Bulk-Edge Correspondence in Fractional Quantum Hall States
Abstract
We substantiate a complete picture of the “bulk-edge correspondence” conjecture for topological phases. By studying the eigenstates in the entanglement spectrum for both the ideal and realistic Coulomb ground state of the fractional quantum Hall system, it is verified that the eigenstates in the universal part of the entanglement spectrum purely lie in the Hilbert space of the edge excitations projected onto the physical Hilbert space of the subsystem itself. Hence, not only the eigenlevels in the entanglement spectrum are in one-to-one correspondence with the eigenenergies of an effective dynamical edge Hamiltonian, but all the eigenstates are confirmed to be the actual (projected) edge excitations of the subsystem. This result also reveals the possibility of extracting the full information of the edge excitations from the state of the subsystem reduced from a geometric cut of the pure ground state of the total system in topological phases.
Since the discovery of the fractional quantum Hall effect Tsui et al. (1982); Gir (1987), a new paradigm of phases of matter which is of a topological nature Wen (1989); Wen and Niu (1990); Nayak et al. (2008) and yet does not fit into the conventional symmetry-breaking picture comes into play. These novel phases of matter exhibit spectacular behaviors, such as fractional charges Laughlin (1983), (non-)Abelian anyons Leinaas and Myrheim (2007); ?; ?, or topologically protected gapless edge excitations. In absence of local order parameters, these phases are usually characterized by certain global properties such as the ground state degeneracy Wen and Niu (1990). Despite intense effort, a complete understanding of such topological ordered phases still remains elusive.
Among various approaches to characterize these intriguing zero-temperature phases of matters, entanglement plays an increasingly crucial role Amico et al. (2008). It is believed that ground states of topological ordered systems are able to build up long range entanglement Chen et al. (2010). One way to study the entanglement in a many-body system is to look at the correlation between a subsystem and its complement in the total system. After the total system has been prepared in a pure ground state , the reduced density matrix of a subsystem is obtained by tracing over the degrees of freedom of the outside subsystem , . The Von Neumann entropy of subsystem A, , usually called the entanglement entropy, captures the entanglement between subsystem A and the rest of the total system. In typical ground states of many-body systems, the entanglement entropy follows an area scaling law Eisert et al. (2010). For a state in a topological phase, besides the usual term that is proportional to boundary area, there emerges a constant sub-leading term that is characteristic of the topological nature of the system. Coined as topological entanglement entropy Hamma et al. (2005); Levin and Wen (2006); Kitaev and Preskill (2006), this quantity is one of the most recognized signatures for states with topological phases.
In subsequent work Li and Haldane (2008), Li and Haldane proposed and numerically substantiated that more information about the underlying topological state can be obtained by studying instead the full spectrum of the reduced density matrix of the subsystem. Writing , the entanglement spectrum is defined as the spectrum of the effective Hamiltonian . The eigenvalues and eigenstates of can be alternatively extracted from the Schmit decomposition of the total state,
(1) |
Though , in general, is not a real dynamical Hamiltonian, it is ultimately related to the physical edge of the subsystem: In the quantum Hall system, the number of the low-lying universal eigenenergies, which are separated by a finite gap Li and Haldane (2008); Thomale et al. (2010) from other generic levels, are in one-to-one correspondence with the edge modes describable by a macroscopic edge theory, at least for small angular momentum excitations.
Both the entanglement entropy and the counting structure of the entanglement spectrum can be naturally interpreted if a relation between the bulk and the physical edge of the subsystem is established. This relation, termed the bulk-edge correspondence Qi et al. (2012); Swingle and Senthil (2012); Dubail et al. (2012a), states that the entanglement Hamiltonian corresponds to an effective dynamical Hamiltonian in an edge theory. The bulk-edge correspondence, originally sketched in the early work of entanglement entropy Kitaev and Preskill (2006), has been extensively studied in various fields Liu et al. (2013); Santos (2013); Ho et al. (2015); Chen et al. (2016); Ho et al. (2017); Koch-Janusz et al. (2017). This correspondence emerged in part from general arguments based purely on the topological properties of the system and on the standard renormalization-group method Qi et al. (2012), in part from observations of geometric aspects and the Lorentz invariance of the emergent effective theory Swingle and Senthil (2012), and in part through study of model wavefunctions specific to the quantum Hall system constructed from conformal blocks Dubail et al. (2012a). These arguments are only valid provided certain assumptions hold in the thermodynamic limit.
In this work, we argue that a complete picture of the bulk-edge correspondence should consist of two pieces. One is that the eigenvalues are in one to one correspondence with the eigenvalues of an effective edge Hamiltonian. The other is that the universal eigenstates that appear in the bulk density matrix are all real edge states of the subsystem
The system studied in this work consists of a few spinless fermions in the lowest Landau level on a two-dimensional disk. The plane geometry is a convenient platform for the discussion of edge physics. System partitions are restricted to the subsets of the single particle magnetic orbitals in the symmetric gauge (with unnormalized analytic wavefunction and angular momentum quantum number ). In the lowest Landau level, these orbitals have localized ring shapes. Thus the orbital partition mimics approximately the real space partition. It is worthwhile to mention that another commonly used scheme, based on a cut of particle number Haque et al. (2007); Zozulya et al. (2007); Sterdyniak et al. (2011), has usually been adopted for the study of quasi-hole excitations. The level counting in the orbital entanglement spectrum and the one in the particle entanglement spectrum, which correspond to quasi-hole excitations in the bulk, were shown Chandran et al. (2011) to be ultimately connected. This relation, also termed as “bulk-edge correspondence”, is not to be confused with the one we are considering in the present work.
In the subsystem, the particle number and total angular momentum are still good quantum numbers; Thus, the reduced density matrix of a subsystem contains distinct sectors which do not couple with each other. Fig. 1 compares the standard entanglement spectrum for the ideal Laughlin state with that of the realistic Coulomb ground state of 10 particles in the lowest Landau level. For entanglement spectra of the quantum Hall states in other geometry and partition schemes, see also Refs. Zozulya et al. (2007); Läuchli et al. (2010); Sterdyniak et al. (2011); Dubail et al. (2012b); Sterdyniak et al. (2012); Rodríguez et al. (2012). The subsystem studied in this work involves the inner most 14-orbitals, which is the largest system that is sufficiently far away from the edge of the total system, in the sense that it is still in the linear region of the entanglement entropy, as is indicated in the inset of Fig. 1. In this region, edge effects are supposed to be removed as much as possible. By analyzing the fraction of each eigenstate that resides in the Hilbert space of edge modes (by the procedure explained in the following), we are able to identify all the universal levels apart from the generic levels in the case of Coulomb interactions. It is observed that the universal and generic parts barely mix with each other, albeit at larger angular momentums they start to mix, in contrast to the low angular momenta, where there is a finite gap between them.
To prove the bulk-edge correspondence, on the “edge” part, we use the well-known series of model wavefunctions Wen (1992) for the (neutral) edges excitations of the Laughlin ground state. These analytic model wavefunctions are generated by the power sum symmetric polynomials , . To build an edge mode with angular momentum (this is the additional angular momentum carried by the edge mode beyond the ground state angular momentum), one generates an integer partition , where are positive integers. The corresponding (unnormalized) edge state is constructed as the product of the Laughlin wavefunction and a symmetric polynomial:
(2) |
where is the filling factor. For a given , the number of different modes is thus the number of integer partitions of . These states are the only zero energy states for the case of p-wave contact interactions (equivalent to the Haldane pseudo-potential Haldane (1983) in the lowest Landau level), and they describe the gapless edge excitations of the Laughlin ground state. In the thermodynamic limit, the Hilbert space of the edge modes generated by the order- power sum polynomials is identical to that generated by the U(1) Kac-Moody algebra Wen (1990a, b, 1992) in a macroscopic theory of the edges physics. However, for finite size systems, when the orders of the symmetric polynomials become larger than the number of particles, these states are no longer linearly independent. In fact, the number of linearly independent symmetric polynomials of order is the number of partitions of into at most parts, where is the number of particles and the number of variables in the polynomial. Denote this restricted partition as . For each such integer partition , one alternative way to construct the edge state is to use the corresponding order- monomial symmetric polynomial , where the summation is taken over all permutations of such that the polynomial is symmetrized. The leads to a correction of the number of edges modes for finite size systems compared with the macroscopic theory for systems in the thermodynamical limit.
Another finite size effect arises when comparing the edge modes with the eigenstates in the entanglement spectrum. That is, the ideal edge modes might involve single particle orbitals that are outside the region of the subsystem under consideration. (Fig.4 shows the actual spectra of the subsystem and the edge state spectra with open boundary condition.) These additional degrees of freedom need to be traced out in formatting the appropriate Hilbert space of the subsystem Dubail et al. (2012a). Precisely speaking, the Fock space spanned by the free edge states are projected onto a subspace involving only orbitals that are resident in the subsystem. This inevitable projection procedure was also discussed in Ref. Dubail et al. (2012a) for the case the real space partitions. In this manner, the projections of the original linearly independent edge modes might become linearly dependent, such that the projected Fock space could have a smaller dimension than the original Fock space of the free edge states. Table 1 lists the number of free edge modes, the number of independent projected edge modes, and the number of observed energy levels in the entanglement spectrum for the 5- and 4-particle sectors. The observed level counting in the primary 5-particle sector of the entanglement spectrum matches precisely the reduced dimension of edge modes Hilbert space at every angular momentum sector. In the large particle number limit, the restricted integer partition reduces to the regular integer partition. This is in agreement with the original conjecture for the counting structure of the entanglement spectrum in Ref. Li (2009).
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | … | |
1 | 1 | 2 | 3 | 5 | 7 | 10 | 13 | 18 | 23 | 30 | 37 | 47 | 57 | 70 | 84 | … | |
1 | 1 | 2 | 3 | 5 | 7 | 9 | 11 | 14 | 16 | 18 | 19 | 20 | 20 | 19 | 18 | … | |
1 | 1 | 2 | 3 | 5 | 7 | 9 | 11 | 14 | 16 | 18 | 19 | 20 | 20 | 19 | 18 | … | |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | … | |
1 | 1 | 2 | 3 | 5 | 6 | 9 | 11 | 15 | 18 | 23 | 27 | 34 | 39 | 47 | 54 | … | |
1 | 1 | 2 | 3 | 5 | 6 | 9 | 11 | 14 | 16 | 19 | 20 | 23 | 23 | 24 | 23 | … | |
1 | 1 | 2 | 3 | 4 | 5 | 7 | 7 | 8 | 8 | 8 | 7 | 7 | 5 | 4 | 3 | … |
In order to prove that the universal part of the entanglement spectrum is indeed spanned by the edge modes, we compute the probability for each eigenstate in the entanglement spectrum to reside in this space. The projection probability is defined as
(3) |
where is the projection operator to the Hilbert space of projected edge modes.
For the ideal Laughlin wavefunction, it is confirmed that all the eigenstates in the entanglement spectrum have identity projection probabilities in the Hilbert space of the projected edge states. For the case of Coulomb ground state, both the ideal edge states and the real edge modes of the subsystem are used to analyze the projection probability Eq.(3). The real edge modes are computed from the Coulomb interaction at the corresponding angular momenta with open boundary condition (without performing single particle orbital cut). In the thermodynamic limit, they are low-lying modes describing the gapless edge excitations. However, in few-body calculations, there are usually no clear gaps separating the edge modes and the bulk excitations. In this case, we identify each edge state with the aid of high (0.97) overlaps with the ideal edge wavefunctions (2). Fig. 2 and Fig. 3 show the projection probability patterns. Both the primary 5-particle sector and the sector with particle number offset (4-particle sector in this case) show clear high probability plateaus for the universal levels in the entanglement spectrum. In the 4-particle sector, since the number of universal levels is smaller than the dimension of the projected edges modes (as was shown in Table.1), the generic levels also have fluctuating non-zero probabilities, in contrast to 5-particle sector, where all the generic levels have nearly zero projection probabilities. They are nevertheless clearly separated from the universal part.
In the primary sector, since all universal eigenstates in the entanglement spectrum have nearly identity probabilities in the corresponding Hilbert space of projected edge modes, it is concluded that the information of all edge excitations is hidden in the pure ground state of the total system, which can be extracted from the bulk state of the subsystem. This can be viewed as a complementary effect to a similar result in Ref. Chandran et al. (2011), where it is observed that the primary sector of the entanglement spectrum under particle partition spans the entire space of quasi-hole excitations of the subsystem.
To further firm up the evidence of the bulk-edge correspondence, we present in Fig. 4 the actual energy spectrum of 5 particles under Haldane interaction with and without boundary conditions. The degenerate zero-energy states for the latter are model edge states. If we did not use the mechanism of projecting the edge states of the subsystem with open boundaries, the nondegenerate low-lying states of the subsystem alone would have been inadequate to explain all the observed universal levels in the entanglement spectrum (Fig. 1).
In conclusion, this work shows that the universal part of the subsystem reduced density matrix, by orbital partition, of the fractional quantum Hall state lie in the Hilbert space of edge modes of the subsystem itself, hence completes the picture of the bulk-edge correspondence conjecture. This result also reveals the possibility of extracting edge excitations from the state of the subsystem reduced by geometric cut of the pure ground state of the total system in topological phases. In the future, it will be informative to apply this method to other topological phases.
The work of BY and CHG have been supported in part by National Science Foundation grant PHY-1607180. R.R.B. was supported by Purdue University startup funds.
Footnotes
- preprint: APS/123-QED
- To be precise, the universal levels stands for the ones appear in the entanglement spectrum of the ideal model wavefunctions, as well as the part of the spectrum having similar structures for the case of generic interactions. By comparing the entanglement spectrum eigenstates with the actual edge excitations, we are able to identify every universal state (See Fig. 1 for an example).
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