Finite CAR quasifree states and covariance

Let $r\ge 1$, let $H_1={\mathbb{C}}^r$ with ordered orthonormal basis $e_1,\dots ,e_r$, and let ${\mathcal{F}}_{-}(H_1)={⨁}_{k=0}^r{\bigwedge }^k{H}_1$ be its finite-mode fermionic Fock space. Let ${\mathcal{A}}_r$ be the unital *-algebra generated by its annihilation and creation operators; it is finite-dimensional. Let $c_j,c_j^{\ast }$ be those operators in this basis, satisfying $c_i{c}_j+c_j{c}_i=0,c_i{c}_j^{\ast }+c_j^{\ast }{c}_i={\delta }_{ij}I,c_i^{\ast }{c}_j^{\ast }+c_j^{\ast }{c}_i^{\ast }=0.$ A state is a complex-linear functional $\omega$ on ${\mathcal{A}}_r$ such that $\omega (I)=1$ and $\omega (X^{\ast }X)\ge 0$ for every $X\in {\mathcal{A}}_r$. Define the gauge automorphism ${\gamma }_{\theta }$ by ${\gamma }_{\theta }(c_j)=e^{-i\theta }{c}_j$ and ${\gamma }_{\theta }(c_j^{\ast })=e^{i\theta }{c}_j^{\ast }$ for $\theta \in \mathbb{R}$; the CAR relations are preserved, so these assignments extend to a one-parameter family of algebra automorphisms. A state is gauge invariant when $\omega ({\gamma }_{\theta }(X))=\omega (X)$ for every $X$ and every $\theta$.

For a gauge-invariant state, define its one-particle covariance matrix $C\in M_r(\mathbb{C})$ by $C_{ji}:=\omega (c_i^{\ast }{c}_j)$. A finite gauge-invariant state is quasifree when $C=C^{\ast }$, $0\le C\le I$, and its normally ordered moments obey the following rule: for $p\mathrm{\ne }q$, $\omega (c_{i_1}^{\ast }\cdots c_{i_p}^{\ast }{c}_{j_q}\cdots c_{j_1})=0,$ and for $p=q$, $\omega (c_{i_1}^{\ast }\cdots c_{i_p}^{\ast }{c}_{j_p}\cdots c_{j_1})=\det \text{\hspace{0.17em}⁣}{[\omega (c_{i_a}^{\ast }{c}_{j_b})]}_{a,b=1}^p=\det \text{\hspace{0.17em}⁣}{\left[C_{j_b{i}_a}\right]}_{a,b=1}^p\mathrm{.}$ The empty moment is $\omega (I)=1$. The inequalities $0\le C\le I$ mean $⟨Cv,v⟩\ge 0$ and $⟨(I-C)v,v⟩\ge 0$ for all $v\in H_1$. Moments of words not in normal order are obtained by reordering with the displayed CAR relations.

For a Hermitian one-particle matrix $h=h^{\ast }$, a real chemical potential $\mu$, and $\beta >0$, define $d\mathrm{\Gamma }(h-\mu I):={\sum }_{i,j=1}^r{c}_i^{\ast }(h-\mu I{)}_{ij}{c}_j,Z:=\mathrm{Tr}\text{\hspace{0.17em}⁣}\left(e^{-\beta d\mathrm{\Gamma }(h-\mu I)}\right),\rho :=Z^{-1}{e}^{-\beta d\mathrm{\Gamma }(h-\mu I)}\mathrm{.}$ The finite grand Gibbs state has covariance $C=(I+e^{\beta (h-\mu I)}{)}^{-1}\mathrm{.}$ Here $h$ and $\mu$ are in joules, $\beta$ is in inverse joules, and $C$ is dimensionless. The existence and uniqueness of the state for every Hermitian contraction $C$, and the displayed finite Gibbs covariance formula, are established by an earlier finite-dimensional result.

Occupation bases and actual CCR/CAR domains

Finite matrix spectral and state calculus

Existence, uniqueness, and Gibbs covariance for finite CAR states ↗

The formula defines finite-mode quasifree states only. It does not assert Wick factorization for interacting states or an infinite-mode density operator.