Quantum entanglement is a fundamental and important tool to probe the properties of a variety of physical systems such as black holes in astrophysics, quantum phase transition in condensed matter physics, and photosynthesis in biophysics [1]. In a bipartite system, one usually calculates the von Neumann entropy or the α-Rényi entropy to quantitatively describe the magnitude of the entanglement between two subsystems. Once the system size is smaller than the correlation length, the entropy is proportional to the volume of the system. For a gapped state, it is generally proportional to the area/length of the interface between two subsystems. This is referred to as the area law in a three-dimensional or perimeter law in a two-dimensional system. Heuristically, this could be understood from the fact that an energy gap gives rise to a finite correlation length which defines the scale on which particles inside the subsystem are correlated with the environment. In gapless critical systems, such as quantum Hall edges or critical spin systems which could be described by the conformal field theory (CFT), it is known that the EE has a logarithmic dependence on the boundary length, and the prefactor is related to the central charge of its underlying CFT [2–5].

After the discovery of topological quantum systems, such as the fractional quantum Hall effects [6, 7], it is known that there is an extra correction of the bulk EE which depends on the quantum dimension of the collective excitation in the bulk. It is referred to as the topological EE γ [8, 9], an important quantity to characterize the nontrivial topology of the long-range entangled quantum many-body states. Moreover, Li and Haldane [10] found that the eigenvalue spectra of the reduced density matrix, named the entanglement spectrum, provides more information about the topology since it could be treated as virtual edge excitation spectra at the bipartite boundary. The mechanism of the bulk-edge correspondence [11, 12] could tell us many of the bulk properties. On the other hand, the quantum Hall edge excitation is usually a chiral gapless mode which could be described by (1 + 1)d chiral CFT. Once the cutting line is along the realistic quantum Hall edge with length l _A , a logarithmic type of α-Rényi EE S edge ≃ c + c ̄ 12 ( 1 + 1 α ) log l A [13] with central charge c is expected. For the chiral edge mode, the anti-holomorphic part is c ̄ = 0 and thus S edge ≃ c 12 ( 1 + 1 α ) log l A .

Up to now, the behavior of the EE mentioned previously is under the assumption that the bi-partition has a smooth boundary, such as a circle or an infinite straight line. Once the boundary has a sharp corner, regardless of whether the system is gapped or not, it was found that the corner on the boundary has an important contribution in the EE which was previously explored [14–27] in two-dimensional quantum critical systems and CFT. Recently, it was extended to the gapped topological system such as the integer quantum Hall states [13, 28, 29]. The corner angle dependence of the EE is found to be universal [30]. Therefore, the complete formula of the EE is S = a A + b ∂ A + γ + S edge + S θ + O 1 / L , (1) in which we include the volume, area, topological, edge, and corner contributions in the first five terms.

In this work, as an example of unification, we consider the quantum Hall state in disk geometry with an open boundary. The fan-shaped bipartite EE with different radius simultaneously gives the contributions from the area law, critical edge mode, and the non-smooth corner. For the integer quantum Hall state, we observed that the corner contribution has similar behavior to the charge cumulation at the cone tip if we put the electrons in a cone-shaped geometry. Moreover, the logarithmic behavior and the central charge are obtained, and the results are immune from the edge reconstruction which conserves the chirality. The robust behavior of the geometric entanglement at the corner could be an explanation of the recent observation that a universal the angle dependence of the particle number fluctuations in a non-smooth bi-partition.

The rest of the article is organized as follows. In Section 2, we revisit the correlation matrix and EE with real space cut in disk geometry. The exact prefactor of the area law is found for α − Rényi entropy with α = 1, 2, 3. In Section 3, the corner contribution is obtained after subtracting the law part by a fan-shaped cut in the bulk. A cone-shaped quantum Hall state reveals the charge cumulation at the cone tip which has similarity to the EE. In Section 4, we consider the fan-shaped bi-partition including the quantum Hall edge. The logarithmic edge contribution could be obtained after subtracting both the area law and corner contribution. The central charge is found robust to any edge reconstructed pattern. Section 5 gives the conclusions and discussions.

For a two-dimensional electron gas in a strong perpendicular magnetic field, the typical length scale is the magnetic length l B = ℏ / e B which we set to one in the following. The single electron wave function in the symmetric gauge is ϕ n , m = c n , m † | 0 〉 = − 1 n n ! 2 π 2 m m + n ! L n m | z | 2 2 z m e − | z | 2 / 4 , (2) where n, m are the Landau level and the angular momentum index, respectively. In the lowest Landau level with n = 0, the mth orbit is a Gaussian wave package in the radial direction which has the most probable radius at r m = 2 m . For a bipartite system, von Neumann entropy is defined as S ₁ (ρ _A ) = −Trρ _A ln ρ _A once we have the reduced density matrix ρ _A for the subsystem. More generally, the α-Rényi entropies [29] are defined as S α ( ρ A ) = 1 1 − α ln ( Tr ρ A α ) which reduce to the von Neumann entropy in the limit α → 1. Their scaling behaviors with increasing the size of the subsystem A have yielded plentiful interesting results for both gapped and critical systems. Generally, the universality of the entanglement appears when the system length scale, such as the boundary length, is much larger than l _B . Therefore, one usually allows a strong finite size effect for small system sizes. For a many-body system, diagonalizing the ρ _A is usually limited to small systems because of the exponential explosion of its dimension as increasing the system size. Fortunately, for a non-interacting fermionic system which has Slater determinants as its eigenstates, it is known [4, 31] that this could be simplified to calculate the eigenvalues of the single particle correlation matrix C i j = Tr ( ρ A c i † c j ) , where c _i is the single particle operator. Its dimension is the number of orbitals which linearly grow as the system size increases. Furthermore, the correlation matrix is naturally diagonal in case the bipartite cutting conserves the symmetry of its parent wave function, i.e., the circular cutting on a disc or latitude cut on a sphere. In this case, the two types of entropy are defined by its diagonal terms, or its eigenvalues {λ _m } as follows: S 1 = ∑ m − λ m ⁡ log λ m − 1 − λ m log 1 − λ m , (3) S α = 1 1 − α ∑ m log λ m α + 1 − λ m α . (4)

For a circular bipartite finite disk as shown in Figure 1A, the electron operator for the mth orbital in the lowest Landau level (n = 0), which could be written as [32–34] c m = α m A m + β m B m , (5) where A _m and B _m are the electron operators in subsystem A and its environment B, respectively. α m 2 and β m 2 are the probabilities in the two subsystems. For a circular cut with radius R _A , they are α m 2 = ∫ 0 R A ∫ 0 2 π | ϕ 0 , m r , θ | 2 d 2 r = 1 − Γ 1 + m , R A 2 2 Γ 1 + m (6) and β m 2 = 1 − α m 2 . For the ν = 1 IQH state which does not have topological term γ, if R _A is much smaller than that of the whole disk, namely, the cut edge is far away from the physical edge at R = 2 N e , the EE contains only the area law term in Eq. 1 as S α = b α l A + O ( 1 / l A ) , where l _A = 2πR _A is the perimeter of the boundary. In Figure 1B, we plot the b _α = S _α /l _A as increasing the R _A for a system with N _e = 171 electrons. The prefactor b _α , shown as constant values in a large range for R _A > 1 and R A < 2 N orb ≃ 18.5 , demonstrates a perfect linear behavior of S _α for a smooth cut in the bulk. The prefactor for S ₁ is b ₁ ≃ 0.203, which is exactly the same as that from a similar study in cylinder geometry where an analytical formula was given as b 1 = ∫ d μ 2 π H [ 1 2 Erfc ( μ ) ] ≃ 0.20329081 in which the integral function is H(x) = −x log(x) − (1 − x) log (1 − x) [35]. We found that the prefactors of the S ₂ and S ₃ are b ₂ = 0.158 and b ₃ = 0.142, respectively. Their analytical results could also be obtained by the corresponding integrate function H α ( x ) = 1 1 − α log ( x α + ( 1 − x ) α ) , which gives b ₂ = 0.15843 and b ₃ = 0.14213. Moreover, the S _α=2,3/l _A saturates faster than S ₁/l _A at small R _A , means the α-Rényi entropy suffers weaker finite size effects than the von Neumann entropy. As a conclusion in this section, we demonstrate that the area law prefactor b _α for IQH is the same for different geometries and could be calculated analytically. This will be applied in the following to subtract the area law term in the non-smooth cut.

In this section, we consider a non-smooth partition as shown in Figure 2. The subsystem A is a fan-shaped region with a corner angle θ at the center of the disk. Supposing the arc-shaped boundary is far away from the physical edge of the disk, then the EE is still in the area law region. The non-smooth cutting gives one corner of θ-degree at the center and two corners of π 2 -angle at the intersections with the arc. Therefore, the EE in this case is S α = b α l A + S α θ + 2 S α π 2 + O 1 / l A (7) with the boundary length l _A = 2R _A + θR _A . In order to screen out the pure angle contribution S _α (θ), we subtract the value at θ = π with the same R _A . In this case, we have S _α (π) = 0 and l _A = 2R _A + πR _A . Therefore, the residual entropy is ΔS _α = b _α (θ − π)R _A + S _α (θ) in which the first term could be exactly obtained from the previous section supposing the area law and corner contributions are independent of each other.

In Figure 3, we plot the S _α (θ) = ΔS _α − b _α (θ − π)R _A as a function of the angle θ. A recent work in Ref. 13 discussed a similar bipartition by calculating the Rényi entropy via the cumulants of the particle number in subsystem S _α = ∑ _m s _α (m)κ _2m , where κ _m are the even cumulants of the particle number distribution in region A. In particular, κ = κ ₂ is the variance of the number of particles in region A which was obtained analytically for the IQH case, namely, κ ( θ ) = N e π 3 / 2 + 1 2 π 2 ln ( N e sin ( θ 2 ) ) − 1 + ( π − θ ) cot ⁡ θ 4 π 2 . The third term is the corner contribution in the second cumulant. We assume the final S _α obeys the same θ-dependence although the prefactor could be non-analytical. We fit the S _α (θ) [36, 37] with the function S _α (θ) = u _α [1 + (π − θ) cot(θ)] and find that u ₁ = −0.0861 (±0.0009), u ₂ = −0.0613 (±0.0007) and u ₃ = −0.0531 (±0.0006). In [28], the corner contribution was recently calculated in cylinder geometry. While θ → 0, it was found that the divergence of the S ₁(θ) behaves as S ₁(θ) = −0.0886 (±0.0004)/θ where the coefficient is consistent with our result of u ₁ for the bulk corner. It is interesting to know that a refined fit formula was also proposed [29] which gives accuracy fits at both asymptotic limits, namely, S α ( θ ) ≃ β 1 ( π − θ ) 2 θ ( 2 π − θ ) − β 2 [ 1 + ( π − θ ) cot ⁡ θ ] which has an extra θ-dependent term. In our fitting process, the asymptotic behavior in the limit of θ → 0 gives S _α ≃ u _α π/θ. The coefficient u _α π ≃ 0.2705 is qualitatively consistent with the result of 0.276 in [29]. The accuracy in this work could be lower due to missing the correction term and possibly mixing different types of EE on a finite disk.

To look more clearly into the corner contribution of the EE, we treat the correlation matrix as entanglement Hamiltonian. The eigenstate which has the most important contribution in the EE is the one that has an eigenvalue near 1/2. In Figure 4, we plot the single particle density for the state which has an eigenvalue with the smallest | λ − 1 2 | . It is interesting to see that for this state, the density mainly concentrates near the boundary of subsystem A, and the sharp corner has a higher density than the smooth edge. We compare two cases with θ = π/3 and θ = 2π/3. It is obvious that the acute angle corner has a much higher density than that of the obtuse angle corner. The behavior of the EE density accumulation at the sharp corner is qualitatively the same as the phenomena of the tip charge accumulation in electromagnetism.

Now we consider a realistic system with a corner. We suppose that the particles live in a fan-shaped geometry and glue it to a cone as shown in Figure 5. The quantum Hall state on a cone has been realized experimentally in synthetic Landau levels for photons [38, 39]. In this case, the Landau wave functions [40–43] come to be ϕ n , m β , z = N n , m L n β m | z | 2 2 z β m e − | z | 2 / 4 , where N n , m = ( − 1 ) n β n ! 2 π 2 β m Γ ( β m + n + 1 ) . The angle of the system θ = 2π/β could be continuously tuned by varying the parameter β. In this geometry, the wave function of the IQH state is | Ψ 〉 = ∏ i < j z i β − z j β exp − ∑ i | z i | 2 / 4 . (8)

In Figure 6, we plot the radial density of the IQH state for different βs. β = 1 corresponds to the original disk geometry which has density ρ ( r ) = 1 2 π l B 2 at the center. While increasing β or decreasing the θ-angle of the system, the density at the corner tip increases dramatically. From the inserted two-dimensional density profiles, it is obvious that the density at the corner tip cumulates gradually and finally separates from the bulk. The occupation number for each orbital is exactly n _m = 1 for ν = 1 IQH. It is easy to analytically calculate the density at the center ρ(0) = ∑ _m n _m |ϕ _0,m (β, 0)|² which is shown in Figure 7. After subtracting the background density with ν = 1, the data are perfectly fitted by f(θ) = 2π/θ − 1 which is the same as that of the EE while θ → 0. Moreover, because of the Pauli exclusive principle of the fermions, the occupation number on each orbital is at most equal to one. Therefore, for other quantum Hall states, such as the fractional quantum Hall states which have ν < 1 at β = 1, the θ → 0 behavior should be universal once the particle number on the 0th orbital reaches one.

Therefore, we obtain the exact corner contribution of the EE via the fan-shaped bi-partition in the bulk. We found the 1/θ divergence of the corner contribution near θ → 0 which has similarity to the charge density cumulation at the tip once we put the system on a cone. The similarity between the EE and the local charge density or its fluctuation has been studied in several systems, either the classical or quantum many-body systems [30, 44].

If we extend the region A in the radial direction to infinity, its arc-shape boundary is the physical edge of the system. As shown in Figure 8, now the EE between A and B contains the area law contribution from the radial boundary, a θ-angle contribution at the center, two π 2 -angle contributions at the intersections, and the edge contribution on the boundary, namely, S α = b α l A + S α θ + 2 S α π / 2 + S edge + O 1 / l A . (9)

Similar to the previous section, after subtracting the EE at θ = π, the area law term and S _α (π/2) are eliminated. Now the residual EE contains the corner and edge contributions. ΔS = S _α (θ) + S _edge (θ) − S _edge (π). The CFT predicts [13, 45, 46] that the EE of the chiral edge is equal to S edge ( θ ) = c 12 ( 1 + 1 α ) ln [ 2 R A ⁡ sin ( θ / 2 ) ] , where 2R _A sin (θ/2) is the chordal distance and the central charge c = 1. Therefore, we expect Δ S α = S α θ + c 12 1 + 1 α ln ⁡ sin θ 2 , (10) which is independent of the R _A . Here, we suppose the previous results of the corner keep invariant as S(θ) = u _α [1 + (π − θ) cot θ] and fit the data of ΔS by function p α ⁡ ln ⁡ sin θ 2 + q α [ 1 + ( π − θ ) cot ⁡ θ ] with parameters q _α and p _α . The results are shown in Figure 9. The result q _α ≃ u _α is expected which shows again that the corner and edge terms are independent. The fitting results p ₁ = 0.166, p ₂ = 0.126, and p ₃ = 0.113 are consistent with the formula p α = 1 12 ( 1 + 1 α ) , and thus c = 1 is verified.

To see how robust the edge contribution of EE is, we consider the edge reconstruction [47–49] pattern as shown in Figure 10. The system contains two unconnected parts. One is the IQH state at the center with N ₁ electrons, the other is the reconstructed part with N ₃ electrons, and we assume that they form the same IQH state at ν = 1. In the middle, there are N ₂ unoccupied orbitals. In this case, although there are three chiral edge modes, the inner edge of the reconstructed stripe has opposite chirality to that of the other two edges. Therefore, the total chirality is not affected by the edge reconstruction. We follow the same logic of subtracting the EE at θ = π as the unreconstructed case. We fix the total number of orbitals N = N ₁ + N ₂ + N ₃ = 171 and consider several combinations of different {N _i }. The fitting parameters are shown in Table 1. Here we only show the results of S ₁, and the results for other S _α s could be expected. It shows that the p ₁ and q ₁ are very robust and consistent with the previous unreconstructed results.

As a conclusion, in the IQH state on a finite disk, we used a simple unified bipartite method to explore the independent EE contributions from the area law, the sharp corner, and the gapless chiral edge contributions. The coefficients of the area law b _α are found to be universal and analytically solvable. With the exact area law term, we obtained the angle dependence of the corner contribution. It has a fixed prefactor, and the behaviors at θ → 0 are consistent with the tip charge accumulation of a realistic IQH liquid on a cone surface. It is similar to recent work of Ref. 30 in which the θ dependence of particle number fluctuations at the corner was found to be universal and has the same 1/θ behavior while θ → 0. While the fan-shaped subsystem contains the edge of the disk, the gapless chiral quantum Hall edge contributes a logarithmic type of EE in which the central charge c in its prefactor is as expected by its underlying CFT. Moreover, we found the edge reconstruction of the IQH does not change any of the prefactors due to the conservation of the chirality.

Here we should note that the correlation matrix method is only applicable to the non-interacting case, such as the IQH state. For the interacting case, namely, the fractional quantum Hall (FQH) states, the direct calculation of the reduced density matrix with breaking the rotational symmetry on a disk is complicated and limited to a small system size. However, as was expected from the charge fluctuation calculations, we believe that our bipartite method is also applicable, and the corner contribution in the FQH states still obeys the same universality, especially in the limit θ → 0 where the charge densities at the tip are the same. The FQH edge also contributes a logarithmic type of EE which has a prefactor with its corresponding central charge.