Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands

Zhu, Wei; Haldane, F. D. M.

doi:10.1103/physrevb.88.035122

Cited by 45 publications

(51 citation statements)

References 41 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The ED calculations further confirm this state on a N = 3 × 4 × 3 cluster by extracting modular transformation matrix5152 from the MESs of two noncontractable cuts (see Supplementary Information for more details).…”

Section: Resultsmentioning

confidence: 63%

See 1 more Smart Citation

Emergent Chiral Spin Liquid: Fractional Quantum Hall Effect in a Kagome Heisenberg Model

Gong

Zhu

Sheng

2014

Sci Rep

Self Cite

258

298

View full text Add to dashboard Cite

The fractional quantum Hall effect (FQHE) realized in two-dimensional electron systems under a magnetic field is one of the most remarkable discoveries in condensed matter physics. Interestingly, it has been proposed that FQHE can also emerge in time-reversal invariant spin systems, known as the chiral spin liquid (CSL) characterized by the topological order and the emerging of the fractionalized quasiparticles. A CSL can naturally lead to the exotic superconductivity originating from the condense of anyonic quasiparticles. Although CSL was highly sought after for more than twenty years, it had never been found in a spin isotropic Heisenberg model or related materials. By developing a density-matrix renormalization group based method for adiabatically inserting flux, we discover a FQHE in a isotropic kagome Heisenberg model. We identify this FQHE state as the long-sought CSL with a uniform chiral order spontaneously breaking time reversal symmetry, which is uniquely characterized by the half-integer quantized topological Chern number protected by a robust excitation gap. The CSL is found to be at the neighbor of the previously identified Z2 spin liquid, which may lead to an exotic quantum phase transition between two gapped topological spin liquids.

show abstract

Section: Resultsmentioning

confidence: 63%

“…To demonstrate the nature of the new quantum phase, we first find the minimum entangled states (MESs) in each topological sector285251, which represent the eigenstates of the Wilson-loop (string-like) operators encircling the cylinder and are the simplest states of the quasiparticles. In Fig.…”

Section: Resultsmentioning

confidence: 99%

Emergent Chiral Spin Liquid: Fractional Quantum Hall Effect in a Kagome Heisenberg Model

Gong

Zhu

Sheng

2014

Sci Rep

Self Cite

258

298

View full text Add to dashboard Cite

show abstract

“…We would like to mention some possible future research directions following the present work. It is possible to further explore or utilize the FCI/FQAH wave functions at hand now, e.g., to compare them with the lattice Laughlin states from conformal field theory [43][44][45] and the Gutzwiller-projected parton wave functions [52][53][54][55], and extracting the quasi-particle fractional statistics via modular matrices [56][57][58][59]. It would be also very interesting to extend the present approach to non-Abelian FCI/FQAH states [60][61][62], FCI/FQAH states in high-Chern-number TFBs [63][64][65], and also hierarchical FCI/FQAH states [66,67].…”

Section: Summary and Discussionmentioning

confidence: 99%

Wave functions for fractional Chern insulators in disk geometry

Luo

Wang

et al. 2015

New J. Phys.

View full text Add to dashboard Cite

Recently, fractional Chern insulators (FCIs), also called fractional quantum anomalous Hall (FQAH) states, have been theoretically established in lattice systems with topological flat bands. These systems exhibit similar fractionalization phenomena to the conventional fractional quantum Hall (FQH) systems. Using the mapping relationship between the FQH states and the FCI/FQAH states, we construct the many-body wave functions of the fermionic FCI/FQAH states in disk geometry with the aid of the generalized Pauli principle (GPP) and Jack polynomials. Compared with the ground state by the exact diagonalization method, the wave-function overlap is higher than 0.97, even when the Hilbert space dimension is as large as 3×10 6 . We also use the GPP and the Jack polynomials to construct edge excitations for the fermionic FCI/FQAH states. The quasi-degeneracy sequences of fermionic FCI/FQAH systems reproduce the prediction of the chiral Luttinger liquid theory, complementing the exact diagonalization results with larger lattice sizes and more particles. OPEN ACCESS RECEIVED exclusion principle corresponds to the Fermi statistics, the GPP corresponds to the fractional statistics of anyons.Recently, finding the suitable trial wave functions of FCI/FQAH states has become an outstanding problem. It is conjectured that there is a one-to-one mapping between IQH and CI/QAH states [10,[34][35][36][37]. In other words, the single-particle orbitals of an LL can be mapped to the ones in a TFB. Based on one-dimensional (1D) maximally localized Wannier functions, Qi et al have used a mapping between the FQH and the FCI/FQAH states to construct generic wave functions of FCI/FQAH states in cylinder geometry with these 1D Wannier functions [10,[34][35][36][37]. In terms of this mapping relationship, the Haldane pseudo-potential [26] for these FCI/ FQAH states can be constructed [38,39] through a suitable gauge choice for the Wannier functions. An improved prescription has been adopted to construct variational wave functions of FCI/FQAH states in torus geometry by utilizing gauge-fixed (non-maximally) localized Wannier states [40][41][42]. From another aspect, conventional FQH states can also be obtained for 2D lattices analytically by using conformal field theory [43][44][45]. In contrast to the above analytical or semi-analytical approaches, we here pursue a very direct yet effective purely numerical prescription to construct FCI/FQAH states in disk geometry, without any variational parameter or adjustable gauge freedom, but just utilizing the powerful GPP and the Jacks structure of FQH states, and the information from exact numerical single-particle orbitals of TFB models.In this paper, we exploit the single-particle wave functions of CI/QAH states (in the Kagomé model [5, 46] and the Haldane [3, 9] model) with TFB parameters in disk geometry, and explore the polynomial structure of the continuum Laughlin wave functions [25] to establish the many-body wave functions of the FCI/FQAH states. We firstly construct finite-size lattices in...

show abstract

“…Therein, many experimental and theoretical investigations [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43] are carried out and various interesting phenomena are discovered. Also, the electron spin polarization is investigated for the FQH states with ] > 2 by the papers [44][45][46][47][48][49].…”

Section: Introductionmentioning

confidence: 99%

Spin Polarization Curve of Fractional Quantum Hall States with Filling Factor Smaller than 2

Sasaki

2013

ISRN Condensed Matter Physics

View full text Add to dashboard Cite

Kukushkin et al. have measured the electron spin polarization versus magnetic field in the fractional quantum Hall states. The polarization curves show wide plateaus and small shoulders. The 2D electron system is described by the total Hamiltonian ( + ). Therein, is the sum of the Landau energies and classical Coulomb energies. is the residual interaction yielding Coulomb transitions. It is proven for any filling factor that the most uniform electron configuration in the Landau states is only one. The configuration has the minimum energy of . When the magnetic field is weak, some electrons have up-spins and the others downspins. Then, there are many spin arrangements. These spin arrangements give the degenerate ground states of . We consider the partial Hamiltonian only between the ground states. The partial Hamiltonian yields the Peierls instability and is diagonalized exactly. The sum of the classical Coulomb and spin exchange energies has minimum for an interval modulation between Landau orbitals. Using the solution with the minimum energy, the spin polarization is calculated which reproduces the wide plateaus and small shoulders. The theoretical result is in good agreement with the experimental data.

show abstract

Minimal entangled states and modular matrix for fractional quantum Hall effect in topological flat bands

Cited by 45 publications

References 41 publications

Emergent Chiral Spin Liquid: Fractional Quantum Hall Effect in a Kagome Heisenberg Model

Emergent Chiral Spin Liquid: Fractional Quantum Hall Effect in a Kagome Heisenberg Model

Wave functions for fractional Chern insulators in disk geometry

Spin Polarization Curve of Fractional Quantum Hall States with Filling Factor Smaller than 2

Contact Info

Product

Resources

About