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arXiv:1710.06631

Topological phase transitions in Chern insulators within three-band models

Paweł Potasz, Błażej Jaworowski, Michał Kupczyński, Marta Brzezińska, Piotr Kaczmarkiewicz, Arkadiusz Wójs

Published Oct 18, 2017• 0 citations
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Abstract

We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from $C=1$ to $C=-1$. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with $1/3$ filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy $α(n_A)$ term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.

DOI
10.48550/arxiv.1710.06631

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We investigate topological phase transitions in Chern insulators within three-band models, focusing on the empty band and lowest band populated by spinless fermions. We consider Lieb and kagome lattices and notice phase transitions driven by the hopping integral between nearest-neighbors, which leads to the change of the lowest band Chern number from C=1C=1C=1 to C=−1C=-1C=−1. In the single-particle picture, different phases are examined by investigating corresponding entanglement spectra and the evolution of the entanglement entropy. Entanglement spectra reveal the spectral flow characteristic of topologically nontrivial systems before and after phase transitions. For the lowest band with 1/31/31/3 filling, Fractional Chern insulator (FCI) phases are identified by examining the ground state momenta, the spectral flow, and counting of the entanglement energy levels below the gap in the entanglement spectrum. A quasilinear dependence of the entanglement entropy α(nA)α(n_A)α(nA​) term is observed for FCI phase, similarly to linear behavior expected for Laughlin phase in the fractional quantum Hall effect. At the topological phase transition, for both empty and partially filled lowest bands, the energy gap closure and a discontinuity in the entanglement entropy are observed. This coincides with the divergence of a standard deviation of the Berry curvature. We notice also a phase transition driven by the nearest-neighbor interaction in the Lieb lattice, where the many-body energy gap closes. The phase transitions are shown to be stable for an arbitrary system size, thus predicted to be present in the thermodynamic limit. While our calculations are performed for an interaction energy far exceeding the gap between the two lowest energy bands, we note that the higher band does not affect the phase transitions, however destabilizes FCI phases.
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.
The stability of the $\ensuremath{\nu}=1/3$ fractional Chern insulator (FCI) phase is analyzed on the example of a checkerboard lattice undergoing a transition into a Lieb lattice. The transition is performed by the addition of a second sublattice, whose coupling to the checkerboard sites is controlled by sublattice staggered potential. We investigate the influence of these sites on the many-body energy gap between three lowest energy states and the fourth state. We consider cases with different complex phases acquired in hopping and a model with a flattened topologically nontrivial band. We find that an interaction with the additional sites either open the single-particle gap or enlarge the existing one, which translates into similar effect on the many-particle gap. By looking at Berry curvature flatness we notice its strong correlation with the magnitude of the many-body energy gap, suggesting that the main mechanism of the FCI stabilization by additional atoms is via their influence on the Berry curvature. Evidence of the FCI phase for a region in a parameter space with larger energy gap is shown by looking at momenta of the threefold degenerate ground state, spectral flow, and quasihole excitation spectrum.
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 four different models of Chern insulators in the presence of strong electronic repulsion at partial fillings. We observe that all cases exhibit a Laughlin-like phase at filling fraction 1/31/31/3. We provide evidence of such a strongly correlated topological phase by studying both the energy and the entanglement spectra. In order to identify the key ingredients of the emergence of Laughlin physics in these systems, we show how they are affected when tuning the band structure. We also address the question of the relevance of the Berry curvature flatness in this problem. Using three-body interactions, we show that some models can also host a topological phase reminiscent of the $\ensuremath{\nu}=1/2$ Pfaffian Moore-Read state. Additionally, we identify the structures indicating cluster correlations in the entanglement spectra.
Fractional Chern insulators have attracted broad interest as lattice analogs of fractional quantum Hall states without Landau levels. However, low-filling fractional Chern insulators are fragile because charge-ordered phases can compete strongly with the fractional topological liquid. Here, we propose a center-decorated kagome model, motivated by geometry-tunable artificial lattices, in which the center-site hopping t2t_2t2​ provides a direct knob for the quantum geometry of an isolated C=1C=1C=1 flat band. Here quantum geometry refers to the Berry curvature and Fubini--Study metric, which determine the form factors of interactions projected into the Chern band. Exact diagonalization shows that tuning t2t_2t2​ away from the flatness-optimized kagome limit reduces the trace-condition deviation, suppresses competing charge order, and enhances the many-body stability at both ν=1/3ν=1/3ν=1/3 and the more fragile ν=1/5ν=1/5ν=1/5 filling. At ν=1/5ν=1/5ν=1/5, this stability-enhanced window persists under nearby interaction profiles, including variations of the dominant third-neighbor repulsion and weak nearest-neighbor admixtures. Low-energy spectra, spectral flow, quasihole and entanglement counting, static structure factors, and the quantized total many-body Chern number Ctot=1C_{\mathrm{tot}}=1Ctot​=1 consistently support Laughlin-like fractional Chern insulators. These results identify quantum-geometry engineering as a route to stabilizing dilute fractional Chern insulators beyond band-flatness optimization alone.
Recent theoretical works have demonstrated various robust Abelian and non-Abelian fractional topological phases in lattice models with topological flat bands carrying Chern number C=1C=1C=1. Here we study hard-core bosons and interacting fermions in a three-band triangular-lattice model with the lowest topological flat band of Chern number C=2C=2C=2. We find convincing numerical evidence of bosonic fractional quantum Hall effect at the $\ensuremath{\nu}=1/3$ filling characterized by threefold quasidegeneracy of ground states on a torus, a fractional Chern number for each ground state, a robust spectrum gap, and a gap in quasihole excitation spectrum. We also observe numerical evidence of a robust fermionic fractional quantum Hall effect for spinless fermions at the $\ensuremath{\nu}=1/5$ filling with short-range interactions.