Fermion-like Occupancy Bound

• Fermion-like occupancy bounds are constraints on one-particle occupation numbers, defined by additional generalized Pauli constraints beyond the standard exclusion principle.
• They form a convex polytope that confines allowed fermionic states, with pinning and quasipinning simplifying many-body wavefunction representations.
• Applications of these bounds include enhanced state tomography and efficient variational methods for quantum many-body systems, reducing computational complexity.

A fermion-like occupancy bound is a fundamental constraint on one-particle occupation numbers that arises from the antisymmetry of fermionic wavefunctions. Beyond the standard Pauli exclusion principle—which restricts natural occupation numbers (NONs) to the interval $[0,1]$—the fermionic exchange symmetry imposes a finite set of additional linear inequalities, referred to as generalized Pauli constraints (GPCs). These constraints define a convex polytope that characterizes all possible NONs accessible to pure $N$-fermion states, leading to profound implications for the structure and physical properties of many-fermion systems. Pinning and quasipinning, describing cases where occupation numbers saturate or nearly saturate these constraints, play a crucial role in simplifying many-body wavefunctions and understanding correlation effects.

1. Mathematical Structure of Fermion-Like Occupancy Bounds

Fermion-like occupancy bounds are formalized in terms of the eigenvalues $(\lambda_i)$ of the one-body reduced density matrix (1RDM) derived from an $N$-fermion pure state in a $d$-dimensional one-particle Hilbert space. The standard Pauli principle restricts these to $0 \leq \lambda_i \leq 1$ and $\sum_i \lambda_i = N$. However, Klyachko’s solution to the quantum marginal problem establishes that any such 1RDM from a pure antisymmetric state must fulfill a finite set of additional linear inequalities (Schilling et al., 2012, Schilling, 2015):

$$D_j(\lambda) = \kappa_j^{(0)} + \sum_{i=1}^d \kappa_j^{(i)} \lambda_i \geq 0, \quad j = 1, \ldots, r_{N,d}$$

where all $\kappa_j^{(i)}$ are integers. The set of all $\lambda = (\lambda_1, ..., \lambda_d)$ obeying these, plus the trivial Pauli simplex constraints, forms the GPC-polytope $P_{N, d} \subset \Sigma$.

In the canonical Borland–Dennis setting $(N=3, d=6)$, the NONs satisfy (Schilling et al., 2012, Tennie et al., 2016):

• $$\lambda_1 + \lambda_6 = \lambda_2 + \lambda_5 = \lambda_3 + \lambda_4 = 1$$
• $2 - (\lambda_1 + \lambda_2 + \lambda_3) \geq 0$

These define a convex three-dimensional polytope strictly smaller than the Pauli simplex.

2. Pinning, Quasipinning, and Physical Interpretations

Pinning refers to the exact saturation of one or more GPCs by the NON vector; i.e., $D_j(\lambda) = 0$ for some $j$. Quasipinning describes the situation where $D_j(\lambda)$ is small but nonzero, so the vector is close to a facet of the polytope (Schilling et al., 2012, Tennie et al., 2016). The degree of (quasi)pinning can be quantified by

$D_{\min}(\lambda) = \min_j D_j(\lambda)$

Pinning and strong quasipinning have robust structural implications:

• The $N$-fermion state is restricted to a lower-dimensional subspace of Slater determinants, drastically reducing the number needed in its configuration interaction (CI) expansion (Tennie et al., 2016, Benavides-Riveros et al., 2017).
• In the Borland–Dennis case, pinning to $2-(\lambda_1+\lambda_2+\lambda_3)=0$ enforces a wavefunction that is a superposition of exactly three determinants.
• Quasipinning leads to approximate collapse onto the corresponding subspace, allowing efficient, physically-motivated multiconfigurational ansätze (Tennie et al., 2016, Benavides-Riveros et al., 2017).

3. Models and Scaling Regimes

Quasipinning has been analyzed in exactly solvable models such as harmonically trapped spinless fermions (Harmonium) in one dimension (Tennie et al., 2016, Schilling et al., 2012):

• NONs approach the GPC facets with a scaling $D_{\min}(\kappa) \sim \kappa^{8}$ for weak coupling parameter $\kappa$, while their Pauli-simplex distance scales as $\kappa^4$. This shows non-trivial proximity to the GPC boundary that is not a consequence of the standard exclusion principle.
• In few-site Hubbard models, tuning the on-site interaction $U$ reveals intervals of exact pinning, sharp pinning-nonpinning transitions, and strong symmetry dependence (Schilling, 2015, Schilling et al., 2012). For example, for three fermions on three sites, pinning persists up to $U_0 \approx 12.86$ and vanishes beyond that value.

For lattice systems such as the Hubbard and Kondo models, rigorous bounds relate the average momentum occupation numbers to their non-interacting values. At finite temperature,

$$|\langle n_{k\sigma} \rangle - f_k| \leq \delta = \frac{|u|}{k_B T}$$

where $f_k$ is the Fermi–Dirac occupation and $u$ is the interaction strength (Lapa, 2021). At $T=0$, deviations decay as the inverse distance in energy from the (interacting) Fermi surface.

4. Wavefunction Structure, Correlation Energy, and Variational Methods

Pinning to (or strong proximity to) a GPC facet imposes a selection rule on the wavefunction: only configurations that themselves saturate the constraint contribute (Benavides-Riveros et al., 2017, Tennie et al., 2016). This enables:

• Generalized Hartree–Fock or minimal MCSCF (multiconfigurational self-consistent-field) ansätze, which expand the many-body state only in the restricted active space determined by the pinned facet.
• Universal geometric bounds on correlation energy residuals for such approximate states:

$\Delta E \leq C\, D_j(\lambda_0)$

with $C$ depending on the excitation gap, and the tightness of the bound assessed via the $l^1$-distance to the Hartree–Fock vertex (Benavides-Riveros et al., 2017).

Pinning thus provides physically-motivated truncations that retain high accuracy while drastically reducing computational complexity for the many-electron problem.

5. Physical Significance and Experimental Realization

The physical relevance of GPC-induced fermion-like occupancy bounds manifests in:

• Dynamical constraints: Pinning restricts the one-body dynamics to the facet, imposing extra conservation laws and kinematical blockades analogous to the Pauli principle (Tennie et al., 2016, Schilling et al., 2012).
• Reduced-density matrix functional theory: Incorporation of GPCs strengthens the representability domain for practical functional development (Tennie et al., 2016).
• Quantum state tomography: GPCs offer sharp tests for the feasible occupation spectra in state-reconstruction experiments (Tennie et al., 2016).
• Experimental scenarios: Proposals exist for direct tests of extended Pauli constraints using quantum-dot arrays, where the Borland–Dennis scenario is realized with high-precision measurement of NONs (Hackl et al., 2021).

6. Beyond Simple Fermions: Composite Particles and Medium Effects

Fermion-like occupancy bounds generalize to composite particles. In dense media:

• For two-fermion (deuteron-like) bound states, Pauli blocking reduces the binding energy and defines a Mott density at which bound states dissolve (Liebing et al., 2014).
• For three-fermion bound states, the interplay of Bose enhancement (from pair correlations) and Pauli blocking yields an in-medium Borromean regime: beyond the two-body Mott density, the three-body state can remain bound due to medium-induced correlations, with the occupancy bound determined by

$1 + \sum_q g_q^2 [n_q^{(D)} - n_q^{(Q)}] > 0$

where $n_q^{(D)}$ (diquark) and $n_q^{(Q)}$ (quark) occupation numbers reflect the competing effects (Liebing et al., 2014).

7. Broader Implications and Future Directions

Fermion-like occupancy bounds, and their generalized Pauli constraints, reveal the deeper structure of many-fermion kinematics, surpassing the exclusion principle in both precision and physical consequence (Schilling, 2015). Their non-perturbative, model-independent origin implies foundational stability of fermionic matter, robustness of occupation profiles against interactions, and enables the development of efficient computational strategies and experimental probing of highly correlated quantum states. An open question remains as to the ubiquity and precise mechanisms underlying strong quasipinning in higher-dimensional and correlated many-body models.

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References

• (Tennie et al., 2016) Pinning of Fermionic Occupation Numbers: General Concepts and One Dimension
• (Schilling et al., 2012) Pinning of Fermionic Occupation Numbers
• (Schilling, 2015) Quantum Marginal Problem and its Physical Relevance
• (Benavides-Riveros et al., 2017) Natural Extension of Hartree-Fock through extremal $1$-fermion information: Overview and application to the lithium atom
• (Schilling, 2015) Hubbard model: Pinning of occupation numbers and role of symmetries
• (Lapa, 2021) Momentum occupation number bounds for interacting fermions
• (Liebing et al., 2014) Composite Fermions in Medium: Extending the Lipkin Model
• (Hackl et al., 2021) Experimental proposal to probe the extended Pauli principle