Gibbs States and Trace-Class Caveats

The expression $\rho_\beta=Z_\beta^{-1}e^{-\beta H}$ defines a density operator only when $0<Z_\beta=\operatorname{Tr}e^{-\beta H}<\infty$. Self-adjointness and a lower energy bound do not guarantee this. This page gives spectral criteria, counterexamples, and free-Fock partition-function conditions, distinguishing a normal Gibbs density in a specified Hilbert space from an algebraic equilibrium state.

Required background. Trace-class operators supplies trace ideals and trace duality. Fock construction defines $d\Gamma(h)$. KMS equilibrium limits explains equilibrium without a global trace.

A heat operator must be summable

Let $H$ be self-adjoint and bounded below on a nonzero separable Hilbert space, and let $\beta>0$. Spectral calculus makes $e^{-\beta H}$ a bounded positive injective operator. If it is trace class, its trace is strictly positive and normalization gives a positive trace-one density operator.

The trace-class condition is equivalent to having an orthonormal eigenbasis of $H$, with eigenvalues $E_j$ repeated according to multiplicity, such that

$$\sum_j e^{-\beta E_j}<\infty.$$

For an infinite-dimensional space the eigenvalues must tend to $+\infty$. To prove necessity, use compactness of the heat operator: its nonzero eigenvalues have finite multiplicity and approach zero, and injectivity makes their eigenvectors complete. Applying the inverse scalar function $-\beta^{-1}\log$ identifies the corresponding eigenvalues of $H$. Sufficiency is the positive diagonal trace criterion.

Thus continuous spectrum or infinitely degenerate finite-energy levels obstruct a trace-class heat operator. Compact resolvent alone is weaker: it gives discreteness, but not the required speed of eigenvalue growth.

After shifting the energy so that $H\geq0$, define the counting function $N(E)=\#\{j:E_j\leq E\}$. Tonelli’s theorem gives, allowing infinity,

$$Z_\beta =\beta\int_0^\infty e^{-\beta E}N(E)\,dE.$$

Every eigenvalue contributes $\beta\int_{E_j}^\infty e^{-\beta E}dE=e^{-\beta E_j}$. This formula makes spectral growth the relevant quantitative input.

Discrete spectrum can still fail at high temperature

On $\ell^2(\mathbb N)$ take

$$He_n=\varepsilon\log(n+1)e_n, \qquad \varepsilon>0.$$

The operator is positive and has compact resolvent. Nevertheless

$$Z_\beta=\sum_{n=1}^{\infty}(n+1)^{-\beta\varepsilon}$$

is finite exactly when $\beta\varepsilon>1$. An energy spectrum that grows only logarithmically supplies too many high-energy states at larger temperatures.

If $Z_{\beta_0}<\infty$, then $Z_\beta<\infty$ for every $\beta>\beta_0$, after the harmless energy shift. The converse need not hold. Finiteness at $\beta_0$ also does not automatically control the mean energy there. At every strictly larger $\beta$, however, the bound $E e^{-(\beta-\beta_0)E}\leq C$ proves $\operatorname{Tr}(He^{-\beta H})<\infty$ for $H\geq0$.

Free motion in infinite space is not a Gibbs density

For $H=-\hbar^2\Delta/(2m)$ on $L^2(\mathbb R^d)$, the heat operator is not compact and therefore not trace class. One direct proof uses translations.

Choose a nonzero compactly supported normalized vector $\psi$ and translations $\psi_n$ escaping to spatial infinity. Then $\psi_n\rightharpoonup0$. A compact operator would send this bounded weakly convergent sequence to zero in norm. But translation invariance gives

$$\|e^{-\beta H}\psi_n\| =\|e^{-\beta H}\psi\|>0.$$

This contradicts compactness. The problem is the infinite spatial degeneracy, not a failure of positivity of the heat kernel.

A finite confining box with suitable boundary conditions can have a finite heat trace. Removing the box requires local or intensive limits; dividing an infinite trace by an infinite partition function is not an operator construction.

Free Fock traces require two distinct controls

Let $h$ have a discrete eigenbasis, be bounded below, and satisfy $\operatorname{Tr}e^{-\beta h}<\infty$. For chemical potential $\mu$, write $k=h-\mu I$ and $x_j=e^{-\beta(E_j-\mu)}$.

On fermionic Fock space the occupation numbers are zero or one, giving

$$\operatorname{Tr}_{\mathcal F_-}e^{-\beta d\Gamma(k)} =\prod_j(1+x_j).$$

The product is finite because $\sum_jx_j<\infty$. Any fixed real $\mu$ is allowed under these hypotheses; only finitely many one-particle levels lie below it.

On bosonic Fock space,

$$\operatorname{Tr}_{\mathcal F_+}e^{-\beta d\Gamma(k)} =\prod_j(1-x_j)^{-1}$$

is finite precisely when $\mu<\inf\sigma(h)$ and $\sum_jx_j<\infty$. Each geometric series requires $x_j<1$, while convergence of the infinite product requires summability. Under the heat-trace assumption the spectral bottom is an eigenvalue, so equality $\mu=\inf\sigma(h)$ already produces a divergent zero-mode occupation sum.

These formulas follow first for finitely many modes by summing occupation numbers. Monotone convergence over the increasing occupation basis justifies the infinite product, including divergent cases. In determinant notation they are $\det(I+e^{-\beta k})$ and $\det(I-e^{-\beta k})^{-1}$.

A positive one-particle gap alone does not guarantee either full Fock trace: infinitely many modes at the same positive energy still make $\sum_jx_j$ diverge. Conversely, the CAR quasi-free state construction can define a thermal state from its two-point function without a vacuum-Fock Gibbs density.

Exercises

An oscillator check. For one bosonic mode with $h=\varepsilon>0$, compute the grand partition function and expected number.

Solution

For $\mu<\varepsilon$, put $x=e^{-\beta(\varepsilon-\mu)}<1$. Then $Z=(1-x)^{-1}$ and $\langle N\rangle=x/(1-x)$. At $\mu=\varepsilon$ all occupations have equal weight and normalization fails. For $\mu>\varepsilon$, the grand Hamiltonian is unbounded below.

Energy shifts. If $H'=H+cI$, determine whether trace-class existence or the normalized Gibbs state changes.

Solution

$e^{-\beta H'}=e^{-\beta c}e^{-\beta H}$ and $Z'_\beta=e^{-\beta c}Z_\beta$. A finite positive scalar changes neither trace-class existence nor the normalized state. This justifies the nonnegative-energy shift used in the counting-function argument.

References

• J. Dereziński, “Introduction to representations of the canonical commutation and anticommutation relations,” in Large Coulomb Systems, Lecture Notes in Physics 695, Springer, 2006, pp. 63–143, doi:10.1007/3-540-32579-4_3, author text.
• B. Simon, Trace Ideals and Their Applications, 2nd ed., Mathematical Surveys and Monographs 120, American Mathematical Society, 2005, doi:10.1090/surv/120.