Topological vacuum bubbles by anyon braiding

According to a basic rule of fermionic and bosonic many-body physics, known as the linked cluster theorem, physical observables are not affected by vacuum bubbles, which represent virtual particles created from vacuum and self-annihilating without interacting with real particles. Here we show that this conventional knowledge must be revised for anyons, quasiparticles that obey fractional exchange statistics intermediate between fermions and bosons. We find that a certain class of vacuum bubbles of Abelian anyons does affect physical observables. They represent virtually excited anyons that wind around real anyonic excitations. These topological bubbles result in a temperature-dependent phase shift of Fabry–Perot interference patterns in the fractional quantum Hall regime accessible in current experiments, thus providing a tool for direct and unambiguous observation of elusive fractional statistics.

In many-body quantum theory 1 , Feynman diagrams are used to compute the expectation value of observables. This approach invokes vacuum bubble diagrams, which describe virtual particles excited from vacuum and self-annihilating without interacting with real particles. According to the linked cluster theorem 1 , each diagram possessing vacuum bubbles comes with, hence is exactly cancelled by, a partner diagram of the same magnitude but of the opposite sign. Consequently, vacuum bubbles do not contribute to physical observables.
In the following, we demonstrate that this common wisdom has to be revised for anyons: a certain class of vacuum bubbles of Abelian anyons does affect observables. These virtual particles, which we call topological vacuum bubbles, wind around a real anyonic excitation, gaining the braiding phase ± 2pn. We propose a realistic setup for detecting them and y* ¼ pn.

Results
Topological vacuum bubble. We illustrate topological vacuum bubbles. In Fig. 1a, a Feynman diagram represents interference a 1 a Ã 2 between processes a 1 and a 2 of propagation of a real particle. In a 1 , a virtual particle-hole pair is excited then self-annihilates after the virtual particle winds around the real particle, forming a vacuum bubble, while it is not excited in a 2 . The winding results in a braiding phase 2pn and an Aharonov-Bohm phase 2pF=F Ã 0 from the magnetic flux F enclosed by the winding path, contributing to the interference signal as e i 2pF=F Ã 9,19 .
The limiting cases of bosons (n ¼ 0) and fermions (n ¼ 1) imply that this bubble diagram appears together with, and is cancelled by, a partner diagram in Fig. 1b. The partner diagram has a bubble not encircling the real particle and involves only 2pF=F Ã 0 . The two diagrams (and their complex conjugates) yield For bosons and fermions, the two diagrams fully cancel each other with sin(pn) ¼ 0 in agreement with the linked cluster theorem; hence, the signal disappears. By contrast, for anyons they cancel only partially, producing the non-vanishing interference in an observable, and are topological as the braiding phase is involved.
Interferometer setup. In Fig. 2a, we propose a minimal setup for observing topological vacuum bubbles. It is a Fabry-Perot interferometer 9,17,20-23 in the n ¼ 1/(2n þ 1) FQH regime, coupled to an additional edge channel (Edge 1) via a quantum point contact (QPC1). At QPCi, there occurs tunnelling of a single anyon (rather than anyon bunching), fulfilled 24 with g i ( k B T; g i is the tunnelling strength and T is the temperature. Gate voltage V G is applied, to change the interferometer loop enclosing Aharonov-Bohm flux F. The interference part I int D 3 of charge current at drain D 3 is measured with bias voltage V applied to source S 1 ; the other S i 's and D i 's are grounded. Together with 'virtual' (thermal) anyon excitations in the interferometer, a voltage-biased 'real' anyon, dilutely injected at QPC1 from Edge 1 to the interferometer, forms topological vacuum bubbles, as shown below. The bubbles contribute to I int D 3 at the leading order ðI int D3 / g 2 1 g 2 g 3 Þ in QPC tunnelling, as Edges 2 and 3 are unbiased. It is noteworthy that in the setups previously studied 9-18 , topological bubbles do not contribute to current at the leading order.
We consider the regime of e Ã V ) k B T\' v p =L, where the size L V ' v p /(e*V) of the dilutely injected anyons is much smaller than interferometer size L and the injection of hole-like anyons at QPC1 is ignored; v p is anyon velocity along the edges and e*V should be much smaller than the FQH energy gap. Because of the dilute injection and L V ( L, anyon braiding is well defined in the interferometer. As shown below, the dependence of I int D3 on F or on V G provides a clear signature of the topological bubbles, consequently, y* ¼ pn in both of the pure Aharonov-Bohm regime (where Coulomb interaction of the edge channels with bulk anyons localized inside the interferometer loop is negligible) and the Coulomb-dominated regime (where the interaction is strong) 22,25 . Below, we first ignore bulk anyons.
Interference current. Employing the chiral Luttinger liquid theory 26,27 for FQH edges and Keldysh Green's functions 10,12 , we compute I int D 3 ð/ g 2 1 g 2 g 3 Þ at the leading order in g. There are four types of the processes mainly contributing to I int we obtain the analytical expression of I int D 3 : The interference current contributed by Type I-1 processes is a 2 a 1 1 a b Figure 1 | Topological vacuum bubble. Feynman diagrams for interference involving a real particle and a virtual particle-hole excitation from vacuum. Full (empty) circles represent particles (holes). Solid (dashed) lines denote propagations of real (virtual) particles. (a) Diagram for the interference a 1 a Ã 2 of two processes: (a 1 , blue) A real particle propagates, a virtual particlehole pair is excited, then the pair self-annihilates after the virtual particle winds around the real one. (a 2 , magenta) A real particle propagates. The entire virtual process constitutes a vacuum bubble. For anyons, the bubble gains a topological braiding phase 2pn from the winding. (b) Partner diagram of a. Here a virtual particle, constituting another bubble, does not encircle a real one and hence gains no braiding phase. The diagrams in a and b contribute to observables for anyons, while they do not for bosons and fermions.
, and those by Type II and III are where g V; T ð Þ/g 2 1 g 2 g 3 VT ð Þ 2 n À 1 e À 2L=L T ; e À 2L=L T is a thermal suppression factor, thermal length L T ' v p /(pnk B T) and Cðn ¼ 1=3Þ ' 0:43; see Methods and the Supplementary Note 3.
Type I-1 processes describe interference between two paths of an anyon moving from S 1 to D 3 via (i) QPC2 and (ii) QPC3, respectively. They were previously studied 9 .
In Type I-2, an anyon injected from S 1 interferes with a particle-like anyon excited at a QPC or annihilates a hole-like anyon; particle-like and hole-like anyons are pairwise excited thermally at QPCs. For example, consider the two following interfering histories: (i) an anyon is injected from S 1 to D 2 , an anyon pair is excited at QPC3 and then the particle-like (hole-like) anyon of the pair moves to D 3 (D 2 ). (ii) An anyon is injected from S 1 to D 3 via QPC2 without any excitations. The hole-like anyon annihilates the injected anyon on Edge 2 in history (i) and the particle-like anyon of (i) interferes with the injected anyon of (ii) on Edge 3. The sum of such interference processes yields I I À 2 D3 / sin 2 pncos 2pF=F Ã 0 À Á . The sin 2 pn factor appears, because relative locations of anyons on Edge 2 or 3 differ between the processes, leading to an exchange phase ± pn, and because a process with an excitation (of a particle-like anyon moving to D 2 and a hole-like one to D 3 ) yields charge current in the opposite direction to another with its particle-hole conjugated excitation (of a particle to D 3 and a hole to D 2 ).
In Types II and III, a real anyon injected from S 1 moves to D 2 and a virtual anyon pair excited at QPC2 interferes with another at QPC3. The interference path effectively encloses the real anyon, forming a topological vacuum bubble (cf. Fig. 1). In Type II, when the real anyon is located on Edge 2 between QPC2 and QPC3, a virtual pair is excited at QPC2 in history (i) and at QPC3 in history (ii). Next, the hole-like (particle-like) anyon of each pair moves, for example, to D 2 (D 3 ). The interference of the two histories corresponds to the winding of a virtual anyon around the real one and F, forming a topological bubble with interference phase AE 2pF=F Ã 0 þ 2pn À Á ; ±depends on whether the hole-like anyon moves to D 2 or D 3 . In the interference, the winding of a virtual anyon around the real one effectively occurs through the exchanges of the positions of the anyons in each of Edges 2 and 3, as relative locations of anyons on Edges 2 and 3 differ between (i) and (ii) (see Supplementary Fig. 2 and Supplementary Note 6). This interference is accompanied by a partner process. The latter has a bubble that winds around F (gaining phase AE 2pF=F Ã 0 ), but not around a real anyon. The two partner processes partially cancel each other, yielding I II ; the remaining sin pn factor in equations (3) and (4) has a similar origin to the sin 2 pn factor of I I À 2 D 3 . In Type III, the two interfering histories are as follows: (i) a virtual pair is excited at QPC2 before a real anyon injected from S 1 arrives at QPC2 and (ii) another pair is excited at QPC3 after the real one arrives at QPC3. The ensuing chronological sequence on Edge 2 is opposite to Type II: an anyon excited at QPC2 arrives at QPC3; the real one arrives at QPC3; a pair is excited at QPC3. The resulting topological bubble effectively winds around the real anyon in the direction opposite to its winding around F, yielding a phase AE 2pF=F Ã 0 À 2pn À Á . Partial cancellation of the bubble and its partner leads to I III (4)) comes from the time window compatible with the chronological sequence on Edge 2.
Type II and III processes of topological bubbles do not affect any observables at n ¼ 1 (fermions), due to full cancellation between partner bubbles (the linked cluster theorem). They are distinct from I-2. I-2 processes produce, for example, non-vanishing current noise hðI int D3 À hI int D3 iÞ 2 i at n ¼ 1, as the particle-hole conjugated excitations (mentioned before) equally contribute to the noise (although the contributions of the conjugations to I I À 2 D 3 cancel each other, leading to I I À 2 D 3 ¼ 0). In the more general regime of e Ã V ) k B T\' v p =L, we employ the parametrization The phase y is determined by competition between the various contributions to I int D3 and contains information about statistics phase pn. At e Ã V ) k B T\' v p =L, I II D 3 þ I III D 3 is much larger than I I À 1 D3 þ I I À 2 D3 and dominates I int D3 , because the interfering anyon of Types I-1 and I-2 is voltage biased and has width L V pV À 1 much narrower than the thermal anyon excitations (whose width L T pT À 1 ) of II and III, showing much weaker interference. From The arctan 2L V /L T term represents an error in the braiding phase 2pn of the topological bubbles. It occurs when the size L V of a real anyon is not sufficiently smaller than the winding radius of a virtual anyon around the real one. It is negligible at e Ã V ) k B T, as the radius is effectively large when L V ( L T ; those corresponding to the error are ignored in equations (3) and (4), and are shown in Supplementary Note 3. For and y-pn. Remarkably, y depends on T, contrary to common practice in electron interferometry 28 .
Coulomb-dominated regime. In experimental situations of a Fabry-Perot interferometer in the FQH regime, it is expected that there exist bulk anyons localized inside the interferometer loop. There are two regimes of Fabry-Perot interferometers, the pure Aharonov-Bohm regime and the Coulomb-dominated regime. In the former regime, Coulomb interaction between the bulk anyons and the edge of the interferometer is negligible, whereas it is crucial in the latter 25 . The Fabry-Perot interferometers of recent experiments 17,[20][21][22][23] in the FQH regime are in the Coulombdominated regime. Below we compute the interference current I D3 in the presence of the Coulomb interaction and show that equation (6) is applicable to both of the pure Aharonov-Bohm regime and the Coulomb-dominated regime.
For e Ã V ) k B T\' v p =L, we numerically compute I int D 3 ð/ g 2 1 g 2 g 3 Þ in Fig. 3, combining our theory with the capacitive interaction model 25 that successfully describes thermally fluctuating bulk anyons and the interaction (see the Method and Supplementary Note 5). We find the gate-voltage dependence of I int D 3 pcos(2pV G /V G,0 þ y) with periodicity V G,0 in the Coulomb-dominated regime and I int D3 / cosð2pF=F 0 þ yÞ in the pure Aharonov-Bohm limit; here the periodicity of the F dependence is F 0 h/e rather the period F Ã 0 of equation (5), because of the fluctuation of the number of bulk anyons 9,19 . In both the regimes, the interference processes discussed before (Fig. 2) appear in the same manner; hence, y satisfies the analytic expression in equation (6) (cf. Fig. 3c).
How to measure the phase h. Experimental measurements of y can be affected by possible side effects, including the externalparameter (magnetic fields, gate voltages and bias voltages) dependence of the size, shape, QPC tunnelling and bulk anyon excitations. Below we propose how to detect y with avoiding the side effects, using the setup in Fig. 2a The fractional statistics phase is directly and unambiguously identifiable in experiments, by observing y-pn at e Ã V ) k B T ) ' v p =L with excluding the side effects as above, or one applies the fit function of arctan [A 1 /(1 þ A 2 /T)] with fit parameters A 1 and A 2 to measured data of y(T) and extracts arctan A 1 ¼ pn from the fit (cf. the second line of equation (6)). Observation of y ¼ pn or y(T) will suggest a strong evidence of anyon braiding and topological bubbles.
The parameters in Fig. 3 are experimentally accessible 17,[20][21][22][23] . For the QPCs, there are constraints (i) that the number of voltagebiased anyons injected through QPC1 is at most one in the interferometer loop at any instance (to ensure that the braiding phase of a topological vacuum bubble is 2pn), (ii) that the anyon tunnelling probabilities at QPC2 and QPC3 are sufficiently small (to ensure that the double winding of an anyon along the interferometer loop is negligible) and (iii) that anyon tunnelling (rather than electron tunnelling) occurs at the QPCs. The constraint (i) is satisfied when the anyon tunnelling probability at QPC1 is o' v p /(2Le*V), which is B0.05 under the parameters. To achieve the constraints (ii) and (iii), each tunnelling probability of QPC2 and QPC3 is typically set to be 0.4 in experiments 22,29 ; then the amplitude of the double winding is smaller than that of the single winding by the factor 0.4 exp( À 2L/L T ), which is B0.04 at 30 mK. With the constraints we estimate the amplitude of I int D3 tðne 2 V=hÞð' v p =ð2Le Ã VÞÞ0:4exp ðÀ 2L=L T Þ, which is 1.5 pA at 30 mK and 0.6 pA at 40 mK under the parameters. It is noteworthy that y ¼ 0.9pn is reached at 30 mK, while y ¼ 0.95pn at 40 mK under the parameters. The estimation is within a measurable range in experiments, where current \0:5 pA is well detectable 30 . We remark that the above strategy of detecting the fractional statistics phase is equally applicable to the more general quantum Hall regime 22 of filling factor n 0 ¼ n þ n 0 , in which the edge channels from the integer filling n 0 are fully transmitted through the QPCs, while the channel from the fractional filling n forms the interferometry in Fig. 2. For this case we compute I int D 3 and I int Ref ; D3 , and find that in both of the pure Aharonov-Bohm regime and the Coulomb-dominated regime the interference-pattern phase shift between them is identical to the phase y of the n 0 ¼ 0 case discussed in equation (6) and Fig. 3 (cf. Supplementary Note 5).
Certain anyonic vacuum bubbles involve topological braiding and affect physical observables surprisingly, contrary to vacuum bubbles of bosons and fermions. They can be detected with current experiment tools, which will provide an unambiguous evidence of anyonic fractional statistics. We expect that they are relevant also for other filling fractions n ¼ p/(2np þ 1), non-Abelian anyons 17,21 and topological quantum computation setups 31 .

Methods
Hamiltonian for the interferometer. We present the Hamiltonian for the setup. We recall the chiral Luttinger liquid theory for FQH edges.
The Hamiltonian H ¼ P i H edge,i þ H tun for the interferometer in Fig. 2a consists of H edge,i for edge channel i and H tun for anyon tunnelling at QPCs. Edge channel 1 is biased by V and its Hamiltonian, employing the bosonization 26 for chiral Luttinger liquids, is given by For the other unbiased channels, Here, e*40,N i is the anyon number operator of channel i and f i (x) is the bosonic field of channel i at position x, which describes the plasmonic excitation of anyons. The tunnelling Hamiltonian is where C y i ðx; tÞ ¼ F y i ðtÞe À if i ðx;tÞ = ffiffiffiffiffiffiffi ffi 2pa p creates an (particle-like) anyon at position x and time t on Edge i, a is the short-length cutoff, g i is the tunnelling strength at QPC i (chosen as real) and the Aharonov-Bohm flux F enclosed by Edges i ¼ 2,3, QPC2 and QPC3 is attached to T # 2 and T # 3 , respectively, under certain gauge transformation; the dynamical phase common to the all edge channels is absorbed to 2pF=F Ã 0 . The Klein factor F y i increases the number of anyons on Edge i by 1 and satisfies F y (1) is described by f i and F i . On Edge i, it is satisfied by The exchange rule between anyons on different edges is achieved with the commutators of F i , A conventional way 32 for obtaining the commutators is to think of an extended edge connecting the different channel segments with no twist (cf. Fig. 4). The connection should preserve the chiral propagation direction of the channels. The exchange rule Cðx 1 ÞCðx 2 Þ ¼ Cðx 2 ÞCðx 1 Þe À ipnsgnðx1 À x2Þ of anyons of the extended edge agrees with equations (11) and (12). We consider the regime of weak tunnelling of anyons and treat H tun as a perturbation on P i H edge,i . Perturbation theory is applicable 24 in the renormalization group sense, when e*V and k B T are higher than Cg 1/(1 À n) , C being a non-universal constant.
The current I D3 is expressed as , into direct current I dir D3 / g 2 1 g 2 2 ; g 2 1 g 2 3 and interference current I int D3 / g 2 1 g 2 g 3 depending on F (the leading-order contribution). Equations (3) and (4)  ' v p 4pn Here, Q bulk ¼ nBA area =F 0 þ nN L À q is the number of the net charges localized within the interferometer bulk (inside the interference loop), A area is the area of the interferometer, N L is the net number of quasiparticles minus quasiholes and q is the number of positive background charges induced by the gate voltage applied to the interferometer. U int is the strength of Coulomb interaction between the charges of the interferometer edge and the charges localized in the interferometer bulk, and U bulk is the strength of interaction between the bulk charges. In the second equality of equation (13), we introduce a boson field f i for each Edge i, ' up U int Q bulk R x À 1 K i ðx 0 Þdx 0 , where K 2 (x) ¼ 1 for doxod þ L, K 3 (x) ¼ 1 for 0oxoL and K i (x) ¼ 0 otherwise. The second term of f describes the charges À 2pn ' up U int Q bulk induced per unit length by the interaction. In equation (13), the Hamiltonian is quadratic in f i and has H bulk ¼ ðU bulk À 2pnL ' up U 2 int ÞQ 2 bulk . It is noteworthy that U bulk À 2pnL ' up U 2 int 40. The main interference signal I int D3 and the reference signal