[Paper Review] Composite Fermions and Quantum Hall Stripes on the Topological Insulator Surface

Ying Ran, Hong Yao+1 more|arXiv (Cornell University)|Mar 4, 2010

Topological Materials and Phenomena3 citations

TL;DR

This paper investigates the interplay of Coulomb and local electron correlations on the surface of a topological insulator at the Dirac point under a quantizing magnetic field. Using Hartree-Fock and large-N approximations, it predicts a novel 'axion stripe' phase—distinct from conventional quantum Hall stripes—where charge density waves coexist with oscillating mass terms (axion fields), stabilized by strong local repulsion (U) and parametrically larger wavelength than the magnetic length. The phase is destroyed by strong Zeeman coupling, with a critical g-factor of order one.

ABSTRACT

We study the problem of a single Dirac fermion in a quantizing orbital magnetic field, when the chemical potential is at the Dirac point. This can be realized on the surface of a topological insulator, such as Bi2Se3, tuned to neutrality. We study the effect of both long range Coulomb interactions (strength alpha=e^2/(epsilon hbar v_F).) and local repulsion U which capture the effect of electron correlations. Interactions resolve the degeneracy of free fermions in the zeroth Landau level at half filling, but in a manner different from that in graphene. For weak interactions, U=0 and alpha<<1, a composite Fermi liquid is expected. However, in the limit of strong local correlations (large U but alpha<<1), a charge density wave phase is predicted, which we term "axion stripe". While reminiscent of quantum Hall stripe phases, its wavelength is parametrically larger than the magnetic length, and the induced fermion mass term (axion) also oscillates with the charge density. This phase is destroyed by sufficiently strong Zeeman coupling. A phase diagram is constructed and consequences for experiments are discussed.

Motivation & Objective

• To understand the ground state of a single Dirac fermion at the Dirac point on a topological insulator surface under a quantizing magnetic field.
• To investigate the interplay between long-range Coulomb interactions (α) and short-range local repulsion (U) in lifting the degeneracy of the zeroth Landau level.
• To determine whether the system forms a homogeneous composite Fermi liquid or a spatially modulated charge density wave phase under strong correlations.
• To identify and characterize a new type of quantum Hall stripe phase—'axion stripe'—where the induced mass term oscillates with charge density and has a wavelength parametrically larger than the magnetic length.
• To assess the stability of the axion stripe phase under Zeeman coupling and its experimental feasibility.

Proposed method

• Uses a Hartree-Fock approximation in the large-N limit to study the ground state of a single Dirac node at half-filling in a magnetic field.
• Models electron interactions via a dimensionless Coulomb coupling α = e²/(εℏvF) and a local Hubbard U term to capture short-range correlations.
• Analyzes the stability of phase separation and charge density wave order by computing the second derivative of the free energy F''(m*) with respect to the induced mass m*.
• Derives the condition for phase separation to be favored: F''(m*) < 0, which depends on the magnetic field strength and ultraviolet cutoff.
• Evaluates the role of Zeeman coupling by redefining the filling factor ν → ν̃ = ν + (πgμB)/U and assessing its impact on the energy scaling of phase separation.
• Estimates the critical g-factor for destruction of the axion stripe phase using order-of-magnitude estimates involving vF, a, and ℏΛ/mₑvF, finding g_c ~ O(1) for realistic parameters.

Experimental results

Research questions

• RQ1 What is the ground state of a single Dirac fermion at the Dirac point on a topological insulator surface when both Coulomb and local electron correlations are present?
• RQ2 How does the interplay between weak Coulomb interactions (α ≪ 1) and strong local repulsion (U) affect the stability of the zeroth Landau level degeneracy?
• RQ3 Can a spatially modulated charge density wave phase emerge, and if so, what are its distinguishing features compared to conventional quantum Hall stripes?
• RQ4 Under what conditions does the 'axion stripe' phase—characterized by oscillating mass terms and a parametrically large wavelength—remain stable?
• RQ5 How does Zeeman coupling influence the formation and stability of the axion stripe phase?

Key findings

• For weak local repulsion (U ≪ Uc), the system forms a homogeneous composite Fermi liquid (CFL), analogous to the Halperin-Lee-Read state.
• For strong local repulsion (U > Uc), a charge density wave phase emerges with a spatially modulated mass term, forming an 'axion stripe' phase with a wavelength λ ~ lB² / (√α ξ), parametrically larger than the magnetic length lB.
• The axion stripe phase is stabilized by a linear energy gain in magnetic field B, which requires opposite mass signs in ν = 1/2 and ν = -1/2 regions, achievable when 4πℏgμB/(eU) < 1.
• The critical Zeeman coupling for destroying the axion stripe phase is estimated to be g_c ~ O(1), depending on microscopic parameters such as vF, a, and Λ.
• Experimental observations of nearly field-independent zeroth Landau level energy in Bi2Se3 up to 11 T suggest a surface g-factor much smaller than the bulk (g ~ 30), implying g_surface ~ O(1), consistent with the existence of the axion stripe phase.
• Phase separation is favored in the absence of Coulomb interactions, but is suppressed by weak long-range Coulomb interactions (α ≪ 1), favoring the CFL or axion stripe phases instead.