We investigate the entanglement entropy of a class of $N$-particle excited state of fermions confined in a two-dimensional harmonic trap rotating at an angular frequency $Ω$. The excited state is constructed by filling a particular set of $N$ single-particle energy levels. We analytically compute the Rényi entropies of order $q$ in a disc of radius $r$ around the centre of the trap, and the cumulants corresponding to number fluctuations of fermions within the disc. We found that the area law scaling of entanglement entropy holds even for a class of excited states. We also verified the well-known series expansion of entanglement entropy in terms of the particle number cumulants for non-interacting fermions. We further derive the centered cumulant generating function, demonstrating that the associated probability distribution function in the disc has identical scaling properties, up to a variable shift, to the known ground-state result. Finally, we extend our result to an annular region, showing that both the Rényi entropy and particle number cumulants of the annulus decompose into sums of the corresponding quantities of the two bounding discs. These additive relations hold as long as the width of the annulus is sufficiently large.
We investigate the entanglement entropy of a class of N-particle excited state of fermions confined in a two-dimensional harmonic trap rotating at an angular frequency Ω. The excited state is constructed by filling a particular set of N single-particle energy levels. We analytically compute the Rényi entropies of order q in a disc of radius r around the centre of the trap, and the cumulants corresponding to number fluctuations of fermions within the disc. We found that the area law scaling of entanglement entropy holds even for a class of excited states. We also verified the well-known series expansion of entanglement entropy in terms of the particle number cumulants for non-interacting fermions. We further derive the centered cumulant generating function, demonstrating that the associated probability distribution function in the disc has identical scaling properties, up to a variable shift, to the known ground-state result. Finally, we extend our result to an annular region, showing that both the Rényi entropy and particle number cumulants of the annulus decompose into sums of the corresponding quantities of the two bounding discs. These additive relations hold as long as the width of the annulus is sufficiently large.
We establish an exact mapping between the positions of N noninteracting fermions in a two-dimensional rotating harmonic trap in its ground state and the eigenvalues of the $N\ifmmode\times\else\texttimes\fi{}N$ complex Ginibre ensemble of random matrix theory (RMT). Using RMT techniques, we make precise predictions for the statistics of the positions of the fermions, both in the bulk as well as at the edge of the trapped Fermi gas. In addition, we compute exactly, for any finite N, the R\'enyi entanglement entropy and the number variance of a disk of radius r in the ground state. We show that while these two quantities are proportional to each other in the (extended) bulk, this is no longer the case very close to the trap center nor at the edge. Near the edge, and for large N, we provide exact expressions for the scaling functions associated with these two observables.
We study the ground state of $N\ensuremath{\gg}1$ noninteracting fermions in a two-dimensional harmonic trap rotating at angular frequency $\mathrm{\ensuremath{\Omega}}>0$. The support of the density of the Fermi gas is a disk of radius Re. We calculate the variance of the number of fermions, NR, inside a disk of radius R centered at the origin for R in the bulk of the Fermi gas. We find rich and interesting behaviors in two different scaling regimes, (i) $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}<1$ and (ii) $1\ensuremath{-}\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1/N)$, where $\ensuremath{\omega}$ is the angular frequency of the oscillator. In the first regime we find that $\mathrm{Var}\phantom{\rule{0.16em}{0ex}}{\mathcal{N}}_{R}\ensuremath{\simeq}(AlnN+B)\sqrt{N}$ and we calculate A and B as functions of R/Re, $\mathrm{\ensuremath{\Omega}}$, and $\ensuremath{\omega}$. We also predict the higher cumulants of NR and the bipartite entanglement entropy of the disk with the rest of the system. In the second regime, the mean fermion density exhibits a staircase form, with discrete plateaus corresponding to filling k successive Landau levels, as found in previous studies. Here, we show that VarNR is a discontinuous piecewise linear function of $\ensuremath{\sim}(R/{R}_{e})\sqrt{N}$ within each plateau, with coefficients that we calculate exactly, and with steps whose precise shape we obtain for any k. We argue that a similar piecewise linear behavior extends to all the cumulants of NR and to the entanglement entropy. We show that these results match smoothly at large k with the above results for $\mathrm{\ensuremath{\Omega}}/\ensuremath{\omega}=O(1)$. These findings are nicely confirmed by numerical simulations. Finally, we uncover a universal behavior of VarNR near the fermionic edge. We extend our results to a three-dimensional geometry, where an additional confining potential is applied in the z direction.
We derive exact relations between the Rényi entanglement entropies and the particle-number fluctuations of (connected and disjoint) spatial regions in systems of N non-interacting fermions in arbitrary dimension. We prove that the asymptotic large- N behavior of the entanglement entropies is proportional to the variance of the particle number. We also consider 1D Fermi gases with a localized impurity, where all particle cumulants contribute to the asymptotic large- N behavior of the entanglement entropies. The particle cumulant expansion turns out to be convergent for all integer-order Rényi entropies (except for the von Neumann entropy) and the first few cumulants provide already a good approximation. Since the particle cumulants are accessible to experiments, these relations may provide a measure of entanglement in these systems.
We revisit the computation of particle number fluctuations and the R\'enyi entanglement entropy of a two-dimensional Fermi gas using multidimensional bosonization. In particular, we compute these quantities for a circular Fermi surface and a circular entangling surface. Both quantities display a logarithmic violation of the area law, and the R\'enyi entropy agrees with the Widom conjecture. Lastly, we compute the symmetry-resolved entanglement entropy for the two-dimensional circular Fermi surface and find that, while the total entanglement entropy scales as RlnR, the symmetry-resolved entanglement scales as RlnR, where R is the radius of the subregion of our interest.
Abstract In the presence of a global symmetry, the entanglement entropy of a quantum system can be decomposed among the individual symmetry sectors, dubbed symmetry-resolved entanglement entropy. For a conformal field theory with Abelian symmetry, it obeys the equipartition theorem in the scaling limit, i.e., at the leading order, the entanglement is distributed equally between different symmetry sectors. In this work, we examine the thermal corrections to a single interval symmetry-resolved Rényi and entanglement entropies for two-dimensional conformal field theories on a circle. Using a low-temperature expansion of the thermal density matrix, we find that in addition to the mass gap and the degeneracy of the first excited state, these universal corrections depend also on the four-point correlation function of the primary fields. We also obtain thermal corrections to the full counting statistics of the ground state and define the probability fluctuations function which scales as e−2πΔψβ/L e − 2 π Δ ψ β / L , where ∆ψ is the scaling dimension of the lowest weight states. As an example, we explicitly evaluate the thermal corrections to the symmetry-resolved entanglement entropy and FCS for the spinless fermions and find a term breaking entanglement equipartition at order (log l)−2.
We consider the entanglement entropy for a line segment in the system of noninteracting one-dimensional fermions at zero temperature. In the limit of a large segment length L , the leading asymptotic behavior of this entropy is known to be logarithmic in L . We study finite-size corrections to this asymptotic behavior. Based on an earlier conjecture of the asymptotic expansion for full counting statistics in the same system, we derive a full asymptotic expansion for the von Neumann entropy and obtain first several corrections for the Rényi entropies. Our corrections for the Rényi entropies reproduce earlier results. We also discuss the entanglement spectrum in this problem in terms of single-particle occupation numbers.