Engineering Yang-Lee anyons via Majorana bound states

Anyons are exotic particles which obey unconventional quantum statistics different from fermions and bosons [1], and emerge as fractionalized quasiparticles in topological phases of two-dimensional systems such as fractional quantum Hall (FQH) states [2, 3, 4], topological superconductors [5, 6], and quantum spin liquids[7]. In last two decades, the application of anyons to fault-tolerant quantum computation, which is referred to as topological quantum computation  [8, 9, 7, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], has been extensively studied. In this scheme, informations are stored as topological charges of anyons in a non-local way, and quantum gates necessary for computation are implemented via the exchange of spatial positions (braiding) of anyons. In particular, non-Abelian anyons for which the braiding operations are non-commutative unitary operations acting on a topologically degenerate ground state of many anyon systems are quite useful for the realization of universal topological quantum computation [7, 10]. One possible candidate of non-Abelian anyons for this purpose is Majorana bound states of topological superconductors [11, 12, 13, 14, 6, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. However, unfortunately, the braiding of Majorana bound states is not sufficient for the construction of universal quantum gates. More promising candidates are Fibonacci anyons, which enable us to realize universal topological quantum computation only via topologically protected braiding manipulations [35, 36, 10, 37]. One of the important properties exploited in topological quantum computation is the fusion rule of anyons, which determines what type of anyon appears, when two anyons are brought together. The fusion rule of Fibonacci anyons is,

$\displaystyle\tau\times\tau=1+\tau,$

(1)

where $\tau$ denotes the Fibonacci anyon, and $1$ denotes the trivial particle (or vacuum). This fusion rule determines the Hilbert space where quantum informations are encoded and, combined with braiding, is the basis of universal topological quantum computation. The fusion rule is also obtained from conformal field theory (CFT) corresponding to the anyon theory. In general, anyons are described by chiral conformal field theory, which corresponds to gapless chiral edge states at open boundaries of two-dimensional (2D) gapped topological phases. In the case of the Fibonacci anyon, the corresponding CFT is the level-1 $G_{2}$ Wess-Zumino-Witten theory with the central charge $c=14/5$. There are several proposals for condensed matter systems where Fibonacci anyons appear; e.g. the Read-Rezayi FQH state [38, 39], superconductor-FQH junction systems[40], a septuple-layer topological superconductor [41], and a Rydberg atom gas [42]. However, it is still enormously challenging to realize these systems experimentally.

On the other hand, there is a close cousin of Fibonacci anyon realizable in a non-hermitian quantum system, which is referred to as Yang-Lee anyon [43, 44]. Yang-Lee anyons obey the same fusion rule as that of Fibonacci anyons given by Eq. (1), and hence, the quantum dimension of a Yang-Lee anyon is equal to the golden mean $\phi=(1+\sqrt{5})/2$, which is the measure of the degree of entanglement. Thus, it is expected that they may have computational power comparable to Fibonacci anyons. However, Yang-Lee anyons obey non-unitary non-Abelain statistics; i.e. the non-Abelian braidings of Yang-Lee anyon are non-unitary, because of the non-unitary character of the underlying CFT (see Sec. II. C). Although this feature is not suitable for the application to unitary quantum computation, it may be utilized for the construction of non-unitary quantum gates. Non-unitary quantum computation has been studied in connection with the measurement-based quantum computation [45, 46, 47, 48]. The systematic realization of non-unitary quantum gates may be useful for simulating non-unitary time-evolution of open quantum systems in a controllable way. The Yang-Lee anyons can be described by the CFT for Yang-Lee edge singularity with the central charge $c=-22/5$. The Yang-Lee edge singularity is realized in the one-dimensional (1D) Ising model with an imaginary magnetic field, which is a non-hermitian system. Because of the non-hermiticity, the corresponding CFT is non-unitary, and the central charge and the conformal weight of a primary field are negative. It is noted that one can not realize a 2D topological phase in a hermitian system which hosts edge states described by a non-unitary CFT [43]. However, instead, we can consider a 1D non-hermitian system which effectively simulates the Yang-Lee edge criticality. Also, 1D anyon models such as the Fibonacci chain model [49], and the Yang-Lee chain model [44], have the same central charge as the chiral counterparts at criticality, implying the existence of anyons.

Motivated by these considerations, in this paper, we propose a scheme based on a 1D superconductor junction system which realizes the Yang-Lee edge criticality. Our system consists of topological superconducting nanowire coupled with a semiconductor which plays the role of dissipative electron bath. There are Majorana bound states at open ends of 1D topological superconductors. The coupling with electron baths gives rise to damping of fermion parity for two Majorana bound states, resulting in a non-hermitian term of the effective Hamiltonian. By using the correspondence between Majorana operators and spin operators, we can map this non-hermitian Majorana systems to the 1D Ising model with an imaginary magnetic field for the Yang-Lee edge singularity.

As mentioned above, this critical state is not topologically protected, and not suitable for the application to topological quantum computation. However, our proposal has some advantages. In the case of the Read-Rezayi FQH state [38, 39] and superconductor-FQH junction systems [40], in addition to Fibonacci anyons, there are several different types of quasiparticles which are described by the $Z_{3}$ parafermion CFT, and it is quite nontrivial how to detect experimentally the topological charge of Fibonacci anyons, discriminating the Fibonacci anyons from other particles. In contrast, in the Yang-Lee edge criticality, there is only one type of a nontrivial particle corresponding to a Yang-Lee anyon, and the field operator for a Yang-Lee anyon is nothing but the scaling limit of the spin magnetization of the non-hermite Ising model, which is given by the fermion parity of Majorana bound states in the topological superconductor nanowire system. Thus, the detection and the manipulation of Yang-Lee anyons in the nanowire system is more feasible than Fibonacci anyons in the $Z_{3}$ parafermion system.

The organization of this paper is as follows. In Sec. II, we summarize important features of the Yang-Lee edge criticality, using the Ising spin model with an imaginary magnetic field, and the CFT analysis. We also explain some basic properties of the Yang-Lee anyon model, focusing on the fusion rule and the braiding rules of anyons. In Sec. III, we present the proposed platform for the Yang-Lee edge criticality based on a topological superconductor nanowire junction system, which realizes a non-hermitian Majorana many-body system. In Sec. IV, we discuss the stability of the Yang-Lee edge criticality against perturbations which arise from quasiparticle excitations in the junction system. In Sec. V, we present the scheme for realizing and detecting non-unitary non-Abelian statistics of Yang-Lee anyons in our proposed system. Conclusion is given in Sec. VI

In this section, we present a scheme of realizing the Yang-Lee edge criticality in heterostructure electron systems. Our designed system is constructed from topological superconductor (TSC) nanowires coupled with metallic substrates which play the role of electron baths, as shown in Fig. 6. The key idea here is that the coupling with electron baths gives rise to a non-hermitian term associated with dissipations.

The system consists of coupled units, each of which is composed of two TSC nanowires and a nanoscale metallic substrate. In each unit, there are four Majorana bound states at the open edges of the two TSC nanowires. We assume that one of them is sufficiently away from the other three, and its effect is negligible. We denote the Majorana fields for the remaining three Majorana bound states as $\gamma_{j}^{a}$, $\gamma_{j}^{b}$, and $\gamma_{j}^{c}$. These three Majorana fields constitute an $s=1/2$ spin operator [64],

$\displaystyle S^{x}_{j}=-\frac{i}{2}\gamma_{j}^{b}\gamma_{j}^{c},\quad S^{y}_{j}=-\frac{i}{2}\gamma_{j}^{c}\gamma_{j}^{a},\quad S^{z}_{j}=-\frac{i}{2}\gamma_{j}^{a}\gamma_{j}^{b}.$

(23)

The Majorana bound states $\gamma^{b}_{j}$ and $\gamma^{c}_{j}$ are coupled via the tunneling term,

$\displaystyle\mathcal{H}_{\Gamma,j}\equiv\frac{1}{2}\Gamma i\gamma^{b}_{j}\gamma^{c}_{j}=-\Gamma S^{x}_{j},$

(24)

which simulates the Zeeman interaction due to a transverse magnetic field. Furthermore, the Majorana bound states $\gamma^{a}_{j}$ and $\gamma^{b}_{j}$ are coupled to electrons in a nanoscale metallic substrate which plays the role of a dissipative electron bath. The coupling Hamiltonian is given by,

	$\displaystyle\mathcal{H}^{\prime}_{j}$	$\displaystyle=\sum_{k,\sigma}iV^{a}_{k\sigma j}\left(c^{\dagger}_{k\sigma}+c_{k\sigma}\right)\gamma^{a}_{j}+\sum_{k,\sigma}V^{b}_{k\sigma j}\left(c^{\dagger}_{k\sigma}-c_{k\sigma}\right)\gamma^{b}_{j}$
		$\displaystyle\qquad+\sum_{k,\sigma}\epsilon_{k\sigma}c^{\dagger}_{k\sigma}c_{k\sigma},$		(25)

where $c^{\dagger}_{k\sigma}(c_{k\sigma})$ is a creation (an annihilation) operator of electrons with momentum $k$, spin $\sigma$ in the metallic substrate, $\varepsilon_{k\sigma}$ is its energy band, and $V^{a,b}_{k\sigma j}$ are the tunneling amplitudes between Majorana bound states $\gamma_{j}^{a}$, $\gamma_{j}^{b}$ and the electron bath. To obtain an effective non-hermitian term corresponding to the imaginary magnetic field of the Yang-Lee model, we switch to the imaginary-time path integral formulation. Since Eq.(25) is quadratic in electron fields $c^{\dagger}_{k\sigma}(c_{k\sigma})$, we can integrate out these fields exactly. Then, we arrive at $2\sum_{k,\sigma}V^{b}_{k\sigma j}\gamma^{b}_{j}G_{0}(\varepsilon_{n},k)iV^{a}_{k\sigma j}\gamma^{a}_{j}$ where $G_{0}(\varepsilon_{n},k)\equiv\frac{1}{i\varepsilon_{n}-\varepsilon_{k\sigma}}$ is the unperturbed Green function of electrons in the metallic substrate with $\varepsilon_{n}$ the fermionic Matsubara frequency. We apply the analytic continuation of the Matsubara frequency $i\varepsilon_{n}\rightarrow\varepsilon+i\delta$ with $\delta$ an infinitesimal quantity, and note $G_{0}(\varepsilon_{n},k)\rightarrow{\rm P}\frac{1}{\varepsilon-\varepsilon_{k\sigma}}-i\pi\delta(\varepsilon-\varepsilon_{k\sigma})$. Assuming that $k$-dependence of $V_{k\sigma j}^{a,b}$ is weak, and that the Fermi energy of the substrate is sufficiently large, and the density of states of electrons in the substrate is constant, which justify the approximation, $\sum_{k,\sigma}\rightarrow\int^{\infty}_{-\infty}d\varepsilon_{k\sigma}$, we have $\sum_{k,\sigma}V^{b}_{k\sigma j}V^{a}_{k\sigma j}{\rm P}\frac{1}{\varepsilon-\varepsilon_{k\sigma}}\approx 0$. Then, we end up with the effective Hamiltonian $\mathcal{H}_{\rm eff}$ with the damping of fermion parity eigenstate:

	$\displaystyle\mathcal{H}_{{\rm eff},j}$	$\displaystyle\sim 2\pi\gamma^{a}_{j}\gamma^{b}_{j}\sum_{k,\sigma}V^{a}_{k\sigma j}V^{b}_{k\sigma j}\delta(\mu-\epsilon_{k\sigma})$
		$\displaystyle=-ihS_{j}^{z},$		(26)

where $h=4\pi\sum_{k,\sigma}V^{a}_{k\sigma j}V^{b}_{k\sigma j}\delta(\mu-\epsilon_{k\sigma})$, and $\mu$ is the Fermi level of the metallic substrate. Equation (26) is indeed non-hermitian, and simulates the Zeeman interaction between an imaginary magnetic field and the $z$-component of the spin. We, furthermore, introduce the tunneling term between Majorana bound states in neighboring units, $\gamma_{j}^{a(b)}$ and $\gamma_{j\pm 1}^{a(b)}$:

$\displaystyle\mathcal{H}_{\rm tun}=\sum_{j}\left[it_{a}\gamma^{a}_{j}\gamma^{a}_{j+1}+it_{b}\gamma^{b}_{j}\gamma^{b}_{j+1}\right].$

(27)

As will be shown below, this term generates the Ising interaction between neighboring spins expressed by Majorana fields. We deal with $\mathcal{H}_{\rm tun}$ as a perturbation to the eigenstates of the Hamiltonian of the decoupled Majorana units $\mathcal{H}_{0}=\sum_{j}[\mathcal{H}_{{\rm eff},j}+\mathcal{H}_{\Gamma,j}]$. The eigen energies of each Majorana unit are given by $E_{\pm}=\pm\frac{1}{2}\sqrt{\Gamma^{2}-h^{2}}$. A key assumption in the following analysis is that the hopping amplitudes of $\mathcal{H}_{\rm tun}$ is much smaller than $|E_{\pm}|$, which justifies the perturbative treatment. We consider the correction to the Hamiltonian due to the second order perturbation, $P\mathcal{H}_{\rm tun}\frac{1}{E_{0}-\mathcal{H}_{0}}\mathcal{H}_{\rm tun}P$ where $E_{0}$ is the eigenenergy of $\mathcal{H}_{0}$ in the ground state, $P$ is the projection to the ground state. The intermediate state is the excited state of $\mathcal{H}_{0}$. The single-Majorana tunneling process induced by $\mathcal{H}_{\rm tun}$ results in the change of the energy $2E_{+}$. The first order correction due to $\mathcal{H}_{\rm tun}$ is suppressed for $t_{a,b}\ll 2E_{+}$. As a result, the second order perturbation with respect to $\mathcal{H}_{\rm tun}$ leads to a four-body interaction among Majorana bound states, which simulates the Ising interaction (see Appendix A for more details),

$\displaystyle\sum_{j}\frac{J}{4}\gamma_{j}^{a}\gamma_{j}^{b}\gamma_{j+1}^{a}\gamma_{j+1}^{b}=-J\sum_{j}S^{z}_{j}S^{z}_{j+1},$

(28)

where $J\sim t_{a}t_{b}/E_{+}$. Collecting all the terms, we obtain the Majorana Hamiltonian equivalent to the Yang-Lee model,

	$\displaystyle\mathcal{H}_{\rm MF}$	$\displaystyle=$	$\displaystyle\sum_{j}\frac{J}{4}\gamma_{j}^{a}\gamma_{j}^{b}\gamma_{j+1}^{a}\gamma_{j+1}^{b}+\frac{h}{2}\sum_{j}\gamma^{a}_{j}\gamma^{b}_{j}+\frac{\Gamma}{2}\sum_{j}i\gamma^{b}_{j}\gamma^{c}_{j}$		(29)
		$\displaystyle=$	$\displaystyle\mathcal{H}_{\rm YL}.$

The above argument implies that $J$ is much smaller than the other energy scales, i.e. $J\ll\sqrt{\Gamma^{2}-h^{2}}<\Gamma$. According to the phase diagram of the Yang-Lee model shown in Fig.(2), in the case of $\lambda=\frac{J}{2\Gamma}\ll 1$, the criticality appears for $h\sim\Gamma$. Thus, the Yang-Lee edge criticality for this parameter region can be realized for the interacting Majorana fermion system (29). For the feasibility of this scenario, it is crucial to suppress the single-Majorana hopping processes in Eq. (27). In the following section, we examine the stability of the Yang-Lee edge criticality with the central charge $c=-22/5$ against this perturbation (27).

Before closing this section, we mention about whether the tunnel term (27) affects the $\mathcal{PT}$ symmetry of the Yang-Lee model or not. We define the $\mathcal{P}$ symmetry and $\mathcal{T}$ symmetry operations of Majorana fields at the $j$-site, which are consistent with those of spin operators as follows:

	$\displaystyle\mathcal{P}$	$\displaystyle:$	$\displaystyle~{}~{}\gamma^{a}_{j}\rightarrow(-1)^{j}\gamma^{a}_{j},~{}\gamma^{b}_{j}\rightarrow(-1)^{j+1}\gamma^{b}_{j},~{}\gamma^{c}_{j}\rightarrow(-1)^{j+1}\gamma^{c}_{j},$
	$\displaystyle\mathcal{T}$	$\displaystyle:$	$\displaystyle~{}~{}\gamma^{a}_{j}\rightarrow(-1)^{j+1}\gamma^{a}_{j},~{}\gamma^{b}_{j}\rightarrow(-1)^{j}\gamma^{b}_{j},~{}\gamma^{c}_{j}\rightarrow(-1)^{j+1}\gamma^{c}_{j},$		(30)
			$\displaystyle i\rightarrow-i.$

Actually, according to these transformation rules, the Majorana Hamiltonian $\mathcal{H}_{\rm MF}$ preserves the $\mathcal{PT}$ symmetry. Also, the tunnel term (27) is indeed invariant under the $\mathcal{PT}$ symmetry operation obtained from Eq. (30). Thus, the perturbations arising from the tunnel term (27) do not break the $\mathcal{PT}$ symmetry of the Yang-Lee model.

Here, we discuss the stability of the Yang-Lee edge criticality against the single-Majorana hopping term (27) via numerical simulations. In this section, we adopt the periodic boundary condition, because of efficiency of numerical computation. The universality class is characterized by the central charge $c=-22/5$ and the scaling dimension $\Delta=-2/5$ of the primary field $\hat{\phi}$. A standard method for the calculation of the central charge is to extract it from the prefactor of the entanglement entropy [65, 66, 67, 68, 69, 70]. However, a naive application of this approach to non-hermitian systems occasionally leads to some difficulty in numerics (for details, see Appendix. C). Thus, we exploit the finite-size scaling method  [71, 72, 73] combined with the numerical exact diagonalization.

For finite-size scaling, the energy spectra of the Yang-Lee model at criticality are, for the periodic boundary condition, given by,

	$\displaystyle E^{\hat{\mathbb{I}}}(n,\overline{n})=\epsilon_{0}L-\frac{\pi v}{6L}c+\frac{2\pi v}{L}\left(n+\overline{n}\right)$		(31)
	$\displaystyle E^{\hat{\phi}}(n,\overline{n})=\epsilon_{0}L-\frac{\pi v}{6L}c_{\rm eff}+\frac{2\pi v}{L}\left(n+\overline{n}\right),$		(32)

where we define the effective central charge $c_{\rm eff}$: $c_{\rm eff}\equiv c-12(h_{1,2}+\overline{h}_{1,2})$, and $v$ is the velocity of the low-energy excitation. The indices $n$, $\bar{n}$ $(\geq 0)$, represent the level of descendant states. The states with $n=\bar{n}=0$ correspond to the primary states. For this system, the physical ground state $\ket{\rm GS}$ is the primary state $\ket{h,\overline{h}}$ with the conformal weights $(h,\overline{h})=(-1/5,-1/5)$, and the energy level $E^{\hat{\phi}}(0,0)$. The first excited state $\ket{\rm 1st}$ is the trivial primary state with the conformal weights $(h,\overline{h})=(0,0)$, which leads to the energy level $E^{\hat{\mathbb{I}}}(0,0)$. The second excited state $\ket{\rm 2nd}$ is the level $1$ descendant state of the ground state $\ket{\rm GS}$ expressed as,

$\displaystyle\ket{\rm 2nd}\sim L_{-1}\ket{\rm GS}\mbox{~{}~{}or~{}~{}}\overline{L}_{-1}\ket{\rm GS},$

(33)

where $L_{-n}$ and $\overline{L}_{-n}$ ($n=1,2,...$) are the Virasoro generators. The energy level of the second excited state are given by $E^{\hat{\phi}}(1,0)$ and $E^{\hat{\phi}}(0,1)$. From the energy spectrum obtained by using exact diagonalization, we obtain the central charge $c$, the scaling dimension $\Delta$ and the velocity $v$. We set the parameters as $(\lambda,h)=(0.011,0.910),(0.008,0.927)$. (see Appendix B) We, first, check the Yang-Lee Edge criticality without Majorana hopping term. From the finite-size scaling and fitting method (Appendix. B), we numerically estimate the central charge and the conformal weight (Tab. 1). Numerical results imply that the universality class of the criticality for these parameters coincides with the Yang-Lee edge singularity.

We, next, examine the stability of the Yang-Lee edge criticality against Majorana hopping term. In Fig. 7, we show the central charge $c$ and the scaling dimension $\Delta$ versus the amplitude of the Majorana hopping term. It is found that the Yang-Lee edge criticality is stable up to $t/J\sim 10^{-4}$. The results indicate that the criticality is stable as long as the Majorana hopping term is sufficiently suppressed, which is achieved by controlling the charging energy $E^{\rm SC}_{C}$ in our TSC junction system.

The non-Abelian anyon statistics is detected by the braiding, fusion, and measurement of anyons. We, here, discuss how to implement these operations of Yang-Lee anyons in our Majorana-based system. Generally, braidings of anyons are achieved for chiral fields of CFTs. However, our system is not chiral, but a purely 1D system, which consists of both holomorphic and anti-holomorphic parts of non-trivial fields, if the periodic boundary condition is imposed. Then, the phase changes of these two parts arising from a braiding operation cancel with each other resulting in trivial braidings only, unless one can manipulate these two parts independently, which is not the case of the Yang-Lee Ising model. However, on the other hand, in the case of open boundary condition, all the primary fields are expressed only by the holomorphic part, and the finite size energy spectrum is given by $E=E_{0}+\frac{\pi v}{L}(h_{1,2}+n)$ where $E_{0}$ is the ground state energy. Thus, $\tau$ is regarded as a chiral field with a conformal spin equal to $\exp(-i\pi h_{1,2})=\exp(i\pi/5)$. We conjecture that the nontrivial field $\hat{\tau}$ in this case obeys the rule described by the $F$-symbol (13) and the $R$-matrix (14) in Sec. II. On the basis of this argument, we consider an array of open chains of the Yang-Lee model which are constructed by using the scheme presented in Sec. III. In Fig. 8(a), we show the schematic of fusion process.s The chains are connected with each other through trivial chains which are also constructed in the same scheme, but gate potentials are tuned to make them in a trivial insulating phase. By controlling the gate potentials, one can change the trivial chains into a topological state supporting Majorana bound states. Then, two Yang-Lee chains connected via a trivial chain can be turned into a single Yang-Lee chain, which realizes the fusion of non-trivial Yang-Lee particles.

The detection of the topological charge of a Yang-Lee anyon can be carried out by the measurement of the ”magnetization” $S_{z}$ of the Yang-Lee chain system. As seen in Sec. II. A, the sign of $\langle S_{z}\rangle$ is different between the non-trivial ground state with an anyon $\tau$ and the trivial vacuum state (the first excited state). This difference can be utilized for the detection of the state with $\tau$. Furthermore, the magnetization exhibits a divergent behavior near the critical point as shown in Fig. 3 because of the non-unitary feature. This unique behavior may be also useful for the experimental detection. The magnetization $S_{z}=\frac{i}{2}\gamma_{a}\gamma_{b}$ in our Majorana-based system is nothing but a fermion parity, and thus, can be measured by the two-terminal conductance measurement [24].

We, finally, comment on how to carry out the braiding operations of Yang-Lee anyons. We consider two different approaches. The first one is the physical transfer of the Yang-Lee anyon state with the use of gate potential control, as proposed for the case of Ising anyons in topological superconductor nanowire systems [74]. By changing trivial regions into a topological state, and vice versa, the region of the Yang-Lee edge criticality can be transferred to other regions(Fig. 8(b)). Using $Y$-shape junctions, as considered in Ref. [74], we can achieve the transposition of two separated Yang-Lee regions The second approach is to use measurement-based braiding. The measurement-based braiding was considered before for general anyon systems [75], and also for Majorana systems [76]. The basic idea is to exploit quantum teleportation induced by projective measurements of topological charges. Following Ref. [75], we apply a sequence of projective measurements of the topological charge, using the above-mentioned procedure, until we obtain desired results, which leads to the transfer of the state with an anyon. Applying this measurement-based transfer to two anyons, one can realize braiding of them.

In this study, we design the platform for Yang-Lee anyons which are non-unitary counterparts of Fibonacci anyons, obeying the same fusion rule. Our proposed systems is a non-hermitian interacting Majorana system constructed from the 1D TSC junctions coupled with dissipative electron baths, which can simulate the Yang-Lee edge criticality. The coupling with electron baths gives rise to damping of fermion parity in a superconducting nanowire, which play the role of an imaginary magnetic field in terms of the spin representation. To stabilize the Yang-Lee edge criticality, it is important to suppress single-Majorana hopping processes in the junction system, which do not exist in the Yang-Lee spin model. We can decrease these undesirable processes by controlling the charging energy of TSC nanowires. We also present the scheme for the braiding, fusion, and detection of Yang-Lee anyons in our TSC junction system.

In the Yang-Lee edge criticality, which is described by the non-unitary CFT with the central charge $c=-22/5$, the ground state is the non-trivial state corresponding to a Yang-Lee anyon, while the first excited state is a trivial vacuum. Because of non-unitary character of the CFT, the non-Abelian braiding operator of Yang-Lee anyons is non-unitary, leading to non-unitary non-Abelian statistics. Although it is not suitable for the application to unitary quantum computation, it may be utilized for simulating non-unitary quantum dynamics in a controllable way.

Our study suggests that non-hermitian Majorana many-body systems embrace rich physics inherent in open quantum systems, and can be realized by exploiting topological superconductor junction systems in a feasible and controllable way. It is a future interesting issue to explore for further exotic phases emerging from non-hermitian Majorana many-body systems.

We would like to thank M. Sato for valuable discussions on non-hermitian quantum systems. T.S. is supported by a JSPS Fellowship for Young Scientists. M.G.Y. is supported by Multidisciplinary Research Laboratory System for Future Developments, Osaka University. This work was supported by JST CREST Grant No.JPMJCR19T5, Japan, and JST PRESTO Grant No.JPMJPR225B, and the Grant-in-Aid for Scientific Research on Innovative Areas “Quantum Liquid Crystals (JP22H04480)” from JSPS of Japan, and JSPS KAKENHI (Grant No.JP20K03860, No.JP20H01857, No.JP21H01039, No.JP22K14005 and No.JP22H01221). A part of the computation in this work has been done using the facilities of the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo.