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Physical review. B./Physical review. B

Quantum phase transitions in a ν=12 bosonic fractional Chern insulator

Wei-Wei Luo, Ai‐Lei He, Yuan Zhou, Yifei Wang, Chang-De Gong

Published Oct 15, 2020• 9 citations
Publisher

Abstract

Interacting quantum many-body systems often generate exotic orders, either topological or conventional competing orders. Understanding these phases and the phase transitions between them are common issues in condensed-matter physics. How to characterize these states in a convincing and convenient way remains ambiguous. Here, we report our numerical study on the interacting $\ensuremath{\nu}=\frac{1}{2}$ hard-core boson system with topological flat bands based on the infinite density matrix group algorithm. We explicitly show that the entanglement spectral flow can be used to diagnose the quantum phases emerging in the hard-core boson systems. In particular, the subsequent charge pumping exhibits a linear, sinelike, and indiscernible dependence by the flux threading in the fractional Chern insulator, supersolid, and solid state, respectively. The real-space boson occupancy in the thermodynamic limit is further calculated for a better understanding of the respective state. We compare the phase diagram obtained here with that obtained by the previously exact diagonalization method, and qualitative differences are observed. Therefore, the infinite density matrix group algorithm provides a convincing and convenient way to study the phase transition in such a strongly correlated topological system.

DOI
10.1103/physrevb.102.155120

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Interacting quantum many-body systems often generate exotic orders, either topological or conventional competing orders. Understanding these phases and the phase transitions between them are common issues in condensed-matter physics. How to characterize these states in a convincing and convenient way remains ambiguous. Here, we report our numerical study on the interacting \ensuremathν=12\ensuremath{\nu}=\frac{1}{2}\ensuremathν=21​ hard-core boson system with topological flat bands based on the infinite density matrix group algorithm. We explicitly show that the entanglement spectral flow can be used to diagnose the quantum phases emerging in the hard-core boson systems. In particular, the subsequent charge pumping exhibits a linear, sinelike, and indiscernible dependence by the flux threading in the fractional Chern insulator, supersolid, and solid state, respectively. The real-space boson occupancy in the thermodynamic limit is further calculated for a better understanding of the respective state. We compare the phase diagram obtained here with that obtained by the previously exact diagonalization method, and qualitative differences are observed. Therefore, the infinite density matrix group algorithm provides a convincing and convenient way to study the phase transition in such a strongly correlated topological system.
The integer quantum Hall state occurs when Landau levels are fully occupied by fermions, while the fractional quantum Hall state usually emerges when the Landau level is partially filled by strongly correlated fermions or bosons. Here, we report two fractional Chern insulating states of hard-core bosons in a multiband lattice model hosting topological flat bands with a high Chern number. The previously proposed $\ensuremath{\nu}=1/3$ fractional Chern insulating state inherited from the high Chern number C=2C=2C=2 of the lowest topological flat band is revisited by the infinite density matrix renormalization group algorithm. In particular, we numerically identify a bosonic 1/21/21/2-Laughlin-like fractional Chern insulating state at the integer fillings. We show two lower topological flat bands jointly generate an effective C=1C=1C=1 Chern band with half filling. Furthermore, we find a strictly particle-hole-like symmetry between the $\ensuremath{\nu}$ and $3\ensuremath{-}\ensuremath{\nu}$ filling in our model. These findings extend our understanding of quantum Hall states and offer a route to realize fractional states in the system with multibands and high Chern numbers.
Recent theoretical papers have demonstrated the realization of fractional quantum anomalous Hall states (also called fractional Chern insulators) in topological flat band lattice models without an external magnetic field. Such newly proposed lattice systems play a vital role in obtaining a large class of fractional topological phases. Here we report the exact numerical studies of edge excitations for such systems in a disk geometry loaded with hard-core bosons, which will serve as a more viable experimental probe for such topologically ordered states. We find convincing numerical evidence of a series of edge excitations characterized by the chiral Luttinger liquid theory for the bosonic fractional Chern insulators in both the honeycomb disk Haldane model and the kagome-lattice disk model. We further verify these current-carrying chiral edge states by inserting a central flux to test their compressibility.
Using the infinite density matrix renormalization group method on an infinite cylinder geometry, we characterize the 1/31/31/3 fractional Chern insulator state in the Haldane honeycomb lattice model at $\ensuremath{\nu}=1/3$ filling of the lowest band and check its stability. We investigate the chiral and topological properties of this state through (i) its Hall conductivity, (ii) the topological entanglement entropy, (iii) the U(1)U(1)U(1) charge spectral flow of the many-body entanglement spectrum, and (iv) the charge of the anyons. In contrast to numerical methods restricted to small finite sizes, the infinite cylinder geometry allows us to access and characterize directly the metal to fractional Chern insulator transition. We find indications it is first order and no evidence of other competing phases. Since our approach does not rely on any band or subspace projection, we are able to prove the stability of the fractional state in the presence of interactions exceeding the band gap, as has been suggested in the literature. As a by-product we discuss the signatures of Chern insulators within this technique.
We investigate an interaction-driven transition between crystalline and liquid states of strongly correlated spinless fermions within topological flat bands at low density (with filling factors $\ensuremath{\nu}=\frac{1}{5}, \frac{1}{7}, \frac{1}{9}$). Using exact diagonalization for finite-size systems with periodic boundary conditions, we distinguish different phases, whose stability depends on the interaction range, controlled by the screening parameter of the Coulomb interaction. The crystalline phases are identified by a crystallization strength, calculated from the Fourier transforms of pair correlation density, while the fractional Chern insulator (FCI) phases are characterized using momentum counting rules, entanglement spectrum, and overlaps with corresponding fractional quantum Hall states. The type of the phase depends on a particular single-particle model and its topological properties. We show that for $\ensuremath{\nu}=\frac{1}{7}$ and 15\frac{1}{5}51​ it is possible to tune between the Wigner crystal and fractional Chern insulator phase in the kagome lattice model with the band carrying the Chern number C=1C=1C=1. In contrast, in the C=2C=2C=2 models, the Wigner crystallization was absent at $\ensuremath{\nu}=\frac{1}{5}$, and appeared at $\ensuremath{\nu}=\frac{1}{9}$, suggesting that C=2C=2C=2 FCIs are more stable against the formation of crystalline order.
We study NNN-component interacting particles (hardcore bosons and fermions) loaded in topological lattice models with SU(N)\mathrm{SU}(N)SU(N)-invariant interactions based on exact diagonalization and density matrix renormalization group method. By tuning the interplay of interspecies and intraspecies interactions, we demonstrate that a class of SU(N)\mathrm{SU}(N)SU(N) fractional quantum Hall states can emerge at fractional filling factors $\ensuremath{\nu}=N/(N+1)$ for bosons [$\ensuremath{\nu}=N/(2N+1)$ for fermions] in the lowest Chern band, characterized by the nontrivial fractional Hall responses and the fractional charge pumping. Moreover, we establish a topological characterization based on the K\mathbf{K}K matrix and discuss the close relationship to the fractional quantum Hall physics in topological flat bands with Chern number NNN.
Chern insulators are band insulators exhibiting a nonzero Hall conductance but preserving the lattice translational symmetry. We conclusively show that a partially filled Chern insulator at 1/3 filling exhibits a fractional quantum Hall effect and rule out charge-density wave states that have not been ruled out by previous studies. By diagonalizing the Hubbard interaction in the flat-band limit of these insulators, we show the following: The system is incompressible and has a 3-fold degenerate ground state whose momenta can be computed by postulating an generalized Pauli principle with no more than 1 particle in 3 consecutive orbitals. The ground state density is constant, and equal to 1/3 in momentum space. Excitations of the system are fractional statistics particles whose total counting matches that of quasiholes in the Laughlin state based on the same generalized Pauli principle. The entanglement spectrum of the state has a clear entanglement gap which seems to remain finite in the thermodynamic limit. The levels below the gap exhibit counting identical to that of Laughlin 1/3 quasiholes. Both the 3 ground states and excited states exhibit spectral flow upon flux insertion. All the properties above disappear in the trivial state of the insulator - both the many-body energy gap and the entanglement gap close at the phase transition when the single-particle Hamiltonian goes from topologically nontrivial to topologically trivial. These facts clearly show that fractional many-body states are possible in topological insulators.