Negative Excess Shot Noise by Anyon Braiding

TL;DR

This paper predicts a unique negative excess shot noise signature in fractional quantum Hall systems, revealing Abelian anyonic statistics through the braiding effect during electrical tunneling at a quantum point contact.

Contribution

It introduces a novel theoretical prediction of negative excess shot noise as a fingerprint of Abelian anyonic fractional statistics in quantum Hall systems.

Findings

Negative excess noise occurs at large voltages, below thermal equilibrium noise.

The negative excess noise results from effective anyon braiding around another anyon.

This signature distinguishes Abelian anyonic statistics from conventional fractional charge detection.

Abstract

Anyonic fractional charges $e^*$ have been detected by autocorrelation shot noise at a quantum point contact (QPC) between two fractional quantum Hall edges. We find that the autocorrelation noise can also show a fingerprint of Abelian anyonic fractional statistics. We predict the noise of electrical tunneling current $I$ at the QPC of the fractional-charge detection setup, when anyons are dilutely injected, from an additional edge biased by a voltage, to the setup in equilibrium. At large voltages, the nonequilibrium noise is {\it reduced} below the thermal equilibrium noise by the value $2 e^* I$. This negative excess noise is opposite to the positive excess noise $2e^* I$ of the conventional fractional-charge detection and also to usual positive autocorrelation noises of electrical currents. This is a signature of the Abelian fractional statistics, resulting from the effective braiding of an anyon thermally excited at the QPC around another anyon injected from the additional edge.

Equations

$$\delta S=-2e^{\ast }I<0\text{at}{e}^{\ast }V\gg k_{B}T.$$

$$I$$

$$\delta S$$

$$\delta S=$$

$$-$$

$$W_{i\to j}\simeq W_{i\to j}^{\text{P}}+W_{i\to j}^{\text{TVB}}\text{at}{e}^{\ast }V\gg k_{B}T.$$

$$C_{3\to 2}^{\text{TVB}}=$$

$$-$$

$$⟨{\mathcal{T}}_{1\to 2}^{†}(t_1){\mathcal{T}}_{3\to 2}^{†}(t_2^{\prime }){\mathcal{T}}_{3\to 2}(t_2){\mathcal{T}}_{1\to 2}(t_1)⟩$$

$$=e^{2i\pi \nu }⟨{\mathcal{T}}_{3\to 2}^{†}(t_2^{\prime }){\mathcal{T}}_{3\to 2}(t_2)⟩⟨{\mathcal{T}}_{1\to 2}^{†}(t_1){\mathcal{T}}_{1\to 2}(t_1)⟩,$$

$$⟨{\mathcal{T}}_{3\to 2}^{†}(t_2^{\prime }){\mathcal{T}}_{3\to 2}(t_2)⟩⟨{\mathcal{T}}_{1\to 2}^{†}(t_1){\mathcal{T}}_{1\to 2}(t_1)⟩$$

$$=e^{i\pi \nu }{D}_2(d,t_2,t_2^{\prime })D_3(0,t_2,t_2^{\prime })D_1(0,t_1,t_1)D_2(0,t_1,t_1).$$

$$W_{3\to 2}^{\text{TVB}}\propto {\gamma }_1^2{\gamma }_2^2{V}^{2\nu -1}{T}^{2\nu -2}\text{Re}[e^{i\pi \nu }(e^{2i\pi \nu }-1)],$$

Full text

Abelian anyons appear in fractional quantum Hall (FQH) systems of filling factor $\nu=1/(2n+1)$, $n=1,2,\cdots$. They obey the fractional exchange statistics Leinaas ; Arovas ; Stern . Two anyons gain the phase $\pm\pi\nu$ when their positions are adiabatically exchanged, and $\pm 2\pi\nu$ when one braids around the other. Proposals Chamon ; Vishveshwara ; Kim06 ; Safi ; Campagnano12 ; Law ; Feldman ; Kane03 ; Rosenow ; An ; Camino ; Willett ; Ofek ; McClure ; Halperin ; Rosenow2 ; Grosfeld for detecting the fractional statistics are based on interferometers or current-current cross-correlations. They involve quantities experimentally inaccessible or affected by unintended setup change or Coulomb interaction. It will be useful to find fractional-statistics effects experimentally feasible.

Shot noise $S$, zero-frequency nonequilibrium fluctuation of electrical current $I$, has valuable information Blanter . Its Poisson value $S=2qI$ in the tunneling regime of a quantum point contact (QPC) was used to detect the charge $q$ of current carriers Reznikov . The fractional charge $e^{*}=\nu e$ of anyons was measured Kane94 ; Goldman ; Picciotto ; Saminadayar ; Dolev ; Reznikov2 ; Griffiths from the ratio $S/I=2e^{*}$ at a QPC between FQH edges; $e$ is the electron charge. The Poisson value originates from uncorrelated transfer of discrete charges. Reduction or enhancement from the value signifies effects such as resonances, diffusive scattering, Cooper pairing, etc Blanter .

In this work, we predict unusual behavior of shot noise, originating from the Abelian fractional statistics of Laughlin anyons, in the setup [Fig. 1(a)] composed of the conventional fractional-charge detection part (Edge2, Edge3, QPC2) and an additional edge (Edge1). Anyons are dilutely injected Comforti ; Comforti2 ; Chung ; Kane_PRB via QPC1 from Edge1, biased by voltage $V$, to the detection part in equilibrium. We find that the zero-frequency autocorrelation noise $S(V,T)$ of tunneling current $I$ at QPC2 is reduced below the thermal equilibrium noise $S(0,T)$ at temperature $T$,

[TABLE]

$\delta S\equiv S(V,T)-S(0,T)$ is the excess shot noise with respect to the thermal noise and $k_{B}$ is Boltzmann constant. The negative excess noise is unusual, since the setup has the conventional Poisson process [Fig. 1(b)] enhancing the noise; it is opposite to the positive noise $2e^{*}I>0$ of the conventional fractional-charge detection Kane94 ; Goldman ; Picciotto ; Saminadayar ; Dolev ; Reznikov2 ; Griffiths . By contrast, in the integer quantum Hall regime at $\nu=1$, the setup shows the positive Poisson noise of $\delta S=2eI>0$, which cannot be extrapolated from Eq. (1) with $e^{*}=e/(2n+1)\to e$.

The negative excess noise results from an interference involving anyon braiding [Fig. 1(c)], which weakens thermal anyon tunneling at QPC2, reducing the noise. The reduction dominates over the enhancement by the Poisson process. Interestingly, for electrons at $\nu=1$, the interference does not exist, as it is described by a pair of disconnected Feynman diagrams that exactly cancel each other, according to the linked cluster theorem Fetter . For anyons, the cancellation is only partial, since the subdiagrams (vacuum bubbles) of one of the disconnected diagrams are linked AlgebraicT by the braiding. This type of anyon processes, vacuum bubbles linked by braiding, is called topological vacuum bubbles (TVBs) Han . Detection of the negative excess noise is experimentally feasible, and will provide a signature of TVBs and the fractional statistics in the case of pristine edges (without edge reconstruction). The signature manifests itself in the leading-order contributions (in QPC tunneling strengths) to the excess noise, thanks to the dilute anyon injection at QPC1.

Excess noise.— We consider the time $t$ average $I=\overline{I(t)}$ of tunneling current $I(t)$ at QPC2, and its zero-frequency noise $S=2\int^{\infty}_{-\infty}dt(I(t)-I)(I(0)-I)$. Employing a perturbation theory based on the chiral Luttinger liquid Wen ; vonDelft , Keldysh Green’s functions, and Klein factors Guyon , we derive $I$ and $\delta S=S(V,T)-S(0,T)$ at voltages $e^{*}V\gg k_{B}T$ in the anyon tunneling regime of $\gamma_{i}T^{\nu-1}\ll 1$, up to the leading order $O(\gamma_{1}^{2}\gamma_{2}^{2})$ of tunneling strength $\gamma_{i}$ at QPC$i$,

[TABLE]

This gives Eq. (1) Correction1 ; Supple . Notice that $I>0$ but $\delta S<0$. The factors having $\pi\nu$ originate from anyon braiding.

The current $I$ and excess noise $\delta S$ are linked to measurable quantities. $I$ equals the average current $I_{3}=\overline{I_{3}(t)}$ at D3, as only S1 is biased. $\delta S$ is obtained Supple by

[TABLE]

The noise $S_{3}(V,T)=2\int^{\infty}_{-\infty}dt(I_{3}(t)-I_{3})(I_{3}(0)-I_{3})$ is measured at D3. $\partial I_{3}(V,T,V_{3})/\partial V_{3}|_{V_{3}=0}$ is measured with the voltage $V_{3}$ applied to S3 in addition to the voltage $V$ at S1, and equals the correlation between the tunneling current $I(t)$ at QPC2 and the current from S3 to QPC2, according to the nonequilibrium fluctuation-dissipation theorem Wang1 ; Wang2 ; Smits .

Main processes.— We discuss the origin of $\delta S<0$. The tunneling current and its excess noise satisfy Feldman2 $I=e^{*}(W_{2\to 3}-W_{3\to 2})$ and $\delta S=2(e^{*})^{2}(W_{2\to 3}+W_{3\to 2})$. $W_{2\to 3}$ ($W_{3\to 2}$) is the change, by the voltage $V$, in the rate for a particle-like (hole-like) anyon to move from Edge2 to Edge3 at QPC2. Two types of processes, Poisson processes and TVBs, make contribution $W^{\textrm{P}}_{i\to j}$ and $W^{\textrm{TVB}}_{i\to j}$, respectively, to $W_{i\to j}$,

[TABLE]

$W_{i\to j}$ is computed in Ref. Supple .

In the Poisson process [Fig. 1(b)] for $W^{\textrm{P}}_{2\to 3}$, a particle-like anyon, biased by the voltage $V$, moves from Edge1 to Edge3 through tunneling at QPC1 and QPC2. This leads to $W_{2\to 3}^{\textrm{P}}\propto\gamma_{1}^{2}\gamma_{2}^{2}V^{4\nu-3}$, as the voltage-biased tunneling probability at QPC$i$ and the current from S1 to QPC1 are proportional to $\gamma_{i}^{2}V^{2\nu-2}$ and $V$, respectively. By contrast, $W^{\textrm{P}}_{3\to 2}=0$, since tunneling of a hole-like anyon from Edge2 to Edge3 is not induced by $V$.

Next, we consider the TVB for $W^{\textrm{TVB}}_{3\to 2}$. It is the interference of two subprocesses $a_{1}$ and $a_{2}$ [Fig. 1(c)]. In $a_{1}$ and $a_{2}$, a particle-like anyon, induced by the voltage $V$, moves from Edge1 to Edge2 via tunneling at QPC1 at time $t_{1}$, and then moves to D2. The operator for the QPC1 tunneling is $\mathcal{T}_{1\rightarrow 2}(t_{1})=\Psi_{2}^{\dagger}(0,t_{1})\Psi_{1}(0,t_{1})$. $\Psi^{\dagger}_{i}(x_{i},t_{1})$ creates an anyon at position $x_{i}$ of Edge$i$; QPC1 is located at $x_{i}=0$. After (before) this anyon passes QPC2, a particle-hole pair is thermally excited at QPC2 at time $t_{2}$ ($t_{2}^{\prime}$) in the subprocess $a_{1}$ ($a_{2}$). Then the particle-like thermal anyon moves to D2 along Edge2, while the hole-like one to D3 along Edge3. The excitation is described by the QPC2 tunneling operator $\mathcal{T}_{3\to 2}(t)=\Psi_{2}^{\dagger}(d,t)\Psi_{3}(0,t)$ at $t=t_{2}$ ($t_{2}^{\prime}$) in $a_{1}$ ($a_{2}$); QPC2 is located at $x_{2}=d$ ($x_{3}=0$) on Edge2 (Edge3).

To illustrate the nontrivial features (topological link by anyon braiding and the partner disconnected process) of the TVB for $W^{\textrm{TVB}}_{3\to 2}$, we consider the $V\to\infty$ limit where the voltage-biased particle-like anyon becomes a point particle (its spatial broadening $\hbar v/(e^{*}V)\to 0$; $v$ is the anyon velocity). In this limit, the correlator

[TABLE]

describes the TVB. $\langle\cdots\rangle$ is the ensemble average with the bare Hamiltonian Supple $H_{i}$ of Edge$i$.

The first term of Eq. (5) shows the interference between the subprocesses $a_{1}$ and $a_{2}$; $\mathcal{T}_{3\to 2}(t_{2})\mathcal{T}_{1\to 2}(t_{1})$ describes $a_{1}$, while $\mathcal{T}_{3\to 2}(t_{2}^{\prime})\mathcal{T}_{1\to 2}(t_{1})$ describes $a_{2}$. This term is factorized Supple into a subcorrelator for the voltage-biased anyon, another for the thermal anyons, and a phase factor $e^{i2\pi\nu}$ (Fig. 2),

[TABLE]

by using the exchange rules of the fractional statistics $\Psi_{i}^{\dagger}(x)\Psi_{i}(y)=\Psi_{i}(y)\Psi_{i}^{\dagger}(x)e^{i\pi\nu\textrm{sgn}(x-y)}$ and $\Psi_{i}^{\dagger}(x)\Psi_{i}^{\dagger}(y)=\Psi_{i}^{\dagger}(y)\Psi_{i}^{\dagger}(x)e^{-i\pi\nu\textrm{sgn}(x-y)}$ (the rules between operators of different edges are constructed, using Klein factors Guyon ; Supple ). The factor $e^{i2\pi\nu}$ is attributed to effective braiding of the thermal anyon around the voltage-biased anyon in the interference $a_{2}^{*}a_{1}$, depicted as the link of two loops in Fig. 2(b); the factorization is equivalent to untying the link. The solid blue loop corresponding to the subcorrelator $\langle\mathcal{T}_{3\to 2}^{\dagger}(t_{2}^{\prime})\mathcal{T}_{3\to 2}(t_{2})\rangle$ for the thermal anyons is formed, although $t_{2}\neq t_{2}^{\prime}$, with the help of the thermal length $\hbar v/(k_{B}T)$; $\langle\mathcal{T}_{3\to 2}^{\dagger}(t_{2}^{\prime})\mathcal{T}_{3\to 2}(t_{2})\rangle$ is nonvanishing for $|t_{2}-t_{2}^{\prime}|\lesssim\hbar/(k_{B}T)$. Similarly, at finite $V$, the dashed red loop representing $\langle\mathcal{T}_{1\to 2}^{\dagger}(t_{1})\mathcal{T}_{1\to 2}(t_{1})\rangle$ for the voltage-biased anyon is formed with $|t_{1}-t_{1}^{\prime}|\lesssim\hbar v/(e^{*}V)$, when the tunneling at QPC1 occurs at $t_{1}^{\prime}$ ($\neq t_{1}$) in $a_{2}$ as described by $\mathcal{T}_{1\rightarrow 2}(t_{1}^{\prime})$. In this case, the braiding occurs for $t_{2}^{\prime}<t_{1}+d/v<t_{2}$ and $t_{2}^{\prime}<t_{1}^{\prime}+d/v<t_{2}$.

The effective braiding ($e^{2i\pi\nu}$) is decomposed into two events of anyon exchange. One exchange ($e^{i\pi\nu}$) occurs in the subprocess $a_{1}$ when the thermal anyon is excited on Edge2 at QPC2 [Fig. 2(a)]. It happens such that the thermal anyon effectively moves from the right side of the voltage-biased anyon to the left on Edge2 Supple . The other ($e^{i\pi\nu}$) occurs in the interference $a_{2}^{*}a_{1}$. The voltage-biased anyon of $a_{2}$ moves back to QPC1 passing the thermal anyon of $a_{1}$ [the top dashed arrow in Fig. 2(b)].

We call the first term of Eq. (5) a TVB since the trajectory (dashed red loop) of the voltage-biased anyon and that (solid blue loop) of the thermal anyon are disconnected to each other in the conventional sense but topologically linked AlgebraicT by the braiding. The TVB is accompanied by a partner disconnected process [Fig. 2(c)] that gives the second term of Eq. (5) and has the same subprocesses as the TVB except the braiding. The TVB and its partner disconnected process (or the correlator in Eq. (5)) appear in our calculation Supple of $W_{i\to j}$. The pairwise appearance is understood by considering electrons at $\nu=1$. For the electrons, the TVB is described by a disconnected Feynman diagram as the braiding link has no meaning, $e^{2i\pi\nu}=1$. Then it must be accompanied and exactly cancelled (leading to $C_{3\to 2}^{\textrm{TVB}}=0$; cf. Eqs. (5) and (6)) by the partner disconnected diagram, following the linked cluster theorem Fetter ; the second term of Eq. (5) has the minus sign for the cancellation; mathematically, the partner diagram appears due in part to the partition function of a Green’s function in its perturbation expansion, hence it does not have the braiding link. For the anyons, the cancellation is partial, because of the braiding.

The common factor of the two terms of Eq. (5) is further factorized with a correlator $D_{i}(x,t,t^{\prime})=\langle\Psi^{\dagger}_{i}(x,t)\Psi_{i}(x,t^{\prime})\rangle$ of each Edge$i$,

[TABLE]

The factor $e^{i\pi\nu}$ comes from exchange of a thermal anyon of $a_{1}$ and another of $a_{2}$ [Figs. 2(b,c)].

The TVB and its partner disconnected process give

[TABLE]

as the thermal (voltage-biased) tunneling probability at QPC2 (QPC1) is proportional to $\gamma_{2}^{2}T^{2\nu-2}$ ($\gamma_{1}^{2}V^{2\nu-2}$) while the current from S1 to QPC1 is proportional to $V$. The phase factors come from $\textrm{Re}[C_{3\rightarrow 2}^{\textrm{TVB}}]\propto\textrm{Re}[e^{i\pi\nu}(e^{2i\pi\nu}-1)]$ in Eqs. (5)-(7). $\textrm{Re}[\cdots]$ is taken, considering $[C_{3\rightarrow 2}^{\textrm{TVB}}]^{*}$.

There is a TVB process for $W^{\textrm{TVB}}_{2\to 3}$. $W^{\textrm{TVB}}_{2\to 3}$ is negligibly small at $e^{*}V\gg k_{B}T$ W23 .

We now compute $\delta S/I$. At $e^{*}V\gg k_{B}T$ and $\nu=1/(2n+1)<1$, the TVB for $W_{3\to 2}^{\textrm{TVB}}$ and its partner disconnected process dominate over the Poisson process for $W_{2\to 3}^{\textrm{P}}$, $W_{3\to 2}^{\textrm{TVB}}\gg W_{2\to 3}^{\textrm{P}}$; cf. Eq. (8) and $W_{2\to 3}^{\textrm{P}}\propto\gamma_{1}^{2}\gamma_{2}^{2}V^{4\nu-3}$. Hence, they determine the current and the excess noise, $I=-e^{*}W_{3\to 2}^{\textrm{TVB}}$ and $\delta S=2(e^{*})^{2}W_{3\to 2}^{\textrm{TVB}}$, leading to Eqs. (1) and (2). We emphasize that the ratio $\delta S/I$ has the negative universal value of $-2e^{*}$. This originates from the TVB for $W_{3\to 2}^{\textrm{TVB}}$ and its partner disconnected process, and equivalently from the anyon braiding. It is nontrivial that the disconnected process contributes to the observables $I$ and $\delta S$; for electrons or bosons, disconnected Feynman diagrams never contribute to observables Fetter .

The above findings are confirmed by numerically computing $\delta S$ Supple . For $\nu=1/3$, $\delta S$ approaches to $-2e^{*}I$ such that $\delta S=-1.8e^{*}I$ at $V=60$ $\mu$V at 50 mK and $-1.99e^{*}I$ at 80 $\mu$V at 50 mK.

Discussion.— The negative excess noise $\delta S<0$ results from the TVB process for $W_{3\to 2}^{\textrm{TVB}}$. It is interpreted as follows. At $V=0$, tunneling of a particle-like or hole-like anyon between Edge2 and Edge3 is thermally induced at QPC2, causing the thermal noise $S(0,T)$. Among those tunneling events, thermal tunneling of a hole-like anyon from Edge2 to Edge3 is weakened by a voltage-biased particle-like anyon injected from Edge1 to Edge2, when the voltage $V$ is applied to Edge1. The weakening is due to the effective braiding of the thermal anyon around the voltage-biased anyon, which results in the partial cancellation between the TVB and its partner disconnected process, $W_{3\to 2}^{\textrm{TVB}}\propto\textrm{Re}[e^{i\pi\nu}(e^{2i\pi\nu}-1)]<0$. The weakening leads to the current $I>0$ and the reduction of the noise $S(V,T)$ below $S(0,T)$. Note that $\delta S<0$ at any $V$, although both the Poisson process and the TVB (and its partner) contribute to $\delta S$ at $e^{*}V\lesssim k_{B}T$.

By contrast, for electrons at $\nu=1$, the Poisson process determines $I=eW_{2\to 3}^{\textrm{P}}$ and $\delta S=2e^{2}W_{2\to 3}^{\textrm{P}}$, leading to $\delta S=2eI>0$ at $e^{*}V\gg k_{B}T$. There is no topological link by the braiding ($e^{2i\pi\nu}=1$), and the TVB becomes a disconnected process and fully cancelled by its partner disconnected diagram, $W_{i\to j}^{\textrm{TVB}}=0$. This is why the excess noise $\delta S=2eI$ of the electrons cannot be extrapolated from Eq. (1) with $e^{*}\to e$.

Measurement of $\delta S$ is feasible, as the setup was experimentally studied in other contexts Comforti ; Chung ; Comforti2 : Typically, the tunneling probability of QPC1 and QPC2 is set to be 0.2, to have anyon tunneling Picciotto . We estimate $I\sim 50$ pA and $\delta S\sim$ $2.7\times 10^{-30}$ A2/Hz at 100 $\mu$V and $\nu=1/3$, which is detectable Comforti ; ChungChoi . When $\delta S$ is measured by using Eq. (3), one has to experimentally determine temperature $T$. The determination accuracy is within $\pm 3$ mK ChungChoi . Then, it is possible to obtain $\delta S=-2e^{*}I(1\pm 0.2)$ at 50 mK, $V=80$ $\mu$V, and $\nu=1/3$.

Our study is generalized to edges with multiple channels or reconstruction (see Ref. Supple ). For example, at filling factor $4/3$ or $7/3$ An ; Baer , the inner fractional edge channel corresponding to $\nu=1/3$ interacts with co-propagating outer channels, and is weakly backscattered at the QPCs. In this case $\delta S$ is still negative. On the other hand, when the $\nu=1/3$ edge channel interacts with an unexpected counter-propagating mode Rosenow_edge_reconst due to edge reconstruction, $\delta S$ is negative only when the interaction is sufficiently weak Saminadayar ; Roddaro . The outer channels at filling factor $4/3$ or $7/3$ are helpful in this case, since they can screen the edge reconstruction. In the above cases of multiple channels or edge reconstruction, detection of $\delta S<0$ may imply the fractional statistics of the quasiparticles deviating from Laughlin anyons due to the interchannel interactions. The quasiparticles become closer to Laughlin anyons for weaker interactions.

In summary, we predict the negative excess autocorrelation noise $\delta S<0$, a signature of the Abelian fractional statistics or the new process (TVB) not existing with fermions or bosons. It is unusual that the excess autocorrelation noise of electrical tunneling current is negative Blanter ; Lesovik .

We suggest that autocorrelation noise can provide signatures Buttiker ; Lee of identical-particle statistics. This is different from the conventional approach Henny ; Oliver ; Jeltes of detecting particle bunching or antibunching with Hanbury Brown-Twiss cross-correlations. It is unnatural to interpret the negative excess autocorrelation noise as deviation (anyonic partial bunching Safi ; Vishveshwara ; Kim06 ; Campagnano12 ; Rosenow2 ) from fermionic antibunching and bonsonic bunching, because it originates from the TVB having no counterpart in fermions or bosons.

We thank Hyungkook Choi, Sang-Jun Choi, Yunchul Chung, Sourin Das, Dmitri Feldman, Bertrand Halperin, Charles Kane, and Bernd Rosenow for valuable discussions, and the support by Korea NRF (SRC Center for Quantum Coherence in Condensed Matter, Grant No. 2016R1A5A1008184).