Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands

Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands

W. Zhu, D. N. Sheng, F. D. M. Haldane Department of Physics and Astronomy, California State University, Northridge, California 91330, USA Department of Physics, Princeton University, Princeton, NJ 08544, USA

We perform an exact diagonalization study of the topological order in topological flat band models through calculating entanglement entropy and spectra of low energy states. We identify multiple independent minimal entangled states, which form a set of orthogonal basis states for the groundstate manifold. We extract the modular transformation matrices () which contains the information of mutual (self) statistics, quantum dimensions and fusion rule of quasiparticles. Moreover, we demonstrate that these matrices are robust and universal in the whole topological phase against different perturbations until the quantum phase transition takes place.


Introduction.— The fractional quantum Hall (FQH) state is the best-known many-body state with topological order discovered in 2D electron systems under strong magnetic field. The most striking features of FQH state are the topological ground state degeneracy on torus and the emerging quasiparticles obeying fractional statistics Laughlin1983 (); Wen1989 (). Recently, it has been demonstrated that FQH states can also be realized in various topological flat-band (TFB) models without Landau levels Haldane1988 (); Tang (); Neupert (); Sun (); DNSheng2011 (); YFWang2011 (); Bernevig2011 (); Bernevig2012 (); YFWang2012 (); Bernevig2013 (); XLQi2012 (); Moller (); DXiao (); Shankar (); Venderbos (); ZLiu (); Kapit (); Shuo (); Sondhi (). In such an interacting system, an explicit demonstration of topological order and quasiparticle statistics is still highly desired, which has attracted lots of recent interests Wen1990 (); Kitaev (); Levin (); Haldane2008 (); Lauchli2010a (); HCJiang (); YZhang2012 (); Vidal (); Pollmann (); Tu ().

Entanglement measurements such as topological entanglement entropy (TEE) Kitaev (); Levin () and entanglement spectrum Haldane2008 () have been identified as powerful tools for detecting topological properties of many-body quantum states. Insightfully, Zhang et al. proposed to extract modular matrix through the entanglement measurement YZhang2012 (), which encodes the complete information of the topological order including quasiparticles quantum dimension and statistics as first described by Wen Wen1990 (). Based on the model wavefunctions for toric code and chiral spin liquid states, they demonstrated that the transformation between the minimal entangled states (MESs) along two interwinding partition directions gives rise to modular matrices. The new route to extract modular matrix through MESs improves the practical implementation for strongly interacting systems as such information is accessible through larger system density matrix renormalization group (DMRG) calculations demonstrated for bosonic FQH state in TFB Vidal () and fermionic FQH states with magnetic fieldPollmann (). However, it remains difficult to access multiple low energy states in DMRG in a controlled way, when there are coupling between different topological sectors induced by interaction or when the groundstates have higher degeneracy, which will be the focuses of our exact diagonalization (ED) study.

In this letter, we present an ED calculation for the TFB model and map out the entanglement entropy profile for superposition states of the near degenerating groundstates. We demonstrate that there are the same number of the MESs as the ground state degeneracy for FQH phase on a torus, which form the orthogonal and complete basis states for modular transformation. Through locating the MESs along two interwinding partition directions, we extract the modular matrices and containing generalized statistics of quasiparticles, which unambiguously demonstrate the fractional quasiparticle statistics in such systems for (bosons), and (fermions) FQH states, respectively. We also analyze the entanglement spectra and obtain TEE from the difference of the maximum and minimum of entanglement entropies of these superposition states. Furthermore, we study the quantum phase transition from FQH phase to the topological trivial phase driven by the disorder scattering or attractive anisotropic interaction. Significantly, the extracted modular matrices remain to be universal containing the same quasi-particle fractional statistics information as theoretical ones for the model FQH states in the whole topological phase until the quantum phase transition takes place. This is distinctly different from following the Berry phase of the ground states, where only the sum of the total Chern number remains invariantDNSheng2003 () due to the lifting of the degeneracy by perturbations for any finite size systems.

We study the Haldane model Haldane1988 () on the honeycomb (HC) lattice:


where creates a hard-core boson (or fermion) at site , is the boson (or fermion) number operator. , and denote the nearest-neighbor (NN), the next-nearest-neighbor (NNN) and the next-next-nearest-neighbor (NNNN) pairs of sites, and is the NN interaction. The last term models the Anderson on-site disorder randomly distributed in . On the HC lattice, we select the parameters , , and the magnitude of the hopping phase , which lead to a topological flat-band with flatness ratio about YFWang2011 (). We consider a finite system of unit cells (total number of sites ) with periodic boundary conditions. The filling factor is , where is the number of particles. We denote the momentum vector with as integer quantum numbers.

The entanglement entropy is defined by partitioning the full system into two subsystems A and B. Tracing out the subsystem , one can obtain the reduced density matrix of subsystem : , where is the many-body state of the full system. The Renyi entanglement entropy is defined as: . Here we focus on two noncontractible bipartitions on a torus, as shown in Fig. 1(a) as cut-I and cut-II, respectively.

Multiple MESs as superpositions of near degenerating groundstates.— In TFB lattice model, it has been identified that there are near degenerating groundstates at filling factor YFWang2011 (); DNSheng2011 () when the interacting system realizes a FQH phase. Let us first consider a HC lattice filled with hard-core bosons at half-filling YFWang2011 (). We set the NN interaction to be zero since hard-core bosons are intrinsically interacting. From ED calculation, we find the two groundstates and both in the same momentum sector . This is the general case as long as the two system lengths and are factors of the particle number . Now we form the general superposition state as,

where and are the real parameter and the relative phase of the state respectively, while . For each state , we construct the reduced density matrix and obtain the corresponding entanglement entropy. In Fig. 1(b), we draw the in the surface and contour plots so that the peaks in entropy show up clearly representing the minimums of . We identify two peak structures in parameter space corresponding to two independent MESs:


The minimal entropies at the two peaks are different with and respectively, indicating the finite size effect. However, we find the relative phase difference between the two MESs is and consequently the two MESs are approximately orthogonal to each other: . Due to the rotation symmetry in the system, the MESs along cut-II are related to as , where is the rotation operator.

Figure 1: (Color online) (a) Haldane model on HC lattice with lattice vectors . The arrow directions on red dotted lines present the signs of the phases in the NNN hopping terms. The NNNN hoppings are represented by the blue dashed lines. The two ways to bipartition the system along dashed lines are labeled as cut-I and cut-II. (b) Surface and contour plots of Renyi entanglement entropy () of wavefunction on HC lattice filled with hard-core bosons. (c) The entropy () of wavefunction on HC lattice with interacting fermions.

Now we further examine the relation between the MESs and the degeneracy of the ground state manifold by studying the TFB model filled with fermions at . We consider a HC lattice with fermions with repulsive NN to stabilize the FQH phase DNSheng2011 (). In the ED study, we find three quasi-degenerating groundstates , (with ) in momentum sectors , respectively. We search for the superposition states in the space of the groundstate manifold with minimal entropy using the following general wavefunctions:

where are real parameters and , are relative phases for the state, while can be obtained using normalization condition. For the bipartition along cut-I, we observe two key points: 1) We can locate three global minimal entropy states in the given parameter space, which always occur when ; 2) The relative phases of two different MESs and satisfy: , for . In Fig. 1(c), we show the surface and contour plots of the entropy of the state as functions of and while other parameters are fixed at so that MESs occur with varying the relative phases. The three MESs are determined as:


with state index . We find , , , corresponding to minimum entropies , respectively. Very importantly, the three MESs we found are nearly orthogonal to each other: , and , which is the necessary condition for these states to form the basis states for modular transformation. The small overlap is a finite size effect as the MESs become the true ground states only in the thermodynamic limit. Since there is no rotation symmetry in lattice, we separately locate three MESs in the parameter space for the partition along cut-II. Here, we find that each groundstate is indeed the MES


In general, if the groundstates have different momentum along the entanglement cut direction, these states are eigenstates with definite number of quasiparticles, and thus any form of mixing will increase the entropy of the state.

We have also studied the TFB model on checkboard (CB) lattice Sun (); DNSheng2011 () and obtained similar results. Interestingly, we also identified a four fold degenerating MESs at filling corresponding to a FQHYFWang2011 (); supplement ().

Figure 2: (Color online) Eigenvalues of the reduced density matrix of two MESs (square) and two maximal entangled states (diamond) for (a) and (b) HC lattice with hard-core bosons at . The number near the dots shows the degeneracy. Crossed blue (purple) dots stand for the combination of as described in the text.
system lattice size GSM
HB on HC Y 0.849 1.232
HB on HC N 0.693 0.999
FM on HC N 1.125 1.024
HB on CB Y 0.774 1.117
HB on CB N 0.693 0.999
FM on CB N 1.128 1.026
Table 1: The comparison of the calculated TEE (for cut-I) and the theoretical values for Laughlin states. HB and FM denote hard-core boson and fermion systems, respectively. HC and CB represent Honeycomb and checkboard lattices. ’Y(N)’ means the groundstates have the same (different) momentum.

Modular transformation matrix based on MESs.— The generalized quasiparticle statistics of a topological ordered state is captured by the modular matrix and as first proposed by Wen Wen1990 (). () determines the mutual (self) statistics of the different quasiparticles as well as the quantum dimension and fusion rules of quasiparticles SDong (); ZHWang (); Fendley (). In general, the relationship between the modular matrices and MESs is , where are integers determined by specific modular transformation on a lattice YZhang2012 ().

Specially, for a HC lattice filled with hard-core bosons, the rotation symmetry leads to the overlap YZhang2012 (); Vidal (). Thus by computing the overlap using states from Eq. Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands, we obtain,

which are nearly identical to the theoretical onesSDong (); ZHWang (); Fendley () for the model FQH state: , . From , we determine the quantum dimension for two type of quasiparticles as , and total quantum dimension (close to ). show that one quasiparticle as a boson while indicates another quasiparticle acquires a phase encircling themselves. Combined with the topological spin from , we identify that these quasiparticles are semions ZHWang (). For HC lattice filled with interacting fermions, the overlap between Eq. 3 and Eq. 4 gives YZhang2012 ():


The obtained result is close to the analytic prediction Wen1990 (); ZHWang (): , where . The extracted mutual statistics between quasiparticles reflects the statistics. Within the same route, we also obtain modular matrix for FQH states on CBsupplement ():

, which is nearly the same as the one representing statistics: .

Figure 3: (Color online) (a-f) Disorder effect on MESs on HC lattice filled with hard-core bosons. (a) Energy spectrum (two lowest eigenvalues are labeled by diamond and cross) and (b) Particle entanglement spectrum (PES) for tracing out particles. There are states below the PES gap for , in good agreement with the counting of quasihole excitations in FQH state. Contour plot of entropy for (c) ; (d) ; (e) ; (f) . (g-l) Anisotropic interaction effect on MESs on HC lattice filled with fermions. (g) Energy spectrum (three lowest eigenvalues are labeled by blue square, red diamond and green cross) and (h) PES for tracing out particles. There are states below the PES gap for . Contour plot of entropy for (i) ; (j) ; (k) ; (l) . Bipartition are all along cut-I direction.

Topological entanglement entropy.— For a topological ordered state, one can also identify a topological term in the entanglement entropy since , where is the length of the smooth boundary between two subsystems and the TEE term is quantized as with as the total quantum dimension Kitaev (); Levin (). Recently, it has been shown that TEE of Abelian FQH state can be extracted through Grover (), , where is the Renyi entanglement entropy corresponding to maximal (minimal) entangled state. To check out this relation, the calculated for different systems are shown in Table 1. Indeed, the obtained gives a good estimate of the quantized theoretical value for Laughlin state on torus Lauchli2010b (). For symmetric system of lattice, we obtained a bigger deviation between and , which may result from the strong coupling among the groundstates in the same momentum sector. To elucidate the physical difference between minimal and maximal entangled states, we further show entanglement spectra of these states in Fig. 2 Haldane2008 (); Lauchli2010a (). We find that the spectra of the maximal entangled states can be exactly recovered by reducing the density matrix eigenvalues by a factor ( for FQH state) for two sets of spectra of MESs and imposing them on top of each other: as shown as cross dots in Fig. 2.

Modular Matrix and Quantum Phase Transition.— Topological order is robust in the presence of any weak local perturbations, which can be used to characterize the topological phase. Here we first consider the disorder effect on bosonic state. As shown in Fig. 3(a-b), the energy spectrum remains two-fold quasi-degenerating protected by a spectrum gap until a disorder strength . Further calculation of particle entanglement spectrum (PES) reveals a gap at small and the number of states below this gap agrees with the number of quasihole excitations in a FQH state Bernevig2011 (). This PES gap disappears at signaling the quantum phase transition from the FQH phase to a topological trivial state. As shown in Fig. 3(c-d), there are two distinguishable valleys I and II in entropy for the states , and the corresponding MESs are always approximately orthogonal to each other. The modular matries obtained for an intermediate disorder strength are and , which remain to be very close to the exact results for bosonic Laughlin state. After the quantum phase transition at as shown in Fig. 3(e), there are still two valleys of MESs near and , however these two states start to have bigger overlap . The corresponding modular matrix and are qualitatively different from exact results for FQH state as: and . In particular, the quasi-particle statistics has changed with the matrix becomes unit matrix, which indicates we are in a topological trivial phase. At shown in Fig. 3(e), there is only one valley left corresponding to one MES state in parameter space indicating the lost of any feature of topological order.

Furthermore, we consider the effect of the anisotropic interaction for fermionic system by tunning the interaction on one NN bond, while keeping the other two at unit strength. Consistent with the geometrical theory of the FQHE Haldane2011 (); Haldane2012 (), we find that the topological state and its modular matrix remain to be universal insensitive to the strength of the additional repulsive interactions with no quantum phase transition. So we turn to the additional attractive interaction on one bond. From both energy spectrum and PES, we identify a quantum phase transition which appears between and as shown in Fig. 3(g-h). As shown in Fig. 3(i-l), in the FQH phase, there are three minimal entropy valleys in parameter space while we take , which are the optimized values for all these systems to minimize the entanglement entropy. In FQH phase, the calculated modular matrix is always nearly identical to the expected theoretic result for Laughlin state. Taking as an example, from the overlap of the MESs we extract . After the phase transition occurs (Fig 3(k-l)) at , we can only locate one minimal entropy valley in parameter space, which demonstrates the disappearance of the FQH phase.

Summary and discussion.— We study the structure of MESs in the space of the groundstate manifold obtained from ED calculations. By calculating the overlap between different MESs, we obtain modular matrices for different FQH systems. The obtained and matrices faithfully represent the quasiparticle dimension and fractional statistics for systems with anisotropic interactions and random disorder scattering until a quantum phase transition takes place.

Acknowledgements. We thank Shoushu Gong for discussions. This work is supported by the US DOE Office of Basic Energy Sciences under Grant No. DE-FG02-06ER46305 (DNS) and NSF under grants DMR-0906816 (WZ) and the Princeton MRSEC Grant DMR-0819860 (FDMH). DNS also acknowledges the travel support by the Princeton MRSEC.


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Appendix A Supplemental material for:“Minimal Entangled States and Modular Matrix for Fractional Quantum Hall Effect in Topological Flat Bands”

In the main test, we focus on the topological flat-band (TFB) model on honeycomb lattice and extract the modular matrix and related quasiparticle statistics through locating the minimal entangled states (MESs). In this supplemental material, we apply the similar route on checkboard lattice and we focus on searching the topological order of fractional quantum Hall (FQH) state at filling factor YFWang2011 ().

The Hamiltonian for checkerboard lattice filled with hard-core bosonsYFWang2011 (); Sun ():


where creates a hard-core boson at site , is the boson number operator. , and denote the nearest-neighbor (NN), the next-nearest-neighbor (NNN) and the next-next-nearest-neighbor (NNNN) pairs of sites. We adopt the parameters , , and , which leads to a TFB with the flatness ratio about . To stabilize the FQH phase at filling factor , we set NN interaction following the previous workYFWang2011 ().

We consider a checkboard lattice with five hard-core bosons. In the exact diagonalization study, there are four near degenerating eigenstates which are separated from higher eigenstates by a finite spectrum gap. The four ground states () lie in momentum sector ,, and , respectively. Now we form the general superposition state from the four quasi-degenerating ground states,

where are the real parameters and are the relative phase of the state respectively. For the bipartition along cut-I, we find that each groundstate is indeed the MES due to four ground states having different quantum number along the cut-I direction:


For the partition along cut-II, it is found that the MESs appear when the four ground states are in equal magnitude superposition: . As shown in Fig. 4, we show the entropy of wavefunction in space by setting . The color of dots represents the magnitude of entropy. For simplicity, we just show the points with entropy smaller than . It is clear that there exist four valleys in space. The four valleys corresponding to four independent MESs as:


with state index and , , , , corresponding to minimum entropies , respectively. The four MESs are nearly orthogonal to each other: , , , , which forms orthogonal basis states for modular transformation.

Figure 4: Top: Checkboard lattice with basis vectors . The arrow directions present the signs of the phases in the NN hopping terms. The two ways to partition the system along the dashed lines are labeled as cut-I and cut-II, respectively. Bottom: The entropy of wavefunction on checkboard lattice with interacting hard-core bosons by setting . Here we only show the entropy smaller than . The calculation is for bipartition system along cut-II direction.

As described in the main text, modular matrix can be obtained through the overlap between MESs along two partition direction: YZhang2012 (). Using Eq.7 and Eq.8, we obtain,

, which is nearly the same as statistics prediction ZHWang () up to correction:

The modular matrix clearly demonstrates topological order of FQH states at . For example, from we determine: (i)There are type quasiparticles in the system labeled by the charges , where ; (ii)The quantum dimension of quasiparticles are all thus the total quantum dimension ; (iii)The fusion rule: , where ZHWang ().

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