Classifying fermionic states via many-body correlation measures

Abstract

Understanding the structure of quantum correlations in a many-body system is key to its computational treatment. For fermionic systems, correlations can be defined as deviations from Slater determinant states. The link between fermionic correlations and efficient computational physics methods is actively studied but remains ambiguous. We make progress in establishing this connection mathematically. In particular, we find a rigorous classification of states relative to $k$-fermion correlations, which admits a computational physics interpretation. Correlations are captured by a measure $ω_k$, a function of $k$-fermion reduced density matrix that we call twisted purity. A condition $ω_k=0$ for a given $k$ puts the state in a class $G_k$ of correlated states. Sets $G_k$ are nested in $k$, and Slater determinants correspond to $k = 1$. Classes $G_{k=O(1)}$ are shown to be physically relevant, as $ω_k$ vanishes or nearly vanishes for truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each $k = O(1)$, we give an explicit ansatz with a polynomial number of parameters that covers all states in $G_k$. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.

Figures (2)

• Figure 1: Mathematical structure of our results. Highlighted in red are the main contributions: the twisted purity $\omega_k$, the set of correlated states $\mathcal{G}_k$ defined by $\omega_k=0$, and the ansatz for states $\ket{v}\in\mathcal{G}_k$. (a) The nested pattern of sets $\mathcal{G}_k$. States in $\mathcal{G}_1$ are the familiar Slater determinants. The entire Hilbert space of $n$ fermions on $l$ modes coincides with $\mathcal{G}_{k=n+1}$. (b) $\omega_k$ generalizes the 1-RDM purity $\omega_1$ to the $k$-body case via 'twisted' $k$-RDM $\tilde{\rho}_k$ (see Eq. \ref{['eq:conj_rho_def']}). Vanishing of twisted purity is equivalent to a generalization of Plücker relations $\Omega\ket{v}\ket{v}=0$. (c) As a key technical step, we find that states in $\mathcal{G}_k$ obey a generalization of Wick's rule (Theorem \ref{['thm:thm_corr_decomposition']}) for the amplitudes $v(S)$, although not for observables $\langle O \rangle$. Unlike in the Slater case, there may be states outside $\mathcal{G}_k$ which follow this generalized Wick's rule --- hence the '$\Rightarrow$' sign. (d) Generalized Wick's rule is equivalent to the ansatz representation for state $\ket{v}$. The explicit form of the ansatz is written in Eqs. \ref{['eq:hyperferm_ansatz_maintext']}-\ref{['eq:ansatz_func_maintext']}. The diagram displays how sets of modes $Q$ and $P$ relate to the set of modes $G$ occupied in the reference Slater state. The polynomial number of parameters $\theta_{P,Q}$ follows from the condition $|Q|=|P|\leq k$. For the Slater states $\ket{v}\in\mathcal{G}_1$, the ansatz reduces to a known parameterization using an exponential of a weight-$2$ operator.
• Figure 2: Twisted purity $\omega_k$ for various $n=6$-fermionic states on $l=12$ modes; note the log scale. It is natural to include $\omega_0$ (equal to $1$ by Eq. \ref{['eq:omega_def']}) into the picture. For the Hubbard model eigenstates, the change of $\omega_k$ over $k$ depends on the relative coupling strength $U/t$ (cf. Eq. \ref{['eq:Hubbard_model_def']}) and excitation energy. For the ground state as a function of $U$, the state changes from being perturbatively close to $\mathcal{G}_1$ ($U=t$) to being $\mathcal{G}_3$-like ($U=3t$) to being far from any $\mathcal{G}_k$ ($U=50t$). For $U=50t$, the curve is remarkably similar to the one of a product of Bell-like states. The pink curve shows an example excited state of the Hubbard model (of energy $\epsilon$ s.t. $(\epsilon - \epsilon_{\mathtt{min}})/(\epsilon_{\mathtt{max}} - \epsilon_{\mathtt{min}})) \sim 0.23$), with exponential decay starting from $k=3$. The ground state of a typical SYK model realization is highly correlated but with small $\omega_{k\geq 5}$, similarly to a Haar random state. Note the even-odd fluctutation in multiple plots. These come from the structure of $\omega_k$ itself; we hypothesize that Eq. \ref{['eq:k_Plücker']} for pairs $k=2r-1$ and $k=2r$ is in fact equivalent ftnt_oddeven.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof