Neutral Fermion Gap in Quantum Hall Systems

• Neutral fermion gap is the minimum energy needed to excite neutral, charge-conserving fermion modes in strongly correlated systems such as quantum Hall states.
• It quantifies excitation thresholds in non-Abelian phases, determined by energy differences between even and odd particle numbers and underpins topological protection.
• Its rigorous analysis aids quantum computation by ensuring system temperatures remain below gap energies to suppress error rates and maintain robust topological order.

The neutral fermion gap is a fundamental energy scale associated with neutral (charge-conserving) excitations in strongly correlated quantum systems, most notably quantum Hall states, superfluids, and topological phases. It generalizes the concept of the excitation gap to situations where the relevant quasiparticles are neutral with respect to charge but may carry other quantum numbers (e.g., fermion parity or topological charge). In fractional quantum Hall systems, especially, the neutral fermion gap characterizes the minimal energy required to create a neutral excitation—such as a Bogoliubov quasiparticle in a paired state or a magneto-roton mode in Laughlin-type states. The existence and magnitude of the neutral fermion gap have direct implications for topological protection, non-Abelian statistics, and the robustness of exotic phases for quantum computation and correlated matter more broadly.

1. Definitions and General Framework

The neutral fermion gap generally refers to the minimal energy required to excite a neutral (charge-conserving) fermion mode above the quantum many-body ground state, in contrast to the charge gap, which is the minimal energy for creating a charged excitation. In a quantum Hall system at fractional filling $\nu$, one typically considers two prominent types of excitation gaps:

• Charge gap ($\Delta_c$): Energy cost to change the total particle number by one (i.e., to add or remove an electron).
• Neutral gap ($\Delta_{\text{n}}$) / Neutral fermion gap ($\Delta_\psi$): Energy difference between the ground state and the lowest excited state at fixed particle number (maximal filling), corresponding to neutral collective excitations (e.g., magneto-roton mode, Bogoliubov quasiparticle).

The neutral fermion gap is defined operationally through energy comparisons between systems differing by fermion parity (even vs. odd particle number) or through excitation spectrum analysis in a given topological sector (for example, by the alternation in ground-state energy between even and odd particle number in the paired Moore–Read state) (Bonderson et al., 2010, Moller et al., 2010). In lattice Hamiltonians or pseudopotential models, the neutral gap is often the spectral gap within the ground-state sector, while the charge gap measures the incompressibility of the phase.

2. Neutral Fermion Gap in Fractional Quantum Hall Systems

2.1. Numerical Evaluation and Physical Interpretation

The neutral fermion gap at $\nu=5/2$ in the Moore–Read state is numerically evaluated by comparing ground-state energies for even and odd electron numbers at fixed flux conditions (on the sphere, $N_\phi=2N_e-3$ for even $N_e$ and $N_\phi=2N_e-3$ for odd $N_e$). The key formula is

$$\Delta_F(N_e) = (-1)^{N_e} \frac{1}{2} \left[ E(N_\phi+2, N_e+1) + E(N_\phi-2, N_e-1) - 2 E(N_\phi, N_e) \right]$$

and, in the thermodynamic limit,

$$\Delta_\psi = \lim_{N_e \to \infty} [E_{\text{odd}} - E_{\text{even}}]$$

where $E_{\text{odd}}$ and $E_{\text{even}}$ are offset constants after subtracting the linear energy-in-particle-number piece (Bonderson et al., 2010).

This gap is found to be $\Delta_\psi \approx 0.027\, (e^2/\epsilon \ell_0)$, comparable to the charge gap, which is crucial for the topological protection in this non-Abelian phase.

2.2. Spectral Structure and Experimental Signatures

In the paired (Moore–Read) state, the neutral fermion excitation spectrum displays striking features:

• For odd particle numbers, the system must host a neutral excitation with fermionic topological charge, leading to a well-defined, gapped dispersion.
• Systems with charged quasiparticles (quasiholes or quasielectrons) allow the neutral fermion mode to become gapless, reflecting the non-Abelian fusion rules of the underlying Ising conformal field theory and the near-degeneracy between different fusion channels ("1" and "$\psi$") (Moller et al., 2010).
• The neutral fermion dispersion exhibits multiple minima, which can be detected via photoluminescence as a two-peak structure in spectroscopic measurements.

2.3. Universal Inequality: Charge Gap vs. Neutral Gap

A rigorous mathematical result establishes that in translation-invariant, charge- and dipole-conserving systems with a fractionally filled ground state, the charge gap strictly dominates the neutral gap for both bosons and fermions (Lemm et al., 15 Oct 2024, Lemm et al., 24 Jul 2025). The main result is, for maximal fractional filling $n_q$ and for fermions: $$\text{gap } H_{n_q+1} \geq \frac{n_q}{n_q + 1 - m_0} \cdot \text{gap } H_{n_q}$$ where $H_n$ is the $n$-particle sector Hilbert space Hamiltonian, $m_0$ is the smallest number of particles in any interaction term, and the neutral gap is the spectral gap within the $n_q$-sector.

This universal result hinges on the "gap comparison method" or "recursive spectral relations," which exploit the symmetries and the algebraic structure of the many-body Hamiltonian, especially the presence of dipole conservation (Lemm et al., 15 Oct 2024, Lemm et al., 24 Jul 2025). It has profound implications for the energetic hierarchy of excitations and the incompressibility in FQHS.

3. Spectral Methods and Induction-on-Particle-Number

The gap structure in FQHS and related lattice models can be analyzed via recursive operator inequalities relating Hamiltonians with different particle numbers: $$H_{n+1} \geq \frac{1}{n+1-m_0} \sum_x a_x^\dagger H_n a_x$$ This leads to recursive relations for both ground-state and excited-state energies (Lemm et al., 24 Jul 2025), enabling induction over particle number. For example, the neutral gap propagates to the next sector according to

$$E_{n+1}^{(1)} \geq \frac{n+1-||F_n||}{n+1-m_0} E_n^{(1)},$$

where $||F_n||$ is a norm of a Gram matrix, controlled in fermionic systems by the anticommutation relations and symmetries, leading to $||G_n|| \leq 1$ (Lemm et al., 15 Oct 2024).

This scheme not only proves the universal charge gap domination but also facilitates gap estimates by iteratively propagating spectral bounds using robust algebraic and symmetry properties.

4. Significance of Symmetries: Dipole Conservation and its Consequences

Dipole conservation—implemented via operators $V = \exp(2\pi i D/L)$, where $D = \sum_j j N_j$—imposes strong constraints on the many-body spectrum:

• Guarantees $q$-fold ground-state degeneracy for filling $p/q$, distinguishing translated copies of the ground state using the eigenvalues of $V$.
• Severely restricts the overlaps between ground states and single-particle excitations constructed by $a_j^\dagger$, leading to block diagonalization of Gram matrices and exact norm bounds (Lemm et al., 15 Oct 2024, Lemm et al., 24 Jul 2025).
• Underpins both the robustness of the neutral sector and the strong incompressibility of the fractional quantum Hall fluid, by making "leakage" into local excitations energetically unfavorable unless accompanied by appropriate collective shifts.

In short, dipole conservation is not a technical formality—it fundamentally shapes the excitation structure, spectral gaps, and emergent topological order in these systems.

5. Physical Implications and Future Directions

The universal hierarchy $\Delta_c > \Delta_{\text{n}}$ signifies that topological phases like the fractional quantum Hall effect are robust to charge fluctuations and that neutral bulk excitations (e.g., magneto-roton, neutral fermion modes) provide the lowest-energy excitation channels at fixed filling. Implications include:

• For topological quantum computation, maintaining system temperatures well below $\Delta_{\text{n}}$ suppresses error rates exponentially, as thermal populations of neutral excitations are exponentially small.
• Incompressibility (a haLLMark of FQHS) is underpinned by the strict energetic separation between incompressible ground states (robust under particle-conserving perturbations) and charge-carrying excitations.
• Mathematical control of these gaps, via symmetry-enforced inequalities and induction-on-particle-number, provides tools for proving uniform positivity of excitation gaps in Hamiltonian families, which is central to establishing the bulk gap conjecture and related "Haldane gap" results.

These dual physical–mathematical insights generalize well beyond the quantum Hall context, with possible applications in other symmetry-protected topological phases, lattice gauge theories, and the rigorous paper of spectral gaps in quantum many-body systems.

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Table: Universal Gap Relations in FQH Systems (Fermionic Case)

Gap Type	Definition (Sector $n$)	Universal Bound
Neutral gap	$\min \mathrm{spec}(H_n)$	---
Charge gap	$\min \mathrm{spec}(H_{n+1})$	$$\geq \frac{n_q}{n_q+1-m_0} \times \text{neutral gap}$$

This bound reflects the rigorous inequality proven under translation, charge, and dipole conservation, with $m_0$ the minimal interaction degree in the Hamiltonian, and $n_q$ the number of particles at maximal filling.

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In summary, the neutral fermion gap is a symmetry-enforced, physically significant energy scale that is universally bounded by and subordinate to the charge gap in symmetric, translation-invariant fractional quantum Hall systems. This reflects deep connections between symmetry, spectral theory, and the topological order underlying quantum Hall states (Lemm et al., 15 Oct 2024, Lemm et al., 24 Jul 2025).