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Universality of the topological phase transition in the interacting Haldane model

Simone Fabbri, Alessandro Giuliani, Robin Reuvers

Abstract

The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and the `topological' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of $e^2/h$: this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities.

Universality of the topological phase transition in the interacting Haldane model

Abstract

The Haldane model is a standard tight-binding model describing electrons hopping on a hexagonal lattice subject to a transverse, dipolar magnetic field. We consider its interacting version for values of the interaction strength that are small compared to the bandwidth. We study the critical case at the transition between the trivial and the `topological' insulating phases, and we rigorously establish that the transverse conductivity on the dressed critical line is quantized at a half-integer multiple of : this is the average of the integer values of the Hall conductivity in the insulating phases on either side of the dressed critical line. Together with previous results, this fully characterizes the nature of the phase transition between different Hall plateaus and proves its universality with respect to many-body interactions. The proof is based on a combination of constructive renormalization group methods and exact lattice Ward identities.

Paper Structure

This paper contains 14 sections, 2 theorems, 94 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The model and the main results
  3. Set-up
  4. Correlation functions and Kubo conductivity
  5. The non-interacting theory
  6. Main results: the Hall conductivity in the critical regime
  7. The critical transverse conductivity
  8. The non-interacting case
  9. The general, interacting, case
  10. Proof of Prop. \ref{['proposition_main']}
  11. Review of the RG construction
  12. The differentiable and singular contributions
  13. Completion of the proof
  14. Conclusion

Key Result

Theorem 2.1

There exists $U_0>0$, independent of $W,\phi$, such that, for any $|U|<U_0$, there exist two functions $\mathfrak{d}(U,W,\phi), \mathfrak{z}(U,W,\phi)$, analytic in $U$, vanishing at $U=0$, and continuously differentiable in $W,\phi$, such that, if the chemical potential $\mu$ is fixed at the value where $m^R_{\pm}(U,W,\phi)\equiv W \pm 3\sqrt{3}t_2 \sin\phi \pm \mathfrak{d}(U,W,\phi)$, with the

Figures (3)

  • Figure 1: The conductivity matrix $\sigma$ in units of $\frac{e^2}{h}$ over the whole phase diagram (critical lines included). At the graphene points $(\phi,W)=(0,0), (\pi,0)$, the conductivity matrix is $\pi/4$ times the identity.
  • Figure 2: The phase diagram of the Haldane--Hubbard model with the corresponding conductivity matrix. The effect of the interaction shows in the renormalization of the critical curves (solid lines) compared to their non-interacting counterparts (dashed lines). According to GMP20, the critical curves are continuously differentiable in $\phi$ and $O(U)$-close to the non-interacting ones (see GJMP16 for the leading-order computation). The conductivity matrix on and away from the dressed critical lines is the same as for the non-interacting model and, in particular, is $U$-independent.
  • Figure 3: The honeycomb lattice of the Haldane model, with the white (resp. black) points corresponding to the sublattice $\Lambda$ (resp. $\Lambda'$). The nearest-neighbor vectors $\vec{\delta}_1, \vec{\delta}_2,\vec{\delta_3}$ and next-to-nearest neighbor vectors $\vec{\gamma}_1,\vec{\gamma}_2,\vec{\gamma}_3$ are also represented. For the latter, the phases associated to the hoppings from a white (resp. black) site to their next-to-nearest neighbor are shown in blue (resp. red); the hoppings in the opposite directions have complex conjugate phases.

Theorems & Definitions (4)

  • Theorem 2.1
  • Proposition 3.1
  • Remark 1
  • Remark 2