(\tau)B(0) \right\rangle_{\beta}.$$

In an eigenbasis of $\mathcal K$,

$$C_{AB}(\tau) = \frac1{\mathcal Z} \sum_{m,n} e^{-\beta\kappa_m} e^{\tau(\kappa_m-\kappa_n)/\hbar} \times \langle m|A|n\rangle \langle n|B|m\rangle.$$

This formula already contains the main finite-temperature ingredients:

• thermal weights $e^{-\beta\kappa_m}$;
• excitation-energy differences $\kappa_n-\kappa_m$;
• matrix elements selecting the observable channel;
• exponential decay or growth along the finite imaginary-time interval.

A growing exponential on this finite interval is accompanied by the appropriate suppressed thermal population. It is not an instability; the two-level example below shows how the growing and decaying branches fit together.

KMS Cyclicity

The trace identity behind equilibrium periodicity is especially important. Since

$$A(\tau+\beta\hbar) = e^{\beta\mathcal K} A(\tau) e^{-\beta\mathcal K},$$

one obtains

$$\begin{aligned} \left\langle A(\tau+\beta\hbar)B \right\rangle_{\beta} &= \frac1{\mathcal Z} \operatorname{Tr} \left[ A(\tau)e^{-\beta\mathcal K}B \right] \\ &= \left\langle BA(\tau) \right\rangle_{\beta}. \end{aligned}$$

This is an imaginary-time form of the Kubo–Martin–Schwinger condition. It exchanges the operator order when one insertion is moved once around the thermal circle.

Two distinctions matter:

1. the trace identity itself does not insert a fermionic minus sign;
2. antiperiodicity of a fermionic Green function arises when KMS cyclicity is combined with graded imaginary-time ordering.

Confusing these steps is a common source of sign errors.

Imaginary-Time Ordering

For operators of definite fermion parity, define

$$\eta = \begin{cases} +1, & \text{bosonic or even operators},\\ -1, & \text{fermionic or odd operators}. \end{cases}$$

For two operators of the same parity,

$$\mathcal T_\tau A(\tau)B(\tau') = \theta(\tau-\tau') A(\tau)B(\tau') + \eta\, \theta(\tau'-\tau) B(\tau')A(\tau).$$

The sign $\eta=-1$ records one exchange of odd operators. A common Green-function convention is

$$\mathcal G_{AB}(\tau-\tau') = - \left\langle \mathcal T_\tau A(\tau)B(\tau') \right\rangle_{\beta}.$$

The overall minus sign is conventional and must be checked for each channel. Response functions, coordinate correlators, spin correlators, and single-particle propagators do not all use identical prefactors.

KMS cyclicity and graded ordering imply

$$\mathcal G_{AB} (\tau+\beta\hbar) = \eta\, \mathcal G_{AB}(\tau).$$

Thus:

$$\begin{aligned} \mathcal G_{\mathrm B} (\tau+\beta\hbar) &= \mathcal G_{\mathrm B}(\tau), \\ \mathcal G_{\mathrm F} (\tau+\beta\hbar) &= - \mathcal G_{\mathrm F}(\tau). \end{aligned}$$

Composite operators inherit the parity of the full operator. A fermion density $\psi^\dagger\psi$ is even and therefore has bosonic thermal boundary conditions, despite being built from fermionic fields.

Matsubara Frequencies

A function on a compact interval has a Fourier series. Periodicity or antiperiodicity quantizes its allowed frequencies.

Bosonic modes

For

$$F(\tau+\beta\hbar) = F(\tau),$$

the angular Matsubara frequencies are

$$\omega_n^{\mathrm B} = \frac{ 2\pi n }{ \beta\hbar }, \qquad n\in\mathbb Z.$$

Fermionic modes

For

$$F(\tau+\beta\hbar) = -F(\tau),$$

the angular Matsubara frequencies are

$$\omega_n^{\mathrm F} = \frac{ (2n+1)\pi }{ \beta\hbar }, \qquad n\in\mathbb Z.$$

In energy units,

$$\nu_n = \hbar\omega_n,$$

so that

$$\begin{aligned} \nu_n^{\mathrm B} &= \frac{2\pi n}{\beta}, \\ \nu_n^{\mathrm F} &= \frac{(2n+1)\pi}{\beta}. \end{aligned}$$

The bosonic spectrum contains a zero mode at $n=0$. The fermionic spectrum does not. This difference has major consequences for long-distance finite-temperature physics.

Fourier pair

Using Matsubara energies, a convenient convention is

$$\mathcal G(i\nu_n) = \frac1\hbar \int_0^{\beta\hbar} d\tau\, e^{i\nu_n\tau/\hbar} \mathcal G(\tau),$$

with inverse

$$\mathcal G(\tau) = \frac1\beta \sum_n e^{-i\nu_n\tau/\hbar} \mathcal G(i\nu_n).$$

The sum runs over the bosonic or fermionic sequence appropriate to the operator channel. In conventions where $\hbar=k_{\mathrm B}=1$, the distinction among $\tau$, $\beta$, angular frequency, and energy is hidden. Restoring units after a calculation requires knowing which convention was used.

Five Equivalent Windows on Equilibrium

Finite-temperature calculations repeatedly move among five representations.

Representation	Basic object	Especially useful for	Main caution
operator trace	$\operatorname{Tr}(e^{-\beta\mathcal K}\cdots)$	exact identities, symmetries, finite spectra	Hilbert-space dimension grows rapidly
imaginary time	$\mathcal G(\tau)$	equilibrium ordering and decay	$\tau$ is not real time
Matsubara frequency	$\mathcal G(i\nu_n)$	convolutions, diagrams, frequency sums	frequencies are discrete and imaginary
spectral form	$\rho(E)$ or $A(E)$	excitations, sum rules, real-frequency bridge	normalization and signs are channel dependent
Euclidean path integral	$\int\mathcal D\phi\,e^{-S_{\mathrm E}/\hbar}$	collective fields, saddle points, Monte Carlo	measure, discretization, and sign problems matter

These are not five different theories. Under their assumptions, they are reorganizations of the same equilibrium trace.

Spectral Representation and Real-Frequency Bridge

The eigenstate sum shows that imaginary-time correlators are built from transition energies. A spectral density packages those transitions into a function of real energy.

For many standard channels, one can define an analytic function of a complex energy $z$,

$$\mathcal G(z) = \int_{-\infty}^{\infty} dE\, \frac{ \rho(E) }{ z-E }.$$

The spectral density and overall sign depend on the channel. The displayed Cauchy transform is the convention for $\mathcal G=-\langle\mathcal T_\tau AB\rangle$ and the corresponding retarded Green function. The physical susceptibility with $+i\theta\langle[A(t),B]\rangle/\hbar$ for a source $-fB$ has the opposite sign. Bosonic correlators can also contain an exact static contribution proportional to $\delta_{n0}$ that must be supplied separately from this dynamical spectral transform.

With the same channel definition on both sides, Matsubara data sample

$$z=i\nu_n,$$

Positive Matsubara frequencies lie in the upper half-plane. Continuing the analytic function from that domain gives the retarded boundary value at

$$z=E+i0^+.$$

This motivates the formal continuation

$$i\nu_n \longrightarrow E+i0^+.$$

The arrow is shorthand for evaluating the same analytic function at a different boundary. It is not a substitution rule for an arbitrary finite table of Matsubara values.

Imaginary time is a smoothed spectral transform

For the normal fermionic single-particle convention used in Green Functions in Many-Body QM, one has for $0\lt\tau\lt\beta\hbar$

$$\mathcal G(\tau) = - \int_{-\infty}^{\infty} dE\, \frac{ e^{-E\tau/\hbar} }{ 1+e^{-\beta E} } A(E).$$

The kernel smooths the spectral function. Narrow peaks and nearby thresholds can produce very similar imaginary-time data, especially after statistical noise and finite sampling are included.

Exact Matsubara values at all frequencies, together with appropriate analyticity and asymptotic information, can determine the real-frequency function in principle. Numerical continuation from finitely many noisy values is ill posed. Analytic Continuation develops this inverse problem in detail. A useful reconstruction should show which features remain stable when covariance, frequency range, and regularization are varied, and test resolution on synthetic data. The spectral-representation derivation explains channel-dependent signs and static terms; spectral functions explains how line shapes relate to measurements.

Equilibrium Path Integrals

The operator exponential can be divided into $M$ short factors:

$$e^{-\beta H} = \left( e^{-\Delta\tau H/\hbar} \right)^M, \qquad \Delta\tau = \frac{\beta\hbar}{M}.$$

Inserting complete sets between the factors and taking $M\to\infty$ gives, for a particle with

$$H = \frac{p^2}{2m} +V(q),$$

the formal Euclidean path integral

$$Z = \int_{q(\beta\hbar)=q(0)} \mathcal Dq\, e^{-S_{\mathrm E}[q]/\hbar},$$

where

$$S_{\mathrm E}[q] = \int_0^{\beta\hbar} d\tau\, \left[ \frac{m}{2} \left( \frac{dq}{d\tau} \right)^2 +V(q) \right].$$

The oscillatory real-time weight $e^{iS/\hbar}$ has become the decaying Euclidean weight $e^{-S_{\mathrm E}/\hbar}$. This often makes equilibrium path integrals suitable for saddle-point expansions and stochastic sampling.

Several qualifications are essential:

• the trace enforces a closed imaginary-time boundary condition;
• a time-sliced definition is needed before the continuum notation is meaningful;
• operator ordering can generate discretization corrections;
• first-quantized identical-particle traces include permutation sectors;
• bosonic coherent-state fields are periodic;
• fermionic coherent-state fields are Grassmann-valued and antiperiodic;
• a Euclidean weight need not be real and nonnegative, so Monte Carlo can face a sign or phase problem.

The properly defined path integral represents the same trace. A finite time step, truncated saddle-point or perturbative expansion, and variational ansatz introduce approximation errors; finite numerical sampling adds statistical uncertainty. A quadratic grand-canonical Gibbs state is Gaussian, but projection onto a fixed particle number generally destroys Wick factorization. The distinction matters when choosing free reference states for thermal diagrams.

Worked Example: A Two-Level Thermal Correlator

Consider

$$H = \frac{\Delta}{2}\sigma_z, \qquad \Delta>0,$$

with

$$Z = 2\cosh \left( \frac{\beta\Delta}{2} \right).$$

Take $A=B=\sigma_x$. Since $\sigma_x$ connects the two energy eigenstates, the eigenbasis formula gives, for $0\leq\tau\leq\beta\hbar$,

$$\begin{aligned} C_{xx}(\tau) &= \left\langle \sigma_x(\tau)\sigma_x(0) \right\rangle_\beta \\ &= \frac{ e^{\beta\Delta/2} e^{-\Delta\tau/\hbar} +e^{-\beta\Delta/2} e^{\Delta\tau/\hbar} }{ 2\cosh(\beta\Delta/2) }. \end{aligned}$$

Equivalently,

$$C_{xx}(\tau) = \frac{ \cosh \left[ \Delta \left( \frac{\beta}{2} -\frac{\tau}{\hbar} \right) \right] }{ \cosh(\beta\Delta/2) }.$$

This compact expression makes several checks immediate:

$$\begin{aligned} C_{xx}(0) &= 1, \\ C_{xx}(\beta\hbar) &= 1, \\ C_{xx}(\beta\hbar-\tau) &= C_{xx}(\tau). \end{aligned}$$

Because $\sigma_x$ is an even observable in this spin problem, the thermal correlator is bosonic. Its Matsubara energies are $\nu_n=2\pi n/\beta$. The transform is

$$C_{xx}(i\nu_n) = \frac{ 2\Delta \tanh(\beta\Delta/2) }{ \Delta^2+\nu_n^2 }.$$

At low temperature and fixed positive $\tau$,

$$C_{xx}(\tau) \longrightarrow e^{-\Delta\tau/\hbar}.$$

The imaginary-time decay rate directly reveals the excitation gap $\Delta$. At finite temperature, propagation can begin from either thermally occupied level, producing the two exponentials in the exact result.

Worked Benchmark: One Fermionic Mode

For

$$\mathcal K = \xi c^\dagger c, \qquad \xi = \epsilon-\mu,$$

the Fermi occupation is

$$f(\xi) = \frac1{ e^{\beta\xi}+1 }.$$

For $0<\tau<\beta\hbar$, the exact branch is

$$\mathcal G(\tau) = - \left[ 1-f(\xi) \right] e^{-\xi\tau/\hbar}.$$

For $-\beta\hbar<\tau<0$,

$$\mathcal G(\tau) = f(\xi) e^{-\xi\tau/\hbar}.$$

They glue antiperiodically and transform to

$$\mathcal G(i\nu_n) = \frac1{ i\nu_n-\xi }.$$

Thermal Green Functions derives the one-sided equal-time limits and contact jump, and compares this result with a free boson.

Choosing a Finite-Temperature Method

The observable and desired output should determine the representation.

Question	Natural starting object	Typical method
free energy or equation of state	$\mathcal Z$ or $\ln\mathcal Z$	exact spectrum, linked expansion, path integral
equal-time expectation	$\operatorname{Tr}(\rho_\beta A)$	operator algebra, diagonalization, Monte Carlo estimator
static susceptibility	imaginary-time integrated connected correlation	derivatives of $\ln\mathcal Z$; covariance alone only in the commuting case
excitation spectrum	real-frequency spectral function	Lehmann representation, analytic continuation, real-time method
perturbative equilibrium correction	Matsubara Green functions	frequency sums and diagrams
strongly coupled equilibrium observable	Euclidean path integral or tensor-network thermal state	Monte Carlo, purification, typicality, specialized solver
transient or driven dynamics	contour-ordered real-time correlator	Keldysh or other nonequilibrium method

No representation is universally superior. Exact diagonalization is transparent but size limited. Matsubara perturbation theory is systematic near a controlled reference but can fail at strong coupling or near infrared singularities. Euclidean Monte Carlo can be nonperturbative but may suffer finite-size, discretization, autocorrelation, and sign problems. Analytic continuation may dominate the uncertainty even when imaginary-time data are precise.

Temperature and Scale Hierarchies

Temperature introduces the energy scale

$$k_{\mathrm B}T = \frac1\beta.$$

Its significance depends on comparison with gaps, bandwidths, interaction energies, finite-size spacings, and probe frequencies.

Low temperature

If a finite system has a unique ground state and gap $\Delta$, then thermal corrections to many equilibrium quantities are suppressed by

$$e^{-\beta\Delta}.$$

The thermal circle becomes long:

$$\beta\hbar \to \infty.$$

Imaginary-time correlators can then display a broad window of ground-state exponential decay. Degenerate ground spaces, gapless systems, and thermodynamic limits require more care.

High temperature

As $\beta\to0^+$, the thermal circle shrinks. A high-temperature expansion organizes powers of $\beta H$ only when the relevant operator products and traces are controlled. In an infinite-dimensional Hilbert space, the formal state proportional to the identity may not be normalizable.

Matsubara spacing

Adjacent bosonic Matsubara energies differ by

$$\Delta\nu = \frac{2\pi}{\beta} = 2\pi k_{\mathrm B}T.$$

At lower temperature the grid becomes denser and approaches a continuum. At higher temperature nonzero modes are widely separated, but the bosonic zero mode remains. Long-distance thermal behavior can therefore be dominated by static bosonic fluctuations.

Finite size versus temperature

Let $\delta_L$ be a characteristic finite-size level spacing. Then:

• $k_{\mathrm B}T\ll\delta_L$ resolves individual low-lying levels;
• $k_{\mathrm B}T\gg\delta_L$ thermally averages over many levels;
• taking $L\to\infty$ before $T\to0$ can differ from reversing the limits.

The order of limits is part of the physical statement, especially near phase transitions and in systems with conserved quantities.

Equilibrium does not require weak interactions

The Gibbs trace, KMS identity, and spectral representations do not require quasiparticles or perturbatively weak coupling. Those are assumptions of particular approximation methods. Conversely, specifying a Gibbs state does not show that an isolated preparation dynamically reaches it, or that its spectrum can be recovered stably from noisy Euclidean data.

Equilibrium Versus Nonequilibrium

The thermal circle encodes a stationary Gibbs ensemble. It is well suited to:

• thermodynamics;
• equilibrium correlations;
• static response;
• imaginary-frequency perturbation theory;
• equilibrium spectral constraints.

It is not sufficient for a general quench, drive, transient, or current-carrying steady state. Nonequilibrium calculations typically require an initial density operator plus forward and backward real-time evolution. Retarded response near equilibrium can be obtained from equilibrium correlations, but nonlinear and far-from-equilibrium dynamics require additional information.

The Kubo formula derives near-equilibrium response. Real-time thermal dynamics explains how a thermal initial state enters a forward-and-backward evolution contour.

Common pitfalls

A field is not a periodic operator. KMS relates correlation functions with reversed operator order; it does not assert $A(\tau+\beta\hbar)=\pm A(\tau)$ as an operator identity. Periodicity or antiperiodicity belongs to the appropriately ordered thermal correlator.

A low Matsubara frequency is not a measured excitation. The frequency grid is fixed by temperature and operator parity. Excitation energies occur in the spectral density; extracting them from that grid is an additional inverse problem.

Static response is not always an equal-time variance. For noncommuting source and Hamiltonian, the equilibrium derivative involves an imaginary-time integral. An exactly conserved fluctuation can contribute to the zero Matsubara component even when its isolated retarded response vanishes.

Continue with a calculation

• Imaginary time develops the operator semigroup and heat equation.
• Matsubara formalism works through frequency sums and thermal perturbation theory.
• Statistical-mechanics path integrals constructs the closed thermal trace and permutation sectors.
• Coherent-state path integrals constructs bosonic and fermionic Fock-space traces.
• Finite-temperature phase transitions examines nonanalyticity and finite-size rounding.

Exercises

1. Derive thermal cyclicity

Starting from

$$A(\tau) = e^{\tau\mathcal K/\hbar} A e^{-\tau\mathcal K/\hbar},$$

show that

$$\left\langle A(\tau+\beta\hbar)B \right\rangle_\beta = \left\langle BA(\tau) \right\rangle_\beta.$$

Identify the only trace property used.

Solution

First,

$$A(\tau+\beta\hbar) = e^{\beta\mathcal K} A(\tau) e^{-\beta\mathcal K}.$$

Multiplying the shifted operator by the Gibbs factor gives

$$e^{-\beta\mathcal K} A(\tau+\beta\hbar) B = A(\tau) e^{-\beta\mathcal K} B.$$

Therefore

$$\left\langle A(\tau+\beta\hbar)B \right\rangle_\beta = \frac1{\mathcal Z} \operatorname{Tr} \left[ A(\tau) e^{-\beta\mathcal K} B \right].$$

Cyclicity of the trace gives

$$\operatorname{Tr} \left[ A(\tau) e^{-\beta\mathcal K} B \right] = \operatorname{Tr} \left[ e^{-\beta\mathcal K} B A(\tau) \right],$$

which proves the result. No fermionic sign was introduced; such a sign enters only when odd operators are exchanged by graded time ordering.

2. Quantize the thermal frequencies

Suppose

$$F(\tau+\beta\hbar) = \eta F(\tau), \qquad \eta=\pm1.$$

For a mode $F(\tau)\propto e^{-i\omega\tau}$, derive the allowed frequencies for both values of $\eta$.

Solution

The boundary condition requires

$$e^{-i\omega\beta\hbar} = \eta.$$

For $\eta=+1$,

$$\omega\beta\hbar = 2\pi n,$$

so

$$\omega_n^{\mathrm B} = \frac{2\pi n}{\beta\hbar}.$$

For $\eta=-1$,

$$\omega\beta\hbar = (2n+1)\pi,$$

so

$$\omega_n^{\mathrm F} = \frac{(2n+1)\pi}{\beta\hbar}.$$

The result depends only on the thermal boundary condition, not on whether the underlying particles in the Hamiltonian are bosons or fermions.

3. Check the two-level transform

Starting from

$$C_{xx}(\tau) = \frac{ e^{\beta\Delta/2}e^{-\Delta\tau/\hbar} +e^{-\beta\Delta/2}e^{\Delta\tau/\hbar} }{ 2\cosh(\beta\Delta/2) },$$

evaluate

$$C_{xx}(i\nu_n) = \frac1\hbar \int_0^{\beta\hbar} d\tau\, e^{i\nu_n\tau/\hbar} C_{xx}(\tau)$$

for $\nu_n=2\pi n/\beta$.

Solution

Because $e^{i\beta\nu_n}=1$, the first exponential contributes

$$\frac{ e^{\beta\Delta/2} \left( 1-e^{-\beta\Delta} \right) }{ 2\cosh(\beta\Delta/2) } \frac1{\Delta-i\nu_n}.$$

Its thermal prefactor is

$$\tanh \left( \frac{\beta\Delta}{2} \right).$$

The second exponential similarly contributes

$$\tanh \left( \frac{\beta\Delta}{2} \right) \frac1{\Delta+i\nu_n}.$$

Adding them first gives

$$C_{xx}(i\nu_n) = \tanh \left( \frac{\beta\Delta}{2} \right) \times \left[ \frac1{\Delta-i\nu_n} +\frac1{\Delta+i\nu_n} \right] .$$

Combining the two fractions,

$$C_{xx}(i\nu_n) = \frac{ 2\Delta \tanh(\beta\Delta/2) }{ \Delta^2+\nu_n^2 }.$$

4. Verify fermionic antiperiodicity

For one fermionic mode, use

$$1-f(\xi) = e^{\beta\xi}f(\xi)$$

to show directly that

$$\mathcal G(\tau+\beta\hbar) = -\mathcal G(\tau)$$

for $-\beta\hbar\lt\tau\lt0$. Explain why $\mathcal G$ has a jump at $\tau=0$.

Solution

For $-\beta\hbar\lt\tau\lt0$,

$$\mathcal G(\tau) = f(\xi)e^{-\xi\tau/\hbar}.$$

The shifted argument is positive, so

$$\begin{aligned} \mathcal G(\tau+\beta\hbar) &= - \left[ 1-f(\xi) \right] e^{-\xi(\tau+\beta\hbar)/\hbar} \\ &= - \left[ 1-f(\xi) \right] e^{-\beta\xi} e^{-\xi\tau/\hbar} \\ &= -f(\xi)e^{-\xi\tau/\hbar} \\ &= -\mathcal G(\tau). \end{aligned}$$

At coincident time,

$$\begin{aligned} \mathcal G(0^+) -\mathcal G(0^-) &= -\left[ 1-f(\xi) \right] -f(\xi) \\ &= -1. \end{aligned}$$

This jump is the thermal Green-function form of the canonical anticommutator $\{c,c^\dagger\}=1$.

5. Classify composite operators

Determine whether each object has bosonic or fermionic Matsubara frequencies:

1. $c_{\mathbf k}$;
2. $c_{\mathbf q}^\dagger c_{\mathbf k}$;
3. $c_{\mathbf k}c_{-\mathbf k}$;
4. $c_{\mathbf k}c_{\mathbf q}c_{\mathbf p}$;
5. a spin operator $S^z$.

Solution

Thermal boundary conditions depend on total fermion parity:

Operator	Fermion parity	Matsubara class
$c_{\mathbf k}$	odd	fermionic
$c_{\mathbf q}^\dagger c_{\mathbf k}$	even	bosonic
$c_{\mathbf k}c_{-\mathbf k}$	even	bosonic
$c_{\mathbf k}c_{\mathbf q}c_{\mathbf p}$	odd	fermionic
$S^z$	even	bosonic

A pair field is therefore bosonic under the thermal boundary condition even though its constituents are fermionic.

6. Decide whether imaginary time is enough

For each task, state whether equilibrium imaginary-time data are a natural endpoint, an intermediate representation requiring additional work, or insufficient:

1. compute the heat capacity;
2. compute an equal-time density correlation;
3. determine an optical conductivity spectrum from Monte Carlo data;
4. predict dynamics after a sudden quench;
5. estimate a static susceptibility.

Solution

1. Heat capacity: imaginary-time or partition-function methods are a natural endpoint because energy fluctuations or derivatives of $\ln Z$ determine it.
2. Equal-time density correlation: an equilibrium imaginary-time calculation is a natural endpoint; evaluate the correlator at equal time with the correct contact convention.
3. Optical conductivity: imaginary-time current correlations are intermediate. One needs analytic continuation or a real-time method, together with diamagnetic terms, sum rules, and uncertainty analysis.
4. Quench dynamics: equilibrium imaginary time is insufficient. The initial state and real-time evolution, usually on a suitable contour or through direct unitary propagation, are required.
5. Static susceptibility: imaginary-time methods are natural because the zero bosonic Matsubara component or a thermodynamic derivative can determine the equilibrium static response, subject to the correct order of limits.

References

• P. Coleman, Introduction to Many-Body Physics, Cambridge University Press (2015) – modern graduate treatment of finite-temperature many-body physics and path integrals.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover (2003 reprint of the 1971 edition) – standard operator, Green-function, and finite-temperature treatment.
• R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems”, Journal of the Physical Society of Japan 12, 570–586 (1957) – equilibrium correlation and response foundations.
• M. Le Bellac, Thermal Field Theory, Cambridge University Press (1996) – a concise bridge from finite-temperature quantum mechanics to thermal field theory.
• G. D. Mahan, Many-Particle Physics, 3rd ed., Springer (2000) – nonzero-temperature Green functions and many-body applications.
• P. C. Martin and J. Schwinger, “Theory of Many-Particle Systems. I”, Physical Review 115, 1342–1373 (1959) – Green functions, spectral structure, and equilibrium many-body identities.
• T. Matsubara, “A New Approach to Quantum-Statistical Mechanics”, Progress of Theoretical Physics 14, 351–378 (1955) – original imaginary-time many-body formalism.
• J. W. Negele and H. Orland, Quantum Many-Particle Systems, CRC Press (2018 reissue) – coherent-state path integrals, functional methods, and stochastic formulations.