Majorana nanowires, Kitaev chains, and spin models

The Majorana zero modes (MZMs) are non-Abelian Ising anyons with potential application to fault-tolerant topological quantum computing because of the robustness of the quantum degenerate ground state against local perturbations protected by a topological gap [1, 2]. One prototypical model that was shown to host boundary MZMs is the Kitaev chain [3], which contains the nearest-neighbor hopping and $p$-wave superconductivity in a spinless system [4]. Given appropriate parameters and conditions, a pair of Majorana zero modes can emerge at both ends of the Kitaev chain by construction. Although, the Kitaev chain is an idealized toy model to show a proof of principle, because spinlessness and $p$-wave superconductivity are hard to find in nature, the model has been influential in focusing attention on non-Abelian MZMs, topological superconductivity, topological quantum phase transition (TQPT), and topological quantum computing using Majorana zero modes. Therefore, realistic experimentally feasible models have been proposed to search for MZMs. The most prevailing scheme used in experiments—see, e.g., Ref. 5 and references therein for a recent effort to experimentally realize Majorana zero modes in semiconductor-superconductor hybrid structures— is the one-dimensional (1D) semiconductor-superconductor (SM-SC) nanowire [6, 7, 8, 9], where the semiconductor nanowire with a large Rashba-type spin-orbit coupling (SOC) acquires $p$-wave superconductivity from an $s$-wave superconductor due to the proximity effect, and a pair of MZMs should appear at the ends of the nanowire given a sufficiently large magnetic field. This realistic SM-SC system is often referred to as a Majorana nanowire since the MZMs are predicted to arise at the two ends of the nanowire under suitable conditions. However, although both Kitaev chain and Majorana nanowire systems can host MZMs, they are two completely different models which, at first sight, appear to have little to do with each other except for their one-dimensionality and the possible existence of MZMs. For example, the Kitaev chain is a lattice model made of spinless fermions whereas the nanowire is a continuum system with real spinful electrons. Also, in the Kitaev chain, $p$-wave superconductivity is assumed a priori in the model whereas in the Majorana nanowire the effective topological $p$-wave superconductivity is emergent under the subtle combined effects of induced $s$-wave SC, Rashba SO-coupling, and Zeeman spin splitting. Historically, the Majorana nanowire proposal arose not motivated by the Kitaev chain idea, but from various 2D proposals involving SO-coupling, $s$-wave SC, and spin splitting [10, 11, 8]. After the fact, however, it seems obvious that the two models, Kitaev chain and Majorana nanowire, should be related since both depend on having an effectively spinless $p$-wave SC leading to MZMs localized at the ends of a 1D system. However, the extent to which they are generically equivalent to each other remains unclear despite some isolated efforts [12, 13, 14, 15] to connect them through specific properties. In particular, References 16, 17 were among the first to have preliminary discussions on possible connections between the Kitaev chain and the Majorana nanowire models.

This lack of clarity sometimes leaves a misconception that these two models, Kitaev chain and Majorana nanowire, are always equivalent, which is incorrect, as we will show later under which circumstances these two models are equivalent and when they cannot be mapped into each other. This lack of complete equivalence between the two models is rather obvious if one considers that the Kitaev model is a spinless one band model whereas the nanowire model is a spinful multiband model. Thus, a claim that the nanowire model can be obtained by taking the continuum limit of the Kitaev chain and projecting onto the low-energy degrees of freedom is technically incorrect since such a low-energy projection is meaningless for the original Kitaev chain model. Thus, we are motivated to study the underlying general connection between the Kitaev chain and the SM-SC nanowire try to see whether we can find an exact Majorana mapping between them. Mapping both onto effective spin models by eliminating the fermionic degrees of freedom enables a direct one-to-one comparison between the models to figure out how they are connected and how they are not. This is what we accomplish in the this paper.

We first recall the duality [3, 18, 19] between the Kitaev chain and transverse-field XY model (which can be further tuned to the transverse-field Ising model given appropriate parameters) as this is the origin of the term Ising anyon for MZMs (which is often used [1]). This mapping of the Kitaev chain to the corresponding transverse-field XY/Ising model is achieved by a nonlocal Jordan-Wigner transformation converting the fermions to bosonic spin operators. For the same reason, to see the connection between the Kitaev chain and SM-SC Majorana nanowire, we should ask first how to map an SM-SC Majorana nanowire to a corresponding spin model through the Jordan-Wigner transformation. However, we find that the resulting spin model is, unlike the relatively simple form of the transverse-field XY model (as for the Kitaev chain), rather complicated — it is a generalized cluster model containing up to 4-spin interactions with staggered couplings, which do not exist in the mapping of the Kitaev chain to the transverse-field XY model. Thus the connection between the two models appears much more obscure in the spin language than in the fermionic language, where at least, both models at their cores are free fermion “band” problems.

We, therefore, approach the SM-SC Majorana nanowire from another angle: The first step is to convert the spinful nanowire into an effective low-energy model by projecting out the upper band of the bare semiconductor with higher energy in momentum space in the limit of the large magnetic field. We can then manually construct a spinless model by ignoring the band index and incorporating the superconducting term. We find that the form of the effective low-energy Hamiltonian in momentum space is already a spinless $p$-wave superconductor that contains a normal SM part plus an antisymmetric SC part, which is superficially similar to the Kitaev chain. To make a more direct comparison, we expand the Hamiltonian in momentum space in the limit of a large magnetic field, and recover the real space Hamiltonian up to the third order of approximation. We find that the first order of this high-field expansion can be exactly mapped into the Kitaev chain, and the higher order approximation will result in longer-range hoppings and pairings. Therefore, the Kitaev chain can be viewed at least as a first-order approximation of the SM-SC nanowire model at a high magnetic field deep in the topological regime. This also makes physical sense because, in the high-field limit, the low-energy Majorana nanowire projection essentially produces a spinless $p$-wave SC as assumed by construction in the Kitaev model. The two models obviously cannot be equivalent at a low magnetic field deep in the trivial phase since the nanowire-induced proximity SC is essentially $s$-wave in the trivial phase whereas by construction the SC is $p$-wave in the Kitaev chain.

This paper is organized as follows. In Sec. II, we state the conventions for the Jordan-Wigner transformation adopted throughout this paper. In Sec. III, we recapitulate and expand on the duality between the Kitaev chain and the transverse-field XY model through the Jordan-Wigner transformation. Then we generalize the Jordan-Wigner transformation for the spinful model, and directly apply it to the SM-SC Majorana nanowire Hamiltonian to obtain a generalized spin cluster model with staggered couplings in Sec. IV.1. Finally, we present the direct mapping between the Kitaev chain and the SM-SC nanowire by projecting out the higher energy band in Sec. IV.2. Our conclusion is in Sec. V.

The Jordan-Wigner transformation [20] is a nonlocal transformation that connects a fermionic operator and a series of nonlocal bosonic operators (in $\frac{1}{2}$-spin systems) while keeping the canonical commutation relation the same. We define the Jordan-Wigner transformation following the conventions as

$$\begin{split}f_{i}^{\dagger}&=\prod_{j=1}^{i-1}\left(-\sigma_{j}^{z}\right)\sigma_{i}^{+},\\ f_{i}&=\prod_{j=1}^{i-1}\left(-\sigma_{j}^{z}\right)\sigma_{i}^{-},\end{split}$$

(1)

where $f_{i}$ ($f_{i}^{\dagger}$) is the fermionic annihilation (creation) operator acting on site $i$ in real space, and $\sigma_{i}^{+}$ ($\sigma_{i}^{-}$) is the bosonic ladder operator that raises (lowers) the spin by one unit at site $i$ in $\frac{1}{2}$-spin systems, which can be expressed in terms of the Pauli matrix:

$$\sigma_{i}^{\pm}=\frac{1}{2}\left(\sigma_{i}^{x}\pm i\sigma_{i}^{y}\right).$$

(2)

It is straightforward to verify that the anti commutating relation of the fermionic operators defined in Eq. (1) is also restored by the commuting bosonic spin operators.

For a general noninteracting fermionic Hamiltonian, the bilinear form of a local product of two fermionic operators can be transformed into the product of a series of local spin operators through the Jordan-Wigner transformation, e.g.,

$$\begin{split}f_{i}^{\dagger}f_{i}&=\frac{1}{2}\left(1+\sigma_{i}^{z}\right),\\ f_{i+1}^{\dagger}f_{i}&=\sigma_{i+1}^{+}\sigma_{i}^{-},\\ f_{i+1}^{\dagger}f_{i}^{\dagger}&=-\sigma_{i+1}^{+}\sigma_{i}^{+},\end{split}$$

(3)

which preserves the locality.

We first briefly review, for the sake of clarity and completeness, the duality between the Kitaev chain and the $s=\frac{1}{2}$ 1D transverse-field XY model [21], which can be mapped into each other through the Jordan-Wigner transformation of Eq. (1). The Kitaev chain [3] is a spinless fermionic lattice tight-binding model with the nearest-neighbor hopping $t$, and $p$-wave superconducting pairing energy $\Delta$, where its Hamiltonian in real space is

$$\hat{H}_{\text{KC}}^{(f)}=\sum_{i=1}^{N}-\mu f_{i}^{\dagger}f_{i}+\left(-tf_{i+1}^{\dagger}f_{i}+\Delta f_{i+1}^{\dagger}f_{i}^{\dagger}+\text{h.c.}\right)$$

(4)

Here, $\mu$ is the onsite chemical potential, $N$ is the total number of sites in the Kitaev chain, and we take the periodic boundary condition for simplicity.

By defining the Fourier transformation $\tilde{f}_{k}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{j}f_{j}^{\dagger}e^{ikj}$, we can rewrite the Hamiltonian in momentum space as

$$\hat{H}_{\text{KC}}^{(k)}=\sum_{k}\left(-2t\cos k-\mu\right)\tilde{f}_{k}^{\dagger}\tilde{f}_{k}-\left(i\Delta\sin k\tilde{f}_{k}^{\dagger}f_{-k}^{\dagger}+\text{h.c.}\right).$$

(5)

To obtain the band structure, we construct the Bogoliubov-de Gennes Hamiltonian in momentum space, $\hat{H}_{\text{KC}}^{(k)}=\frac{1}{2}\sum_{k}\tilde{\bm{f}}_{k}^{\dagger}H_{\text{KC}}^{\text{BdG}}(k)\tilde{\bm{f}}_{k}-2t\cos k-\mu$, where $\tilde{\bm{f}}_{k}=\left(\tilde{f}_{k},\tilde{f}_{-k}^{\dagger}\right)^{\intercal}$ and

$$H_{\text{KC}}^{\text{BdG}}(k)=(-2t\cos k-\mu)\tau_{z}+2\Delta\sin k\tau_{y}.$$

(6)

Therefore, by squaring the BdG Hamiltonian, the band structure for the Kitaev chain in Eq. (4) is $E_{\text{KC}}^{2}(k)=\left(2t\cos k+\mu\right)^{2}+4\Delta^{2}\sin^{2}k$ .

By applying the Jordan-Wigner transformation Eq. (1), we obtain the corresponding bosonic spin model by substituting Eq. (3) into Eq. (4),

$$\begin{split}\hat{H}_{\text{KC}}^{(\sigma)}&=\sum_{i=1}^{N}-\frac{\mu}{2}\left(1+\sigma_{i}^{z}\right)-\left(t\sigma_{i+1}^{+}\sigma_{i}^{-}+\Delta\sigma_{i+1}^{+}\sigma_{i}^{+}+\text{h.c.}\right)\\ &=-\frac{\mu N}{2}-\sum_{i=1}^{N}\frac{t+\Delta}{2}\sigma_{i+1}^{x}\sigma_{i}^{x}+\frac{t-\Delta}{2}\sigma_{i+1}^{y}\sigma_{i}^{y}+\frac{\mu}{2}\sigma_{i}^{z},\end{split}$$

(7)

which is the $s=\frac{1}{2}$ 1D transverse-field XY model. Note that Eq. (7) may be easily converted to the simpler transverse-field Ising model by making special choices for $t$, $\Delta$, etc., but the generic mapping of the Kitaev chain is always to the transverse-field XY model, a distinction (i.e., Ising or XY) often overlooked in discussions of the Kitaev chain.

The direct mapping from the Kitaev chain in Eq. (4) to the transverse-field XY model in Eq. (7) indicates that they have the same band structures and wave functions. Furthermore, the two phases in the Kitaev chain also correspond to the two different phases in the transverse-field XY model, where the phase transition happens at the same point. In both cases, there are two gapped phases, separated by a critical point. For the Kitaev chain, the band gap closes at $2\absolutevalue{t}=\absolutevalue{\mu}$ where two different phases, trivial and topological, exist for $2\absolutevalue{t}<\absolutevalue{\mu}$ and $2\absolutevalue{t}>\absolutevalue{\mu}$, respectively. For the transverse-field XY model, if we can reduce it to a transverse-field Ising model [22] by setting $t=\pm\Delta$ (as we can tell that the change of $\Delta$ cannot close the band gap from the fermionic Hamiltonian), the two phases, the paramagnetic and ferromagnetic state, exist for $2\absolutevalue{t}<\absolutevalue{\mu}$ and $2\absolutevalue{t}>\absolutevalue{\mu}$, respectively. This paramagnetic to ferromagnetic transition is, however, not a topological quantum phase transition (TQPT)– it is a standard breaking of the SU(2) spin symmetry at a phase transition. Ising model does not possess Ising anyons! Despite their dual nature revealed by the Jordan- Wigner transformation converting one to the other, the physics of the two models is qualitatively different. The Kitaev chain manifests a topological phase transition protected by the particle-hole symmetry throughout the entire change of parameters, while the transverse-field XY model is a conventional spontaneous symmetry breaking phase transition, where the spin $\mathbb{Z}_{2}$ symmetry breaks as $\absolutevalue{t}$ increases. The spin symmetry is not protected as any random local magnetic field fluctuation can break it as was already clearly stated in a footnote in Ref. 3.

All of the above are standard procedures to show the duality between the Kitaev chain and transverse-field XY (or Ising) model via the Jordan-Wigner transformation. However, the Kitaev chain is a mere toy model to establish the proof of principle that $p$-wave spinless superconductor can support Majorana zero modes, which was in fact already pointed out in Ref. 4 in the context of 2D systems and fractional quantum Hall effects. In experiments, the SM-SC nanowire is a real spinful electronic 1D Majorana nanowire utilizing a semiconductor with a large Rashba-type SOC in proximity to a conventional $s$-wave superconductor in the presence of a magnetic Zeeman spin splitting field to implement an effective spinless $p$-wave superconductor. The minimal model of a one-dimensional single channel SM-SC nanowire can be described in continuum real space as

$$\hat{H}_{\text{NW}}=\int_{0}^{L}\left[\sum_{s,s^{\prime}}f_{s}^{\dagger}(x)\left(-\frac{\hbar^{2}\partial_{x}^{2}}{2m^{*}}-\mu-i\alpha\sigma_{y}\partial_{x}+V_{\text{Z}}\sigma_{x}\right)_{s,s^{\prime}}f_{s^{\prime}}(x)+\left(\Delta f_{\uparrow}^{\dagger}(x)f_{\downarrow}^{\dagger}(x)+\text{h.c.}\right)\right]\differential{x},$$

(8)

where $L$ is the total length of the nanowire, $f_{s}(x)^{\dagger}$ [$f_{s}(x)$] creates (annihilates) an electron at the position $x$ with spin s ($s=\uparrow$/$\downarrow$), $m^{*}$ is the effective mass of electrons in the conduction band, $\mu$ is the chemical potential, $\alpha$ is the magnitude of the Rashba-type SOC which is aligned perpendicular to the direction of the 1D nanowire, $V_{\text{Z}}$ is the Zeeman splitting energy arising from the magnetic field applied along the direction of the 1D nanowire, and $\Delta$ is the proximitized $s$-wave SC pairing term. In Hamiltonian (8), the term in the first parentheses accounts for the semiconductor denoted as $\hat{H}_{\text{SM}}$, and the term in the second parenthesis is for the superconductor denoted as $\hat{H}_{\text{SC}}$. Note that although Eq. (8) is a realistic Hamiltonian for experimental SM-SC hybrid nanowires, it is still a free fermion Hamiltonian within the mean-field BdG description of the induced SC.

Although the SM-SC nanowire serves as a realistic model of the Kitaev chain to host a pair of Majorana zero modes at the wire ends, the connection between the Kitaev chain and the SM-SC nanowire remains nonobvious. For example, the SM-SC nanowire is established in a spinful system with more terms in the normal (SM) part than the Kitaev chain. Therefore, we wonder how these two models are connected, e.g., whether there exists any duality between the two models that allows us to map one model into the other. To answer this question, we consider two approaches: First, we follow the same procedure as in the Kitaev chain by converting the fermionic Hamiltonian of the SM-SC nanowire into the bosonic spin Hamiltonian through the Jordan-Wigner transformation, and comparing the converted spin model with the transverse-field XY model (which is what the Kitaev chain (4) is formally equivalent to) in the hope of finding some hints for how they might be connected. Alternatively, we can first project the complete spinful Hamiltonian for the SM-SC nanowire onto an effective low-energy Hamiltonian to obtain an effective spinless system, and then establish the connection since both systems are spinless now. We will explain the details of the first approach in Sec. IV.1, and the second approach in Sec. IV.2.

Since we establish the Kitaev chain in a lattice model Eq. (4), we need to study the lattice version of the continuous Hamiltonian for the SM-SC nanowire in Eq. (8) for a direct comparison. The discretization procedure is standard and used extensively. By replacing the differential operator with the finite difference (i.e., $\partial_{x}f_{s}^{\dagger}(x)\mapsto\left[f_{s}^{\dagger}\left(x+a\right)-f_{s}^{\dagger}(x-a)\right]/{2a}$, where $a$ is the fictitious lattice constant in the discretization), we obtain a lattice model for the SM-SC nanowire as

$$\hat{H}_{\text{NW}}^{(f)}=\sum_{i=1}^{N}\sum_{s,s^{\prime}}\left[f_{i,s}^{\dagger}\left(2t-\mu+V_{\text{Z}}\sigma_{x}\right)_{s,s^{\prime}}f_{i,s^{\prime}}+f_{i,s}^{\dagger}\left(-t-i\alpha_{R}\sigma_{y}\right)_{s,s^{\prime}}f_{i+1,s^{\prime}}+\text{h.c.}\right]+\left(\Delta f_{i,\uparrow}^{\dagger}f_{i,\downarrow}^{\dagger}+\text{h.c.}\right)$$

(9)

where $f_{i,s}^{\dagger}=f_{s}^{\dagger}(ia)$ and $f_{i,s}=f_{s}(ia)$ are the creation and annihilation operators defined at the lattice site $i$, effective nearest neighbor hopping $t=\frac{\hbar^{2}}{2m^{*}a^{2}}$, and reduced SOC $\alpha_{R}=\frac{\alpha}{2a}$. Here, for a direct comparison between the lattice model for the SM-SC nanowire and the Kitaev chain in Eq. (4), we impose the periodic boundary condition and drop the factor of $\frac{L}{N}$ before Eq. (9) which arises from the discretization.

Like the Kitaev chain, the SM-SC nanowire also carries two topologically distinct phases that cannot be adiabatically connected, which requires us to first solve the band structure of the SM-SC nanowire to find the vanishing of the band gap. Therefore, we rewrite the Hamiltonian (9) in momentum space by imposing the periodic boundary condition as

$$\begin{split}&\hat{H}_{\text{NW}}^{(k)}=\sum_{k}\left[2t\left(1-\cos k\right)-\mu\right]\left(\tilde{f}_{k,\uparrow}^{\dagger}\tilde{f}_{k,\uparrow}+\tilde{f}_{k,\downarrow}^{\dagger}\tilde{f}_{k,\downarrow}\right)\\ &+\left[\left(V_{\text{Z}}-2i\alpha_{R}\sin k\right)\tilde{f}_{k,\uparrow}^{\dagger}\tilde{f}_{k,\downarrow}+\Delta\tilde{f}_{k,\uparrow}^{\dagger}\tilde{f}_{-k,\downarrow}^{\dagger}+\text{h.c.}\right]\end{split}$$

(10)

where $\tilde{f}_{k,s}^{\dagger}=\frac{1}{\sqrt{N}}\sum_{j}f_{j,s}^{\dagger}e^{ikj}$, and we set the fictitious lattice constant $a=1$ for notational simplicity. To diagonalize Hamiltonian (10), we construct an effective Bogoliubov-de Gennes (BdG) Hamiltonian, $\hat{H}_{\text{NW}}^{(k)}=\sum_{k}\frac{1}{2}\Psi(k)^{\dagger}H_{\text{NW}}^{\text{BdG}}(k)\Psi(k)+2t\left(1-\cos k\right)-\mu$ , where

$$\begin{split}H_{\text{NW}}^{\text{BdG}}(k)&=\left[2t\left(1-\cos k\right)-\mu+V_{\text{Z}}\sigma_{x}+2\alpha_{R}\sin k\sigma_{y}\right]\tau_{z}\\ &-\Delta\sigma_{y}\tau_{y},\end{split}$$

(11)

and $\Psi(k)=\left(\tilde{f}_{k,\uparrow},\tilde{f}_{k,\downarrow},\tilde{f}_{-k,\uparrow}^{\dagger},\tilde{f}_{-k,\downarrow}^{\dagger}\right)^{\intercal}$. Here, $\bm{\sigma}$ and $\bm{\tau}$ are vectors of Pauli matrices acting on the spin and particle-hole space respectively. Therefore, the band structure of the SM-SC is

$$\begin{split}E_{\text{NW},\pm}^{2}(k)&=\xi^{2}(k)+V_{\text{Z}}^{2}+\Delta^{2}+\alpha^{2}(k)\\ &\pm 2\sqrt{V_{\text{Z}}^{2}\Delta^{2}+V_{\text{Z}}^{2}\xi^{2}(k)+\alpha^{2}(k)\xi^{2}(k)},\end{split}$$

(12)

where $\xi(k)=2t\left(1-\cos k\right)-\mu$, and $\alpha(k)=2\alpha_{R}\sin k$. By tracking the vanishing of the band gap at $k=0$, we find that the TQPT happens at $V_{\text{Z}}^{2}=\mu^{2}+\Delta^{2}$ with the trivial (topological) phase being $V_{\text{Z}}^{2}<\mu^{2}+\Delta^{2}$ ($V_{\text{Z}}^{2}>\mu^{2}+\Delta^{2}$) to be contrasted with $2\absolutevalue{t}<\absolutevalue{\mu}$ (trivial) and $2\absolutevalue{t}>\absolutevalue{\mu}$ (topological) in the Kitaev chain.