Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 47 (2014)

Phase structure of two-dimensional topological insulators by latticestrong-coupling expansion†
Y. Araki∗1 and T. Kimura∗2
Topological insulators have recently attracted agreat interest in the field of materials physics, which
are characterized by the topologically protected gapless modes localized on the boundary of the system.
The effect of electron correlation in such an electronic system has always been an important problem.
Even in non-topological Dirac fermion systems, suchas graphene, it has been proposed that a sufficiently
strong electron-electron interaction can lead to a spontaneous breaking of some symmetries of the system anda dynamical generation of band gap.
0.8
0.7
0.6(a-2)

0.5

Normal AF

0.4(b-2)

0.3

Tilted AF

0.2

ydar

unboNG mode?
Normal AF

My

Tilted AF

Mx(σ2)
Fig. 2. Schematic picture of the order parameters derived
in this study. It is expected that a gapless NG mode
appear when the full SU(2) is restored.
g2∞∞
Normal AF(b-1)

Tilted AF

Normal AF(Aoki phase)

Normal AF

Normal AF
Topological

Topological(Aoki phase)g2 Tilted AF(Aoki phase)

(a-1)

0.1
0.0
0.00

al p

gicolotopehasMz(σ1)

Topological
0.01

0.02

0.03

0.04

0.05

t’

0.06

0.07

0.08

0.09

-3√3t’

0.10

0

3√3t’

m

-3√3t’

0

3√3t’

m

Fig. 1. The phase diagram of the Kane-Mele model in the(t , σ1 )-space. One of the Dirac cones loses its band gap
at the phase boundary.

Fig. 3. The schematic phase structure of a graphene-likesystem with the spin-orbit interaction, conjectured inanalogy to that of lattice QCD with the Wilson fermion
formalism, for (left) t < tC and (right) t > tC .

In this report, we study the effect of a sufficiently
strong electron-electron interaction on the topologicalphase structure of 2D quantum spin Hall insulators.
By the techniques of strong-coupling expansion of lattice gauge theory, we observe the behavior of the spontaneous antiferromagnetic (AF) order in the strongcoupling limit of the interaction. As a result, we findthat the topological phase structure is modified fromthat of the noninteracting system by the emergence ofa new “tilted AF” phase in-between the normal insulator and the topological insulator phases as shown inFig. 1. Here we use the “modified” mass σ1 , instead of
the bare mass (staggered magnetic field) m, and t isthe spin-orbit counpling constant. As a consequenceof the interplay between the electron-electron interaction and the spin-orbit interaction, there appears a
new “tilted antiferromagnetic (AF)” phase, where theimaginary part of the order parameter becomes nonzero (σ2 = 0), between the normal AF phase and thetopological phase.
The AF order is not parallel to the direction pointed
by the spin-orbit interaction and the staggered magnetic field in the spin SU(2) space in the tilted AF

phase. σ1 and σ2 are antiferromagnetic (AF) orders
corresponding to two directions in the remnant U(1)spin space, which we denote Mz and Mx here. If weextend this argument to the full SU(2) spin space, another direction My is restored, so that the tilted AF
acquires U(1) degree of freedom in choosing its direction, which may result in a massless Nambu-Goldstone
mode (Fig. 2).
We also show the analogy between the phase structure of topological insulators and that of the strongly
coupled lattice QCD with the Wilson fermion formalism in Fig. 3. In this analogy, σ1 and σ2 correspond toψ̄ψ and iψ̄γ5 ψ. The tilted AF phase is similar tothe so-called “Aoki phase” in lattice QCD in that bothof them are characterized by an order parameter orthogonal to the external source term in the continuous
symmetry space.1) Such an analogy may help us understand the behavior of topological insulators with anelectron-electron interaction from the strong-coupling
to the weak-coupling regime.

†∗1∗2

Condensed from the article in Phys. Rev. B87 (2013) 205440
Department of Physics, University of Texas at Austin
RIKEN Nishina CenterReferences
1) S. Aoki, Phys. Rev. D30 (1984) 2653; Nucl. Phys. B314
(1989) 79.

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