Apr 11, 2017 - exhibiting effective couplings between four Majorana zero modes â the nonuniform Ising-Kitaev chain, containing two âtopologicalâ regions ...
Majorana fermions in the nonuniform Ising-Kitaev chain: exact solution B.N. Narozhny
arXiv:1609.00925v1 [cond-mat.mes-hall] 4 Sep 2016
Institut f¨ ur Theorie der Kondensierten Materie, Karlsruher Institut f¨ ur Technologie, 76128 Karlsruhe, Germany and National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), Kashirskoe shosse 31, 115409 Moscow, Russia (Dated: September 6, 2016) A quantum computer based on Majorana qubits would contain a large number of zero-energy Majorana bound states. This system can be modelled as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces. The low-energy sector of the theory describing such a network can be formulated in terms of leading-order couplings between the Majorana zero modes. I consider a minimal model exhibiting effective couplings between four Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Solving the model exactly, I show that inversion symmetry leads to delocalization of the Majorana zero modes between two interface points. The low-energy sector of the theory can still be formulated in terms of localized Majorana fermions, but the couplings between them are counterintuitively independent of their separations. I also show that the delocalization may occur in the case of a “weak” barrier and the variant of the model, where one of the chain ends is coupled to one of the intermediate sites forming a T-junction.
Physicists have been fascinated with Majorana fermions ever since their discovery1,2 in 1937, when Ettore Majorana found a completely real (i.e. not containing complex coefficients) representation of the Dirac equation. The solutions of the Majorana equation describe neutral fermions – particles that obey the Fermi statistics, but at the same time are their own antiparticles. Whether they exist in nature as elementary particles is still an open question. It has been hypothesized that neutrinos might be Majorana fermions. This hypothesis could be experimentally confirmed by observation of an elusive process known as the neutrinoless double beta decay3 , which is the focus of considerable experimental efforts. At the same time, the Majorana concept can be applied to elementary excitations in solids. In condensed matter theory, Majorana fermions proved to be an efficient tool to describe quantum criticality4,5 . More recently, the focus of the community shifted towards the Majorana zero modes in novel, topologically nontrivial systems. In fact, signatures of Majorana bound states were observed in several recent experiments6–9 . Even though the interpretation of the experimental data is not unambiguous10,11 , there is little doubt in the community that solid state devices supporting Majorana bound states can be engineered in a modern laboratory12 . Arguably the main driving force behind the pursuit of Majorana zero modes in solids is the possibility of applications to quantum computing13–15 . The basic building block of a quantum computer, the qubit, can be realized as a coupled system of four spatially separated zero-energy Majorana bound states12,16,17 . It is expected that a Majorana qubit would have a rather long coherence time due to its topological nature16,18 . Quantum computer can then be envisioned as a connected network of such qubits. Certain logical operations in such a computer can also be performed topologically by means of braiding (or adiabatic interchange) of Majorana
FIG. 1: A three-dimensional illustration of the Majorana bound states in the Kitaev model with a T-junction. The chain contains two “topological” regions (the blue line, with N1 sites, and the purple line, with N2 sites) and one “trivial” region (the red line, with M sites). The T-junction is located at the site N0 . The peaks represent the absolute values of the real-space amplitudes of the Majorana zero modes calculated for N1 = M = 20, N2 = 10, and N0 = 11. The model exhibits four Majorana zero modes (corresponding to the nearly fourfold degeneracy of the ground state). Two of them are localized at a single interface point: the dark blue at the site N1 and the green at the T-junction. Note, that this amplitude is spread over only two (out of three) branches at the junction. The remaining two zero modes are delocalized between two interface points, the sites 1 and N1 +M . One of them is illustrated by the red peaks.
fermions16,19 . A paradigmatic model exhibiting zero-energy Majorana bound states is the one-dimensional (1D) lattice model of a p-wave superconductor proposed by Kitaev13 . In the case of open boundary conditions, the model exhibits a quantum phase transition between two gapful (massive) phases, known as the “trivial” and “topological” phases. The “trivial” phase is characterized by a single non-degenerate ground state, while the “topologi-
2 cal” phase possesses a ground state that is nearly doubly degenerate: for any finite-size chain the difference between the energies of the lowest-lying excited state and the ground state is exponentially small, ∝ exp(−L/`0 ), in the length of the chain L (here `0 is a certain correlation length defined below). The energy difference vanishes in the thermodynamic limit and the ground state becomes truly degenerate. This is a manifestation of a well known theorem in statistical physics20 : spontaneous symmetry breaking and the corresponding vacuum degeneracy may only occur in the thermodynamic limit, L → ∞. The experiments6–11 targeting the zero-energy Majorana bound states in solids were motivated by theoretical suggestions of a semiconductor-based nanowire device hosting a pair of localized Majorana fermions21–23 . While the physics of such systems is rather complex, the effective low-energy Hamiltonian describing the nanowire is essentially that of the 1D p-wave superconductor, i.e. the continuous limit of the Kitaev model19 . Alternatively, one can search for Majorana fermions in manifestly discreet systems24–27 . For instance, one may engineer the Majorana bound states using Josephson qubits28,29 to build an artificial spin chain24,25 that is designed to be an experimental realization of the 1D quantum Ising model4,5,30–32 . The quantum Ising chain with open boundary conditions is formally equivalent to the Kitaev chain13,15,25,31,32 . This equivalency is based on the Jordan-Wigner transformation33 that is commonly used in 1D theories to express the spin-1/2 operators in terms of creation and annihilation operators of spinless fermions4 . In fact, the original solution32 of the 1D quantum Ising model was based on the consequent application of the Jordan-Wigner transformation and the Bogolyubov transformation34 , mapping the model onto a system of free fermions31 . The simplicity of the resulting physical picture may be deceptive, since both the Jordan-Wigner and Bogolyubov transformations are nonlocal. Although the original Hamiltonian contains only nearest-neighbor couplings, the model may develop longrange correlations. In fact, the ground state of the openended chain is characterized by the “end-to-end” correlation function32 that vanishes in the “trivial” phase, but remains finite in the “topological” phase in the thermodynamic limit, L → ∞. This result can be interpreted in terms of a nonlocal fermion operator that is a linear combination of the Jordan-Wigner fermions at both ends of the chain. The lowest excited state of the openended chain in the “topological” phase (i.e. the state that is nearly degenerate with the ground state) possesses a similar structure. The wave function of this state decays exponentially away from the chain ends and hence can be represented as a linear combination of the two bound states localized at either end of the chain. The existence of such edge states has been known for a very long time31 , but they were not interpreted in terms of Majorana fermions and related to the quantum information theory before the work of Kitaev13 . A quantum computer based on Majorana qubits would
contain a large number of zero-energy Majorana bound states. Whether the actual device will be built using the nanowires21–23 or the artificial spin chains24,25 , one can envision the effective model of the system as a connected network of the Ising-Kitaev chains alternating the “trivial” and “topological” regions, with the zero-energy Majorana fermions localized at their interfaces14 . The low-energy sector of such a theory can be formulated in terms of leading-order couplings between the Majorana zero modes14,17,35 . These couplings are often chosen based on physical intuition. Given the nonlocal relation between the Majorana zero modes and the Kitaev (or Jordan-Wigner) fermions, it is desirable to test that intuition against a rigorous solution of a representative microscopic model. Such a test is the principle goal of the present work. In this paper I consider a minimal model exhibiting effective couplings between Majorana zero modes – the nonuniform Ising-Kitaev chain, containing two “topological” regions separated by a “trivial” region. Based on the common intuition, one would expect that this model possesses four Majorana zero modes, each localized at one of the four interface points of the chain14,35 (i.e. the two chain ends and two edges of the “trivial” region). I present the exact solution of the model and identify the region of model parameters where the above expectation is indeed fulfilled. However, the exact solution also exhibits situations where the intuitive expectation is not fulfilled. In particular, inversion symmetry (in the case where the two “topological” regions are identical) leads to delocalization of the Majorana zero modes between two interface points. While one can use a basis rotation to express the low-energy sector of the theory in terms of four localized Majorana operators, the corresponding states will no longer be the eigenstates of the model. More importantly, the effective couplings between these modes are completely counterintuitive for they are independent of the distance between them. I also demonstrate that the symmetric case in not the only situation exhibiting the delocalization of the Majorana bound states. As an example, I show that the delocalization may also occur in the variant of the model, where one of the chain ends is coupled to one of the intermediate sites forming a Tjunction (or a Y-junction), see Fig. 1.
I.
THE NONUNIFORM ISING-KITAEV CHAIN
The open-ended, nonuniform quantum Ising chain is described by the Hamiltonian b = −J H
L−1 X n=1
x σ ˆnx σ ˆn+1 −
L X
hn σ ˆnz ,
(1a)
n=1
where σ ˆni are the Pauli matrices corresponding to a spin 1/2 residing on the site i. Using the Jordan-Wigner transformation4,33 , this model can be mapped onto a
3
1
𝜆𝜆1 < 1
𝑁𝑁1
𝜆𝜆2 > 1
𝑁𝑁1 + 𝑀𝑀
𝜆𝜆3 < 1
A.
𝐿𝐿
FIG. 2: The nonuniform Ising-Kitaev chain split into two “topological” (dark green) and one “trivial” (red) regions. The first “topological” region is characterized by the parameter λ1 < 1 and occupies the left part of the chain, 1 6 n 6 N1 . The next M sites are occupied by the “trivial” phase with λ2 > 1. The remaining portion of the chain of the length N2 = L−N1 −M is occupied by the second “topological” region with λ3< 1.
variant of the Kitaev chain4,13,15 b = −J H
L−1 X
cˆ†n −ˆ cn
cˆ†n+1 +ˆ cn+1
(1b)
n=1
−2
L X
hn cˆ†n −ˆ cn cˆ†n +ˆ cn .
Diagonalization of the Ising-Kitaev Hamiltonian
The fermionic Hamiltonian (1b) can be diagonalized exactly using the method30 suggested by Lieb, Schultz, and Mattis for the 1D XY model31 and used by Pfeuty to solve the uniform quantum Ising model32 (hn = h). The method is well known in the theory of superconductivity34 and is based on the transformation X X gkn cˆ†n +hkn cˆn , (4) ηˆk = gkn cˆn +hkn cˆ†n , ηˆk† = n
n
where gkn and hkn are real coefficients and the resulting operators ηˆk satisfy fermionic commutation relations. The latter requirement leads to the fact that the coefficients gkn and hkn form a complete, orthonormal basis in the L-dimensional Euclidean vector space. The coefficients gkn and hkn can be found by assuming the diagonal form of the Hamiltonian in terms of the b As operators ηˆk and considering the commutator [ˆ ηk , H]. a result, the diagonal form of the Hamiltonian (1b) is
n=1
The complete model originally considered by Kitaev13 maps onto the variant of the quantum Ising model cony taining also the σ ˆny σ ˆn+1 couplings (the XY model in a transverse field30,31 ). However, it is well known5,30 that as long as the exchange constants in the xx and yy terms are not identical, the two models are in the same universality class. The models (1) exhibit all of the essential features of the original Kitaev chain and constitute representative models for studies of the zero-energy Majorana bound states15 . In this paper I focus on the minimal model supporting effective couplings between Majorana zero modes choosing the applied field hn to be piece-wise uniform (see Fig. 2 for illustration) h1 < J, 1 6 n 6 N1 , hn = h2 > J, N1 +1 6 n 6 N1 +M, h < J, N +M +1 6 n 6 L. 3 1
b = 2J H
L X
Ek ηˆk† ηˆk
−J
k=1
L X k=1
Ek − J
L X
λn .
(5)
n=1
The first term in Eq. (5) describes the excitation spectrum of the model in terms of free fermion operators ηˆk . The two remaining terms yield the ground state energy. The outlined diagonalization procedure, as well as Eq. (5), is applicable to an arbitrary quadratic Hamiltonian. However, analytic solution is manageable only in a few relatively simple cases. Fortunately, the model (1b) with the specific choice (2) of the applied fields is one of them. The exact single-particle energies Ek of this model can be expressed as E 2 = 1 + λ21 + 2λ1 cos ϑ1
(6)
= 1 + λ22 + 2λ2 cos ϑ2 = 1 + λ23 + 2λ3 cos ϑ3 , (2)
in terms of nontrivial solutions, ϑi , to the equation λ1 D1 (N1 )D2 (M, N2 ) = λ2 D1 (N1 +1)D2 (M +1, N2 ), (7a)
In this case, the chain is split into three regions such that the two “topological” regions (of the length N1 and N2 = L−N1 −M ) are separated by the “trivial” region of the length M . Since physical properties of the model are determined by the ratios of the applied fields to the exchange coupling J, it is convenient to factor out the exchange constant J introducing the parameters λi = hi /J,
λ1 , λ3 < 1,
λ2 > 1.
(3)
Below I present the exact solution of the finite-size, openended lattice model (1) with the specific choice (2) of the applied field in terms of the parameters (3) and the numbers N1 , N2 , M , and L.
where D1 (N1 ) = λ1 sin N1 ϑ1 +sin(N1 −1)ϑ1 ,
(7b)
D2 (M, N2 ) = λ3 sin(N2 +1)ϑ3 sin M ϑ2
(7c)
−λ2 sin N2 ϑ3 sin(M −1)ϑ2 . The two latter equalities in Eq. (6) provide additional constraints on ϑi , which guarantee the uniqueness of the solution of the problem. Consider now the energy (6) as a function of ϑi , regardless of which values of ϑi are allowed by Eq. (7). For
4 ing cases. The symbolic expression O(λN ) hereafter denotes the omitted subleading, exponentially small terms.
1.
FIG. 3: Energy eigenvalues E1,2 of the two lowest-lying excited states of the Ising-Kitaev chain (1) in the piece-wise uniform applied field (2) as a function of λ1 = λ3 with λ2 = 4 for N1 = 10, M = 20, and N2 = 14. The solid curves represent the result of the exact numerical diagonalization. The dashed lines represent the analytic solutions to Eqs. (6) and 2M 1 = λ− , in this (7). The vertical grid line corresponds to λ2N 1 2 particular case, λ1 = 1/16. On the right side of this line the green dashed line corresponds to Eq. (9a) and the purple – to Eq. (9b). On the left side the green dashed line represents Eq. (9d), the purple – Eq. (9c). The horizontal grid line M −20 corresponds to E = λ− . 2 =4
real ϑi , this function exhibits a minimum at ϑi = π. The minimum value of the energy gives a reasonable lower bound for the bulk gap of the model ∆ ≈ 2J min |1 − λi |. i=1,2,3
(8)
Hence, any subgap states including nearly zero-energy states are described by complex solutions to Eq. (7).
B.
Nearly degenerate ground states
The Ising-Kitaev chain split into two “topological” and one “trivial” region possesses two single-particle excitations (herafter denoted by k = 1, 2) that are nearly degenerate with the ground state. Each of these states corresponds to a complex solution to Eq. (7). As long as the parameters λi are not too close to unity and the sizes of the three regions are not too small, such that the three 1 2 quantities λ2N , λ2N , and λ−2M are exponentially small, 1 3 2 the two complex solutions can be found analytically with arbitrary accuracy. Already the leading-order expression shows excellent agreement with the exact numerical diagonalization of the model as illustrated in Figs. 3 and 4. The visible discrepancy between the analytic and numerical results for λ1 ∼ 1 is to be expected: there the above parameters cease being exponentially small and the approximate analytic expressions become invalid. Without specifying the relation between the three exponentially small parameters, even the leading-order expression for the two eigenvalues E1,2 is rather cumbersome. Therefore here I focus on two representative limit-
Asymmetric chain
If the two “topological” regions of the chain are not symmetric, then compact expressions for the energies E1,2 can be found under following assumptions. 2 1 λ−2M , the two > λ2N (i) “Strong barrier”. If λ2N 3 2 1 nearly zero-energy states are determined by the two “topological” regions of the chain, independently of the size of the “trivial” region. The first excited state has the energy s E1 =
(1−λ23 )
λ22 −1 N2 λ + O(λN ), λ22 −λ23 3
(9a)
while the energy of the second excited state is s E2 =
(1−λ21 )
λ22 −1 N1 λ + O(λN ), λ22 −λ21 1
(9b)
These results are illustrated in Fig. 3 by the dashed lines to the right of the vertical grid line (marking the end of 1 the above parameter region at λ2N = λ−2M ). Vanishing 1 2 of the energies (9a) and (9a) at the point λ1 = λ3 = 1 is the artifact of the approximation. As the parameters λi approach unity, the approximate expressions reported here become invalid (while it is possible to write down exact expressions for E1,2 that are valid also near the critical point, their algebraic complexity renders them practically useless). 1 2 (ii)“Weak barrier”. In the case λ2N , λ2N λ−2M , the 1 3 2 larger eigenvalue E2 is determined by the “trivial” region of the chain s (1−λ21 )(1−λ23 ) E2 = (λ22 −1) λ−M + O(λN ), (9c) (λ22 −λ21 )(λ22 −λ23 ) 2 while the energy of the lowest excited state is determined by the two “topological” regions combined E1 =
q
N 1 N2 M (1−λ21 )(1−λ23 ) λN 1 λ3 λ2 + O(λ ).
(9d)
These results are illustrated in Fig. 3 by the dashed lines on the left side of the vertical grid line.
2.
Symmetric chain
N2 1 In the symmetric case, λN 1 = λ3 , the two energies (9b) and (9a) coincide. In this case, one has to consider the subleading terms neglected so far, as the eigenvalues of a finite-size chain (1b) are never truly degenerate.
5 Together with the fermionic form (1b) of the Hamiltonian, this expression invites introduction of the lattice Majorana fermions4 ζˆn = cˆ†n +ˆ cn , ξˆn = −i cˆ†n −ˆ cn , (12b) In terms of the operators (12b), the creation operator ηˆk† has the form ηˆk† = FIG. 4: Energy eigenvalues E1,2 of the two lowest-lying excited states of the symmetric Ising-Kitaev chain (1) in the piece-wise uniform applied field (2) as a function of λ1 = λ3 with λ2 = 4 for N1 = N2 = 10 and M = 20. The solid curves represent the result of the exact numerical diagonalization. The dashed lines represent the analytic solutions to Eqs. (6) 2M 1 = λ− , and (7). The vertical grid line corresponds to λ2N 1 2 in this particular case, λ1 = 1/16. On the right side of this line the green and purple dashed lines corresponds to the two eigenvalues in Eq. (10). On the left side the green dashed line represents Eq. (9d), the purple – Eq. (9c). The horizontal M −20 grid line corresponds to E = λ− . 2 =4 1 (i) “Strong barrier”. Assuming λ2N λ−2M , the re1 2 sulting energies are given by s λ22 −1 N1 sym 2 λ (10) E1(2) = (1−λ1 ) λ22 −λ21 1 s " # 1 −N1 −M λ22 −1 × 1 ∓ λ1 λ2 + O(λN ). 2 λ22 −λ21
This result is illustrated in Fig. 4 to the right of the vertical grid line (the exponentially small difference between the two energies (10) is indistinguishable even on the loglog scale). 1 (ii) “Weak barrier”. In the limit, λ2N λ−2M , no spu1 2 rious degeneracy occurs and hence the expressions (9c) and (9d) are still valid, see Fig. 4 (to the left of the vertical grid line).
II.
MAJORANA BOUND STATES
The excitation spectrum of the Ising-Kitaev chain can be interpreted in terms of Majorana fermions4,13 . Introducing linear combinations of gkn and hkn [cf. Eq. (4)] αkn = gkn +hkn ,
βkn = gkn −hkn ,
(11)
L 1 X αkn cˆ†n +ˆ cn +βkn cˆ†n −ˆ cn . 2 n=1
(12a)
(12c)
The two linear combinations (α)
γˆk =
L X
(β)
αkn ζˆn ,
γˆk =
n=1
L X
βkn ξˆn ,
(12d)
n=1
are themselves Majorana operators in the sense that they satisfy the Majorana commutation relations n o (α) (β) γˆk , γˆk = 0,
(α)
γˆk
2
2 (β) = γˆk = 1.
(12e)
The latter property follows from the fact that the vectors αkn and βkn are normalized. By definition, the Majorana operators (12d) are nonlocal linear combinations13 of the more conventional4 Majorana fermions (12b). Typically, these combinations involve all sites of the chain13 . However, for the two lowest excited states (9) the Majorana amplitudes α1(2)n and β1(2)n exhibit exponential decay away from the interface points of the chain, allowing one to treat the nearly (α) (β) zero-energy Majorana states γˆ1(2) and γˆ1(2) as essentially localized13 , see Figs. 5 and 6.
3.
Asymmetric chain
1 (i) “Strong barrier”. In the limit λ2N > λ32N2 λ−2M , 1 2 the leading behavior of the energy eigenvalues is given by Eqs. (9a) and (9b). The corresponding amplitudes α1(2)n and β1(2)n can also be written in compact from, again retaining only the leading exponential terms. The first excited state (9a) is characterized by the Majorana amplitudes O(λN ), 1 6 n 6 N1 , 1−M α1n = (−1)n c3 λn−1−N , 1 6 n−N1 6 M, 2 n−1−N1−M c3 λ 3 , 1+N1 +M 6 n 6 L,
the creation operators of the Bogolyubov fermions can be written as ηˆk† =
L i 1 Xh αkn ζˆn +iβkn ξˆn . 2 n=1
(13a) β1n = (−1)n+1
N O(λ ), 1 6 n 6 N1 +M, s λL−n , 1+N +M 6 n 6 L, 3 3 1
6 plitudes α2n = (−1)n−1
n−1 N s1 λ1 +O(λ ), 1 6 n 6 N1 , O(λN ),
1 6 n−N1 6 L,
(13c) N1−n N c1 λ1 +O(λ ), 1 6 n 6 N1 , 1−n β2n = (−1)n c1 λN +O(λN ), 1 6 n−N1 6 M, 2 O(λN ), 1+N1 +M 6 n 6 L.
FIG. 5: Majorana amplitudes of the two nearly zero-energy eigenstates of the Ising-Kitaev chain (1) with N1 = 10, M = 20, and N2 = 14 in the piece-wise uniform applied field (2). The red dots represent the result of the exact numerical diagonalization. The curves represent the analytic solutions. Top row: the amplitudes |α1n | and |α1n | in the strong barrier case, λ1 = λ3 = 1/2, λ2 = 4. The curves are given in Eqs. (13a) and (13c). Middle row: the amplitudes |α1n | and |α1n | in the weak barrier case, λ1 = λ3 = 1/20, λ2 = 4, exhibiting weak delocalization. The curves are given in Eqs. (13d) and (13e). Bottom row: the amplitudes |β1n | and |β1n | for λ1 = λ3 = 1/2, λ2 = 4. The curves are given by either Eqs. (13a) and (13c) or Eqs. (13d) and (13e).
The corresponding excitation ηˆ2 is confined to the first “topological”region of the chain. The amplitudes (13c) are illustrated in the two bottom panels in Fig. 5. The results (13a) and (13c) confirm that in the limit 2 1 λ−2M the two lowest-energy excitations of > λ2N λ2N 3 2 1 the model behave similarly to those of the two independent “topological” regions. In particular, they exhibit four nearly zero-energy Majorana fermions localized at the edges of the “topological” regions. 1 2 (ii) “Weak barrier”. In the limit, λ2N , λ2N λ−2M , 1 3 2 the structure of the wave-functions of the two lowest excited states is significantly different. The first excited state (9a) is characterized by the Majorana amplitudes n−1 1 6 n 6 N1 , λ1 , 1 n−1−N1 α1n = (−1)n s1 λN , 1 6 n−N1 6 M, 1 λ2 N1 M n−1−N1−M λ1 λ2 λ3 , 1+N1 +M 6 n 6 L, (13d)
where the symbolic expression O(λN ) denoting the subleading terms is omitted in some lines for brevity and
s cj =
(1−λ2j )(λ22 −1) , λ22 −λ2j
sj =
q
1−λ2j .
(13b)
Hence with exponential accuracy, the lowest-energy excitation of the model can be described by the single fermion, ηˆ1 , confined to the second “topological”region of the chain, cf. Eq. (12c). The amplitudes (13a) are illustrated in the two top panels in Fig. 5. The spreading of the localized Majorana states over several lattice sites exhibited by Eq. (13a) is a generic feature13 that can be seen also in the continuum limit19 . The second eigenvalue (9b) is characterized by the am-
β1n = (−1)n+1
N O(λ ), 1 6 n 6 N1 +M, s λL−n , 1+N +M 6 n 6 L, 3 3 1
Here the amplitude β1n is identical with Eq. (13a), but the amplitude α1n has changed. In the first “topological” region of the chain, it behaves as the corresponding amplitude of the second excited state (13c) of the strong barrier case. Moreover, there is a nonzero probability to find this quasiparticle also at the interface between the “trivial” and the second “topological” regions, see Fig. 5, i.e. the corresponding Majorana fermion is essentially delocalized! The delocalization of the Majorana amplitude α1n in Eq. (13d) is rather weak. For following choice of values of the parameters (3), λ1 = λ3 = 1/20, λ2 = 4, and the sizes of the chain segments N1 = 10, N2 = 14, M = 20, the peak values of the amplitude α1n are |α1,1 | = 0.999 and |α1,31 | = 0.107. Whether this feature survives the thermodynamic limit depends on what happen to the value 1 M of the product of a small and large parameters λN 1 λ2 in the limiting procedure.
7 The second eigenvalue (9b) is characterized by the amplitudes 2 M N1+n−1 , 1 6 n 6 N1 , −(s1 /c3 )λ2 λ1 1−M α2n = (−1)n c3 λn−1−N , 1 6 n−N1 6 M, 2 n−1−N1−M c3 λ 3 , 1+N1 +M 6 n 6 L, (13e) N1−n N c1 λ1 +O(λ ), 1 6 n 6 N1 , 1−n β2n = (−1)n c1 λN +O(λN ), 1 6 n−N1 6 M, 2 O(λN ), 1+N1 +M 6 n 6 L. Again, the amplitude β2n remains unchanged from its value in the strong barrier case, Eq. (13c), while the amplitude α1n exhibits weak delocalization. For the same choice of parameters (λ1 = λ3 = 1/20, λ2 = 4 and N1 = 10, N2 = 14, M = 20), I find |α2,1 | = 0.967 and |α2,31 | = 0.111. In contrast to the strong barrier case, the wavefunction of lowest-energy fermion ηˆ1 is now mostly spread between the two outer edges of the chain, with a small weight at the interface between the “trivial” and the second “topological” region. The second excitation ηˆ2 is mostly confined to the edges of the “trivial” region, with the small weight at the beginning of the chain, see Fig. 5.
FIG. 6: Majorana amplitudes (13f) of the two nearly zeroenergy eigenstates of the symmetric Ising-Kitaev chain (1) in the piece-wise uniform applied field (2) with λ1 = λ3 = 1/2, λ2 = 4, N1 = N2 = 10, and M = 20. The red dots represent the result of the exact numerical diagonalization. The curves represent the analytic solutions given in Eqs. (13f).
In the weak barrier case, the Majorana amplitudes are still described by Eqs. (13d) and (13e). It is worth emphasizing, that although all energies (9) vanish in the thermodynamic limit, Ni , M → ∞, the structure of the Majorana amplitudes (13) remains independent of the chain length (witht he possible exception of the weak delocalization in the chain with the weak barrier). In particular, if the inversion symmetry survives the thermodynamic limit, then the Majorana fermions described by the amplitudes (13f) remain delocalized between two infinitely separated points. The exponential decay of the Majorana amplitudes (13) can be described in terms of a correlation length, which is specific to each of the three regions of the chain `0i ∼ 1/| ln λi |.
4.
Symmetric chain
The two lowest-energy excitations of the symmetric 1 2 chain with the strong barrier, λ2N = λ2N λ−2M , are 1 3 2 characterized by the amplitudes, see Fig. 6 (the symbolic expression O(λN ) denoting the subleading terms is omitted for brevity) n−1 1 6 n 6 N1 , ∓s1 λ1 , (−1)n n−1−N1−M sym α1(2),n = √ c1 λ2 , 1 6 n−N1 6 M, 2 n−1−N1−M c1 λ1 , 1+N1 +M 6 n 6 L, (13f) 1−n ∓c1 λN , 1 6 n 6 N1 , 1 sym β1(2),n = (−1)n ∓c1 λ2N1−n , 1 6 n−N1 6 M, s1 λ1L−n , 1+N1 +M 6 n 6 L. † In this case, the excitations ηˆ1,2 are no longer confined to one of the two “topological” regions of the chain, but are spread symmetrically over both of them.
(14)
In experiments on discreet systems24–27 , the realistic values of λi might not be extreme and hence the correlation lengthes (14) might not be very small. In such case, even the localized Majorana fermions are spread over several lattice sites as illustrated in Figs. 5 and 6. III.
EFFECTIVE LOW ENERGY THEORY
Applications to quantum computation14 involve adiabatic manipulations of the Majorana zero modes. In any case, this means that any external perturbation applied to the system should be slow enough to avoid exciting higher-energy gapped states. The remaining low-energy sector of the theory consists of the ground state |GSi and the nearly degenerate excitations that can be interpreted in terms of Majorana zero modes. For the specific model considered in this paper, the low-energy sector contains four states |GSi,
η1† |GSi,
η2† |GSi,
η1† η2† |GSi,
(15)
where the last state is the two-particle excitation. These four states can be further split into two groups of mutually orthogonal states, belonging to the two parity sectors of the model (where the total fermion number is either even or odd).
8 Projecting the Hamiltonian (1b) onto either of the above sectors, one finds the effective low-energy theory. In the one-fermion (odd) sector, the effective Hamiltonian has the simplest form in the basis of the Majorana (α) (β) fermions γˆ1(2) and γˆ1(2) . In the asymmetric chain with the strong barrier, i.e. 2 1 λ−2M , the localized Majorana > λ2N in the limit λ2N 3 2 1 fermions describe the exact eigenstates of the model. Hence, the projected Hamiltonian in the basis the four Majorana bound states (counted from left to right) has the block-diagonal structure 0 −iE2 0 0 iE2 0 0 0 b eff ∝ H . (16a) 0 0 0 −iE1 0 0 iE1 0 Note the absence of any coupling between the two pairs of the Majorana zero modes (which is the consequence of their direct relation to the exact eigenstates of the model). In contrast, in the case of the symmetric chain with the 1 2 strong barrier, λ2N = λ2N λ−2M the Majorana ampli1 3 2 tudes (13f) are not localized to any single interface point in the chain. While it is still possible to represent the effective Hamiltonian in the basis of the localized Majorana fermions, these quasiparticles are no longer associated with the exact eigenstates and hence are no longer decoupled. Using an obvious basis rotation and introducing a short-hand notation for the eigenvalues (10) sym E1(2) = ± δ,
δ ,
I find the following Hamiltonian 0 −i 0 −iδ i 0 −iδ 0 b eff ∝ H . 0 iδ 0 −i iδ 0 i 0
(16b)
(16c)
The Hamiltonian (16c) appears to be counterintuitive: the effective couplings for the two pairs the Majorana zero modes are exactly the same despite the large difference in separation between the Majorana fermions in each pair. This result is the consequence of the nonlocal nature of the eigenstates of the model. Finally, in the weak barrier case, the structure of the low-energy Hamiltonian change since the low-energy excitations are no longer confined to the “topological” regions of the chain. In order to take into account the weak delocalization of the Majorana amplitudes α1(2)n , I de1 M note the ratio of the peak values α1(2)n as κ1(2) ∼ λN 1 λ2 . Then, the low-energy Hamiltonian takes the form 0 −iκ2 E2 0 −iE1 iκ2 E2 0 −iE2 0 b eff ∝ H . (16d) 0 iE2 0 −iκ1 E1 iE1 0 iκ1 E1 0 Without weak delocalization (i.e. in the limit κ1(2) → 0), the Hamiltonian (16d) can be made block-diagonal by
FIG. 7: The nonuniform Ising-Kitaev chain with a T-junction. The chain contains two “topological” (blue and purple) regions and one “trivial” (red) region. The first “topological” region is characterized by the parameter λ1 < 1 and contains the first N1 sites. The next M sites are occupied by the “trivial” phase with λ2 > 1. The remaining portion of the chain of the length N2 = L−N1 −M is occupied by the second “topological” region with λ3 < 1. The T-junction is located at the site N0 . In this paper, I report results for the case where the T-junction is approximately in the middle of the first “topological” region of the chain.
renumbering the localized Majorana operators. The structure of Eq. (16d) reflects the fact that the first excited state is mostly spread between the two outer edges of the chain [unlike the case of the asymmetric chain with the strong-barrier, described by Eq. (16a), where the first excited state is confined to the second “topological” region], while the second excited state is mostly confined to the edges of the “trivial” region. Taking into account weak delocalization of the Majorana amplitudes α1(2)n , one finds effective couplings between the two pairs of Majorana bound states. IV.
KITAEV CHAIN WITH A T-JUNCTION
Consider now a modified model where one of the chain ends is coupled to an intermediate site forming a Tjunction (sometimes also referred to as a Y-junction), see Fig. 7. Such a model can be easily formulated in the fermion language by adding another coupling term to the Hamiltonian (1b) b = −J H
L−1 X
cˆ†n −ˆ cn
cˆ†n+1 +ˆ cn+1
(17)
n=1
−J cˆ†L −ˆ cL cˆ†N0 +ˆ cN0 −
L X
hn cˆ†n −ˆ cn cˆ†n +ˆ cn ,
n=1
where 1 < N0 < L. A similar modification is also possible for the Hamiltonian (1a) in the language of the Ising spins. However,
9
FIG. 8: Majorana amplitudes of the two nearly zero-energy eigenstates of the modified Kitaev chain (17) with a Tjunction in the piece-wise uniform applied field (2) with λ1 = λ3 = 1/2, λ2 = 4, N1 = M = 20, N2 = 10, and N0 = 11. The red dots represent the result of the exact numerical diagonalization. The solid curves are presented for comparison. The sym amplitudes α1,2 shown on the left two panels are compared with Eq. (13f). The amplitude β1sym appears to be well described by Eq. (13c). The amplitude β2sym of the Majorana zero mode localized at the junction point is given by Eq. (18).
such model cannot be mapped onto a quadratic fermionic Hamiltonian similar to (17) due to non-cancellation of the Jordan-Wigner strings at the junction point. As a result, the Ising chain with a T-junction turns out to be a much more complicated, albeit a very interesting model, which is beyond the scope of the present paper. The fermionic model (17) can be solved exactly by the same methods discussed in this paper. Given the more complicated topology and the larger number of parameters, the model (17) exhibits more parameter regimes as compared to the chain (1b). A detailed analysis of the model will be presented elsewhere36 . Here I will report the results of exact numerical diagonalization of the model (17) demonstrating delocalization of the Majorana zero modes, similar to Eq. (13f), focusing on the most surprising points. As a representative example, I consider a nearly symmetric configuration with λ1 = λ3 < 1, λ2 > 1, where I use the exact numerical diagonalization to find the Majorana amplitudes for the following particular values of the model parameters: N1 = 20, M = 20, N2 = 10, and the Tjunction point being at N0 = 11, see Fig. 7. The results are presented in Fig. 8 by the red dots and illustrated in Fig. 1. The chosen configuration is almost symmetric – the inversion symmetry (of the loop region of the chain) would hold if the numbers of sites in the two “topological” regions would satisfy the condition N2 = N1 −N0 . Even though the symmetry condition is violated by the above choice of parameters, the Majorana amplitudes sym α1,2 are delocalized between the edge point of the chain and one of the borders of the “trivial” region. Moreover, the amplitude α2sym appears to be in perfect agreement with Eq. (13f), while the amplitude α1sym shows a barely
perceptible deviation (see the solid curves in the two left panels in Fig. 8). I postpone the analysis of such tiny effects for a future publication36 and emphasize the more dramatic differences in the behavior of the nearly zeroenergy Majorana fermions in the T-junction model (17) as compared to the open chain (1b). sym Now the Majorana amplitudes β1,2 are no longer desym localized. The amplitude β1 is localized at the edge of the “trivial” region and is perfectly described by Eq. (13c), the top right panel in Fig. 8. Similarly, the amplitude β2sym is localized at the T-junction point, see the bottom right panel in Fig. 8, and can be described analytically by 0−1−n , 1 6 n 6 N0 −1, s1 λN 1 (−1)n sym = √ β2n (18) O(λN ), N0 6 n 6 N1 +M, 2 −s1 λL−n 1 , N1 +M +1 6 n 6 L. Again, this result is completely counterintuitive – one would expect that the Majorana fermion localized at the T-junction will be spread out over all three branches of the junction. Instead, the amplitude β2sym is spread over only two of them! Similarly to the case of the chain, one might formulate the low-energy effective theory in a rotated basis of localized Majorana fermions, see Eq. (16). In the absence of the exact symmetry, the structure of the effective couplings between the localized modes will be similar to Eq. (16c), but with generic matrix elements. V.
DISCUSSION
In this paper I presented the exact analytic solution of the nonuniform Ising-Kitaev chain with open boundary conditions. The motivation for this work was two-fold. Firstly, I was motivated by the proposal24,25 of experimental realization of zero-energy Majorana bound states in an artificial spin chain engineered using Josephson qubits. Such a system would be discreet and, given current technological limitations, contain not too many qubits. Hence, it is reasonable to describe such a system by an effective finite-size lattice model. Secondly, I wanted to reach a better understanding of the common lore in the field regarding the effective coupling between the zero-energy Majorana bound states in networked systems used as paradigmatic examples of possible applications to quantum computing14 . The model solved in this paper represents the first step in reaching the above goals. The exact solution not only defines the parameter regions where the model exhibits four localized Majorana zero modes at the interface points between the “trivial” and “topological” sections of the system, but also yields several unexpected results. (i) Delocalization. In the presence of an additional symmetry, the zero-energy Majorana bound states may
10 delocalize between two well separated interface points in the chain, see Eq. (13f) and Fig. 6. The modified model with a T-junction exhibits this effect even if the symmetry is slightly broken, see Figs. 1 and 8. In finite-size chains, the delocalization may appear in the “weak barrier” case even without the extra symmetry, see Eqs. (13d) and (13e). (ii) Effective couplings between the Majorana fermions, which are not related to the exact eigenstates of the model, may be independent of the separation between them, see Eq. (16c). (iii) The Majorana fermion localized at the T-junction spreads between only two branches at the junction. (iv) The above results are not finite-size effects and persist in the thermodynamic limit. The results reported in this paper are relevant for experimentalists designing small systems hosting multiple zero-energy Majorana bound states24–27 . In particular, in systems involving relatively few Josephson qubits with conservative parameter values the spreading of the Majorana bound states over a few qubits and their delo-
1 2 3
4
5
6
7
8
9
10
11
12
13 14
15
16
17
E. Majorana, Nuovo Cimento 14, 171 (1937). F. Wilczek, Nature Physics 5, 614 (2008). F. T. Avignone, S. R. Elliott, and J. Engel, Rev. Mod. Phys. 80, 481 (2008). A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, 1998). A. M. Tsvelik, Quantum field theory in condensed matter physics (Cambridge University Press, 2003). V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Science 336, 1003 (2012). L. P. Rokhinson, X. Liu, and J. K. Furdyna, Nature Physics 8, 795 (2012). A. Das, Y. Ronen, Y. Most, Y. Oreg, M. Heiblum, and H. Shtrikman, Nature Physics 8, 887 (2012). M. T. Deng, C. L. Yu, G. Y. Huang, M. Larsson, P. Caroff, and H. Q. Xu, Nano Letters 12, 6414 (2012). H. O. H. Churchill, V. Fatemi, K. Grove-Rasmussen, M. T. Deng, P. Caroff, H. Q. Xu, and C. M. Marcus, Phys. Rev. B 87, 241401 (2013). E. J. Lee, X. Jiang, M. Houzet, R. Aguado, C. M. Lieber, and S. De Franceschi, Nature Nanotechnology 9, 79 (2014). C. Beenakker, Annual Review of Condensed Matter Physics 4, 113 (2013). A. Y. Kitaev, Physics-Uspekhi 44, 131 (2001). J. Alicea, Reports on Progress in Physics 75, 076501 (2012). A. Y. Kitaev, in Exact Methods in Low-dimensional Statistical Physics and Quantum Computing, edited by J. Jacobsen, S. Ouvry, V. Pasquier, D. Serban, and L. Cugliandolo (Oxford Univeristy Press, 2010), vol. 89 of Lecture Notes of the Les Houches Summer School, arXiv:0904.2771. C. Nayak, S. H. Simon, A. Stern, M. Freedman, and S. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008). P. Zhang and F. Nori, Phys. Rev. B 92, 115303 (2015).
calization between two well separated points are generic effects that need to be taken into account while interpreting the experimental data and especially when planning any kind of manipulation of the Majorana bound states by some external bias. Delocalization of the Majorana bound states and the corresponding separationindependent effective coupling between localized Majorana zero modes can also be observed in experimental samples built around long nanowires in the presence of additional symmetries incorporated in the system design.
Acknowledgments
The author would like to thank Alexander Shnirman and Victor Gurarie for numerous helpful discussions. This research was funded by the German Science Foundation (DFG) through Grants No. SCHO 287/7-1 and No. SH 81/2-1 and the EU Network Grant InterNoM.
18 19
20
21 22
23
24
25
26
27
28
29 30
31
32 33
34
35
36
A. Stern, Annals of Physics 323, 204 (2008). B. I. Halperin, Y. Oreg, A. Stern, G. Refael, J. Alicea, and F. von Oppen, Phys. Rev. B 85, 144501 (2012). M. N. Barber (Academic Press, London, 1983), vol. 8 of Phase transitions and critical phenomena. L. Fu and C. L. Kane, Phys. Rev. Lett. 100, 096407 (2008). R. M. Lutchyn, J. D. Sau, and S. Das Sarma, Phys. Rev. Lett. 105, 077001 (2010). Y. Oreg, G. Refael, and F. von Oppen, Phys. Rev. Lett. 105, 177002 (2010). L. S. Levitov, T. P. Orlando, J. B. Majer, and J. E. Mooij (2001), cond-mat/0108266 (unpublished). J. Q. You, Z. D. Wang, W. Zhang, and F. Nori, Scientific Reports 4, 5535 (2014). I. C. Fulga, A. Haim, A. R. Akhmerov, and Y. Oreg, New Journal of Physics 15, 045020 (2013). P. Zhang and F. Nori, New Journal of Physics 18, 043033 (2016). L. B. Ioffe, M. V. Feigel’man, A. Ioselevich, D. Ivanov, M. Troyer, and G. Blatter, Nature 415, 503 (2002). J. Q. You and F. Nori, Nature 474, 589 (2011). S. Suzuki, J. Inoue, and B. K. Chakrabarti, Quantum Ising phases and transitions in transverse Ising models (Springer, 2013). E. Lieb, T. Schultz, and D. Mattis, Annals of Physics 16, 407 (1961). P. Pfeuty, Annals of Physics 57, 79 (1970). P. Jordan and E. Wigner, Zeitschrift f¨ ur Physik 47, 631 (1928). N. N. Bogolyubov, Izv. Acad. Nauk USSR 11, 77 (1947), [J. Phys. (USSR) 11, 23 (1947)]. F. J. Burnell, A. Shnirman, and Y. Oreg, Phys. Rev. B 88, 224507 (2013). B. N. Narozhny, Kitaev chain with a T-junction (2016), (to be published).
