⇦Chinese Physics Letters, 2019, Vol. 36, No. 8, Article code 087301 Quasihole Tunneling in Disordered Fractional Quantum Hall Systems * Min Lu (卢旻)1, Na Jiang (江娜)1, Xin Wan (万歆)1,2** Affiliations 1Zhejiang Institute of Modern Physics, Zhejiang University, Hangzhou 310027 2CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190 Received 17 April 2019, online 22 July 2019 *Supported by the National Natural Science Foundation of China under Grant No 11674282, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No XDB28000000, and the National Basic Research Program of China under Grant No 2015CB921101.
**Corresponding author. Email: xinwan@zju.edu.cn Citation Text: Lu M, Jiang N and Wan X 2019 Chin. Phys. Lett. 36 087301    Abstract Fractional quantum Hall systems are often described by model wave functions, which are the ground states of pure systems with short-range interaction. A primary example is the Laughlin wave function, which supports Abelian quasiparticles with fractionalized charge. In the presence of disorder, the wave function of the ground state is expected to deviate from the Laughlin form. We study the disorder-driven collapse of the quantum Hall state by analyzing the evolution of the ground state and the single-quasihole state. In particular, we demonstrate that the quasihole tunneling amplitude can signal the fractional quantum Hall phase to insulator transition. DOI:10.1088/0256-307X/36/8/087301 PACS:73.43.-f, 73.43.Jn, 73.43.Cd, 73.43.Nq © 2019 Chinese Physics Society Article Text The fractional quantum Hall (FQH) effect is a remarkable demonstration of the topological order in quantum many-body systems.[1] The ground state is a quantum liquid, which supports quasiparticles with fractional charge.[2] The minimum energy to create an isolated collective excitation in the bulk is the energy gap, which varies continuously with the Hamiltonian parameters, such as the strength of disordered potential.[3,4] For weak disorder, quasiparticle excitations in the bulk are localized, hence the quantized Hall conductance persists. For a sufficiently strong disorder, the system becomes a normal insulator. Quasiparticles propagating along the FQH edge can carry charge currents.[5] They can also tunnel across the edges that are brought together by, for example, a quantum point contact (QPC).[6] By measuring the fluctuations of the current due to quasiparticle tunneling across the QPC, de-Picciotto et al.[7] and Saminadayar et al.[8] have demonstrated the charge fractionalization in the $\nu=1/3$ Laughlin state. Quasiparticle tunneling has also been an indispensable method to reveal the non-Abelian nature of the $\nu=5/2$ FQH state.[9–11] The calculation of the quasiparticle tunneling amplitude[12–14] has also been important in understanding the interference effect in the non-Abelian state.[15,16] The edge of a Laughlin state can be described by the chiral Luttinger liquid theory.[5] The quasihole propagator develops an anomalous exponent, which is related to the conformal dimension of the quasiholes. This exponent also appears in the scaling behavior of the quasihole tunneling amplitude in a pure system.[17] In particular, Hu et al. have found an exact solution that matches the numerical results in the conformal limit, which can be used to extract the conformal dimension of the quasiholes accurately.[17] Motivated by the recent development in quasiparticle tunneling theories and experiments to detect the topological order of the FQH states, we study the disorder driven FQH-insulator transition in a quasihole-tunneling setup. We theoretically set up a model that includes quasiparticle trapping and tunneling in a disordered potential. We show that the ground state wave function overlap and the quasihole tunneling amplitude develop a broad distribution for large disorder. Nevertheless, the properly chosen disorder-averaged tunneling amplitude exhibits qualitatively different behaviors and can be used to detect the FQH to insulator transition. Our results demonstrate the robustness of quasiparticle tunneling, especially its scaling behavior associated with the topological order of the system. We study a two-dimensional microscopic system of $N$ electrons with long-range Coulomb interaction on disk geometry, confined by uniform neutralizing background charge of radius $R=\sqrt{2N/\nu}$ at a setback distance $d$ above the electrons. For simplicity, we set the magnetic length $l_{\rm B}=1$. The density of the background charge $\rho_0=\nu e/(2\pi)$ is fixed according to a Landau level (LL) filling factor $\nu=1/3$. The Hamiltonian of the system is $$ H=\frac{1}{2}\sum_{mnl} V^{l}_{mn} c^†_{m+l} c^†_{n} c_{n+l} c_{m} + H^{\rm bg} + H^{\rm d},~~ \tag {1} $$ where $c^†_{m}$ and $c_{m}$ are the electron creation and annihilation operators of orbital $m$, with single-particle wave function $\phi_m({\boldsymbol r})$, and $V^{l}_{mn}$ are the corresponding Coulomb matrix elements, $$ V^{l}_{mn}\!\!=\!\!\!\int d^{2}r_{1} d^{2}r_{2} \phi^{*}_{m+l}({\boldsymbol r}_{1})\phi^{*}_{n}({\boldsymbol r}_{2}) \frac{e^{2}}{\epsilon r_{12}} \phi_{n+l}({\boldsymbol r}_{2})\phi_{m}({\boldsymbol r}_{1}). $$ The realistic background confinement potential is[18] $H^{\rm bg}=\sum_m U^{\rm bg}_m c_m^† c_m$ with $$ U^{\rm bg}_m=- {\rho_0 e \over \epsilon} \int d^{2} r \int _{r_1 \leq R} d^{2} r_1 \frac{\vert \phi_m({\boldsymbol r}) \vert^2}{\sqrt{ \vert {\boldsymbol r}-{\boldsymbol r}_1 \vert ^{2}+d^{2}}}.~~ \tag {2} $$ For this study, we fix $d=0.5$ such that the system stays away from edge reconstruction.[18,19] We introduce a disorder potential $H^{\rm d}=\sum_{m} U^{\rm d}_{m} c^†_{m} c_{m}$, where $U^{\rm d}_{m}$ is distributed uniformly within the interval $[-W, W]$. The random potential preserves the rotational symmetry and can be thought to be already averaged over polar angle $\theta$; as a result, it provides fluctuations to the confinement potential. We calculate the ground state $\vert {\it \Psi}_{\rm gs} \rangle$ with $M_{\rm tot}^{\rm gs}=3N(N-1)/2$, the total angular momentum of the corresponding Laughlin state, by the Lanczos diagonalization. To obtain the state $\vert {\it \Psi}_{\rm qh} \rangle$ with an additional quasihole at the origin, we apply a Gaussian trapping potential $H^{\rm tr}=t\sum_{m} \exp(-m^{2}/2s^{2})c^†_{m}c_{m}$, where $s$ and $t$ are the width and the strength of the trapping potential, respectively.[20] We choose $s=2$, which is comparable to the size of a quasihole. For sufficiently strong trapping potential, the global ground state is expected to be the quasihole state $\vert {\it \Psi}_{\rm qh} \rangle$, whose total angular momentum is $M^{\rm qh}_{\rm tot}=M_{\rm tot} ^{\rm gs}+N$. To study the quasiparticle tunneling across a QPC, we introduce a tunneling potential $T=V_{\rm t} \sum_i \delta(\theta_i)$, which breaks the rotational symmetry and allows a quasiparticle originally trapped at the droplet center to tunnel to the edge. The quasihole tunneling amplitude on disk geometry is then $$ {\it \Gamma}=\frac{\vert \langle {\it \Psi}_{\rm qh} \vert T \vert {\it \Psi}_{\rm gs}\rangle \vert }{\sqrt{\langle {\it \Psi}_{\rm qh} \vert {\it \Psi}_{\rm qh} \rangle } \sqrt{\langle {\it \Psi}_{\rm gs} \vert {\it \Psi}_{\rm gs} \rangle}}.~~ \tag {3} $$ For an ideal quasihole of charge $q$ in the Laughlin state, we expect that, in the thermodynamic limit $$ {\it \Gamma}^{\rm q} (N, d) \propto N^{1-2 {\it \Delta}^{q}},~~ \tag {4} $$ where ${\it \Delta}^{\rm q}$ is the conformal dimension of the quasihole. The power-law contribution can be obtained by scaling analysis, while additional contributions due to the form factor of the disk geometry appear in the prefactor.[17]
cpl-36-8-087301-fig1.png
Fig. 1. Histograms of the overlap between the Laughlin state and the ground states at disorder strengths $W=0.05$, 0.12, 0.15 and 0.20. The system has $N=8$ electrons at $\nu=1/3$.
We start by exploring the evolution of wave functions in the presence of disorder. We generate $N_{\rm s}=1000$ samples of $N=8$ electrons at $\nu= 1/3$ with random potential for disorder strength ranging from $W=0.05$ to $W=0.22$. Figure 1 shows the histograms of the overlap between the pure Laughlin state and the ground states at various disorder strengths. For small disorder strength $W=0.05$, the overlap is peaked at 1, which implies that small randomness cannot destroy the topological order of the Laughlin state. As disorder strength increases, the peak shifts to smaller overlap with a broad asymmetric shape. For $W=0.12$, the smallest overlap is approaching 0, while the largest remains close to 1. A second peak at zero overlap starts to develop at $W=0.15$, while the main peak continues to diminish its peak value. For $W=0.2$, a singular peak dominates at 0, but the population with significant overlap is still large. The overlap analysis of the ground state wave function suggests that the Laughlin topological order persists in the microscopic system with $N=8$ up to about $W=0.15$, which is consistent with previous studies.[21] However, this is not strong enough evidence to define critical disorder strength in the thermodynamic limit, because the overlap for sufficiently large systems would eventually drop to zero even in the pure case. We turn, instead, to the tunneling amplitude of a quasiparticle across edges with increasing tunneling distance. For weak disorder, we expect to see the power-law scaling behavior resulting from the chiral Luttinger liquid theory. For a sufficiently large disorder, we expect to see an exponential decay, signaling the breakdown of the topological order over large distances.
cpl-36-8-087301-fig2.png
Fig. 2. Histograms of the logarithmic value of quasihole tunneling amplitudes, as in Eq. (3), at disorder strengths $W=0.05$, 0.12, 0.15, 0.20. The system has $N=8$ electrons at $\nu=1/3$.
To generate a quasihole in disordered samples, we choose a trapping potential strength $t$ that is stronger than the threshold value in a pure sample.[20] The wave function with the lowest energy in the quasihole momentum sector is then the quasihole state in the presence of disorder. Together with the ground state wave function in the absence of the trapping potential, we can calculate the tunneling amplitude according to Eq. (3). The tunneling amplitude, again, is a fluctuating quantity in disordered samples, thus we need to study its distribution for various disorder strengths. Due to the non-Gaussian distribution of the tunneling amplitude similar to that of the ground state overlap in Fig. 1, we plot the histograms of $\ln {\it \Gamma}$ instead in Fig. 2. On the logarithmic scale, the histograms appear reasonably bell-shaped with an asymmetric tail on the left for sufficiently large disorder. This skewness is associated with the breakdown of topological order in finite systems, as also revealed in the histograms of the ground state overlap. The long tail on the logarithmic scale appears to emerge simultaneously with the second peak at zero overlap in Fig. 1. The topological properties of FQH systems are robust against weak disorder. In the case of quasihole tunneling, the robustness is reflected by the result that the peak of ${\it \Gamma}(W=0.05)$ remains close to the exact value ${\it \Gamma}(W=0)$ in the clean limit. When disorder increases, ${\it \Gamma}$ deviates further away from the exact value. Two questions arises here. What is the proper way to average the tunneling amplitude over disorder? And, how should we analyze the averaged value in the presence of disorder? By comparing Figs. 1 and 2, we can conclude that the average of $\ln {\it \Gamma}$, which is related to the geometric mean of ${\it \Gamma}$, is the more appropriate quantity to explore. At large disorder, the geometric mean captures the distribution that is broad even on the logarithmic scale and can distinguish power-law scaling from exponential decay. The average tunneling amplitude is, therefore, defined as $$ \bar{{\it \Gamma}}=\exp \Big[\frac{1}{N_{\rm s}}\sum^{N_{\rm s}}_{i=1} \ln {\it \Gamma}_{i}\Big].~~ \tag {5} $$ To separate exponential decay from power-law decay, we calculate the normalized tunneling amplitude $\bar{{\it \Gamma}}/{\it \Gamma}_0$, where ${\it \Gamma}_0$ is the tunneling amplitude in the clean case. Recall that ${\it \Gamma}_0$ exhibits the power-law behavior associated with the topological order of the ground state. We expect that $\bar{{\it \Gamma}}/{\it \Gamma}_0$ approaches a constant when the ground state is in the FQH phase, while it decays exponentially when the topological order is destroyed. We plot the disorder averaged normalized tunneling amplitude $\bar{{\it \Gamma}}/ {\it \Gamma}_0$ for systems of 2–8 electrons at $\nu=1/3$ on a semi-logarithmic scale in Fig. 3. For $W=0.05$, $\bar{{\it \Gamma}}/{\it \Gamma}_0$ remains close to 1 for all system sizes, reflecting the robustness of topological order against disorder. As $W$ increases to 0.1, $\bar{{\it \Gamma}}/{\it \Gamma}_0$ exhibits visible decay for $2 \leq N \leq 5$, whereas saturates to about 0.69 for $N \geq 6$. The saturation value continues to decrease to 0.58 and 0.43 for $W=0.12$ and 0.14, respectively. We find no saturation in $\bar{{\it \Gamma}}/{\it \Gamma}_0$ for $W \geq 0.15$ within the finite system sizes that we can access. For $W=0.2$, $\bar{{\it \Gamma}}/{\it \Gamma}_0$ exhibits a clear exponential decay for $N=2$–8, indicating insulating behavior. The stark contrast between the saturation and decay clearly classifies the systems with different $W$ and $N$ into two categories. Sufficiently large systems with smaller disorder belong to the FQH phase, in which $\bar{{\it \Gamma}}$ develops a power-law scaling, just as that of their disorder-free counterpart. On the other hand, small systems with sufficiently large disorder belong to the trivially insulating phase, in which $\bar{{\it \Gamma}}$ is dominated by an exponential decay. For the system sizes that we can acquire significant statistics, the transition between the FQH and insulating phases occurs at about $W=0.145$. However, this cannot be taken as the critical $W$ because $\bar{{\it \Gamma}}$ for larger $W$ may bend up and saturate at larger systems beyond our reach. We therefore postulate that the saturation value continues to decrease as $W$ increases, and at the critical $W$, the value drops to zero.
cpl-36-8-087301-fig3.png
Fig. 3. Disorder averaged tunneling amplitude $\bar{{\it \Gamma}}$, normalized by the amplitude ${\it \Gamma}_0$ in the pure case, for various numbers of electrons and disorder strengths. For large disorder, $\bar{{\it \Gamma}}/{\it \Gamma}_0$ decays exponentially with the number of electrons. In weakly disordered systems, $\bar{{\it \Gamma}}/{\it \Gamma}_0$ bends up for large enough system sizes and appears to saturate, indicating the survival of the FQH topological order.
cpl-36-8-087301-fig4.png
Fig. 4. The saturation value of $\bar{{\it \Gamma}}/{\it \Gamma}_0$ as a function of $W$. We try to fit the data by a critical form (Eq. (6)) and a power-law form (Eq. (7)). Their intercepts with the $x$-axis indicate the critical disorder strength $W_{\rm c}=0.155$ and 0.181, respectively, for the FQH to insulator transition.
Figure 4 shows the saturation value $\bar{{\it \Gamma}}_{\rm s}/{\it \Gamma}_0$ as a function $W$, which converges to unity in the small disorder limit. We fit the data by the following two formulas to estimate the critical disorder. Firstly, we attempt to fit data at $W \ge 0.1$ with a critical behavior $$ \bar{{\it \Gamma}}_{\rm s}/{\it \Gamma}_0=(1-W/W_{\rm c})^{\beta}.~~ \tag {6} $$ As shown in Fig. 4, we obtain $W_{\rm c}=0.155 \pm 0.004$, which is slightly larger than 0.145 that is roughly corresponding to the separatrix of the two distinct behaviors in Fig. 3. The result of the critical exponent is $\beta=0.37 \pm 0.03$. An alternative fit, which takes into account of the small-disorder data, is $$ \bar{{\it \Gamma}}_{\rm s}/{\it \Gamma}_0=1-(W/W_{\rm c})^{\alpha}.~~ \tag {7} $$ In this fit we obtain $W_{\rm c}=0.181 \pm 0.004$, which can be regarded as the upper bound for the critical point. Interestingly, the curve almost has a parabolic form with $\alpha=2.12 \pm 0.10$. We note that the critical disorder is determined by data with $W < 0.15$. According to Fig. 1, the disordered ground states still have rather large overlap with the Laughlin state (e.g., peaked around 0.7 at $W= 0.15$), thus concepts such as the ground state topological order and quasihole, as well as its tunneling, are well defined. This justifies our approach of quasihole tunneling amplitude. Motivated by experiments, we have calculated the quasiparticle tunneling amplitude in the $\nu=1/3$ system with disorder. The amplitude for weak disorder and large system size exhibits scaling behavior, which is reminiscent of the behavior in the pure system dictated by the underlying conformal field theory. The deviation from the scaling behavior can be explored to determine the critical disorder of the FQH-insulator transition. Disorder induced by impurities is ubiquitous in experimental samples. From experimental perspectives, we have demonstrated the robustness of the scaling behavior in quasiparticle tunneling, which is a topological property, in the presence of disorder. Our calculation suggests that for moderate disorder, the scaling behavior can be observed for sufficiently large systems, even though the overall strength of the scaling behavior is reduced by a significant factor. This guarantees the observation of the topological properties in dirty experimental samples, as long as the experimental precision allows. References Two-Dimensional Magnetotransport in the Extreme Quantum LimitAnomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged ExcitationsDisorder-Driven Collapse of the Mobility Gap and Transition to an Insulator in the Fractional Quantum Hall EffectMobility gap in fractional quantum Hall liquids: Effects of disorder and layer thicknessElectrodynamical properties of gapless edge excitations in the fractional quantum Hall statesNonequilibrium noise and fractional charge in the quantum Hall effectDirect observation of a fractional chargeObservation of the e / 3 Fractionally Charged Laughlin QuasiparticleObservation of a quarter of an electron charge at the ν = 5/2 quantum Hall stateQuasi-Particle Properties from Tunneling in the Formula Fractional Quantum Hall StateCompeting ν = 5/2 fractional quantum Hall states in confined geometryInterferometric signature of non-Abelian anyonsQuasiparticle tunneling in the Moore-Read fractional quantum Hall stateScaling analysis of the non-Abelian quasiparticle tunneling in ${{\mathbb Z}}_k$ FQH statesMeasurement of filling factor 5/2 quasiparticle interference with observation of charge e/4 and e/2 period oscillationsAlternation and interchange of e/4 and e/2 period interference oscillations consistent with filling factor 5/2 non-Abelian quasiparticlesScaling and non-Abelian signature in fractional quantum Hall quasiparticle tunneling amplitudeEdge reconstruction in the fractional quantum Hall regimeReconstruction of Fractional Quantum Hall EdgesTrapping Abelian anyons in fractional quantum Hall dropletsBULK AND EDGE QUASIHOLE TUNNELING AMPLITUDES IN THE LAUGHLIN STATE
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