Edge Reconstruction of a Time-Reversal Invariant Insulator: Compressible-Incompressible Stripes

Two-dimensional (2D) topological electronic insulators are known to give rise to gapless edge modes, which underlie low energy dynamics, including electrical and thermal transport. This has been thoroughly investigated in the context of quantum Hall phases, and time-reversal invariant topological insulators. Here we study the edge of a 2D, topologically trivial insulating phase, as a function of the strength of the electronic interactions and the steepness of the confining potential. For sufficiently smooth confining potentials, alternating compressible and incompressible stripes appear at the edge. Our findings signal the emergence of gapless edge modes which may give rise to finite conductance at the edge. This would suggest a novel scenario of a nontopological metal-insulator transition in clean 2D systems. The incompressible stripes appear at commensurate fillings and may exhibit broken translational invariance along the edge in the form of charge density wave ordering. These are separated by structureless compressible stripes.

Figure 1

Changes in the edge structure driven by variations of slope of the confining potential ($w^{-1}$). (a)–(b) Average charge density as a function of the distance from the edge ($y$) for (a) sharp ($w=0.5$) and (b) smooth ($w=8.5$) confining potentials. Regions at large negative (positive) $y$ are part of the insulator (vacuum) and are doubly occupied (empty). (c)–(d) Local compressibility (defined in the text) of the states presented in panels (a)–(b) respectively. As seen smoothening the potential leads to the formation of compressible stripes and an incompressible half-filled plateau. The parameters used here are $t=V=1$ and $U=15$.

Figure 2

(a)–(d) The $y$ dependent order parameters (defined in the text) in the four edge structures found for smooth confining potentials. An identical confining potential ($w=50$) is used for all panels, while the interaction strengths ($U$ and $V$) vary. Moving from (a) to (b), $V$ is increased while $U$ is fixed. Moving from (b) to (d), $U$ is decreased at a constant $V$. Black curves (empty and filled squares) show the average occupation [$\rho (y)$]. Filled [empty] squares denote regions with a CDW of period 2 [a uniform charge density] along the edge (along $x$). The blue [red] line (triangles [circles]) shows the average ferromagnetic (F) [antiferromagnetic (AF)] order parameter. The empty (filled) triangles and circles denote magnetic order along the $s_z$ ($s_x$) direction. This direction is chosen arbitrarily for the moments on the half-filled stripe. All edge structures feature a central plateau at half filling ($\rho =1$). This central plateau has (a) AF order (in the limit $U\gg V\gg t$) or (b) CDW order (in the limit $V\gg U$). (c),(d) Additional plateaus at $\rho =1/2$ and $3/2$ can appear when both $U$ and $V$ are large. These plateaus feature a CDW order of period 2. Since the (average) spin on each partially occupied site in these regions is equal, the average F and AF order parameters are identical. The additional plateaus do not change the order on the central plateau which can be either (c) AF or (d) CDW. (e)–(h) The local compressibility in the states depicted in panels (a)–(d) respectively. The presence of alternating compressible and incompressible stripes is clear. The values of $U$ and $V$ used here are shown in Fig. 4.

Figure 3

Ordered stripes at the edge of a 2D insulator. The color map shows the average occupation at each site. The black arrows represent the average spin (along $s_z$) at each site. Only the first eight sites along the edge ($x$ direction) are shown. The region around $y=0$ features single occupied sites with AF ordering [the central plateau in Fig. 2]. The surrounding regions host a CDW (marked by the alternating colors), forming the quarter-filled plateaus in Fig. 2. This is the result for 30 chains (along $y$ direction) with $w=12$. The parameters $U$, $V$, and $t$ are the same as those used for Fig. 2.

Figure 4

Phase diagram of the possible structures at the edge for a smooth confining potential ($w=10$) as a function of the interaction strengths $U$ and $V$. A half-filled incompressible plateau exists at all parameters. Additional one and three quarter-filled plateaus (also incompressible) exist roughly in the region above the solid black line. The half-filled plateau features AF (CDW) order in the region above (below) the dashed black line. The points $A$, $B$, $C$, and $D$ mark the parameters $U$, $V$ used for panels (a), (b), (c), and (d) of Fig. 2 respectively.

Figure 5

Tunneling density of states (${\rho }_e$) in the different structures at the edge as a function of energy $\omega$ and position $y$. The white color denotes zero density. The regions with darker colors have a larger density of states. Panels (a)–(d) correspond to the states shown in Figs. 2 respectively. Comparing with Fig. 2, we see that the incompressible plateaus feature a gap in the tunneling density of states, i.e., no states at $\omega \sim 0$. By contrast, the compressible regions feature a finite density of states even at $\omega \approx 0$. Therefore the incompressible (compressible) regions are expected to be insulating (metallic) in a clean system.