Entanglement Scaling and Full Counting Statistics in Excited States of Two-Dimensional Rotating Fermions

Technical Summary: Entanglement Scaling and Full Counting Statistics in Excited States of Two-Dimensional Rotating Fermions

Problem Statement
The paper investigates the entanglement entropy (EE) and full counting statistics (FCS) of particle number fluctuations for a specific class of excited states in a system of $N$ non-interacting, spinless fermions confined in a two-dimensional rotating harmonic trap. While the area law scaling of EE is well-established for ground states of gapped systems and certain gapless systems (scaling as $\sim L^{d-1} \ln L$), high-energy or high-temperature states in generic systems are typically expected to exhibit volume-law scaling ($\sim L^d$). The authors address the question of whether the area law persists for specific, structured excited states in this rotating trap model, a regime where the system exhibits analogies to quantum Hall systems.

Methodology
The study focuses on a many-body state constructed by filling $N$ consecutive single-particle energy levels within the lowest Landau level (LLL, $l=0$) sector. Specifically, the authors consider a "sliding window" of quantum numbers $k = M, M+1, \dots, M+N-1$, where $M \ge 0$ determines the excitation level. The single-particle wavefunctions in this sector are rotationally symmetric, allowing for exact analytical treatment.

The core methodology relies on the overlap matrix formalism for Gaussian states. For non-interacting fermions, both EE and FCS are fully determined by the eigenvalues of the correlation matrix restricted to a spatial domain. The authors exploit the fact that for radially symmetric domains (discs and annuli) and the specific $l=0$ states chosen, the overlap matrix is diagonal.

1. Overlap Matrix: The matrix elements $A_{n_1, n_2}$ for a disc of radius $r$ are shown to be diagonal, with eigenvalues $a_s(r)$ expressible in terms of the incomplete gamma function.
2. Scaling Limits: The analysis is performed in the limit of large $M$ and $N$, with the ratio $\mu = M/N$ fixed. In this limit, the discrete sums over eigenvalues are converted to integrals using saddle-point approximations.
3. Quantities Computed:
   • Rényi Entropy ($S_q$): Calculated using the eigenvalues $a_s$ via the standard formula $S_q = \frac{1}{1-q} \sum \ln(a_s^q + (1-a_s)^q)$.
   • Full Counting Statistics: The centered cumulant generating function (CCGF) is derived to compute particle number cumulants $\langle \hat{N}_D^p \rangle_c$.
   • Density Profile: The radial density $\rho(r)$ is derived to identify the bulk, inner edge, and outer edge regimes of the fermion cloud.

Key Results

1. Persistence of Area Law in Excited States:
   The primary finding is that the area law scaling of entanglement entropy holds for this class of excited states. For a disc of radius $r$ located in the bulk region ($\sqrt{M} \ll r \ll \sqrt{M+N}$), the Rényi entropy scales linearly with the perimeter (circumference) of the disc:
   $$S_q \sim \sigma_q \times (2\pi r)$$
   where $\sigma_q$ is a coefficient dependent on the Rényi index $q$. This confirms that sub-volume entanglement can exist in high-energy pure states, contrary to the generic expectation of volume-law scaling.

2. Scaling of Cumulants and FCS:
   The particle number cumulants $\langle \hat{N}_r^p \rangle_c$ for $p \ge 2$ also exhibit area-law scaling in the bulk, growing linearly with the radius $r$. The authors derive the centered cumulant generating function (CCGF) and show that the probability distribution function (PDF) of the particle number in the disc, up to a variable shift determined by $\mu = M/N$, possesses identical scaling properties to the ground state ($M=0$) studied in previous literature. The distribution exhibits three regimes: typical fluctuations ($O(N^{-3/4})$), intermediate fluctuations ($O(N^{-1/2})$), and large deviations ($O(1)$).

3. Verification of Cumulant Expansion:
   The paper verifies the known series expansion relating EE to particle number cumulants for non-interacting fermions in Gaussian states:
   $$S_q(D) = \sum_{p=1}^{\infty} \beta_p(q) \langle \hat{N}_D^p \rangle_c$$
   Numerical analysis demonstrates that this series converges rapidly for $q \ge 2$, providing a direct link between the entanglement entropy and the full counting statistics in this excited state regime.

4. Additivity in Annular Regions:
   For an annular region defined by inner radius $r_1$ and outer radius $r_2$ (with sufficient width such that the boundaries do not interact), the authors prove that both the Rényi entropy and the particle number cumulants are additive. Specifically:
   $$S_q(A_{r_1, r_2}) \approx S_q(D_{r_1}) + S_q(D_{r_2})$$
   $$\langle \hat{N}_{r_1, r_2}^p \rangle_c \approx \langle \hat{N}_{r_1}^p \rangle_c + \langle \hat{N}_{r_2}^p \rangle_c$$
   This decomposition arises from the reflection symmetry of the overlap matrix eigenvalues near the edges of the annulus.

5. Edge Behavior:
   Near the inner ($r \approx \sqrt{M}$) and outer ($r \approx \sqrt{M+N}$) edges of the fermion cloud, the scaling behavior deviates from the linear bulk law. The authors provide universal scaling functions (involving the complementary error function) that describe the crossover from the bulk area law to the edge behavior.

Significance and Claims
The paper claims to provide a concrete analytical example where the area law for entanglement entropy persists in high-energy excited states of a non-interacting fermionic system. By utilizing the specific structure of the rotating harmonic trap and the $l=0$ sector, the authors extend previous ground-state results (where $M=0$) to arbitrary excited states ($M > 0$).

The significance lies in:

• Demonstrating that volume-law scaling is not inevitable for high-energy pure states; specific constructions can maintain area-law scaling.
• Establishing a rigorous connection between the entanglement entropy and full counting statistics for these excited states, verifying the cumulant expansion relation.
• Showing that the statistical properties of particle number fluctuations in these excited states are structurally identical to the ground state, merely shifted in the variable space.
• Providing an exact analytical framework (via the diagonal overlap matrix) that can be extended to more general states with non-zero angular momentum, provided the angular momentum indices remain distinct.

The authors do not propose new experimental setups but suggest that their analytical framework could be applied to investigate entanglement in other excited state configurations within similar trapping potentials.