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arXiv:1206.5773 · Physical Review B

Gauge-fixed Wannier wave functions for fractional topological insulators

Yang-Le Wu, Nicolas Regnault, Bogdan Andrei Bernevig

Published Aug 20, 2012• 93 citations
arXivHTMLPDF

Abstract

We propose an improved scheme to construct many-body trial wave functions for fractional Chern insulators (FCI), using one-dimensional localized Wannier basis. The procedure borrows from the original scheme on a continuum cylinder, but is adapted to finite-size lattice systems with periodic boundaries. It fixes several issues of the continuum description that made the overlap with the exact ground states insignificant. The constructed lattice states are translationally invariant, and have the correct degeneracy as well as the correct relative and total momenta. Our prescription preserves the (possible) inversion symmetry of the lattice model, and is isotropic in the limit of flat Berry curvature. By relaxing the maximally localized hybrid Wannier orbital prescription, we can form an orthonormal basis of states which, upon gauge fixing, can be used in lieu of the Landau orbitals. We find that the exact ground states of several known FCI models at $\ensuremath{\nu}=1/3$ filling are well captured by the lattice states constructed from the Laughlin wave function. The overlap is higher than $0.99$ in some models when the Hilbert space dimension is as large as $3\ifmmode\times\else\texttimes\fi{}{10}^{4}$ in each total momentum sector.

DOI
10.1103/physrevb.86.085129

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We propose an improved scheme to construct many-body trial wave functions for fractional Chern insulators (FCI), using one-dimensional localized Wannier basis. The procedure borrows from the original scheme on a continuum cylinder, but is adapted to finite-size lattice systems with periodic boundaries. It fixes several issues of the continuum description that made the overlap with the exact ground states insignificant. The constructed lattice states are translationally invariant, and have the correct degeneracy as well as the correct relative and total momenta. Our prescription preserves the (possible) inversion symmetry of the lattice model, and is isotropic in the limit of flat Berry curvature. By relaxing the maximally localized hybrid Wannier orbital prescription, we can form an orthonormal basis of states which, upon gauge fixing, can be used in lieu of the Landau orbitals. We find that the exact ground states of several known FCI models at \ensuremathν=1/3\ensuremath{\nu}=1/3\ensuremathν=1/3 filling are well captured by the lattice states constructed from the Laughlin wave function. The overlap is higher than 0.990.990.99 in some models when the Hilbert space dimension is as large as 3\ifmmode×\else\texttimes\fi1043\ifmmode\times\else\texttimes\fi{}{10}^{4}3\ifmmode×\else\texttimes\fi104 in each total momentum sector.
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.
In this paper we construct fully symmetric wave functions for the spin-polarized fractional Chern insulators (FCIs) and time-reversal-invariant fractional topological insulators (FTIs) in two dimensions using the parton approach. We show that the lattice symmetry gives rise to many different FCI and FTI phases even with the same filling fraction $\ensuremath{\nu}$ (and the same quantized Hall conductance ${\ensuremath{\sigma}}_{xy}$ in the FCI case). They have different symmetry-protected topological orders, which are characterized by different projective symmetry groups. We mainly focus on FCI phases which are realized in a partially filled band with Chern number 1. The low-energy gauge groups of a generic ${\ensuremath{\sigma}}_{xy}=1/m\ifmmode\cdot\else\textperiodcentered\fi{}{e}^{2}/h$ FCI wave function can be either SU(m)\text{SU}(m)SU(m) or the discrete group Zm{Z}_{m}Zm​, and in the latter case the associated low-energy physics are described by Chern-Simons-Higgs theories. We use our construction to compute the ground-state degeneracy. Examples of FCI/FTI wave functions on honeycomb lattice and checkerboard lattice are explicitly given. Possible non-Abelian FCI phases which may be realized in a partially filled band with Chern number 2 are discussed. Generic FTI wave functions in the absence of spin conservation are also presented whose low-energy gauge groups can be either $\text{SU}(m)\ifmmode\times\else\texttimes\fi{}\text{SU}(m)$ or ${Z}_{m}\ifmmode\times\else\texttimes\fi{}{Z}_{m}$. The constructed wave functions also set up the framework for future variational Monte Carlo simulations.
We devise local lattice models whose ground states are \emph{ideal fractional Chern insulators}---Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Zk\mathcal{Z}_kZk​ parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1)\nu=k/(\mathcal{C}+1)ν=k/(C+1) and (ii) nematic states at ν=k/2\nu=k/2ν=k/2, where C\mathcal{C}C is the Chern number of the lowest band in our models. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1)(k+1)(k+1)-body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well-suited for the study of novel phenomena involving e.g. twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
Fractional Chern insulator (FCI) states have been proposed in singular lattices with arbitrary nnn-fold rotational symmetry. Particles occupy non-defect-core orbitals in previously reported FCIs. However, the role of defect-core orbitals in the singular-lattice FCI states has seldom been discussed. In this paper, we report a series of singular-lattice FCI states with one particle occupying the defect-core orbital. This singular-lattice FCI state can be identified with the aid of the edge excitations, the angular momentum of the ground state (GS), and the optimal trial wave functions. Edge excitations and GS angular momentum are obtained using exact diagonalization. Optimal trial wave functions are constructed on the basis of the generalized Pauli principle, the Jack polynomials, and single-particle states. The present FCI state is identified as a $\ensuremath{\nu}=1/2$ FCI with high values of wave function overlap, where one particle occupies the low-energy defect-core orbital. Our studies reveal the intriguing interplay between the singular-lattice FCI and the defect-core orbital and provide an effective approach for future researches on FCIs with various defects or disorders.
We devise local lattice models whose ground states are model fractional Chern insulators-Abelian and non-Abelian topologically ordered states characterized by exact ground state degeneracies at any finite size and infinite entanglement gaps. Most saliently, we construct exact parent Hamiltonians for two distinct families of bosonic lattice generalizations of the Z_{k} parafermion quantum Hall states: (i) color-entangled fractional Chern insulators at band filling fractions ν=k/(C+1) and (ii) nematic states at ν=k/2, where C is the Chern number of the lowest band. In spite of a fluctuating Berry curvature, our construction is partially frustration free: the ground states reside entirely within the lowest band and exactly minimize a local (k+1) body repulsion term by term. In addition to providing the first known models hosting intriguing states such as higher Chern number generalizations of the Fibonacci anyon quantum Hall states, the remarkable stability and finite-size properties make our models particularly well suited for the study of novel phenomena involving, e.g., twist defects and proximity induced superconductivity, as well as being a guide for designing experiments.
The many-body ground state wave function of a fractional Chern insulator (FCI) can be constructed by mapping a fractional quantum Hall state to a FCI state through a substitution of the Bloch state of the flat Chern band for the magnetic Bloch state of the Landau level. There is a gauge freedom in choosing the single particle Bloch basis of the flat Chern band. Instead of considering only one form of interaction in FCI when choosing the gauge as done in previous works, we determine the optimal gauges for FCIs with different forms of interaction, including a short-range interaction, the Coulomb interaction, and an interpolation between them, by applying the variational principle proposed by Zhang et al. [Phys. Rev. B 93, 165129 (2016)]. We find that the optimal gauge strongly depends on the form of interaction. It contradicts with the common belief that the wave function of FCI is not sensitive to the interaction. In comparing the optimal gauge with the previous gauges proposed by Qi [Phys. Rev. Lett. 107, 126803 (2011)] and Wu et al. [Phys. Rev. B 86, 085129 (2012)], we find Wu et al.'s gauge is close to the optimal one when the interaction is a certain mixture of the Coulomb interaction and the short-range interaction, while Qi's gauge is qualitatively different from the optimal gauge in all the cases.