Entanglement Entropy of Random Fractional Quantum Hall Systems

The entanglement entropy of the $\nu = 1/3$ and $\nu = 5/2$ quantum Hall states in the presence of short range random disorder has been calculated by direct diagonalization. A microscopic model of electron-electron interaction is used, electrons are confined to a single Landau level and interact with long range Coulomb interaction. For very weak disorder, the values of the topological entanglement entropy are roughly consistent with expected theoretical results. By considering a broader range of disorder strengths, the fluctuation in the entanglement entropy was studied in an effort to detect quantum phase transitions. In particular, there is a clear signature of a transition as a function of the disorder strength for the $\nu = 5/2$ state. Prospects for using the density matrix renormalization group to compute the entanglement entropy for larger system sizes are discussed.


I. INTRODUCTION
This paper is a numerical study, using direct diagonalization, of the entanglement entropy of fractional quantum Hall systems in the presence of a delta correlated random potential.
The entanglement entropy, quite distinct from the thermodynamic entropy, is the Von Neumann entropy of the reduced density matrix of a subsystem and is a quantitative measure of the entanglement of the subsystem with the system. Our interest in this subject is two-fold; firstly, it has been proposed that entanglement entropy can be used as a tool to characterize fractional quantum Hall states. More precisely, Kitaev and Preskill 1 and Levin and Wen 2 have shown, for a topologically ordered state, that the entanglement entropy of a subsystem obeys an asymptotic relation where L is the linear size of the subsystem (the area law ) and γ is a universal quantity, the topological entanglement entropy, the natural logarithm of the quantum dimension.
For this scaling law to apply, the system must be very large and the subsystem must be large (compared to a cutoff, but the subsystem must be small compared to the system).
This is a rather formidable numerical requirement, however, there has been some success numerically 3-8 using (1) to extract the topological entanglement entropy of quantum Hall states. One may hope, that by adding weak randomness, there may be less system size dependence and hence it will be easier to obtain the topological entanglement entropy. Of course, by adding randomness, momentum conservation is destroyed and one cannot treat as large systems by direct diagonalization. In any case, it is of interest to see if the topological entanglement entropy can be calculated in the presence of weak disorder and to see if the values obtained are consistent with previous numerical estimates.
The second motivation to undertake this study, is to see whether the entanglement entropy can be used to detect transitions between phases of quantum Hall systems. For example, experimentally, it is well known that fractional quantum Hall states are particularly sensitive to disorder. Can this sensitivity be detected in the entanglement entropy? The two questions discussed above will be studied for 2 filling factors ν = 1/3 in the lowest Landau level, representative of Laughlin states, and the 5/2 th state in the second Landau level. Currently, there is good evidence both experimentally and numerically 9 that the essential physics of the 5/2 state is given by the Moore-Read wave function and thus the 5/2 state is representative of the more exotic states with non abelian statistics.
The paper is then organized as follows: in the second section, the model and the numerical method are briefly described and the results for the topological entanglement entropy for weak disorder are discussed. In the third section, the entanglement entropy is calculated as a function of disorder strength for a wider range of disorder to determine whether transitions between phases of Hall systems can be detected. In the fourth section, some preliminary results using the density matrix renormalization group to calculate the entanglement entropy are described. The fifth section is a summary and gives conclusions. In the final section, a recent alternative method 23 to obtain the topological entanglement entropy on the torus is discussed.

WEAK DISORDER
The numerical method we have used is direct diagonalization applied to square (aspect ratio 1) clusters with periodic boundary conditions (the square torus geometry). The Landau gauge is used for the vector potential. Spin polarized electrons are confined to a single Landau level and interact with a pure Coulomb potential. One can approach the limit of very large system sizes through clusters of any fixed aspect ratio and since we are concerned with quantum liquid states, aspect ratio one has been chosen. This numerical approach has previously been used to study the entanglement entropy without a disorder potential 5,7 . The random potential 10 U(r) is taken to be delta correlated i.e. < U(r)U(r ′ ) >= U 0 δ(r − r ′ ) and the disorder strength will be given in terms of a parameter U R = 3U 0 /2. Since momentum is not conserved, one is limited to smaller system sizes then for a disorder free system. In particular, the largest system size treated for ν = 1/3 is 10 electrons in 30 orbitals with a state space of approximately 30X10 6 and 14 electrons in 28 orbitals for ν = 5/2 with a state space of approximately 40X10 6 . (This is in contrast to the disorder free case, ν = 1/3 13 electrons in 39 orbitals , and ν = 5/2 18 electrons in 36 orbitals, are relatively straightforward to treat).
To calculate the entanglement entropy, we take a subsystem consisting of l adjacent orbitals (recall in the Landau gauge, these orbitals consist of strips oriented along, say the y-axis, of width of order the magnetic length). The reduced density matrix is straightforward to compute from the ground state wave function. It is then diagonalized giving the eigenvalues λ j from which the l-orbital entanglement entropy S(l) , S(l) = − j λ j ln λ j is obtained.
This procedure is done for every realization of the random potential, the results are then averaged to give < S(l) > where <> denotes average over the random potential. The position of the subsystem has been fixed, that is, for say S(l = 3) the subsystem always consists of the 1st , 2nd and 3rd orbitals. coming from the 2 boundaries of the subsystem. However, as will be discussed below, the excellent agreement may be fortuitous in that for the small system sizes considered γ tends to be overestimated at this filling.
The dependence of the topological entanglement entropy on system size for filling 1/3 is shown in figure 3 for U R = 0.01. In this figure γ is plotted vs 1/N (N=number of orbitals).
Clearly, it would be desirable even with disorder, to be able to treat larger system sizes.
Another approach to obtain the topological entanglement entropy is, for a given < S(l) >, to do a linear extrapolation in 1/N yielding < S * (l) >. < S * (l) > is then plotted vs √ l , a linear least squares fit is performed and the y intercept gives -γ. A plot of < S * (l) > vs.
√ l is shown in figure 4, for ν = 1/3 using systems with 21 to 30 orbitals to get the extrapolations.
The negative of the y intercept is given by 1. 30  γ 1/3 is considered. Using the < S * (l) > method (< S * (l) > obtained from the 3 largest system sizes) γ Laughlin ≈ 1.89. In any case, it appears the expected more generic behavior with weak randomness is unable to overcome the advantage of additional system sizes available to disorder free calculations. That is, by using the S*(l) method, and 2 more system sizes (without disorder), In an effort to characterize possible phase transitions with disorder, we have calculated the variance < S(l) 2 > − < S(l) > 2 . In figures 9 and 10, the variance for l = 12 is plotted for ν = 5/2 and ν = 1/3, respectively. For ν = 5/2, figure 9, the variance is nominal through the transition region (other then for the anomalous behavior of 10 electrons in 20 orbitals).
In contrast, for ν = 1/3, figure 10, there is a general increase of the variance starting at news is that this depends on the exponential of the √ s , however,the good news is that it does not depend on the exponential of s as in direct diagonalization. Hence, at least in principle, one should (if one can avoid being stuck in local minimum) be able to treat larger system sizes for quantum Hall systems by dmrg [16][17][18][19] . In particular, reference 18 was able to accurately calculate ground state energies for ν = 1/3 for up to 20 electrons and up to 26 electrons for ν = 5/2 in the spherical geometry. In the spherical geometry 14 electrons at ν = 1/3 and 20 electrons at ν = 5/2 are accessible to direct diagonalization. However, the excitation gap, a more difficult numerical quantity at ν = 5/2 was only accurately calculable by dmrg for up to 22 electrons, 1 "non-aliased" system size larger then that accessible to direct diagonalization. In this section, dmrg will be used to calculate the entanglement entropy for quantum Hall systems without disorder. We will be content, in this preliminary study, to use dmrg to study a large system size still accessible to direct diagonalization, that is, 12 electrons in 36 orbitals in the n=0 and n=1 Landau levels.
In table I we display, the ground state energy vs. m, the number of states kept in the block; the first column is for the lowest Landau level, the second for the second Landau level. (the Madelung energy, which can be calculated exactly, is not included).
One sees for the lowest Landau level a fairly accurate result can be obtained even without  with experiment, taken with due caution in that a quantitative comparison likely requires considering longer range disorder. In our study, one number, the entanglement entropy has been used to characterize the reduced density matrix. There is possibly additional information in the full spectrum of the reduced density matrix 14 , which has been shown to be related to the conformal field theory describing the one dimensional edge state of the quantum Hall state 8,14,24 . It would definitely be of interest 11 to study the entanglement spectrum in the present system. Even if the topological entanglement entropy (derived from the entanglement entropy) is a complete invariant 25 , numerically it may well be easier to see transitions using the entire spectrum 8,14 . Finally, we have displayed some preliminary results using dmrg to compute the entanglement entropy. These results indicate dmrg holds some promise in calculating the entanglement entropy in the lowest Landau level; it appears more difficult to do calculations in the second Landau level and to go much beyond systems that one can treat by direct diagonalization. This may indicate that potentially more powerful numerical methods, for example, tensor network states 21 or the methods of reference 28 , will prove useful.

VI. FINAL REMARKS
After this manuscript was posted at arXiv.org, we became aware of an interesting paper that calculates the topological entanglement entropy using a different method in the flat torus geometry. (We thank Dr. Haque for bringing this reference to our attention.) In essence, ref. 23 , calculates the entanglement entropy S(N/2) taking the subsystem to be half the system size. The scaling law S(N/2) ∼ c 1 N α − 2γ is then used where α is the aspect ratio and N is the number of orbitals in the system; this approach was also used by Shibata 20 . In the method described in section II (see also 5 i.e. l 2π N so the width goes to zero as 1 N . However, at the same time the width goes to zero, the length goes as √ 2πN. Although the width and length are both "singular" as N goes to infinity, the area is perfectly well defined, 2πl (again in units of the magnetic length squared). Since the area law relates the entanglement entropy to a linear dimension of the subsystem, it is reasonable that S(l) scales as the square root of the area, S(l) ∼ c √ l, and this is verified by explicit calculations.
It should be emphasized that neither approach is fully justified by the considerations in ref. 1,2 . At least for the current state of knowledge, the best justification for either method is that they give reasonable results where the physics is well understood, Laughlin states. This is true for both techniques, hence in principle, either technique can be used to calculate the topological entanglement entropy. That being said, since system sizes are limited, one technique may well be superior depending on the filling fraction in question. In particular, the method of reference 20,23 allows one to extract γ more accurately for ν = 1/3 from finite size calculations. As an illustration of this method, in figure 13, S(17) is plotted vs.