[Paper Review] The Haldane model under quenched disorder

Miguel Gonçalves, Pedro Ribeiro+1 more|arXiv (Cornell University)|Jul 30, 2018

Topological Materials and Phenomena2 references6 citations

TL;DR

This study investigates the Haldane model under quenched disorder, demonstrating that disorder stabilizes topologically nontrivial phases in regions where the clean system is trivial. Using numerical diagonalization and self-consistent Born approximation, it shows that both Anderson and binary disorder induce topological Anderson insulator phases, with binary disorder exhibiting a reentrant topological transition, and finds that gapless topological phases persist even at strong disorder, extending beyond perturbative predictions.

ABSTRACT

We study the half-filled Haldane model with Anderson and binary disorder and determine its phase diagram, as a function of the Haldane flux and staggered sub-lattice potential, for increasing disorder strength. We establish that disorder stabilizes topologically nontrivial phases in regions of the phase diagram where the clean limit is topologically trivial. At small disorder strength, our results agree with analytical predictions obtained using a first order self-consistent Born approximation, and extend to the intermediate and large disorder values where this perturbative approach fails. We further characterize the phases according to their gapless or gapped nature by determining the spectral weight at the Fermi level. We find that gapless topological nontrivial phases are supported for both Anderson and binary disorder. In the binary case, we find a reentrant topological phase where, starting from a trivial region, a topological transition occurs for increasing staggered potential $η$, followed by a second topological transition to the trivial phase for higher values of $η$.

Motivation & Objective

• To map the complete phase diagram of the half-filled Haldane model under quenched disorder, including topological, trivial, gapped, and gapless phases.
• To determine how disorder—specifically Anderson and binary types—affects the stability and emergence of topological phases.
• To extend beyond perturbative approaches by analyzing intermediate and strong disorder regimes where the self-consistent Born approximation fails.
• To characterize phases by their spectral weight at the Fermi level, distinguishing gapped from gapless behavior.
• To identify nontrivial topological transitions, including reentrant behavior in the binary disorder case.

Proposed method

• Numerical diagonalization of the Haldane Hamiltonian on finite lattices with quenched disorder to compute the full phase diagram.
• Use of the self-consistent Born approximation (SCBA) to analytically predict topological phase boundaries at weak disorder, with renormalized topological mass and energy shift.
• Mapping the low-energy effective Hamiltonian of the Haldane model to that of an HgTe quantum well for analytical tractability.
• Disorder averaging via the self-energy equation: Σ = (3√3/2)(σ/2π)²⟨[G₀⁻¹ − Σ]⁻¹⟩ over the Brillouin zone.
• Calculation of the Chern number via sgn(m′₊) − sgn(m′₋), where m′ is the renormalized topological mass.
• Comparison of numerical results with SCBA predictions to assess breakdown of perturbation theory at strong disorder.

Experimental results

Research questions

• RQ1 Does quenched disorder stabilize topologically nontrivial phases in regions where the clean Haldane model is trivial?
• RQ2 How does the phase diagram evolve with increasing disorder strength, particularly beyond the perturbative regime?
• RQ3 What differences exist between Anderson and binary disorder in driving topological phase transitions?
• RQ4 Are gapless topological phases robust against strong disorder, and how do they manifest in the spectral weight at the Fermi level?
• RQ5 Does the binary disorder case exhibit reentrant topological transitions, and if so, what is the mechanism?

Key findings

• Disorder stabilizes topologically nontrivial phases in regions of the phase diagram that are trivial in the clean limit, particularly for both Anderson and binary disorder.
• At small disorder, numerical results agree with first-order self-consistent Born approximation predictions, confirming the renormalization of the topological mass.
• For intermediate and strong disorder, the growth rate of the topological phase window exceeds SCBA predictions, indicating breakdown of perturbation theory.
• Gapless topological nontrivial phases are supported under both Anderson and binary disorder, as confirmed by non-zero spectral weight at the Fermi level.
• In the binary disorder case, a reentrant topological phase is observed: a topological transition occurs at intermediate staggered potential η, followed by a second transition back to trivial at higher η.
• The phase diagram for Anderson disorder shows striking similarity to experimental measurements of differential drift velocity in ultracold fermion systems, suggesting disorder as a key factor in observed deviations.