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arXiv:1112.4872 · Physical Review B

Interaction-driven quantum phase transitions in fractional topological insulators

Hua Chen, Kun Yang

Published May 9, 2012• 28 citations
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Abstract

We study two species of (or spin-1/2) fermions with short-range intraspecies repulsion in the presence of opposite (effective) magnetic field, each at Landau level filling factor 1/3. In the absence of interspecies interaction, the ground state is simply two copies of the 1/3 Laughlin state, with opposite chirality, representing the fractional topological insulator (FTI) phase. We show this phase is stable against moderate interspecies interactions. However, strong enough interspecies repulsion leads to phase separation, while strong enough interspecies attraction drives the system into a superfluid phase. We obtain the phase diagram through exact diagonalization calculations. The FTI-superfluid phase transition is shown to be in the (2 $+$ 1)-dimensional XY universality class, using an appropriate Chern-Simons-Ginsburg-Landau effective field theory.

DOI
10.1103/physrevb.85.195113

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We study two species of (or spin-1/2) fermions with short-range intraspecies repulsion in the presence of opposite (effective) magnetic field, each at Landau level filling factor 1/3. In the absence of interspecies interaction, the ground state is simply two copies of the 1/3 Laughlin state, with opposite chirality, representing the fractional topological insulator (FTI) phase. We show this phase is stable against moderate interspecies interactions. However, strong enough interspecies repulsion leads to phase separation, while strong enough interspecies attraction drives the system into a superfluid phase. We obtain the phase diagram through exact diagonalization calculations. The FTI-superfluid phase transition is shown to be in the (2 +++ 1)-dimensional XY universality class, using an appropriate Chern-Simons-Ginsburg-Landau effective field theory.
We propose a simple microscopic model to numerically investigate the stability of a two-dimensional fractional topological insulator (FTI). The simplest example of an FTI consists of two decoupled copies of a Laughlin state with opposite chiralities, or double-semion phase. We focus on bosons at half filling. We study the stability of the FTI phase upon addition of two coupling terms of different nature: an interspin interaction term, and an inversion-symmetry-breaking term that couples the copies at the single-particle level. Using exact-diagonalization and entanglement spectra, we numerically show that the FTI phase is stable against both perturbations. We compare our system to a similar bilayer fractional Chern insulator. We show evidence that the time-reversal-invariant system survives the introduction of interaction coupling on a larger scale than the time-reversal-symmetry-breaking one, stressing the importance of time-reversal symmetry in the FTI phase stability. We also discuss possible fractional phases beyond $\ensuremath{\nu}=1/2$.
We propose a simple microscopic model to numerically investigate the stability of a two dimensional fractional topological insulator (FTI). The simplest example of a FTI consists of two decoupled copies of a Laughlin state with opposite chiralities. We focus on bosons at half filling. We study the stability of the FTI phase upon addition of two coupling terms of different nature: an interspin interaction term, and an inversion symmetry breaking term that couples the copies at the single particle level. Using exact diagonalization and entanglement spectra, we numerically show that the FTI phase is stable against both perturbations. We compare our system to a similar bilayer fractional Chern insulator. We show evidence that the time reversal invariant system survives the introduction of interaction coupling on a larger scale than the time reversal symmetry breaking one, stressing the importance of time reversal symmetry in the FTI phase stability.
We study quantum phase transitions in Bose-Fermi mixtures driven by interspecies interaction in the quantum Hall regime. In the absence of such an interaction, the bosons and fermions form their respective fractional quantum Hall (FQH) states at certain filling factors. A symmetry-protected topological (SPT) state is identified as the ground state for strong interspecies interaction. The phase transitions between them are proposed to be described by Chern-Simons-Higgs field theories. For a simple microscopic Hamiltonian, we present numerical evidence for the existence of the SPT state and a continuous transition to the FQH state. It is also found that the entanglement entropy between the bosons and fermions exhibits scaling behavior in the vicinity of this transition.
We study two models for spinless fermions featuring topologically nontrivial bands characterized by Chern numbers C=±1 at fractional filling. Using exact diagonalization, we show that, even for infinitely strong nearest-neighbor repulsion, the ground states of these models belong to the recently discovered class of quantum liquids called fractional Chern insulators (FCI). Thus, we establish that FCI states can arise even if interaction strengths are arbitrarily larger than the noninteracting band gap, going beyond the limits in which FCI states have been previously studied. The strong-coupling FCI states, therefore, depart from the usual isolated-band picture that parallels the fractional quantum Hall effect in Landau levels and demonstrate how a topologically ordered state can arise in a truly multiband system.
We present numerical evidence of an interaction-driven quantum Hall plateau transition between a ∣C∣>1|C|>1∣C∣>1 Chern Insulator (CI) and a ν=1/3ν= 1/3ν=1/3 Laughlin state in the Harper-Hofstadter model. We study the model at flux densities p/qp/qp/q, where the lowest Landau level (LLL) manifold comprises ppp magnetic sub-bands. For weak interactions, the model realises integer CIs corresponding to filled sub-bands, while strongly interacting candidate states include fractional quantum Hall (FQH) states at LLL filling fractions ν=r/tν=r/tν=r/t. These phases may compete at the same particle density when p=tp=tp=t. As a concrete example, we numerically explore the physics at flux density nϕ=3/11n_ϕ = 3/11nϕ​=3/11, where we show evidence that a direct transition occurs between a CI and a ν=1/3ν= 1/3ν=1/3 Laughlin state, which we characterise in terms of its critical, topological and entanglement properties. We also show that strong interactions generically stabilise a ν=1/3ν= 1/3ν=1/3 Laughlin state even when the LLL is split into multiple bands, and introduce a powerful methodology to extract its topological entanglement entropy by exploiting the scaling of magnetic length with nϕn_ϕnϕ​.
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.