Majorana differential shot noise and its universal thermoelectric crossover

Topological superconductors interacting with nanoscopic setups provide a feasible technological platform to entangle a specific quantum core of these setups with non-Abelian Majorana bound states (MBSs) Yu. Kitaev (2001); Alicea (2012); Leijnse and Flensberg (2012); Sato and Fujimoto (2016); Aguado (2017); Lutchyn et al. (2018); Val’kov et al. (2022) imitating particle-antiparticle paradigm of the Abelian Majorana fermions Majorana (1937) known in the particle physics. Such Majorana entangled setups are understood (without a strict relation to the specific meaning of ”entanglement” in quantum information) as those where Majorana and non-Majorana degrees of freedom are coupled via a certain quantum mechanical mechanism. They represent a special class of condensed matter systems which are very attractive both from theoretical and experimental perspectives. On one side, they admit an observation of diverse physical phenomena governed essentially by Majorana entangled states and, on the other side, they may function as elementary blocks integrated in various fault tolerant topological quantum computing Yu. Kitaev (2003) schemes designed to process the quantum nonlocality supported by MBSs.

Nanoscopic setups where MBSs are entangled with the quantum degrees of freedom involved in experimental measurements reveal various remarkable characteristics many of which may be accessed in quantum transport experiments. Such experiments deal with nonequilibrium states which may be generated by bias voltages $V$ or temperature differences $\Delta T$, expressed equivalently through the corresponding thermal voltages $V_{T}$, defined as $eV_{T}\equiv k_{B}\Delta T$. Here Majorana features are predicted within the framework of mean currents $I(V)$ induced by mostly bias voltages Liu and Baranger (2011); Fidkowski et al. (2012); Prada et al. (2012); Pientka et al. (2012); Lin et al. (2012); Lee et al. (2013); Kundu and Seradjeh (2013); Vernek et al. (2014); Ilan et al. (2014); Cheng et al. (2014); Lobos and Das Sarma (2015); Peng et al. (2015); Sharma and Tewari (2016); van Heck et al. (2016); Sarma et al. (2016); Lutchyn and Glazman (2017); Liu et al. (2017); Huang et al. (2017); Liu et al. (2018); Lai et al. (2019); Tang and Mao (2020); Zhang and Spånslätt (2020); Chi et al. (2021); Wang and Huang (2021); Galambos et al. (2022); Jin and Li (2022); Zou et al. (2023a) or, to a lesser extent, mean currents $I(V,V_{T})$ induced by also temperature differences Leijnse (2014); López et al. (2014); Khim et al. (2015); Ramos-Andrade et al. (2016); Smirnov (2020); He et al. (2021); Giuliano et al. (2022); Buccheri et al. (2022); Majek et al. (2022); Bondyopadhaya and Roy (2022); Zou et al. (2022). Experimental efforts on mean currents $I(V)$ induced by bias voltages Mourik et al. (2012); Nadj-Perge et al. (2014); Wang et al. (2022) are aimed to measure the differential conductance $\partial I(V)/\partial V$. Of particular interest here is the zero bias limit of the differential conductance (linear conductance) which should attain a certain universal value predicted theoretically for a specific Majorana entangled setup. Although such mean current experiments are well developed and should be performed in the first place, unfortunately, they may be controversial Yu et al. (2021); Frolov (2021) in detecting MBSs and, as a consequence, other types of quantum transport measurements, or perhaps sequences of measurements Ziesen et al. (2023), are currently in demand. Particularly, one is interested in those physical observables which demonstrate in a given setup a Majorana driven behavior which is qualitatively different from the behavior of the mean currents measured in the same Majorana entangled setup.

An attractive quantum transport alternative to the mean value of a current flowing through a nanoscopic setup is to study the random deviations of this current from its mean value, that is the current fluctuations, characterized, for example, by the shot noise $S^{>}$. Here majority of Majorana shot noise proposals assume nonequilibrium states originating from bias voltages Liu et al. (2015a, b); Haim et al. (2015); Valentini et al. (2016); Zazunov et al. (2016); Smirnov (2017); Jonckheere et al. (2019); Smirnov (2019a); Manousakis et al. (2020); Feng and Zhang (2022); Smirnov (2022) and explore the behavior of $S^{>}(V)$ at small and large $V$. As in mean current experiments, where the differential conductance $\partial I(V)/\partial V$ provides an access to an averaged Majorana universality, the differential shot noise $\partial S^{>}(V)/\partial V$ allows one to reveal a universal fluctuation behavior governed by Majorana entangled states.

One may also avoid resorting to nonequilibrium behavior and address Majorana entangled states in corresponding equilibrium nanoscopic setups by means of quantum thermodynamic tools such as the entropy of these setups Smirnov (2015); Sela et al. (2019); Smirnov (2021a, b); Ahari et al. (2021). Recent experimental and theoretical activities Hartman et al. (2018); Kleeorin et al. (2019); Pyurbeeva and Mol (2021); Child et al. (2022a); Han et al. (2022); Pyurbeeva et al. (2022); Child et al. (2022b) on the entropy of nanoscale and mesoscale systems demonstrate that this engrossing approach may become a powerful and univocal technique which will not be subject to further controversy similar to the one about the Majorana differential conductance.

Nevertheless, presently quantum transport is a more appealing framework within which experimentalists have at their disposal well established technologies verified in diverse nanoscopic setups for a long period of time. Moreover, quantum transport techniques have a wider space of control due to numerous additional parameters utilized to maintain various kinds of nonequilibrium states in which a broad spectrum of physical observables is available for performing experiments. Thus applying the quantum transport framework to Majorana entangled nanoscopic setups provides vast freedom in exploring Majorana phenomena in nonequilibrium. In particular, among measurements of other physical observables, shot noise experiments in various nonequilibrium states are expected to qualitatively enrich the existing results on mean currents in Majorana entangled nanoscopic setups.

Here we focus on the shot noise in nonequilibrium states produced by bias voltages $V$ and thermal voltages $V_{T}$ in a quantum dot (QD) whose degrees of freedom are entangled with MBSs of a topological superconductor. Specifically, we explore the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ which, as has been discussed above, inspects universal Majorana fluctuation behavior. So far it is not much known about this physical observable when both $V$ and $V_{T}$ excite competing current flows in a Majorana setup. Indeed, whereas the differential thermoelectric shot and quantum noise, $\partial S^{>}(V,V_{T})/\partial V_{T}$, have been addressed Smirnov (2018, 2019b) in presence of both bias voltages and temperature differences, the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ remains to a large extent unexplored for Majorana entangled setups in nonequilibrium states driven by both $V$ and $V_{T}$. It should be noted that in nonequilibrium states induced only by bias voltages $V$ the differential shot noise has been studied in combination with the differential conductance. In particular, in Ref. Cao et al. (2023) it is demonstrated that in presence of MBSs a dip of the differential shot noise is always accompanied by a peak of the differential conductance. This behavior has also been observed earlier in Ref. Smirnov (2022) (see its Fig. 4, namely, the insets of the upper panel). As mentioned above, it is important to find for a given Majorana entangled setup physical observables whose behavior has a character qualitatively different from the one of the mean current or its derivative physical quantities such as the differential conductance $\partial I(V,V_{T})/\partial V$ or differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$ whose behavior may be obtained in the same setup. We demonstrate that the differential shot noise is one of such physical observables which is distinguished by the presence of a crossover from a thermoelectric to pure thermal nonequilibrium behavior. It is shown that whereas the differential shot noise passes through its crossover, the differential conductance and differential thermoelectric conductance do not exhibit any crossover or any other peculiarity. Thus, in contrast to $\partial I(V,V_{T})/\partial V$ and $\partial I(V,V_{T})/\partial V_{T}$, the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ brings a nonequilibrium energy scale having a pure fluctuation nature meaning that it cannot be revealed within measurements limited only by the mean current. Besides being of fundamental interest, the energy scale associated with the nonequilibrium crossover in $\partial S^{>}(V,V_{T})/\partial V$ is shown to be of practical importance in expressing quantitatively the fluctuation universality of Majorana entangled states via a number of measurable universal ratios which would be of interest for future experiments.

The paper is organized as follows. In Section II we discuss a theoretical model of an experimentally feasible nanoscopic setup where MBSs are entangled with a QD whose nonequilibrium states are generated by both a bias voltage and thermal voltage. Results of numerical analysis performed with high accuracy for the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$, differential conductance $\partial I(V,V_{T})/\partial V$ and differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$ are presented in Section III where it is demonstrated that, in contrast to $\partial I(V,V_{T})/\partial V$ and $\partial I(V,V_{T})/\partial V_{T}$, one observes in $\partial S^{>}(V,V_{T})/\partial V$ a crossover from a thermoelectric to pure thermal nonequilibrium behavior. The energy scale where this crossover takes place and a number of universal Majorana ratios involving this energy scale are also shown in this section. Finally, with Section IV we make conclusions and discuss possible outlooks.

We start with a description of a setup which, on one side, is technologically feasible Deng et al. (2016, 2018) and, on the other side, involves a basic mechanism of a Majorana entanglement which is sufficient to demonstrate a remarkable nonequilibrium behavior of the differential shot noise in presence of both bias voltages and temperature differences. To this end, let us consider a noninteracting QD,

$$\hat{H}_{QD}=\epsilon_{d}d^{\dagger}d.$$

(1)

The QD is nondegenerate and its energy level $\epsilon_{d}$ is tunable by a proper gate voltage. The choice of a setup with a noninteracting QD is quite a realistic assumption to explore universal Majorana phenomena at low energies. Indeed, the spin degeneracy is assumed to be removed by an external magnetic field which excludes a possible interfering of an interaction induced Kondo universal behavior, well-known in experiment and theory of spin-degenerate QDs Ralph and Buhrman (1994); Goldhaber-Gordon et al. (1998); Meir et al. (1993); Wingreen and Meir (1994); Smirnov and Grifoni (2011a, b); Niklas et al. (2016), with the low-energy Majorana universal effects, which are of interest in this work. Numerical renormalization group calculations Ruiz-Tijerina et al. (2015) have demonstrated that interacting spin-degenerate QDs in external magnetic fields behave similar to noninteracting nondegenerate QDs exhibiting, for example, the linear conductance $e^{2}/2h$ which results entirely from the Majorana entangled states. Thus Eq. (1) is a proper model to explore low-energy Majorana quantum transport, in particular, in nonequilibrium states resulting from bias voltages and temperature differences López et al. (2014).

Two normal noninteracting metallic contacts, denoted below as left ($L$) and right ($R$),

$$\hat{H}_{C}=\sum_{l=\{L,R\}}\sum_{k}\epsilon_{k}c_{lk}^{\dagger}c_{lk},$$

(2)

are connected to the QD via tunneling processes,

$$\hat{H}_{QD-C}=\sum_{l=\{L,R\}}\mathcal{T}_{l}\sum_{k}c_{lk}^{\dagger}d+\text{H.c.}$$

(3)

In Eq. (2) the continuous energy spectrum $\epsilon_{k}$ gives rise to a density of states of the contacts $\nu_{C}(\epsilon)$ which is in general energy dependent. However, around the Fermi energy one usually with a good accuracy assumes that its energy dependence plays no essential role for quantum transport and thus $\nu_{C}(\epsilon)\approx\nu_{0}/2$. In Eq. (3) one additionally assumes that the tunneling matrix elements do not depend on the set $k$ of the quantum numbers used to describe the states in the contacts, $\mathcal{T}_{kl}\approx\mathcal{T}_{l}$. The relevant energy scales brought about by the tunneling between the QD and contacts are $\Gamma_{l}=\pi\nu_{0}|\mathcal{T}_{l}|^{2}$. By proper gate voltages one may achieve the symmetric coupling $\Gamma_{L}=\Gamma_{R}=\Gamma/2$ which will be assumed below.

Each contact is assumed to be in its own equilibrium state with the corresponding Fermi-Dirac distribution,

$$f_{l}(\epsilon)=\frac{1}{\exp\bigl{(}\frac{\epsilon-\mu_{l}}{k_{B}T_{l}}\bigr{)}+1}.$$

(4)

Here the chemical potentials,

$$\mu_{L,R}=\mu_{0}\pm eV/2,$$

(5)

are specified by the bias voltage $V$ such that $eV<0$ and the temperature of the left contact is higher than the temperature of the right contact, that is

$$\begin{split}&T_{L}=T+\Delta T\quad\text{(hot contact)},\\ &T_{R}=T\quad\text{(cold contact)},\end{split}$$

(6)

assuming $T,\Delta T\geqslant 0$. The QD is out of equilibrium when either $V\neq 0$ or $\Delta T\neq 0$.

A topological superconductor hosting two MBSs $\gamma_{1,2}$ at its ends,

$$\hat{H}_{TS}=\frac{1}{2}i\xi\gamma_{2}\gamma_{1},\quad\gamma_{1,2}^{\dagger}=\gamma_{1,2},\quad\{\gamma_{i},\gamma_{j}\}=2\delta_{ij},$$

(7)

interacts with the QD,

$$\hat{H}_{QD-TS}=\eta^{*}d^{\dagger}\gamma_{1}+\text{H.c.},$$

(8)

implementing a direct entanglement of the QD’s degrees of freedom with the Majorana mode $\gamma_{1}$ of the topological superconductor. In Eq. (7) the parameter $\xi$ is an energetic measure of how strong the two Majorana modes overlap with each other. When $\xi$ is small the MBSs are well separated whereas large values of $\xi$ model a situation where the two MBSs merge into a single Dirac fermion. In Eq. (8) the Majorana tunneling amplitude $|\eta|$ specifies the strength of the Majorana entanglement.

A schematic summary of the above theoretical formulation of the setup, based on Eqs. (1)-(8), is illustrated in the inset of Fig. 1.

The Hamiltonian of the setup, $\hat{H}=\hat{H}_{QD}+\hat{H}_{C}+\hat{H}_{QD-C}+\hat{H}_{TS}+\hat{H}_{QD-TS}$, allows us to formulate the problem in terms of the Keldysh field integral Altland and Simons (2010), a convenient tool to calculate various correlation functions. Other technical tools based, e.g., on quantum master equations Xu et al. (2022) may also be considered as alternative approaches to the problem. Within the Keldysh field integral formalism one may straightforwardly derive the shot noise from the Keldysh generating functional,

$$Z[J_{lq}(t)]=\int\mathcal{D}[\bar{\psi},\psi;\bar{\phi},\phi;\bar{\zeta},\zeta]e^{\frac{i}{\hbar}S_{K}[J_{lq}(t)]},$$

(9)

which is a field integral over the Grassmann fields of the QD ($\bar{\psi}_{q}(t),\psi_{q}(t)$), contacts ($\bar{\phi}_{lkq}(t),\phi_{lkq}(t)$) and topological superconductor ($\bar{\zeta}_{q}(t),\zeta_{q}(t)$) whose temporal arguments are on the real axis and $q=\pm$ specifies, respectively, the forward or backward branch of the Keldysh contour. At zero source fields the Keldysh generating functional is determined by the Keldysh action $S_{K}^{(0)}\equiv S_{K}[J_{lq}(t)=0]$ and is equal to unity, $Z[J_{lq}(t)=0]=1$. The Keldysh action $S_{K}[J_{lq}(t)]$,

$$\begin{split}&S_{K}[J_{lq}(t)]=S_{QD}[\bar{\psi},\psi]+S_{C}[\bar{\phi},\phi]+S_{TS}[\bar{\zeta},\zeta]\\ &+S_{QD-C}[\bar{\psi},\psi;\bar{\phi},\phi]+S_{QD-TS}[\bar{\psi},\psi;\bar{\zeta},\zeta]\\ &+S_{O}[\bar{\psi},\psi;\bar{\phi},\phi;J_{lq}(t)],\end{split}$$

(10)

is the sum of, respectively, the actions describing the QD, contacts, topological superconductor, tunneling between the QD and contacts, tunneling between the QD and topological superconductor and the source action added to generate an observable of interest, in particular, the mean current and shot noise. The actions $S_{QD}$, $S_{C}$ and $S_{TS}$ are of the standard matrix form Altland and Simons (2010) in the retarded-advanced space. The actions $S_{QD-C}$, $S_{QD-TS}$ and $S_{O}$ have the following form:

$$S_{QD-C}=-\!\int_{-\infty}^{\infty}\!\!\!\!\!dt\sum_{l=\{L,R\}}\sum_{k,q}[\mathcal{T}_{l}q\bar{\phi}_{lkq}(t)\psi_{q}(t)+\text{G.c.}],$$

(11)

$$\begin{split}S_{QD-TS}&=-\int_{-\infty}^{\infty}dt\{\eta^{*}\sum_{q}q[\bar{\psi}_{q}(t)\zeta_{q}(t)\\ &+\bar{\psi}_{q}(t)\bar{\zeta}_{q}(t)]+\text{G.c.}\},\end{split}$$

(12)

$$S_{O}=-\int_{-\infty}^{\infty}dt\sum_{l=\{L,R\}}\sum_{q}J_{lq}(t)I_{lq}(t),$$

(13)

where G.c. denotes the Grassmann conjugated terms and $I_{lq}(t)$ is the current operator in the Grassmann representation,

$$I_{lq}(t)=\frac{ie}{\hbar}\sum_{k}\bigl{(}\mathcal{T}_{l}\bar{\phi}_{lkq}(t)\psi_{q}(t)-\text{G.c.}\bigr{)}.$$

(14)

The form of the source action in Eq. (13) implies that one derives the mean current and current-current correlations via proper differentiations,

$$\langle I_{lq}(t)\rangle_{0}=i\hbar\frac{\delta Z[J_{lq}(t)]}{\delta J_{lq}(t)}\biggl{|}_{J_{lq}(t)=0},$$

(15)

$$\langle I_{lq}(t)I_{l^{\prime}q^{\prime}}(t^{\prime})\rangle_{0}=(i\hbar)^{2}\frac{\delta^{2}Z[J_{lq}(t)]}{\delta J_{lq}(t)\delta J_{l^{\prime}q^{\prime}}(t^{\prime})}\biggl{|}_{J_{lq}(t)=0},$$

(16)

where

$$\begin{split}&\langle\prod_{i}I_{l_{i}q_{i}}(t_{i})\rangle_{0}\\ &\equiv\int\mathcal{D}[\bar{\psi},\psi;\bar{\phi},\phi;\bar{\zeta},\zeta]e^{\frac{i}{\hbar}S_{K}^{(0)}}\prod_{i}I_{l_{i}q_{i}}(t_{i}).\end{split}$$

(17)

Choosing the left contact as the one where measurements of the mean current

$$I(V,V_{T})=\langle I_{Lq}(t)\rangle_{0},$$

(18)

and correlations

$$S^{>}(t,t^{\prime};V,V_{T})=\langle\delta I_{L-}(t)\delta I_{L+}(t^{\prime})\rangle_{0}$$

(19)

of the current fluctuations

$$\delta I_{Lq}(t)=I_{Lq}(t)-I(V,V_{T})$$

(20)

are performed, one obtains the shot noise $S^{>}(V,V_{T})$ in the left contact as the zero frequency Fourier transform of $S^{>}(t,t^{\prime};V,V_{T})=S^{>}(t-t^{\prime};V,V_{T})$,

$$\begin{split}&S^{>}(\omega;V,V_{T})=\int_{-\infty}^{\infty}dt\,e^{i\omega t}S^{>}(t;V,V_{T}),\\ &S^{>}(V,V_{T})=S^{>}(\omega=0;V,V_{T}).\end{split}$$

(21)

As it has already been mentioned in Section I, majority of quantum transport experiments deal with mean currents, Eq. (18), specifically, with their differential characteristics such as the differential conductance or, less often, differential thermoelectric conductance, $\partial I(V,V_{T})/\partial V$ or $\partial I(V,V_{T})/\partial V_{T}$, respectively. These quantities have universal units of $e^{2}/h$ and thus provide direct access to universal properties of MBSs. Likewise, experiments dealing with shot noises, Eq. (21), and their derivatives, may access universal fluctuation behavior of Majorana entangled states via, for example, the differential shot noise, $\partial S^{>}(V,V_{T})/\partial V$, having universal units of $e^{3}/h$. Whereas the Majorana universality of $\partial S^{>}/\partial V$ is still an experimental challenge for Majorana entangled setups, the differential shot noise has already been successfully measured to probe other types of fluctuation universality, for example the universality of the Kondo noise in quantum dots Basset et al. (2012). Although experiments on current fluctuations are more complicated than those measuring mean currents, results of such noise measurements provide a much more detailed microscopic structure of various nanoscopic setups.

Below we obtain the differential conductance $\partial I(V,V_{T})/\partial V$, differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$ and differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ by means of numerical calculations based on finite differences used to approximate the corresponding derivatives. Here we would like to emphasize that although the above theoretical model is noninteracting, numerical calculations of $S^{>}(V,V_{T})$ and $I(V,V_{T})$ are nevertheless necessary. The point is that after obtaining closed analytic expressions for $S^{>}(V,V_{T})$ and $I(V,V_{T})$, which is possible because the Keldysh field integral is quadratic in the fermionic fields, it still remains to perform integrals in the energy domain (see the Appendix in Ref. Smirnov (2018)) in these analytic expressions. These integrals are hard to calculate analytically, especially, at finite temperature differences (characterized by finite thermal voltages $V_{T}$), that is when the Fermi-Dirac distributions in Eq. (4) cannot be approximated by step-like functions. In general, calculations of the differential shot noise are more time consuming than those which would be necessary to get just the shot noise. Whereas a certain numerical accuracy may be sufficient to get curves looking smooth enough for $S^{>}(V,V_{T})$, using the same numerical data to calculate $\partial S^{>}(V,V_{T})/\partial V$ may result in numerical errors leading to a chaotic dataset. Thus to obtain $\partial S^{>}(V,V_{T})/\partial V$ with an accuracy that allows to identify dependence on various parameters as well as corresponding coefficients, the calculation of $S^{>}(V,V_{T})$ should be done with a proper precision. Clearly, a higher degree of numerical accuracy leads to a notable increase of the computational time but still makes it possible to perform for the setup described in this section a detailed analysis of $\partial S^{>}(V,V_{T})/\partial V$, in particular, its universal Majorana thermoelectric crossover discussed thoroughly in the next section.

In this section we present numerical results for the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ and demonstrate that, in contrast to the differential conductance $\partial I(V,V_{T})/\partial V$ and differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$, it exhibits a crossover from one type of nonequilibrium behavior to a qualitatively different one. This crossover occurs in the regime

$$\Gamma\gg eV_{T}\gg|eV|\gg\xi,$$

(22)

and in the most part of this section we focus on quantum transport in this regime except for the last part where we show that the crossover disappears for large values of the Majorana overlap energy $\xi$.

In Fig. 1 we show numerical results obtained for the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ as a function of the thermal voltage $V_{T}$ for different values of the bias voltage $V$. As can be seen, at a certain value of the thermal voltage $V_{T}=V_{T,\text{min}}$ each of the three curves has a characteristic minimum (shown by the corresponding circle) which represents a crossover from a thermoelectric to pure thermal nonequilibrium behavior. Indeed, the decreasing, or thermoelectric, branch depends on both the electric driving $V$ and thermal driving $V_{T}$ with the asymptotic behavior shown by the inclined dashed line whereas the increasing, or pure thermal, branch depends only on the thermal driving $V_{T}$ and does not

depend on the electric driving $V$. Our numerical analysis shows that the asymptotics of the thermoelectric and pure thermal nonequilibrium branches are, respectively, given by the following analytic expressions:

$$\begin{split}&\frac{\partial S^{>}(V,V_{T})}{\partial V}=\frac{e^{3}}{h}\frac{1}{8}\frac{|eV|}{eV_{T}},\\ &\text{for $V_{T}$:}\quad|eV|\ll eV_{T}\ll eV_{T,\text{min}},\end{split}$$

(23)

and

$$\begin{split}&\frac{\partial S^{>}(V,V_{T})}{\partial V}=\frac{e^{3}}{h}\frac{1-\ln(2)}{4}\frac{\epsilon_{d}(eV_{T})}{\eta^{2}},\\ &\text{for $V_{T}$:}\quad eV_{T,\text{min}}\ll eV_{T}\ll\Gamma,\end{split}$$

(24)

both of which we are able to reproduce with any desired numerical accuracy. Similarly, in the whole range of the thermal voltage $V_{T}$, restricted by the regime specified in Eq. (22), our numerical calculations show that the analytic expression for the differential shot noise is given by the sum of Eqs. (23) and (24),

$$\begin{split}&\frac{\partial S^{>}(V,V_{T})}{\partial V}=\frac{e^{3}}{h}\biggl{[}\frac{1}{8}\frac{|eV|}{eV_{T}}+\frac{1-\ln(2)}{4}\frac{\epsilon_{d}(eV_{T})}{|\eta|^{2}}\biggr{]},\\ &\text{for $V_{T}$:}\quad|eV|\ll eV_{T}\ll\Gamma.\end{split}$$

(25)

An analytic derivation of Eq. (25) is a complicated task which we would like to address in a separate paper. We note that a proper analytic analysis may provide corrections to Eq. (25) and show under which conditions these corrections start to play an essential role. Also using the Sommerfeld expansion Ashkroft and Mermin (1976), one may analytically derive the differential shot noise in the complementary regime where $eV_{T}\ll|eV|$. However, within the specified regime, Eq. (22), the analytic expression in Eq. (25) has been confirmed with any desired numerical precision. This means that the stronger the inequalities in Eq. (22) are fulfilled, the more digits after the decimal point are reproduced numerically for any given value obtained analytically from Eq. (25).

From Eq. (25) we find that $eV_{T,\text{min}}$ depends on the bias voltage $V$, gate voltage $\epsilon_{d}$ and Majorana tunneling amplitude $|\eta|$,

$$eV_{T,\text{min}}=\biggl{\{}\frac{1}{2[1-\ln(2)]}\frac{|eV||\eta|^{2}}{\epsilon_{d}}\biggr{\}}^{\frac{1}{2}}.$$

(26)

The differential shot noise at $V_{T}=V_{T,\text{min}}$ is obtained from Eqs. (25) and (26) which lead to the following result:

$$\frac{\partial S^{>}(V,V_{T})}{\partial V}\bigg{|}_{V_{T}=V_{T,\text{min}}}=\frac{e^{3}}{h}\biggl{[}\frac{1-\ln(2)}{8}\frac{\epsilon_{d}|eV|}{|\eta|^{2}}\biggr{]}^{\frac{1}{2}}.$$

(27)

Note, that the thermoelectric branch, Eq. (23), is universal because it does not depend on the gate voltage $\epsilon_{d}$ and depends only on the ratio between the electric and thermal driving, $V$ and $V_{T}$, respectively.

According to Eqs. (26) and (27) both the location of the crossover, $V_{T,\text{min}}$, and the value of $\partial S^{>}(V,V_{T})/\partial V$ at $V_{T}=V_{T,\text{min}}$ depend on $V$. This suggests that the crossover results from an interplay between the electric and thermal driving. Moreover, the dependence of the crossover on $\epsilon_{d}$ in Eqs. (26) and (27) (see also Fig. 2) is another indication that the crossover emerges from a competition of the two flows excited, respectively, by the electric and thermal driving. Obviously, due to the particle-hole symmetry, the current cannot be excited exclusively by the thermal driving $V_{T}$ when $\epsilon_{d}=0$. The pure thermal driving $V_{T}$ induces a finite current only when $\epsilon_{d}\neq 0$.

The straight solid line in Fig. 1 shows both the locations $V_{T,\text{min}}$ of the crossovers and the values of $\partial S^{>}(V,V_{T})/\partial V$ at $V_{T}=V_{T,\text{min}}$ parameterized by the bias voltage $V$ according to Eqs. (26) and (27). The figure clearly shows that the crossovers (highlighted by the circles), obtained from the numerical calculations, reside perfectly on the analytic straight solid line. As expected, at low energies, $eV_{T}\ll|eV|$ (that is outside the regime in Eq. (22)), the differential shot noise reaches its universal asymptotic unitary value $e^{3}/4h$ shown by the horizontal dashed line (see also Refs. Liu et al. (2015a); Smirnov (2017)).

The numerical results presented in Fig. 2 show the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ as a function of the thermal voltage $V_{T}$ for different values of the gate voltage $\epsilon_{d}$. As in Fig. 1, each of the three curves possesses a crossover (shown by the corresponding circle) from a thermoelectric to pure thermal nonequilibrium behavior. As mentioned above, on the left side of the crossover the decreasing, or thermoelectric, branch is universal: it depends on both the electric driving $V$ and thermal driving $V_{T}$ via their ratio and Fig. 2 explicitly demonstrates that it does not depend on the gate voltage $\epsilon_{d}$. On the right side of the crossover the increasing, or pure thermal, branch, which is driven only by the thermal voltage $V_{T}$, is obviously not universal. Indeed, the figure clearly shows that this branch depends on the gate voltage $\epsilon_{d}$. The inclined dashed line shows the asymptotic behavior of the pure thermal nonequilibrium branch, in particular, its dependence on the gate voltage $\epsilon_{d}$. Note, that in fact this pure thermal nonequilibrium branch is not universal because of two reasons. The first reason, the dependence on $\epsilon_{d}$, was already mentioned above. The second reason is that this branch additionally depends on the Majorana tunneling amplitude $|\eta|$ as it is also shown in its asymptotic behavior.

In addition to the dependence on $V$, both the location of the crossover, $V_{T,\text{min}}$, and the value of $\partial S^{>}(V,V_{T})/\partial V$ at $V_{T}=V_{T,\text{min}}$ depend on $\epsilon_{d}$. The straight solid line shows both the locations $V_{T,\text{min}}$ of the crossovers and the values of $\partial S^{>}(V,V_{T})/\partial V$ at $V_{T}=V_{T,\text{min}}$ parameterized by the gate voltage $\epsilon_{d}$ according to Eqs. (26) and (27). As in the case of the parametric dependence on $V$, one clearly sees that the numerically obtained crossovers, marked by the circles, also reside perfectly on the analytic straight line resulting from the parametric dependence on $\epsilon_{d}$. Here the universality of the differential shot noise at low energies, $eV_{T}\ll|eV|$, is explicitly visible: the asymptotic low-energy behavior is obviously independent of $\epsilon_{d}$ and is characterized by the unitary value $e^{3}/4h$ shown by the horizontal dashed line.

The nonequilibrium Majorana crossover in the differential shot noise has a number of universal properties which may quantitatively be expressed via a number of ratios taking universal values. For example, according to Eqs. (26) and (27) the ratio

$$R_{1}\equiv\frac{eV_{T,\text{min}}}{|eV|}\frac{\partial S^{>}(V,V_{T})}{\partial V}\bigg{|}_{V_{T}=V_{T,\text{min}}}$$

(28)

is independent of $V$ and takes the universal value

$$R_{1}^{(M)}=\frac{e^{3}}{4h}$$

(29)

for bias voltages satisfying Eq. (22).

For the mean current at low bias voltages in Ref. Smirnov (2020) it has been found that

$$\frac{\partial I(V,V_{T})}{\partial V_{T}}=\frac{e^{2}}{h}\frac{\pi^{2}}{12}\frac{\epsilon_{d}(eV_{T})}{|\eta|^{2}}.$$

(30)

Using Eqs. (26), (27) and (30) one finds that the dimensionless ratio

$$R_{2}\equiv\frac{\frac{\partial S^{>}(V,V_{T})}{\partial V}\bigg{|}_{V_{T}=V_{T,\text{min}}}}{eV_{T,\text{min}}\frac{\partial^{2}I(V,V_{T})}{\partial V_{T}^{2}}}$$

(31)

becomes universal,

$$R_{2}^{(M)}=[1-\ln(2)]\frac{6}{\pi^{2}}$$

(32)

under the conditions in Eq. (22).

If one takes two values of the thermal voltage, $V_{T_{1}}$ and $V_{T_{2}}$, such that $V_{T_{1}}$ belongs to the thermoelectric branch and $V_{T_{2}}$ belongs to the pure thermal branch, then from Eqs. (23), (24) and (27) it follows that the dimensionless ratio

$$R_{3}\equiv\frac{V_{T_{1}}}{V_{T_{2}}}\frac{\frac{\partial S^{>}(V,V_{T})}{\partial V}\bigg{|}_{V_{T}=V_{T_{1}}}\frac{\partial S^{>}(V,V_{T})}{\partial V}\bigg{|}_{V_{T}=V_{T_{2}}}}{\biggl{(}\frac{\partial S^{>}(V,V_{T})}{\partial V}\bigg{|}_{V_{T}=V_{T,\text{min}}}\biggr{)}^{2}}$$

(33)

is universal,

$$R_{3}^{(M)}=\frac{1}{4},$$

(34)

that is independent of $V_{T_{1}}$ ($V_{T_{2}}$) at fixed $V_{T_{2}}$ ($V_{T_{1}}$). This is demonstrated in Fig. 3 where the fixed values of $V_{T_{1}}$ (lower panel) and $V_{T_{2}}$ (upper panel) are respectively chosen such that $|eV|\ll eV_{T_{1}}\ll eV_{T,\text{min}}$ and $eV_{T,\text{min}}\ll eV_{T_{2}}\ll\Gamma$. As one can see in the upper panel

of Fig. 3, on the left (thermoelectric) side of the crossover the numerically obtained dimensionless ratio $R_{3}$ has a perfect plateau with $R_{3}=1/4$ (shown by the horizontal dashed line) as expected in the range $|eV|\ll eV_{T_{1}}\ll eV_{T,\text{min}}$ for any fixed value of $V_{T_{2}}$ taken from the range $eV_{T,\text{min}}\ll eV_{T_{2}}\ll\Gamma$. As demonstrated in the lower panel of Fig. 3, also on the right (pure thermal) side of the crossover the numerical curve develops a clear plateau on which $R_{3}=1/4$ (horizontal dashed line) in the range $eV_{T,\text{min}}\ll eV_{T_{2}}\ll\Gamma$ for any fixed value of $V_{T_{1}}$ taken from the range $|eV|\ll eV_{T_{1}}\ll eV_{T,\text{min}}$. In contrast, when $V_{T_{1}}$ moves away from the thermoelectric branch (left side of the crossover), that is when $eV_{T_{1}}\lesssim|eV|$ or $V_{T_{1}}\gtrsim V_{T,\text{min}}$, the ratio $R_{3}$ must deviate from the value $1/4$. The numerical results (solid curve) in the upper panel show that indeed deviations from the plateau $R_{3}=1/4$ occur when $eV_{T_{1}}\lesssim|eV|$ or $V_{T_{1}}\gtrsim V_{T,\text{min}}$. Similarly, when $V_{T_{2}}$ is not located on the pure thermal branch (right side of the crossover), that is when $V_{T_{2}}\lesssim V_{T,\text{min}}$ or $eV_{T_{2}}\gtrsim\Gamma$, the ratio $R_{3}$ must also shift away from the plateau on which it reaches the value $1/4$. As anticipated, the solid curve resulting from numerical calculations demonstrates that its plateau-like behavior in the lower panel breaks in the domains where $V_{T_{2}}\lesssim V_{T,\text{min}}$ or $eV_{T_{2}}\gtrsim\Gamma$.

Now let us consider only the pure thermal nonequilibrium branch of the differential shot noise. From Eqs. (24) and (30) one finds that the ratio

$$R_{4}\equiv\frac{\frac{\partial S^{>}(V,V_{T})}{\partial V}}{\frac{\partial I(V,V_{T})}{\partial V_{T}}}$$

(35)

becomes universal,

$$R_{4}^{(M)}=e\frac{3[1-\ln(2)]}{\pi^{2}},$$

(36)

in the range of the pure thermal nonequilibrium branch. In Fig. 4 we present numerical results for the ratio $R_{4}$. From the upper and middle panels one immediately sees the qualitative difference between the behavior of respectively the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ and differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$. Indeed, whereas the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ exhibits in its minimum a crossover separating a thermoelectric branch (left side of the crossover) from a pure thermal branch (right side of the crossover), the differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$ does not demonstrate any crossover and has only one, pure thermal, nonequilibrium branch. As has been discussed above, in contrast to the universal thermoelectric branch of $\partial S^{>}(V,V_{T})/\partial V$, its pure thermal branch is not universal because of its dependence on the gate voltage $\epsilon_{d}$ and the Majorana tunneling amplitude $|\eta|$ as can be seen in Eq. (24). Comparing the two non-universal pure thermal branches of the differential shot noise and differential thermoelectric conductance, Eqs. (24) and (30), respectively, one sees that both of them depend linearly on the thermal voltage $V_{T}$ and have identical parametric dependence on the gate voltage $\epsilon_{d}$ and Majorana tunneling amplitude $|\eta|$. Thus, although the pure thermal branches of the differential shot noise and differential thermoelectric conductance are not universal when considered separately from each other, their ratio $R_{4}$ in Eq. (35), must be universal, that is it must be independent of the thermal voltage $V_{T}$, gate voltage $\epsilon_{d}$ and Majorana tunneling amplitude $|\eta|$. Moreover, according to Eq. (36), one expects that in the range of $V_{T}$ corresponding to the pure thermal branch of $\partial S^{>}(V,V_{T})/\partial V$ the ratio $R_{4}$ must be equal to $3[1-\ln(2)]/\pi^{2}$ in the universal units of the elementary charge $e$. The numerical results presented in the lower panel confirm this expectation: on the right (pure thermal) side of the crossover of $\partial S^{>}(V,V_{T})/\partial V$ the ratio $R_{4}$ exhibits a plateau-like behavior in the range $eV_{T,\text{min}}\ll eV_{T}\ll\Gamma$ with $R_{4}=3e[1-\ln(2)]/\pi^{2}$ on the plateau. At this point we would also like to note that, similarly to the differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$, the differential conductance $\partial I(V,V_{T})/\partial V$ does not exhibit any crossover. Our numerical calculations show that it remains almost independent of $V_{T}$ and retains its Majorana fractional value $\partial I(V,V_{T})/\partial V=e^{2}/2h$ up to $eV_{T}\sim\Gamma$ where it starts to decrease and becomes strongly suppressed, i.e. $\partial I(V,V_{T})/\partial V\ll e^{2}/2h$, for $eV_{T}\geqslant\Gamma$.

To see what happens when the two MBSs are not well separated, we have performed numerical calculations for larger values of the Majorana overlap energy $\xi$. Our results show that the above discussed crossover and universal values of the ratios $R_{1,2,3,4}$ disappear. For example, Fig. 5 shows the ratio $R_{4}$ as a function of $\xi$. For well separated MBSs the values of the parameters are chosen to drive the system into the regime where it stays within the plateau shown in the lower panel of Fig. 4, that is when the ratio between the pure thermal branch of the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ and differential thermoelectric conductance $\partial I(V,V_{T})/\partial V_{T}$ takes its universal value, $R_{4}^{(M)}=3e[1-\ln(2)]/\pi^{2}$. As Fig. 5 clearly demonstrates, for small values of the Majorana overlap energy $\xi$ the ratio between the pure thermal nonequilibrium branches of $\partial S^{>}(V,V_{T})/\partial V$ and $\partial I(V,V_{T})/\partial V_{T}$ is equal to its universal Majorana value $R_{4}^{(M)}$. However, when $\xi$ grows, the two MBSs significantly merge into a single Dirac fermion and cannot be probed separately anymore. In this situation the universal nonequilibrium Majorana behavior breaks. As a consequence, the ratio $R_{4}$ significantly deviates from its universal Majorana plateau $R_{4}^{(M)}$. Moreover, for large values of the Majorana overlap energy $\xi$ both the thermoelectric and pure thermal nonequilibrium branches of the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ are destroyed and the notion of the crossover discussed above loses its sense as one would expect for a phenomenon having a Majorana nature.

Finally, to demonstrate that the universal Majorana thermoelectric crossover in the differential shot noise represents a specific behavior strikingly distinct from what is observed in conventional systems without coupling to MBSs, we have computed $\partial S^{>}(V,V_{T})/\partial V$ in the absence of the topological superconductor. The Hamiltonian of the setup without the topological superconductor is obtained from our Hamiltonian if one sets $|\eta|=\xi=0$. The results obtained for this setup are shown in Fig. 6. As can be seen, in the absence of MBSs the dependence of the differential shot noise on $V_{T}$ is qualitatively different from the Majorana induced behavior in two respects.

First, as Fig. 6 shows, the differential shot noise becomes a monotonic function of $V_{T}$ for $|eV|\ll eV_{T}\ll\Gamma$. Its monotonically decreasing character makes $\partial S^{>}(V,V_{T})/\partial V$ negative at some point (that is why we avoid using the logarithmic scale for the $y$-axis). In this sense the differential shot noise is not qualitatively singled out because the differential electric and thermoelectric conductances are also monotonically decreasing functions of $V_{T}$ for $|eV|\ll eV_{T}\ll\Gamma$ in the absence of MBSs. Specifically, for $|eV|\ll eV_{T}\ll\Gamma$ the differential electric conductance is almost independent of $V_{T}$ (and is equal to a value which depends on the gate voltage $\epsilon_{d}$) up to $eV_{T}\sim\Gamma$ where it starts to quickly decrease, whereas the differential thermoelectric conductance is, unlike the Majorana case, always negative and decreases linearly with $V_{T}$ (that is its absolute value grows). In contrast, when MBSs are present, the differential shot noise is qualitatively singled out by its nonmonotonic behavior characterized by a minimum, specifying the thermoelectric crossover, as opposed to the differential electric and thermoelectric conductances having monotonic behavior exhibiting no minima or maxima for $|eV|\ll eV_{T}\ll\Gamma$.

Second, the three curves in Fig. 6, corresponding to three different values of the gate voltage $\epsilon_{d}$, demonstrate that in the whole range of the thermal voltage $V_{T}$ the differential shot noise is not universal, that is in the absence of MBSs $\partial S^{>}(V,V_{T})/\partial V$ depends on $\epsilon_{d}$ for any value of $V_{T}$. This nonuniversal behavior of $\partial S^{>}(V,V_{T})/\partial V$ is qualitatively different from what has been demonstrated in Fig. 2 where coupling to MBSs makes the thermoelectric branch (left side of the crossover) of $\partial S^{>}(V,V_{T})/\partial V$ universal that is independent of $\epsilon_{d}$. In contrast, without coupling to MBSs even small variations of the gate voltage $\epsilon_{d}$ produce large changes in the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ in the whole range of the thermal voltage $V_{T}$ as it is clearly seen in Fig. 6.

We have explored the differential shot noise $\partial S^{>}(V,V_{T})/\partial V$ in a Majorana entangled QD device driven out of equilibrium by both the bias voltage $V$ and thermal voltage $V_{T}$. The numerical analysis of high precision has been used to reveal the existence of a crossover in the behavior of $\partial S^{>}(V,V_{T})/\partial V$ as a function of $V_{T}$ and identify its analytic form. In particular, it has been shown that this crossover results from an interplay between the two types of nonequilibrium fluctuations induced by respectively $V$ and $V_{T}$ and separates thermoelectric nonequilibrium behavior of the differential shot noise from its pure thermal nonequilibrium behavior. The energy scale of the crossover as well as its nonequilibrium fluctuation nature invisible for mean current probes have been identified and the crossover dependences on the gate voltage, bias voltage and Majorana tunneling amplitude have been explicitly shown. Additionally, various universal Majorana ratios $R_{1,2,3,4}$ involving the energy scale of the crossover have been provided for a future experimental access to universal fluctuation behavior of Majorana entangled states within either pure noise measurements, ratios $R_{1,3}$, or in combination with measurements of mean currents, ratios $R_{2,4}$. It has been found that the crossover is destroyed when the two MBSs of the topological superconductor start to overlap and merge into a single Dirac fermion. This results in a disappearance of the universal Majorana plateaus in the ratios $R_{1,2,3,4}$ as has been exemplified via numerical calculations for $R_{4}$. Finally, we have demonstrated that whereas for Majorana entangled states the differential shot noise has a nonmonotonic behavior characterized by a minimum with universal properties, in conventional systems without coupling to MBSs the differential shot noise is a monotonic and nonuniversal function in the whole range of $V_{T}$. Thus, in contrast to Majorana entangled states, in setups without MBSs the monotonic differential shot noise is not qualitatively different from the differential electric and thermoelectric conductances which are also monotonic functions of $V_{T}$ in the absence of MBSs.

For an experimental verification of the theoretical results presented in this work one might consider the devices studied in Refs. Deng et al. (2016, 2018). These devices are based on InAs nanowires covered by an Al layer grown by molecular beam epitaxy. The Al layer is the superconductor which is used to induce a topological superconducting state in the InAs nanowire whose ends are assumed to host MBSs $\gamma_{1,2}$. To couple $\gamma_{1}$ to a QD the Al layer is etched on one end of the InAs nanowire. This bare part of InAs is the place where one forms a QD coupled to the Majorana state $\gamma_{1}$ with the coupling strength $|\eta|$. As explained in Ref. Deng et al. (2018), the occupancy (or the energy level $\epsilon_{d}$ in our context) of the QD is tuned by proper gate voltages. In addition to the setup in Refs. Deng et al. (2016, 2018), one may also form two independent normal metallic contacts coupled to the QD with the coupling strength $\Gamma$. These two independent normal metallic contacts may, in general, have different chemical potentials $\mu_{L,R}$ and different temperatures $T_{L,R}$. To measure the differential shot noise one could try to adapt, for example, the technology from Ref. Basset et al. (2012) based on coupling of a setup to a quantum noise detector. Here for the quantum noise detector one also uses Al as a superconductor and thus it might be compatible with the above technology Deng et al. (2016, 2018) for topological superconductivity. The setup in Ref. Basset et al. (2012) is a carbon nanotube. It may be replaced with the InAs nanowire from Refs. Deng et al. (2016, 2018). One possible problem here is that measurements in Ref. Basset et al. (2012) assume finite frequencies. Nevertheless, one may still measure the differential shot noise if the resonant frequencies in Ref. Basset et al. (2012) are made smaller than all the relevant energy scales of our setup. This might be achieved, for example, by increasing the length of the transmission lines in Ref. Basset et al. (2012) or by other relevant techniques.

Among possible outlooks we would like to mention setups with Aharonov-Bohm fluxes Zou et al. (2023b) or setups where both MBSs are directly entangled with a QD whose nonequilibrium states are governed by bias voltages and temperature differences. Majorana interference effects in such setups will emerge through the Majorana tunneling phases forming a complex interplay with the two competing flows induced by respectively $V$ and $V_{T}$ and the fate of the Majorana crossover in $\partial S^{>}(V,V_{T})/\partial V$ in this situation is an interesting and important problem. The results presented in this work have been obtained assuming that interactions between the Majorana entangled setup and its environment are sufficiently weak. Under certain circumstances, however, such interactions may have a significant impact on the shot noise via corresponding inelastic processes Krainov et al. (2022) and thus represent a challenge for future models where MBSs are coupled to an external environment. Another possibility is to study the differential shot noise in nonequilibrium setups with poor man’s MBSs Tsintzis et al. (2022); Dvir et al. (2023) which may arise inside QDs when one fine-tunes parameters of such setups to locate their states as close as possible to their sweet spots.

The author thanks Reinhold Egger for valuable comments.