A study of topological characterisation and symmetries for a quantum-simulated Kitaev chain

Abstract

An attempt is made to quantum simulate the topological characterisations, such as winding number, geometric phase and symmetry properties for a quantum-simulated Kitaev chain. We find that $\alpha$ (ratio between the spin–orbit coupling and magnetic field) and the range of momentum space of consideration, play crucial roles in topological classification. We show explicitly that, for this quantum-simulated Kitaev chain, the topological quantum phase transition occurs for finite range of momentum. We observe that the quasiparticle mass of the fermion plays a significant role in topological quantum phase transition. We also show that the symmetry properties of the simulated Kitaev chain is the same as that of the original Kitaev chain. The exact solution of the simulated Kitaev chain is given. This work provides a new perspective on new emerging quantum simulator and also for the topological state of matter.

Appendices

Appendix A. Derivation of Kitaev chain for a quantum nanowire

Kitaev limit can be achieved in the presence of strong magnetic field. Energy spectrum splits into twoparabolic spectra for two different spin species, up and down and the chemical potential is inside the gap. The lower energy state is for the up-spin. The kinetic energy contribution is

$$\begin{aligned} H_{\mathrm {kin}} = \left( \frac{k^2 }{2m} - (B + \mu ) \right) \tau _z . \end{aligned}$$

(A.1)

At first we introduce six important operators.

$$\begin{aligned} \tau _x= & {} \left( \begin{matrix} 0 &{}&{} \quad 0 &{}&{} \quad 1 &{}&{} \quad 0\\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 1 \\ 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 1 &{}&{} \quad 0 &{}&{} \quad 0 \\ \end{matrix}\right) ,\\ \tau _y= & {} \left( \begin{matrix} 0 &{}&{} \quad 0 &{}&{} \quad -i &{}&{} \quad 0\\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad -i \\ i &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad i &{}&{} \quad 0 &{}&{} \quad 0 \\ \end{matrix}\right) ,\\ \tau _z= & {} \left( \begin{matrix} 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0\\ 0 &{}&{} \quad 1 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 0 &{}&{} \quad -1 &{}&{} \quad 0 \\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad -1 \\ \end{matrix}\right) . \end{aligned}$$

Similarly, there are operators $\sigma _x$, $\sigma _y$ and $\sigma _z$ acting on the spin space.

$$\begin{aligned} \sigma _x= & {} \left( \begin{matrix} 0 &{}&{} \quad 1 &{}&{} \quad 0 &{}&{} \quad 0\\ 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 1 \\ 0 &{}&{} \quad 0 &{}&{} \quad 1 &{}&{} \quad 0 \\ \end{matrix}\right) ,\\ \sigma _y= & {} \left( \begin{matrix} 0 &{}&{} \quad -i &{}&{} \quad 0 &{}&{} \quad 0\\ i &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad -i \\ 0 &{}&{} \quad 0 &{}&{} \quad i &{}&{} \quad 0 \\ \end{matrix}\right) ,\\ \sigma _z= & {} \left( \begin{matrix} 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0\\ 0 &{}&{} \quad -1 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 0 &{}&{} \quad 1 &{}&{} \quad 0 \\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad -1 \\ \end{matrix}\right) . \end{aligned}$$

These operators $\tau _x$, $\tau _y$ and $\tau _z$ are acting on the particle–hole space. Similarly, there are operators $\sigma _x$, $\sigma _y$ and $\sigma _z$ acting on the spin space. These six operators are mainly used for calculating the topological state of matter. Here we are simulating the Kitaev model for the spin-less fermion system. Therefore, we will use the operators $\tau$’s. In the next step, one can consider the pairing term. The low energy space of the BdG equation is spanned by the spin-up electron $|e \rangle = {(1,0,0,0)}^T$ and the spin-up hole $|h \rangle = { (0,0,0,1) }^T$. In this subspace of energy, there is no pairing term. The matrix elements, $\langle e| \Delta \tau _x |e\rangle = \langle h| \Delta \tau _x |e \rangle = \langle e | \Delta \tau _x |h \rangle = \langle h | \Delta \tau _x |h\rangle = 0$.

$$\begin{aligned} \langle h | \Delta \tau _x |h\rangle= & {} \Delta (0,0,0,1) \left( \begin{matrix} 0 &{}&{} \quad 0 &{}&{} \quad 1 &{}&{} \quad 0\\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 1 \\ 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 1 &{}&{} \quad 0 &{}&{} \quad 0 \\ \end{matrix}\right) \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ \end{matrix}\right) \\= & {} \Delta (0,0,0,1) \left( \begin{matrix} 0 \\ 1 \\ 0 \\ 0 \\ \end{matrix}\right) = 0.\\ \langle e | \Delta \tau _x |e\rangle= & {} \Delta (1,0,0,0) \left( \begin{matrix} 0 &{}&{} \quad 0 &{}&{} \quad 1 &{}&{} \quad 0\\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 1 \\ 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 1 &{}&{} \quad 0 &{}&{} \quad 0 \\ \end{matrix}\right) \left( \begin{matrix} 1 \\ 0 \\ 0 \\ 0 \\ \end{matrix}\right) \\= & {} \Delta (1,0,0,0) \left( \begin{matrix} 0 \\ 0 \\ 1 \\ 0 \\ \end{matrix}\right) = 0.\\ \langle e | \Delta \tau _x |h\rangle= & {} \Delta (1,0,0,0) \left( \begin{matrix} 0 &{}&{} \quad 0 &{}&{} \quad 1 &{}&{} \quad 0\\ 0 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 1 \\ 1 &{}&{} \quad 0 &{}&{} \quad 0 &{}&{} \quad 0 \\ 0 &{}&{} \quad 1 &{}&{} \quad 0 &{}&{} \quad 0 \\ \end{matrix}\right) \left( \begin{matrix} 0 \\ 0 \\ 0 \\ 1 \\ \end{matrix}\right) \\= & {} 0 = \langle h | \Delta \tau _x |e\rangle . \end{aligned}$$

Therefore, it is revealed from this study that the spin singlet pairing cannot induce proximity superconductivity in a perfectly spin-polarised system. This is also physically consistent because the spin singlet is possible only when the band is populated with up and down spin states.

Therefore, to get the finite contribution of superconductivity, we must have to consider the spin–orbit coupling, which modifies the energy spectrum and populates it with both up and down spins.

Now the spinors become

$$\begin{aligned} |e \rangle = {\bigg (1, -\frac{uk}{2 B},0,0\bigg )}^T \end{aligned}$$

and

$$\begin{aligned} t |h \rangle = { \bigg (0,0,-\frac{uk}{2 B},1\bigg ) }^T. \end{aligned}$$

In this subspace, one can obtain

$$\begin{aligned} \langle h| \Delta \tau _x |e \rangle = \langle e | \Delta \tau _x |h \rangle = -\frac{uk}{2B} \Delta \end{aligned}$$

and other matrix elements are zero.

$$\begin{aligned}&\langle e | \Delta \tau _x |e\rangle \\&= \Delta \left( 0,\frac{-u k}{2 B},0,0\right) \left( \begin{matrix} 0 &{}&{} 0 &{}&{} 1 &{}&{} 0\\ 0 &{}&{} 0 &{}&{} 0 &{}&{} 1 \\ 1 &{}&{} 0 &{}&{} 0 &{}&{} 0 \\ 0 &{}&{} 1 &{}&{} 0 &{}&{} 0 \\ \end{matrix}\right) \left( \begin{matrix} 0 \\ \dfrac{-uk}{2 B} \\ 0 \\ 0 \\ \end{matrix}\right) \\&= \Delta \left( 0,\dfrac{-u k}{2 B},0,0\right) \left( \begin{matrix} 0 \\ 0 \\ 1 \\ -\dfrac{uk}{2 B} \\ \end{matrix}\right) = 0= \langle h | \Delta \tau _x |h\rangle . \\&\langle h | \Delta \tau _x |e\rangle \\&\quad = \Delta \left( 0,0,\frac{-u k}{2 B},0\right) \left( \begin{matrix} 0 &{}&{} 0 &{}&{} 1 &{}&{} 0\\ 0 &{}&{} 0 &{}&{} 0 &{}&{} 1 \\ 1 &{}&{} 0 &{}&{} 0 &{}&{} 0 \\ 0 &{}&{} 1 &{}&{} 0 &{}&{} 0 \\ \end{matrix}\right) \left( \begin{matrix} 0 \\ \dfrac{-uk}{2 B} \\ 0 \\ 0 \\ \end{matrix}\right) \\&\quad = \Delta \left( 0,0,\frac{-u k}{2 B},0\right) \left( \begin{matrix} 0\\ 0\\ 1 \\ -\dfrac{uk}{2 B} \\ \end{matrix}\right) .\\&\quad = - \Delta \frac{uk}{2 B}= \langle e | \Delta \tau _x |h\rangle . \end{aligned}$$

Fig. 8

These figures present the exact solution for the quantum-simulated Kitaev chain with respect to m. The left panel represents the BZ with a boundary from $-\pi$ to $\pi$. The right panel represents the BZ with a boundary from $-\pi /2.2$ to $\pi /2.2$. Each figure consists of three curves for different values of $\Delta$.

Fig. 9

These figures present the behaviour of geometric phase with varying values of chemical potential. The left panel represents the original Kitaev chain and the right panel represents the simulated Kitaev chain. Each figure consists of three curves for different values of t as depicted in the figures. Here we consider $\Delta =0.1$ and $m=1$.

Therefore, the final form of the model Hamiltonian is

$$\begin{aligned} H \simeq \left( \frac{k^2 }{2 m} - \mu \right) \tau _z - \frac{uk}{2B} \Delta \tau _x. \end{aligned}$$

This is the analogous form of the BdG Hamiltonian of a spin-less p-wave superconductor with the effective pairing ${\Delta }_{\mathrm {eff}} = {u \Delta }/{2B}$. Therefore, we conclude that effective p-wave pairing is due to the presence of spin–orbit coupling and it becomes weak when Zeeman field is large [17].

Appendix B. Exact solution of the simulated Kitaev chain

It is well known that the Kitaev chain has exact solution for $\mu = 0$, for $\Delta = t$. For this limit, the system is always in the topological state without any transition. One can also understand this constant topological state without any transition for $\mu =0$ from the parametric relation $\mu = 2 t$.

Therefore, it is also a challenge to check the existence of exact solution for the simulated Kitaev chain. The exact solution of the winding number is

$$\begin{aligned} W_{\mathrm {exact}} = \frac{1}{\alpha \pi } \cot ^{-1}\left( \frac{2 m \alpha \Delta }{a \pi } \right) . \end{aligned}$$

(B.1)

In figure 8, we present the exact result of W with the varying parameter m. Each panel consists of three figures for different values of $\Delta$. The left panel has a BZ limit which runs from $-\pi$ to $\pi$. Here, we do not observe any topological state with integer winding number. In the right panel, we present exact solution with a reduced BZ limit from $-\pi /2.2$ to $\pi /2.2$. Here, we observe that the system is in the topological state with winding number very close to $W_{\mathrm {exact}}=0.5$ for $\alpha =1$. We observe that for higher values of $\Delta$, the topological state do not remain constant with increasing values of m. It is clear from this study that one can also reproduce the exact solution of the Kitaev chain for smaller values of $\Delta$ by using the simulated Kitaev chain. This can be considered as an advantage with quantum-simulated Kitaev chain.

Appendix C. Results of geometric phase with physical explanation

At first, we describe very briefly the basic aspect of geometric phase. During the adiabatic time evolution of the system, the state vector acquires an extra phase over the dynamical phase, $|\psi (R(t) )\rangle = \mathrm {e}^{\alpha _n} |\phi (R(t)\rangle$, where $\alpha _n = \theta _n + \gamma _n$. Here $\theta _n = \frac{-1}{h} \int _{0}^{t} E_n (\tau ) \mathrm {d} \tau$ and $\gamma _n$ are the dynamical and geometric phases respectively. For a system, the geometric phase is given by the cyclic evolution of state vectors over a closed curve. It is evident from the analytical expression of Berry phase that it depends on the geometry of the parameter and loop (C) therein.

$$\begin{aligned} {\gamma _n} (C) = i \int _{C} \langle \phi (x)| \nabla | \phi (x) \rangle \mathrm {d}x . \end{aligned}$$

The geometric (Zak) phase is an important concept for the topological characterisation of low-dimensional quantum many-body system [13, 32, 33]. Zak has considered the one-dimensional Brillouin zone and the cyclic parameter is the crystal momentum (k). The geometric phase in the momentum space is defined as

$$\begin{aligned} \gamma _n = \int _{-\pi }^{\pi } \mathrm {d}k \langle u_{n,k} | i \partial _k | u_{n,k} \rangle , \end{aligned}$$

(C.1)

where $|u_{n,k} \rangle$ are the Bloch states which are the eigenstates of the nth band of the Hamiltonian. The simulated Kitaev chain possesses Z-type topological invariant and also the anti-unitary particle hole symmetry. For this system, the analytical expression of the Zak phase [13, 32, 33] is

$$\begin{aligned} \gamma = W \pi , \mod (2 \pi ) . \end{aligned}$$

(C.2)

In figure 9, we present the behaviour of $\gamma$ with respect to $\mu$. The left panel represents the variation of geometric phase for the original Kitaev chain with respect to the chemical potential where each curve represents different values of t. With increasing values of a, it is not possible to observe the topological phase transitions (Here a is the number associated with the limit of BZ. In general $a=1$ for the original Kitaev chain and takes different values for simulated Kitaev chain). The right panel represents the variation of geometric phase for the simulated Kitaev chain with respect to the chemical potential. Here, each curve represents different values of $\alpha$ and it is also clear from this study that as we increase the value of $\alpha$ the quantisation condition for $\gamma$ smeared out and there is no topological quantum phase transition for the simulated Kitaev chain.

It is revealed from the study of the right panel that $\gamma$ shows the same behaviour of the original Kitaev chain when we consider the momentum space region $-\frac{\pi }{2.2}< k< \frac{\pi }{2.2}$. For the original Kitaev chain, the Bloch state traverse in the whole BZ whereas for the simulated Kitaev chain, the Bloch state traverses in the reduced momentum space as we see from the dispersion. For the consideration of momentum space region $-\pi /2.2< k <\pi /2.2$ for the simulated Kitaev chain, gives the same parametric relation of the original Kitaev chain for the topological phase transition.

Appendix D. An extensive derivation of symmetries for the simulated Kitaev chain

Among the vast variety of topological phases, one can identify an important class called symmetry-protected topological (SPT) phase, where two quantum states have distinct topological properties protected by certain symmetry. Under this symmetry constraint, one can define the topological equivalent and distinct classes. Hamiltonians which are invariant under the continuous deformation into one another preserving certain symmetries are the topological equivalent classes.

Different SPT states can be well understood with the local (gauge) non-spatial symmetries such as, time reversal (TR), particle–hole (PH) and chiral. In general, non-interacting Hamiltonians can be classified in terms of symmetries into ten different symmetry classes [34,35,36,37]. A particular symmetry class of a Hamiltonian is determined by its invariance under time-reversal, particle–hole and chiral symmetries. Apart from that, we also study the parity (P) symmetry, parity–time (PT) symmetry, charge conjugation–parity–time (CPT) symmetry, CP symmetry and CT symmetry. In this section, we present symmetry properties of the simulated Kitaev chain and also check how much it is equivalent with the original Kitaev chain. Here we present the final results of the symmetry operations for this quantum-simulated model Hamiltonian.

1.1 1: Time-reversal symmetry

Time-reversal symmetry operation is ${\hat{\Theta }}$. We use the properties of ${\chi }_z (k) = {\chi }_z (-k)$ and ${\chi }_y (k) = - {\chi }_y (-k)$.

$$\begin{aligned}&{\hat{\Theta }}^{\dagger } \hat{H_{BdG}}(k) {\hat{\Theta }}= {\hat{K}}^{\dagger } \hat{H_{BdG}}(k) {\hat{K}}\cdot {\hat{\Theta }}^{\dagger } \hat{H_{BdG}}(k) {\hat{\Theta }}\\&\quad = {\hat{K}} \begin{bmatrix} {\chi }_{z} (k) &{}&{} i {\chi }_{y} (k)\\ -i {\chi }_{y} (k) &{}&{} - {\chi }_{z} (k) \\ \end{bmatrix} {\hat{K}} = H (k) \\&{\chi }_{z} (k) = \frac{k^2}{2m} - \mu ,\quad {\chi }_{y} (k) = \alpha (= u/B) \Delta k . \end{aligned}$$

Thus, the Hamiltonian obeys time-reversal symmetry.

1.2 2: Charge-conjugation symmetry

This symmetry operator is ${\hat{\Xi }}$.

$$\begin{aligned} {\hat{\Xi }}^{\dagger } {\hat{H}}(k) {\hat{\Xi }}= & {} (\sigma _x{\hat{K}})^{\dagger }{\hat{H}}(k)(\sigma _x {\hat{K}}) = {\hat{K}}^{\dagger }\sigma _x{\hat{H}}(k)\sigma _x {\hat{K}},\\ {\hat{\Xi }}^{\dagger } {\hat{H}}(k) {\hat{\Xi }}= & {} {\hat{K}}^{\dagger } \left( \begin{matrix} 0 &{}&{} 1\\ 1 &{}&{} 0 \\ \end{matrix}\right) \left( \begin{matrix} {\chi }_z (k) &{}&{} i {\chi }_y (k)\\ -i {\chi }_y (k) &{}&{} -{\chi }_ (k)\\ \end{matrix}\right) \\&\times \left( \begin{matrix} 0 &{}&{} 1\\ 1 &{}&{} 0 \\ \end{matrix}\right) {\hat{K}},\\ {\hat{\Xi }}^{\dagger } {\hat{H}}(k) {\hat{\Xi }}= & {} {\hat{K}} \left( \begin{matrix} -{\chi }_z (k) &{}&{} -i {\chi }_y (k)\\ i {\chi }_y (k) &{}&{} {\chi }_z (k)\\ \end{matrix}\right) {\hat{K}}\\= & {} \left( \begin{matrix} -{\chi }_z (k) &{}&{} -i {\chi }_y (k)\\ i {\chi }_y (k) &{}&{} {\chi }_z (k)\\ \end{matrix}\right) = -{\hat{H}} (k). \end{aligned}$$

Thus, the Hamiltonian obeys charge-conjugation symmetry.

1.3 3: Chiral symmetry

This symmetry operator is given by ${\hat{\Pi }}$.

$$\begin{aligned}&{\hat{\Pi }}^{\dagger } \hat{H_{BdG}}(k) {\hat{\Pi }} = \sigma _x {\hat{H}}(k)\sigma _x \\&{\hat{\Pi }}^{\dagger } {\hat{H}}(k) {\hat{\Pi }} =\left( \begin{matrix} 0 &{}&{} 1\\ 1 &{}&{} 0 \\ \end{matrix}\right) \left( \begin{matrix} {\chi }_z (k) &{}&{} i {\chi }_y (k)\\ -i {\chi }_y (k) &{}&{} - {\chi }_z (k)\\ \end{matrix}\right) \left( \begin{matrix} 0 &{}&{} 1\\ 1 &{}&{} 0 \\ \end{matrix}\right) \\&\qquad \qquad \quad \quad = - H (k). \end{aligned}$$

Thus, the Hamiltonian also obeys chiral symmetry.

1.4 4: Parity symmetry

This symmetry operator is given by ${\hat{P}}$.

$$\begin{aligned} PH(k)P^{-1}= & {} \sigma _z H(k) \sigma _z\\= & {} \left( \begin{matrix} 1 &{}&{} 0\\ 0 &{}&{} -1 \\ \end{matrix}\right) \left( \begin{matrix} \chi _z (k) &{}&{} i \chi _y (k)\\ -i \chi _y (k) &{}&{} \chi _z (k) \\ \end{matrix}\right) \left( \begin{matrix} 1 &{}&{} 0\\ 0 &{}&{} -1 \\ \end{matrix}\right) \\= & {} H(-k). \end{aligned}$$

Thus, the Hamiltonian obeys parity symmetry.

1.5 5: PT symmetry

This symmetry operator is given by ${\hat{PT}}$.

$$\begin{aligned} \begin{aligned}&PTH(k)(PT)^{-1}=\sigma _zKH(k)K^{-1}\sigma _z =\sigma _zH(k)\sigma _z\\&\quad = \left( \begin{matrix} 1 &{}&{} 0\\ 0 &{}&{} -1 \\ \end{matrix}\right) \left( \begin{matrix} \chi _z (k) &{}&{} i \chi _y (k) \\ -i \chi _y (k) &{}&{} -\chi _z (k) \\ \end{matrix}\right) \left( \begin{matrix} 1 &{}&{} 0\\ 0 &{}&{} -1 \\ \end{matrix}\right) \\&\quad = \left( \begin{matrix} \chi _z (k) &{}&{} -i \chi _y (k)\\ i \chi _y (k) &{}&{} -\chi _z (k) \\ \end{matrix}\right) \ne H(k). \end{aligned} \end{aligned}$$

Thus, the Hamiltonian does not obey PT symmetry.

1.6 6: CP symmetry

This symmetry operator is given by ${\hat{CP}}$.

$$\begin{aligned} \begin{aligned}&CP H(k)(CP)^{-1}=\sigma _x K \sigma _zH(k)\sigma _z K^{-1}\sigma _x\\&\quad = \sigma _x K \left( \begin{matrix} 1 &{}&{} 0\\ 0 &{}&{} -1 \\ \end{matrix}\right) \left( \begin{matrix} \chi _z (k) &{}&{} i \chi _y (k)\\ -i \chi _y (k) &{}&{} -\chi _z (k)\\ \end{matrix}\right) \\&\qquad \times \left( \begin{matrix} 1 &{}&{} 0\\ 0 &{}&{} -1 \\ \end{matrix}\right) K^{-1}\sigma _x\\&\quad = \sigma _x K \left( \begin{matrix} \chi _z (k) &{}&{} -i \chi _y (k)\\ i \chi _y (k) &{}&{} -\chi _z (k)\\ \end{matrix}\right) K^{-1}\sigma _x \\&\quad =- H(-k). \end{aligned} \end{aligned}$$

Thus, the Hamiltonian obeys CP symmetry.

1.7 7: CT symmetry

This symmetry operator is given by ${\hat{CT}}$.

$$\begin{aligned} \begin{aligned} CTH(k)(CT)^{-1}&=\sigma _xH(k)\sigma _x = - H(k). \end{aligned} \end{aligned}$$

Thus, the Hamiltonian obeys the CT symmetry.

1.8 8: CPT symmetry

This symmetry operator is given by ${\hat{\alpha }}$.

$$\begin{aligned} \begin{aligned}&\alpha H(k) {\alpha }^{-1}=\sigma _x \sigma _z KH(k)K^{-1} \sigma _z\sigma _x\\&\quad = \left( \begin{matrix} 0 &{}&{} 1\\ -1 &{}&{} 0 \\ \end{matrix}\right) \ne -H(k). \end{aligned} \end{aligned}$$

Thus, the Hamiltonian does not obey the CPT symmetry. Therefore, from this detailed derivation it is clear that the symmetry properties of the quantum-simulated Kitaev chain as well as original Kitaev chain are the same.