Nonequilibrium Majorana fluctuations

Nonequilibrium physics of random events, or fluctuations, is a unique fingerprint of a given system. Here we demonstrate that in noninteracting systems, whose dynamics is driven by Majorana states, the effective charge $e^*$, characterizing the electric current fluctuations, is fractional. This is in contrast to noninteracting Dirac systems with the trivial electronic charge, $e^*=e$. Quite the opposite, in the Majorana state we predict two different fractional effective charges at low and high energies, $e^*_l=e/2$ and $e^*_h=3e/2$, accessible at low and high bias voltages, respectively. We show that while the low energy effective charge $e^*_l$ is sensitive to thermal fluctuations of the current, the high energy effective charge $e^*_h$ is robust against thermal noise. A unique fluctuation signature of Majorana fermions is, therefore, encoded in the high voltage tails of the electric current noise easily accessible in experiments on strongly nonequilibrium systems even at high temperatures.


Introduction
The physics of fluctuation phenomena, or noise, dating back to Brownian [1] motion has received a systematic scientific framework after the Einstein's [2] and Smoluchowski's [3] conceptual theoretical breakthrough proven experimentally by Svedberg [4] and Perrin [5]. Spontaneous or externally excited fluctuations are an extremely insightful tool known as the fluctuation spectroscopy. Due to their sensitivity fluctuations scan the microscopic structure in much more detail than mean values.
In equilibrium, nevertheless, kinetics of a given system makes a clever link between random deviations of its physical quantities from mean values and the mean values themselves. This link dating back to the Nyquist's [6] and Callen's and Welton's [7] fundamental discovery is known as the fluctuation-dissipation theorem [8].
In nonequilibrium the fluctuation-dissipation theorem breaks and for a given system its fluctuation physics deviates from the mean value description. Here nonequilibrium noise might be comparable to or, in fact, become stronger than the equilibrium noise. It is therefore a reliable and comprehensive method to conclusively reveal the microscopic structure of a system when measurements of its mean quantities are physically inconclusive. This is what currently happens in dealing with materialization of a particle cloning its own antiparticle. Namely, via unpairing Majorana [9] fermions, composing a single Dirac fermion, by means of implementations [10,11,12,13] of the Kitaev's [14] model it is hoped to detect a single Majorana state [15,16,17]. Here experiment mainly focuses on measurements [18] of mean quantities such as the differential conductance which should exhibit a peak equal to one-half of the Dirac unitary limit [19]. This is inconclusive because such a peak might result, e.g., from the Kondo effect [20] in an asymmetric mesoscopic system. This problem is inherent to Majorana's transport experiments dealing with mean values. Nevertheless, it is possible to get a conclusive signature of Majorana fermions from the mean value description of both Majorana transport [21,22,23,24,25,26] and Majorana thermodynamics [27].
The freedom to involve nonequilibrium noise [28,29,30,31] in the fluctuation spectroscopy of Majorana fermions triggers transport experiments on Majorana physics to a new azimuth and makes it more interesting. This is because, as mentioned above, fluctuations are usually conclusive on the microscopic structure of a system and at the same time these are transport experiments which are in general simpler than thermodynamic ones. So far Majorana noise has mainly been discussed in linear response. However, the real beauty of nonequilibrium noise is still to be explored beyond linear response. Here fluctuations of the electric current may be characterized by the socalled effective charge e * which is not directly related to a particle's elementary charge but rather characterizes backscattering processes [32]. Modern experiments [33] have already reached a remarkable accuracy and enabled one to measure the noise of the electric current providing e * as a unique fluctuation fingerprint of the system.
In the present work we explore strongly nonequilibrium fluctuations of the electric current flowing through a noninteracting quantum dot coupled to a topological superconductor supporting at its ends two Majorana bound states implemented via the Kitaev's chain model. It is well known that in the absence of Majorana fermions the effective charge for a noninteracting quantum dot is trivial and identical to the electronic charge, e * = e. Here we demonstrate that in the presence of Majorana fermions 1) the effective charge fractionalizes to 2) e * l = e/2 at low energies, to 3) e * h = 3e/2 at high energies and show that 4) even when the low energy effective charge e * l = e/2 is washed out by, e.g., thermal noise, the high energy effective charge e * h = 3e/2 is robust and persists up to very high temperatures providing a simple and reliable experimental platform for a unique signature of Majorana fermions out of strongly nonequilibrium fluctuations.
The paper is organized as follows. In Section 2 we present a Majorana setup suitable for experiments on nonequilibrium noise and explore it using the Keldysh field integral framework. The results on nonequilibrium noise, in particular, on the effective charge are shown and discussed in Section 3. We conclude with Section 4. Appendix A and Appendix B provide details on the Keldysh field integral in the presence of Majorana fermions.

Theoretical setup and its Keldysh field integral description
Let us consider a setup similar to the one of Ref. [27]. It represents a noninteracting quantum dot coupled via tunneling interaction to two (L and R) noninteracting contacts. In contrast to the equilibrium setup of Ref. [27], here the contacts may be used to apply a bias voltage V to the quantum dot, V L = −V R = V /2, V L − V R = V , as it is schematically shown in Fig. 1 for the case V < 0. The quantum dot has a one single particle level which is spin nondegenerate as may be experimentally implemented, e.g., via the Zeeman splitting which also filters out the Kondo effect [20,34]. Note also that below we explore strongly nonequilibrium states which are accessed at very high bias voltage V so that the Kondo state is totally ruined [35] in any case and, therefore, does not lead to any experimental ambiguity. Finally, similar to Ref. [27], the quantum dot interacts via another tunneling mechanism with a grounded topological superconductor supporting at its ends two Majorana bound states.
To give the problem a concrete and mathematically convenient treatment we formulate it in terms of the quantum many-particle Keldysh Lagrangian L K (q, p) = i i q iṗi − H K (q, p) which constitutes the basis for the Keldysh action, S K . Here the momenta p i and coordinates q i are the fermionic coherent states of the system and their conjugate partners, respectively. The Lagrangian formulation is fully equivalent to the quantum Hamiltonian H K (q, p) formulation but has a certain technical advantage in calculating strongly nonequilibrium fluctuations of the electric current via the Keldysh field integral [36] which we employ below to obtain the current-current correlation function.
For the quantum dot (p = ψ, q =ψ), contacts (p = φ, q =φ) and tunneling  Figure 1. Quantum dot with a single particle nondegenerate energy level d is linked via tunneling mechanisms to two normal contacts and to one end of a one-dimensional topological superconductor supporting two Majorana bound states, γ 1 , γ 2 , at its ends. Here Γ L , Γ R characterize the tunneling strength between the left (L) and right (R) contacts while η is the strength of the tunneling between the quantum dot and the Majorana bound state γ 1 . A bias voltage V may be applied to the contacts and induce an electric current flowing in the direction of arrows. The electric current < I(t) > and its noise < I(t)I(t ) > may be measured in one of the contacts, e.g., in the left contact, < I L (t) >, < I L (t)I L (t ) >. between them we, respectively, have: where d is the quantum dot energy level and the real time t runs along the Keldysh closed contour C K , t ∈ C K , where below we will assume identical quantum numbers k l in both L and R contacts as well as large contacts so that their spectrum lk l is continuous and their density of states ν C is constant in the vicinity of the Fermi energy, We use for the Dirac tunneling the standard assumption that the tunneling matrix elements weakly depend on the contacts quantum numbers, T lk l ≈ T l . This allows to characterize the tunneling coupling by the energy scales Γ l = πν C |T l | 2 or Γ ≡ Γ L + Γ R . Finally, the topological (p = ζ, q =ζ) part of the Hamiltonian is given as the sum of the Hamiltonians of the topological superconductor and its tunneling interaction with the quantum dot: where ξ is the energy originating from the overlap of the Majorana bound states ζ 1 and ζ 2 (in particular, ξ = 0 if there is no overlap as in sufficiently long Kitaev's chains), where the Majorana tunneling entangles the Dirac fermions of the quantum dot with only one Majorana state, ζ 1 , and is characterized by the energy scale |η|.
Due to the fundamental property of the Majorana fields,ζ j (t) = ζ j (t), j = 1, 2, and the canonic fermionic anticommutation relations the field integral for the Keldysh partition function, is a functional integral with the constraintsζ j (t) = ζ j (t), ζ 2 j (t) = 1 which might be viewed as fermionic constraints imposed at any given discrete time of the Keldysh closed contour. Nevertheless, since H K is quadratic in all p i , q i , this field integral may be solved exactly as in many standard textbooks [36] as explained in detail in Appendix A and Appendix B.
Using an imaginary time field theory, it has been rigorously proven [27] by entropic reasoning that a macroscopic state of the above setup is Majorana dominated at low temperatures. Therefore, it must exhibit various fractionalizations [17] of its observables. Here, in particular, we are interested in fractionalizations of the electric current fluctuations.
To this end we introduce suitable sources into the Keldysh partition function (6) turning it into the Keldysh generating functional: where J l (t) is the source field and I l (t) is the electric current field. The mean current and current-current correlator are then obtained by proper functional differentiations of Eq. (7) with respect to the source field. Here, calculating the current-current correlator, one should remember that due to the topological superconductor (or fermionic constraints) various anomalous expectation values do not vanish, i.e., in general ψ 1 ψ 2ψ3ψ4 also includes the term ψ 1 ψ 2 ψ 3ψ4 . Below we are interested in the so called greater current-current correlator S > (t, t ) ≡ δI L (t)δI L (t ) (as opposed to the lesser one, and finally obtains the ratio e * /e = S > K (V )/e|I K (V )| at small bias voltages. In particular, for noninteracting quantum dots one gets the trivial result, e * = e.
We generalize the above definition of the effective charge in such a way that it reproduces not only the standard definition at small bias voltages but also provides a unique fluctuation fingerprint of a system far from equilibrium where the system's dynamics is highly nonlinear and the expansion in powers of V makes no sense at all. To this end we note that at small values of |V | the nonlinear parts of both the shot noise and the mean current are cubic in V . Therefore, at low voltages the second derivatives d 2 S > (V )/dV 2 and d 2 I(V )/dV 2 are linear in V and thus linearly depend on each other with the ratio [ Therefore, we define the effective charge as: which is applicable when the second derivatives linearly depend on each other. Note, that the linear dependence of d 2 S > (V )/dV 2 on d 2 I(V )/dV 2 does not necessarily imply a linear dependence of these derivatives on the bias voltage. In fact, at large bias voltages expansions in powers of V do not exist while d 2 S > (V )/dV 2 may still linearly depend on d 2 I(V )/dV 2 and, therefore, the effective charge in Eq. (8) makes sense even at extremely high bias voltages. Note also that the definition in Eq. (8) is highly consistent because, as shown below, in the absence of Majorana fermions it gives for the noninteracting case e * = e both at low and very large bias voltages. Importantly, this definition is also highly relevant Refs. [28,29] (Note that here we calculate the greater noise and not the symmetrized noise as in Refs. [28,29]. For zero frequency they differ by a factor of 2).
for experiments because each of the second derivatives may be measured with sufficient accuracy already at present.

Results and discussion
Let us consider the situation when |η| > Γ. We also currently assume d = 0 and Γ L = Γ R . Fig. 2 shows d 2 S > (V )/dV 2 as a function of d 2 I(V )/dV 2 when |η| = 8Γ for two different values of the overlap energy ξ. The black curve is for ξ/Γ = 10 2 . In this case the Majorana fermions strongly overlap forming a single Dirac fermion leading to the current fluctuations with a trivial effective charge equal to the electronic charge both at low (e|V | Γ) and high (e|V | > Γ) energies. Indeed, the curve is linear near the origin both at its starting point and at its ending point with the tangent lines having unit absolute slope resulting in e * l = e * h = e. However, when ξ/Γ = 10 −4 , Majorana bound states overlap weakly and the fluctuation physics is governed by fractional degrees of freedom leading to fractional effective charges at low and high energies. In this case the curve is linear near the origin both at its starting point and at its ending point with the tangent lines having, respectively, absolute slopes equal to 1/2 and 3/2 resulting in e * l = e/2 at low energies (e|V | Γ) and e * h = 3e/2 at high energies (|η| > e|V | > Γ). At voltages e|V | |η| the Majorana state is ineffective and the curve acquires the trivial linear character with e * = e which is not visible in Fig. 2 because both of the second derivatives become very small at the high voltage tails of S > (V ) and I(V ) shown in Fig. 3 for the case ξ/Γ = 10 −4 . However, it becomes visible when the effective charge is plotted as a function of V (see Fig. 5 below). In Fig. 4 we show d 2 S > (V )/dV 2 as a function of d 2 I(V )/dV 2 for |η| = 8Γ, ξ/Γ = 10 −4 for different temperatures. Since the overlap energy is small, the current fluctuations are essentially governed by the Majorana degrees of freedom. Here we increase the bias voltage of the starting points of the high temperature curves to stay in the regime e|V | > k B T in order to avoid high values of the thermal Majorana noise which is not in the focus of the present research. At high temperatures (red, blue and green curves) the low energy effective charge e * l = e/2 is completely washed out by thermal fluctuations of the electric current. However, the high energy effective charge e * h = 3e/2 is robust against thermal noise and persists up to very high temperatures, k B T /Γ = 10 −1 (green curve). Let us estimate the temperature at which the fractional high energy effective charge e * h = 3e/2 might be observed in experiments. If the induced superconducting gap is taken from Ref. [19], ∆ = 250 µeV, and |η| ≈ ∆, then we obtain T ≈ 36 mK which is easily reachable in modern experiments. If the induced superconducting gap is taken from Ref. [37], ∆ = 15 meV, then T ≈ 2 K which is even more reachable. The fractional high energy effective charge e * h = 3e/2 is perfectly achieved only at |η| > e|V | Γ which requires |η| Γ. However, according to our numerical analysis we estimate that for |η| > Γ (and small ξ) it weakly deviates from the value 3e/2. Namely, from numerical fitting we get e * h ≈ [3/2 − 2(Γ/η) 2 ]e. So that |η| = 8Γ gives e * h ≈ 1.47e, |η| = 20Γ gives e * h ≈ 1.495e and |η| = 50Γ gives e * h ≈ 1.4992e. Importantly, by means of a gate voltage one may easily in realistic experiments increase d so that d > 0, | d | > Γ. In this case the quantum dot is in the empty orbital regime [20] opposite to the Kondo one. In this way one fully eliminates [38] the Kondo effect. At the same time e * l and e * h do not change as soon as the quantum dot is in the Majorana universal regime, |η| > max{| d |, Γ, e|V |}. Since Γ and/or |η| may be easily varied in modern experiments [39], the Majorana universal regime is readily reachable in modern laboratories. Therefore, one may unambiguously observe in realistic experiments the universal plateaus e * l = e/2 and e * h = 3e/2 in the empty orbital and Majorana universal regime as it is shown in Fig. 5 for the case |η|/Γ = 10 3 and d /Γ = 8. These plateaus are universal and do not depend on d as soon as | d | < |η|. Also for |η|/Γ = 10 2 the plot on Fig. 5 is almost unchanged. Moreover, we find that the e * h plateau survives up to very high temperatures, k B T ∼ 10 −2 |η|, i.e., up to k B T = 10 Γ for the present case as shown by the black dashed line in Fig. 5. As one can see, although the plateau e * h becomes very narrow at such a high temperature, it is still visible and it almost reaches the value 3e/2 even at k B T = 10 Γ.
Another important aspect is the universality of the effective charge plateaus e * l and e * h when the quantum dot is asymmetrically coupled to the left and right contacts. This asymmetry may be characterized by the quantities γ L ≡ Γ L /Γ, γ R ≡ Γ R /Γ, which satisfy γ L + γ R = 1. The symmetric setup discussed above corresponds to the case γ L = γ R = 0.5. In a general setup γ L = γ R . Nevertheless, the effective charge e * in Eq. (8) is characterized by two different universal plateaus e * l and e * h at low and high bias voltages, respectively. In this general asymmetric situation when γ L = γ R and when ξ is small, i.e., the two Majorana bound states are well separated, the low energy and high energy plateaus of the effective charge are: We obtain these values with any desired numerical precision which means that Eq. (9) is the numerically exact result. Its analytical proof is a complicated task especially in the case of e * h taking place at high voltages where the dynamics is nonlinear. This analytical proof could be based on a semiclassical picture [40] and will be a challenge for our future research which should, in particular, explain the physical meaning of the high energy effective charge e * h predicted currently by different numerical techniques with very high precision.
From Eq. (9) one can see that e * h − e * l = 2(1 − γ L ). Only when γ L → 1, one gets a unique effective charge both at low and high energies, e * l = e * h = 2e. However, as soon as 0 γ L < 1, the unique value, 2e, of the effective charge splits into two different values. In the symmetric case, γ L = γ R = 0.5, one gets from Eq. (9) the result discussed above, e * l = e/2, e * h = 3e/2. However, when, for example, γ L = 0.8, γ R = 0.2, one gets from Eq. (9) e * l = 7e/5, e * h = 9e/5, as shown in Fig. 5. Once again, we emphasize that the low energy, e * l , and high energy, e * h , effective charge plateaus, given by Eq. (9), are universal for all possible values of the asymmetries γ L , γ R . In particular they do not depend on d , i.e., they do not depend on the gate voltage.
On the other side, when ξ is large, i.e., when the two Majorana bound states strongly overlap forming a single Dirac fermion, we obtain with any desired numerical precision that e * l = e * h = e for all γ L , γ R except for a small vicinity of the point γ L = 1, γ R = 0, where e * l and e * h are sharply peaked to the value e * l = e * h = 2e reached exactly at the point γ L = 1, γ R = 0. Therefore, in the case of large ξ, when the Majorana fermions form a single Dirac fermion, the effective charge plateaus, given by Eq. (9), do not appear for any degree of asymmetry described by the values of γ L and γ R .
This shows that the presence of two different universal effective charges at low and high energies, e * l and e * h , respectively, whose values are given by Eq. (9), is a unique fluctuation signature of the presence of Majorana fermions in the topological superconductor independent of the asymmetry in the coupling of the quantum dot to the left and right contacts. An experimental detection of at least one of these effective charges is enough to conclusively claim that the topological superconductor in this setup supports Majorana fermions.
Note also one practical aspect of Eq. (9). As soon as one of the effective charges, e * l or e * h , is detected in an experiment, the asymmetries γ L and γ R immediately follow in a simple way from Eq. (9). This simple way of extraction of γ L and γ R is definitely a practical advantage since usually in experiments it is difficult to measure the values of γ L and γ R . At the same time it is often necessary to know the values of γ L and γ R to theoretically describe realistic experiments. We would like to emphasize that the low energy effective charge e * l is obtained from Eq. (8) at k B T e|V | Γ. Although the voltage is small here, e|V | Γ, it is still finite to make thermal noise insignificant, e|V | k B T . Therefore, the system is not in equilibrium. To understand how far it is from the equilibrium and to which extent its equilibrium macroscopic states may still govern the behavior of e * l we compare the behavior of e * l at k B T e|V | Γ with the behavior of the system's entropy at V = 0. Here it has been rigorously proven [27] that the macroscopic state of the present setup is characterized by the entropy plateau S = ln(2)/2. This shows that the macroscopic state consists of non-integer number of microscopic states namely it consists of one-half of the Dirac fermion state. That the Majorana equilibrium macroscopic state indeed governs the behavior of e * l is clear from the following fact. When ξ grows, the two Majorana fermions combine into a single Dirac fermion and Eq. (8) gives a transition from the plateau e * l = e/2 to the plateau with the integer electronic charge e * = e. At the same time, when ξ grows, the Majorana plateau S = ln(2)/2 is fully ruined to the trivial plateau S = 0 as shown in Fig. 6.
Concerning the high energy effective charge we would like to note that its presence is also a unique signature of the Majorana fermions for all 0 < γ L < 1. This is particularly clear in the case γ L = γ R = 0.5. Here the noise properties characterized by e * h = 3e/2 cannot be induced by two particle processes as one would expect from the standard point of view where an effective charge is usually associated with backscattering processes at V → 0. From this traditional perspective one would conclude that e * h = 3e/2 is the result of a combination of single particle and two particle processes due to Andreev reflection. However, this traditional approach is usually applied at V → 0 [32,33] and its adequacy at large bias voltages, where the dynamics is highly nonlinear, would be a question for future research, especially in connection with the definition of the effective charge given by Eq. (8). In the present case, however, this traditional point of view is definitely inapplicable because for γ L = γ R = 0.5 the Andreev current, the only possible source of two particle processes here, is equal to zero [29]. This shows that the traditional explanation of the high energy effective charge in terms of combination of different processes is meaningless and the value e * h = 3e/2 is of pure Majorana nature. Likewise, when γ L = γ R , the effective charge e * h is also of Majorana nature although the Andreev current may be finite in this situation. Indeed, when γ L = γ R the Majorana nature of the high energy effective charge is obvious from the fact that it is fractional for small values of ξ, when the Majorana modes are well separated, but takes the trivial value e * = e as soon as the two Majorana fermions combine into a single Dirac fermion at large values of ξ as has been discussed above.
Once more we would like to note that in the present research the name "effective charge" in this high voltage nonlinear regime is used just by analogy with the low voltage regime and the precise meaning is given by the ratio in Eq. (8). However, as mentioned above, this ratio is experimentally relevant and can be measured with high precision at any voltage.

Conclusion
In conclusion, we have explored strongly nonequilibrium Majorana fluctuations of the electric current. It has been shown that in general these fluctuations are characterized by two fractional effective charges e * l and e * h at low and high energies, respectively. We have demonstrated that the low energy effective charge e * l might be washed out by thermal noise but the high energy effective charge e * h is robust and persists up to very high temperatures. The latter, thus, represents a challenge for modern experiments on noise phenomena in quantum dots since it is protected by high bias voltage V from all the perturbations whose strengths are smaller than e|V |. In particular, electronelectron interactions and disorder will not change the high energy effective charge if their characteristic energy scales, V e−e , V dis , are smaller than e|V |, that is if V e−e < e|V | and V dis < e|V |. Of course, in future our research should be improved with more realistic models to test the robustness of the high energy effective charge and to predict its value when, e.g., the density of states in the contacts is not constant or multiple levels in the quantum dot are involved in the transport. However, the model we have explored in the present research is already quite standard and is often applied in many other contexts to successfully describe modern experiments. We, therefore, believe that our results, in particular, the high energy effective charge may become a reliable platform for a unique signature of Majorana fermions out of strongly nonequilibrium fluctuations.
In terms of the Dirac operator fieldsχ, χ, the Hamiltonian H TS (q, p) from the main text takes the form at a given discrete time i: The constant term in Eq. (A.5) cancels out on the forward, "+", and backward, "−", branches of the Keldysh closed time contour and, therefore, plays no role.

Appendix B. Keldysh action
Since the overlap of any two fermionic coherent states, |ψ and |φ has the form [36]: 2) bring the generators χ i−1 at the discrete times neighboring to the discrete times i, i.e, from the Grassmann algebras at the discrete times (i − 1). Therefore, in the calculation of the matrix elements the Majorana operator fields, ζ 1,i and ζ 2,i , bring, respectively, in the continuum limit the factors (χ(t) + χ(t)) and i(χ(t) −χ(t)), t ∈ C K , while the Hamiltonian in Eq. (A.5) brings the factor ξχ(t)χ(t), where the constant term in Eq. (A.5) is dropped out as explained above.
As a result, the Keldysh action S K from the main text may be written as: where S 0 is the conventional noninteracting (quadratic) action of the isolated quantum dot, contacts and topological superconductor and S T is the action which describes the tunneling interaction between the quantum dot and contacts as well as between the quantum dot and topological superconductor. It has the following form: 3) such asψ(t)χ(t). One deals with these terms in the same way as in the field integral theory of superconductivity [36] where the particle-hole space is introduced via the Nambu spinors. The additional particle-hole index, however, is technically inessential because the whole action is still quadratic and, therefore, is exactly solvable.