Incoherent transport on the $\nu =2/3$ quantum Hall edge

We study edge state transport for the ${\nu }_{\text{bulk}}=2/3$ edge in the fully incoherent transport regime. To do so, we use a hydrodynamic approximation for the calculation of voltage and temperature profiles along the edge of the sample. Within this formalism, we study two different bare mode structures with tunneling: the edge model (1) consisting of two counterpropagating modes with filling factor discontinuities (from the bulk to the edge) $\delta \nu =-1/3$ and $\delta \nu =+1$, and the more complicated model (2) consisting of four modes with $\delta \nu =-1/3,+1,-1/3,$ and $+1/3$. We find that the topological characteristics of transport (quantized electrical and heat conductance) within these models are intact, with finite-size corrections which are determined by the extent of equilibration. In particular, our calculation of conductance for edge model (2) in a double quantum point contact geometry reproduces conductance results of a recent experiment [R. Sabo et al., Edge reconstruction in factional quantum Hall states, Nat. Phys. 13, 491 (2017)], which are inconsistent with the edge model (1). Our results can be explained in the charge/neutral mode picture, with incoherent analogs of the renormalization fixed points of the edge models. Additionally, we find diffusive $(\sim 1/L)$ conductivity corrections to the heat conductance in the fully incoherent regime for both models of the edge.

Figure 1

(a) Schematic plot of the filling factor $\nu \left(y\right)$ as a function of the coordinate $y$ perpendicular to the edge for the MacDonald edge structure. The boundary between regions of different filling factor are where the edge modes reside. (b) Filling factor $\nu \left(y\right)$ as a function of the coordinate $y$ perpendicular to the edge for the WMG edge structure. Due to the shallow confining potential, a wide $\nu =1/3$ incompressible strip is created near the edge of the sample.

Figure 2

(a) Discrete tunneling bridge for calculation of the steady-state current/voltage distribution. The dashed line indicates the impurity-mediated tunneling, with associated currents $I_{\tau ,i},J_{\tau ,i}$. The length of each segment $i$ is $a$. The voltage and temperature of the probes are adjusted in such a way that no net electrical or thermal current flows between the probes and the edge. In this way the system equilibrates in between tunneling events, such that it can be characterized by a local chemical potential and temperature. (b) A line junction consisting of counterpropagating $\delta \nu =+1$ and $\delta \nu =-\nu$ edge modes, formed at the interface between ${\nu }_0=1$ and $\nu =1/3$ FQH fluids. The heat/energy current can be injected at $S_1$ or $S_{\nu }$ and is measured at the drains. The dashed lines are schematic, indicating the continuation of the QH fluid not shown in the figure. The short dotted lines represent tunneling between the counterpropagating modes.

Figure 3

(a) Voltage profiles for the 1-1/3 line junction, with $\ell =10a,L=100a$. We see explicitly that the equilibration of the two channels takes place within the length $\ell$. (b) Plot of the temperature profiles, with $\mathrm{\Delta }T=1,T_0=0.1,\overline{\ell }=5.55a,L=100a$. (c) Plot of the ratio of the net heat current to the total injected energy current $\left({\pi }^2/3\right)[T^2\left(x\right)-T_0^2]/{\left(e{V}_0/k\right)}^2$ for $\ell =10a,L=100a$. This ratio is independent of the experimental parameters $T_0,V_0$.

Figure 4

Geometry of a low-density constriction.

Figure 5

(a) Discrete tunneling bridge for the WMG edge structure. The inner three channels are characterized by the tunneling probability $g_1$, while the outer-channel processes are characterized by the tunneling parameter $g_2$. (b) WMG edge structure in the QPC geometry (tunneling not shown).

Figure 6

(a) Linear-log plot of the conductances $G_{S_1{D}_2},G_{S_1{D}_3}$ in the single QPC as a function of $\alpha ={\ell }_1/{\ell }_2$ for ${\ell }_1=100a,L=1000a,L_V=5a,$ and $\alpha =0.01\text{–}1000$. (b) Renormalized edge modes of the single QPC at the intermediate fixed point. This renormalized edge structure picture produces conductance results qualitatively similar to our model in the regime that ${\ell }_2>L$ (the decay of the neutral modes is not shown). (c) Renormalized edge modes of the single QPC in the KFP (low-temperature) fixed point, with a $\nu =1/3$ density QH droplet in the constriction. This renormalized edge structure picture produces conductance results qualitatively similar to our model in the limit that $\alpha \to \infty$ (the decay of the neutral mode is not shown).

Figure 7

(a) Log-log plot of the thermal conductances $K_{S_1{D}_2},K_{S_1{D}_3},K_{S_1{D}_4},K_{\mathrm{back}}$ in the single QPC as a function of $L$ for $\alpha =0.03,{\ell }_1=10a,L_V=0a,$ and $L={10}^{2}a\text{–}{10}^{5}a$. We note that there is a marked change in the behavior of the conductance curves around $L\sim {\overline{\ell }}_2\sim 560a$, beyond which the power law in $L$ emerges; this is an indication of equilibration between edge modes. (b) Plot of the derivative $d(\text{ln}\left(K_{S_1{D}_3}\right))/d(\text{ln}\left(L\right))$, which characterizes the onset of the power law, as a function of $L$ and $\alpha$ for $L=0\text{–}5000a$. We see that the onset of $1/L$ behavior for $K_{S_1{D}_3}$ occurs when ${\overline{\ell }}_2\lesssim L$, which happens when $\alpha \gtrsim \frac{\left({\ell }_1/\tilde{\gamma }\right)}{L}$. We overlay the curve $\alpha =\frac{\left({\ell }_1/\tilde{\gamma }\right)}{L}$, which matches qualitatively with the onset of $1/L$ behavior.

Figure 8

(a) Bare modes of the double QPC in the “winding” geometry. The width of the constriction is $L_V$, the length of an arm of a QPC is $L$, and the circumference of the winding edge states in between the two QPCs is $2L$. There are additional edge structures for this geometry, in analogy with (b)–(d) of this figure. (b) Bare modes of the double QPC in the “no-winding” geometry. The length of the path that the edge states travel between the two QPCs is $3L$. (c) Incoherent analog of the WMG intermediate fixed point in the “winding” geometry (the decay of the neutral modes is not shown). This picture reproduces the same conductance results as our model in the regime with ${\ell }_2>L$. (d) Incoherent analog of the KFP fixed point in the “winding” geometry, with $\nu =1/3$ QH droplets in the constrictions (the decay of the neutral modes is not shown). This picture reproduces the same conductance results as our model in the limit that $\alpha \to \infty$.

Figure 9

(a) Log-linear conductance plot of the conductances $G_{S_1{D}_2},G_{S_1{D}_3}$ as a function of $\alpha$ for $L=100a,L_V=10a,{\ell }_1=5a$; the dashed lines correspond to the “no-winding” geometry, and the solid lines correspond to the “winding” geometry. (b) Log-linear conductance plot of the conductances $G_{S_1{D}_2},G_{S_1{D}_3}$ as a function of $\alpha$ in the limit of a narrow constriction with $L=100a,L_V=0,{\ell }_1=5a$; the convention is the same as in (a).

Figure 10

(a) Single-QPC geometry with points of boundary condition matching shown (red dots); there are 22 such points. The single-QPC geometry consists of six different regions ($A,B,C,D,E,F$ as shown in the figure) where the solutions are matched; these regions are indicated with translucent colorings. (b) Winding geometry with points of boundary condition matching shown (red dots); there are 36 such points. The different regions where the solutions are matched are indicated with translucent colorings. (c) No-winding geometry with points of boundary condition matching shown (red dots); there are 40 such points. The different regions where the solutions are matched are indicated with translucent colorings.