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True spin and pseudo spin entanglement around Dirac Points in graphene with Rashba spin–orbit interaction
JID:PLA AID:25379 /SCO Doctopic: General physics
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True spin and pseudo spin entanglement around Dirac Points in graphene with Rashba spin–orbit interaction
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Zheng Liu , Chao Zhang , J.C. Cao
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Key Laboratory of Terahertz Solid-State Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, China b School of Physics, University of Wollongong, New South Wales 2522, Australia
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Keywords: Rashba spin–orbit interaction Entangled spin states Spin texture
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We analytically obtained the Schmidt decomposition of the entangled state between the pseudo spin and the true spin in graphene with Rashba spin–orbit coupling. The entangled state has the standard form of the Bell state, where the SU(2) spin symmetry is broken. These states can be explicitly expressed as the superposition of two nonorthogonal, but mirror symmetrical spin states entangled with the pseudo spin states. Because of the closely locking between the pseudo spin and the true spin, it is found that the orbit curve in the spin-polarization parameter space for the fixed equi-energy contour around Dirac points has the same shape as the δk-contour. Due to the spin–orbit coupling that cause the topological transition in the local geometry of the dispersion relation, the new equi-energy contours around the new emergent Dirac Points can be obtained by squeezing the one around the original Dirac point. The spin texture in the momentum space around the Dirac points is analyzed under the Rashba spin–orbit interaction and it is found that the orientation of the spin polarization at each crystal momentum k is independent of the Rashba coupling strength. © 2018 Published by Elsevier B.V.
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Article history: Received 26 September 2018 Received in revised form 25 October 2018 Accepted 31 October 2018 Available online xxxx Communicated by M.G.A. Paris
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The spin–orbit (SO) interaction plays an important role in Dirac Fermion properties of graphene. The spin–orbit coupling in graphene can be intrinsic or extrinsic. They affect the electronic properties of graphene in different ways. The intrinsic SO(ISO) which respects the inversion symmetry of the graphene lattice is in the range of 1–50 μeV and therefore it could only be probed at very low temperatures [1, 2]. It can lead to a finite gap between the conduction and valence bands [3] and turn a single layer graphene into a topological insulator [4]. On the other hand, the extrinsic Rashba SO(RSO) interaction, which originates from the interface between graphene and substrate and is generated by the structure inversion asymmetry of the graphene sheet, acts in the opposite direction and tends to close the gap. Compared with the ISO, the Rashba coupling can be tuned to much higher values with an external electric gate voltages, electrostatic interaction with the substrate [1,2,5] or impurities [6]. The experiments on epitaxial graphene grown on a Ni(111) substrate showed that the Rashba coupling can reach values up to 0.2 eV [7] and also Varykhalov et al. reveal a substantial extrinsic spin–orbit coupling of the scale of 10 meV and the strong momentum-dependent in-plane spin polarization with spin- and angular-resolved photoemission spectroscopy (SARPE) techniques [5]. The RSO effect is manifested through the anisotropic spin-splitting of the bands at the K (K ) points [7,8] and split the original Dirac point into four gapless points [8–10]. One of them remains at the K (K ) point and other three link up a equilateral triangle around the K (K ) point. It causes a topological change of the low energy band structure at the K (K ) points of the graphene’s Brillouin zone (BZ), known as the trigonal warping [8,11–14]. This topological change of the energy dispersion directly affects the linear [15,16] and nonlinear [18] optical conductivity. The types of gapless band structures are topologically nontrivial and had been studied early in the pioneering work of Volovik [17]. Recently many theoretical works have been devoted to the studies on the electronic structure resulting from the RSO [9,10,19,20]. In Ref. [9], band split accompanied by the appearance of additional gapless points is confirmed by exactly solving the tight-binding model Hamiltonian. Meanwhile, it is also found that the eigenstates of spin–orbit coupled graphene are polarized in-plane and are perpendicular to the crystal momentum k. The magnitude of spin polarization S at the Bloch eigenstate |k vanishes when k → 0 [19]. Moreover, it was proven that photoconversion of a quasiparticle inside a crystal into a
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Corresponding author. E-mail addresses: [email protected] (C. Zhang), [email protected] (J.C. Cao).
https://doi.org/10.1016/j.physleta.2018.10.049 0375-9601/© 2018 Published by Elsevier B.V.
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AID:25379 /SCO Doctopic: General physics
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ˆ RSO = i H
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l∈ T
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δ
can be written as
⎛
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ϕ0∗
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where |ϕk (r ) = √1
N
ν
λR
i ϕ− 0
∗ − i ϕ+
√
lν ∈ T ν
1 + 2e
eigenvalues of Eq. (2) is given by
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E η1 ,η2 (k) = η1 th
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where W = |ϕ0 |2 + 2
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|ψ± = √
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,
ϕ+ = e
iak y
√
3
1 − 2e
W 2 − |ϕ + 2 ϕ− ϕ+ |2 , 2 0
2
and
|ϕk B ⊗ |θ B , φ B ± |ϕk A ⊗ |θ A , φ A
bA =
c3 , |a A |2 1+|c 2 |2
√ − 12 i 3ak y
+ |b A | = 1. The pertinent parameters are following 2
c1 = −
sin
akx 2
+ π6
,
ϕ− = e
iak y
√
3
2
c3 =
) ∗)
(2)
c4 = 1
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1 + 2e
√ − 12 i 3ak y
sin
akx 2
− π6
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. The
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(3)
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(4) c2 1+|c 2 |2
, bB =
c4 1+|c 2 |2
c1 , 1+|c 2 |2
, |a B |2 +|b B |2 = 1 and a A =
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∗
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− ϕ0 (ϕ+ 2 + ϕ− ϕ0∗ 2 (ϕ+ ) ∗ 2 |ϕ− | 2 − E − E − (ϕ− (ϕ0
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(ϕ+ ) ∗ 2 |ϕ− | 2 − E − + ϕ− ϕ0 2 ∗ 2 i 2 ϕ0 (ϕ+ ϕ− ) ∗ + |ϕ0 | 2 − E − ϕ0 c2 = 2 ∗ 2 2 (ϕ+ ) |ϕ− | − E − + ϕ− ϕ0∗ 2 )∗
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i E − −2 |ϕ− | 2 − |ϕ0 | 2 + E −
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(5)
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Note that the defined spin quantum state |θ A ( B ) (k), φ A ( B ) (k) is the pure spin state and the average of the spin vector σˆ (also known as the spin polarization) in the two states are the unit vectors on the surface of the bloch shere (as shown in Fig. 3), which point to the some specific directions. The directions can be determined by calculating that θ A ( B ) , φ A ( B ) |σˆ z |θ A ( B ) , φ A ( B ) ≡ σˆ z A ( B ) ≡ cos θ A ( B ) and σˆ y A ( B ) /σˆ x A ( B ) ≡ tan φ A ( B ) . Using the relations |c 1 |2 = |c 4 |2 = 1, and |c 2 |2 = |c 3 |2 , it is found that cos θ B = − cos θ A and tan φ A = tan φ B ,
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which means that θ B = π − θ A ≡ π − θ(k), φ B = φ A ≡ φ(k). The normalized eigenstate Eq. (4) can be explicitly rewritten as
|ϕk B ⊗ |π − θ, φ ± |ϕk A ⊗ |θ, φ ,
1
|ψ± = √
2
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where |θ B , φ B ≡ a B (k)| ↑ + b B (k)| ↓, |θ A , φ A ≡ a A (k)| ↑ + b A (k)| ↓ with a B =
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|ϕ+ |2 +|ϕ− |2
cos
akx 2
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ϕ0
normalized eigenvector for the ± E − branches
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, ϕ0 ⎠
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η1 , η2 = ±1 for the different energy branches. In the work we are only interested in the low energy branch where η2 = −1, η1 = ±1. Neglecting the subscript η1 and denote | E ±,− | as E − (k) and after some algebras we get the
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W + η2
∗ − i ϕ−
√ − 12 i 3ak y
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,
i ϕ+ 0 ⎟ ⎟
=
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(1)
e ik·lν φ2p z (r − lν ) (ν = A , B ) with φ2p z (r ) usually chosen as the 2p z orbit wave function of the carbon atom and
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th
ϕ0 = e
iak y
⎞
0
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0
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ϕ0
0
⎜ ˆ = th ⎜ H ⎝
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l +δ
l
where the subscript σ represents the spin index, σˆ are the Pauli matrices δ is the three nearest-neighbor vectors in real space [9, λR λR δ = − d zˆ × δ = − d −δ y , δx , 0 , u δ · σˆ = λdR (δy σˆ x − δx σˆ y ), d0 = √a and λ R is the RSO coupling parameter. In the non-coupling 21], u 0 0 0 3
ˆ representation of the true and pseudo spin where the basis is |ϕk ⊗ | ↑, |ϕk ⊗ | ↑, |ϕk ⊗ | ↓, |ϕk ⊗ | ↓ , the full Hamiltonian H B B A A
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δ · σˆ ⊗ Cˆ † Cˆ l+δ − u δ · σˆ ⊗ Cˆ † Cˆ l , u
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photoelectron can be accompanied with a dramatic change of its spin polarization, up to a total spin flip with the model of the spin–orbit coupled graphene [20]. In this paper we demonstrate a rare phenomenon of true spin-pseudo spin entanglement in graphene with RSC. We emphatically analyze the quantum states based on the tight-binding model with RSO and show that these states can be explicitly expressed as the superposition of two nonorthogonal, but mirror symmetrical spin states entangled with the pseudo spin states. Under the linear approximation of the relative crystal momentum the intrinsic interconnection between the geometry of the equi-energy contour and the orbit curve in the spin-polarization parameter space is revealed. Furthermore, the effect on the entangled states due to the coupling strength is investigated. • Schmidt decomposition of entangled eigenstates – For the monolayer graphene with RSO the Hamiltonian is given as Hˆ = Hˆ 0 + ˆ 0 is the Hamiltonian of a normal monolayer graphene Hˆ 0 = i , j th Cˆ † Cˆ j σ with the nearest-neighbor hopping energy ˆ H R S O , where H iσ th ≈ 2.7 eV [21], and H R S O is the Hamiltonian of Rashba SO interaction [9,22–24]
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where |θ(k), φ(k) =
− cos θ2 e
sin
θ 2
e
iφ 2
iφ 2
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JID:PLA AID:25379 /SCO Doctopic: General physics
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cos β − sin β sin β cos β
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is the rotation matrix by an angle β
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and β K A = 0, β K B = π3 , β K C = − π3 . The dispersion relation Eq. (8) gives the equation of an ellipse with -independent eccentric-
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ity 2 2/3 and also implies that the electronic excitation around the satellite Dirac points behaves like the photon in the anisotropic
√
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ˆ (β) = where L = K A , K B , K C and δk˜ L = (k − k L )/a, E˜ = E /th are dimensionless. D
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triangle in Fig. 1) relative to the point K is independent of the coupling strength . In Fig. 1 it is noted that the equi-energy contours around the satellite Dirac points K A , K B , K C are not strictly circular. To explore the properties of the equi-energy contours around these points we expand the square of the energy around these points with respect to the parameters δk x , δk y , respectively. Up to the second order of these parameters the square of the energy around the degeneracy points is written uniformly as
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pseudo spin states (orbit states). The renormalized states in Eq. (6) has exactly the same form of the Bell states which is extensively used in the study of the quantum information [25–28]. • Energy dispersion relation and the spin states around the Dirac points – RSO causes the topological change of the low energy band structure and gives rise to the trigonal warping shown in Fig. 1. The coordinates of the satellite Dirac points are k K A = (2γ , 0) a−1 , 2 2π 2π −3γ k K B ( K C ) = 2π − γ , ± √ a−1 , where the angle γ () is defined as γ () ≡ cos−1 22 −1 with lim γ = , which implies that 2 +1 3 →0 3 k K ( B ,C ) → k K = ( 4π , 0)a−1 when → 0, recovering the initial topological configuration. The side length of the equilateral triangle comA 3
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decomposition of the entangled energy eigenvector of the spin–orbit-coupled system Eq. (2) and it implies that for each fixed k point in the 2D Brillouin Zone (in x–y plane) the entangled eigenvector of the system H (k) can be decomposed as the superposition of the two direct product states between the pure true spin state the polarization orientation of which is in the direction θ(k), φ(k) and the
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Here the angle θ is also the angle between the spin polarization vector σˆ θ,φ and the polar axis which points the direction of the spin polarization in the state | ↑. In this paper the polarization orientation of the basis state | ↑ is taken as the direction of the z axis that is perpendicular to the plane of the graphene. It is worth pointing out that |θ, φ, |π − θ, φ are not orthogonal since π − θ, φ|θ, φ = sin θ . However the direct product vectors |ϕk ⊗ |π − θ, φ, |ϕk ⊗ |θ, φ may be normalized and orthogonal if it is taken that ϕk |ϕk ≈ δν ,ν . For now we obtained the Schmidt
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∗ −ϕ i (ϕ+ − ϕ− ) ϕ0∗ + ϕ0 ϕ+ c 1 c ∗ − c 1∗ c 3 a A b∗A − a∗A b A σˆ y A − ∗ tan φ = =i = i 3∗ = ∗ ∗ ∗ ∗ ∗ σˆ x A aAbA + aAbA c1 c3 + c1 c3 (ϕ− + ϕ+ ) ϕ0 − ϕ0 ϕ− + ϕ+ ∗ 2 2 2 ∗ − 2E 2 ϕ 4 + ϕ0 4 |ϕ− | 4 + |ϕ0 | 2 2 |ϕ− | 2 − 2 |ϕ+ | 2 − 2E − ϕ− ϕ+ + ϕ02 + 2 (ϕ− ) ∗ ϕ02 ϕ+ − − + E− cos θ = 2 ∗ ϕ 2 ϕ ∗ + 2E 2 ϕ 4 + ϕ0∗ 2 ϕ02 − 2 ϕ− ϕ+ − 2 ϕ− 4 |ϕ− | 4 + |ϕ0 | 2 32 |ϕ− | 2 + 2 |ϕ+ | 2 − 2E − − − + E− 0 +
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Fig. 1. The equi-energy contours around the Dirac points at = 0.2 and the red arrow denote the spin polarization. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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2 k2 k homogeneous-medium with the dispersion relation ωc = x + y . The denominators in Eq. (8) play the role of the effective permittivity y x 2
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parameter it is a remarkable fact that the direction φ(k) of the spin polarization at each k is independent of the coupling strength and the amplitude of the polarization n T (k, ) is tuned by the factor sin θ(k, ). To find the distribution of the orientation of the T (k, ) in the 2D crystal momentum space we expand the quantity n T · δk˜ around K point where spin polarization (spin texture) n δk˜ 3 −3δk˜ 2 δk˜ y δk˜ = (k − k K )/a and find that n T · δk˜ ≈ y 4 x + o(δk˜ 5 ). For the small δk˜ and under the linear approximation it can be taken as that T · δk˜ ≈ 0, i.e. the spin polarization at wavevector δk relative to K point is perpendicular to δk, which is in accordance with the result in n ˜ Ref. [19], where the Hamiltonian around the Dirac point is linearized about the parameter δk. For the pure spin state |θ, φ we can find (k). At Dirac points K , K A , K B , K C the eigen the dependence of the spin state |θ, φ on the crystal momentum k by exploring the vector n energy is zero and the degenerate eigenstates can be expressed as the same form in Eq. (6) with φ| K A = 32π , φ| K B = 56π , φ| K C = π6 , θ| K = 0,
θ| K A = θ| K B = θ| K C ≡ α , where cos α
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numerically calculated and shown in Fig. 4. It can be found that the geometry of the two kinds of curve are very similar, indicating the fine linear relationship between the intrinsic spin polarization and the relative crystal momentum which results from the spin-momentum locking caused by the Rashba SOC. For the calculation of macroscopic physical quantities such as conductivity, susceptibility, the free electron gas model or mean field approximation may be valid when the electron–electron interaction can be ignored or averaged. In this physical picture for the system where the RSO is involved, the quantum state of the many-body system is characterized by the reduced density matrix, a situation may be encountered in the field of quantum information and computing. To explore the possible application of the entangled quantum system with true-pseudo spin and to further understand the physical meaning of the angle θ we calculate the reduced density matrix of the eigenstates in Eq. (6) for the true spin part by taking the partial trace over the pseudo spin space. It is found,
ρˆtrue (k, ) = Trpseudo (|ψ+ ψ+ |) = Trpseudo (|ψ− ψ− |) = Moreover, we obtain
Tr
Tr
1 + sin2 θ
2
2 ρˆtrue =
ρˆtrue σˆ = n T (k, )
1 2
1 i φ(k) e sin θ(k, ) 2
1 −i φ(k) e sin θ(k, ) 2 1 2
(10)
.
ρˆ (t ) =
2π 2
f (k, t )ρˆtrue dk2 ,
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where f (k, t ) is the distribution function in the non-equilibrium state. Considering that the parameter θ reflects the coherence between
the two true spin states in the mixed state Eq. (10), we expand n z (δk˜ L ) around the Dirac points to the linear order of δk˜ L and the second order of to obtain the following results,
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which corresponds to the pure spin state on the equator of the Bloch sphere (see Fig. 3). The off-diagonal matrix element 12 e ±i φ sin θ describe the degree of coherence between the two true spin states and the magnitude of spin polarization in the mixed states, which is exactly the physical meaning of the angle θ . The density matrix of the whole ensemble under the RSO interaction may be expressed as,
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2 2 The trace 1/2 ≤ Tr ρˆtrue ≤ 1 implies that the reduced quantum true spin states are mostly the mixed states except θ = π /2, Tr ρˆtrue = 1,
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T (δk˜ L ) = Dˆ −1 n T − ζL is The equi-energy contours of δk˜ L around the Dirac points K , K B and the corresponding spin-polarization curve n
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T (δk L ) and the crystal momentum δk L relative to the Dirac Points k L implies that there may exist a similarity in-plane spin polarization n between the geometry of the equi-energy contour and the one of the curves of the spin polarization on the same equi-energy contour.
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1−2 sin
where L = K , K A , K B , K C , ζL () is the spin offset vector. Moreover, in the linear it is found that at each Dirac point, √ approximation ˜ ˜ 2 2 ζL = ψ(k L )|σ |ψ(k L ) T are given as ζ K = 0, ζ K A = (0, −2) 1 − 2 , ζ K B (C ) = ∓ 3, 1 1 − 2 . The linear relationship between the
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ˆ (π /2)δk˜ L + ζL () T (δk˜ L ) √ n D 3
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T (k) around the Dirac points K , K A , K B , K C with respect to the parameters δk˜ x , δk˜ y , respectively. By keeping δk˜ and −1 to the linear n order we obtain,
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γ + π6 2 = 2 2 2 2 . For the spin states in Eq. (3) we expand the in-plane spin polarization 9 + +1 1−2 sin γ + π6
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the one around the K point and it may be called the squeezing effect in analogy with the trigonal wrapping effect. |ψ± = θ, φ|σ |θ, φT , where ˆI s is The spin polarization of the electron at the electronic eigenstate |ψ± (k) defined as ψ± | ˆI s ⊗ σ the unit operator in the pseudo space and the subscript T denote the in-plane component of the vector. The mean value of the Pauli |θ, φ = sin θ(cos φe x + sin φe y ) + cos θ(k)e z ≡ n(k) ≡ n T + n z (k)e z . Since the factor tan φ in Eq. (7) does not include the spin θ, φ|σ
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eff,x =
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√
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For a typical value δk˜ x = 0.01, δk˜ y = 0 the corresponding n z δk˜ L = cos θ δk˜ L for the L = K A and L = K with respect to parameter are compared in Fig. 2 It is found that at the k˜ = k˜ L , cos α and cos θ δk˜ K A decrease increases and cos θ δk˜ K is nearly constant with θ δk˜ K ≈ 0. It shows that the coherence between the two true spin states for a fixed δk˜ L around points K A , K B , K C can be enhanced by
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increasing the coupling strength. • Summary and conclusions – Using the tight-binding model we obtained the Schmidt decomposition of the entangled eigenstate massless Dirac Fermions in the presence of the Rashba spin–orbit (RSO) interaction. It has the standard form of the Bell state. The results show that the entangled eigenstates can be exactly decomposed as the superposition of the pseudo spin states entangled respectively with the two mirror symmetrical pure spin states the direction of which is dependent on the crystal momentum and the coupling strength. The spin texture around the Dirac points in the 2D momentum space has been explored in detail and it is rather remarkable that the orientation of the spin polarization of the eigenstate at each crystal momentum k is independent of the coupling strength , which seems to be a little counterintuitive. The stability of the orientation can be used to resist the perturbation from the coupling strength. We guess that the stability may has something to do with some kind of topological invariant in the momentum space. Moreover, the amplitude of the spin polarization is tuned by the coupling strength . Because of the nonzero coupling strength , which leads to the topologically change of the local geometry of the dispersion relation, the new equi-energy contours around the new emergent Dirac points can be obtained by squeezing the one around the original Dirac point. The locking between the relative crystal momentum and the spin polarization is also analyzed quantitatively and an exact relation between them has been obtained. The similarity between the shape of
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Fig. 3. The spin states at the Dirac Points K , K A , K B , K C in the Bloch sphere, where φ| K A =
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Fig. 2. Different n z − curves at a typical value δk˜ x = 0.01, δk˜ y = 0.
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T (red line): (a) around the K point Fig. 4. The equi-energy contours of δk˜ L (blue line) around the Dirac points at = 0.2 and the corresponding spin-polarization curve n with the energy E ( K ) = 0.01th , (b) around the K B point with energy E ( K ) = 0.004th .
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the euqi-energy contour and the shape of the corresponding curve of spin polarization geometrically demonstrate the close locking of spin-momentum. The -independence of the orientation of the equilateral triangle composed of Dirac points and the eccentricity of the contour ellipse implies that the nonzero coupling strength may lead to a geometrically transforming effect in the energy–momentum space that is similar to the conformal transform. By taking the partial trace we obtain the reduced density matrix for the true spin part and the off-diagonal elements reveal a possible physical meaning of the zenith angle θ in the Bloch sphere that characterizes the coherence between the true spin states.
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This work was supported by the 973 Program of China (Grant No. 2014CB339803), the National Natural Science Foundation of China (Grant No. 61404150), and the Shanghai Municipal Science and Technology Commission (Project No. 15JC1403800).
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True spin and pseudo spin entanglement around Dirac Points in graphene with Rashba spin–orbit interaction
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Zheng Liu , Chao Zhang , J.C. Cao
a,∗
a
Key Laboratory of Terahertz Solid-State Technology, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, 865 Changning Road, Shanghai 200050, China b School of Physics, University of Wollongong, New South Wales 2522, Australia
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Keywords: Rashba spin–orbit interaction Entangled spin states Spin texture
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We analytically obtained the Schmidt decomposition of the entangled state between the pseudo spin and the true spin in graphene with Rashba spin–orbit coupling. The entangled state has the standard form of the Bell state, where the SU(2) spin symmetry is broken. These states can be explicitly expressed as the superposition of two nonorthogonal, but mirror symmetrical spin states entangled with the pseudo spin states. Because of the closely locking between the pseudo spin and the true spin, it is found that the orbit curve in the spin-polarization parameter space for the fixed equi-energy contour around Dirac points has the same shape as the δk-contour. Due to the spin–orbit coupling that cause the topological transition in the local geometry of the dispersion relation, the new equi-energy contours around the new emergent Dirac Points can be obtained by squeezing the one around the original Dirac point. The spin texture in the momentum space around the Dirac points is analyzed under the Rashba spin–orbit interaction and it is found that the orientation of the spin polarization at each crystal momentum k is independent of the Rashba coupling strength. © 2018 Published by Elsevier B.V.
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Article history: Received 26 September 2018 Received in revised form 25 October 2018 Accepted 31 October 2018 Available online xxxx Communicated by M.G.A. Paris
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The spin–orbit (SO) interaction plays an important role in Dirac Fermion properties of graphene. The spin–orbit coupling in graphene can be intrinsic or extrinsic. They affect the electronic properties of graphene in different ways. The intrinsic SO(ISO) which respects the inversion symmetry of the graphene lattice is in the range of 1–50 μeV and therefore it could only be probed at very low temperatures [1, 2]. It can lead to a finite gap between the conduction and valence bands [3] and turn a single layer graphene into a topological insulator [4]. On the other hand, the extrinsic Rashba SO(RSO) interaction, which originates from the interface between graphene and substrate and is generated by the structure inversion asymmetry of the graphene sheet, acts in the opposite direction and tends to close the gap. Compared with the ISO, the Rashba coupling can be tuned to much higher values with an external electric gate voltages, electrostatic interaction with the substrate [1,2,5] or impurities [6]. The experiments on epitaxial graphene grown on a Ni(111) substrate showed that the Rashba coupling can reach values up to 0.2 eV [7] and also Varykhalov et al. reveal a substantial extrinsic spin–orbit coupling of the scale of 10 meV and the strong momentum-dependent in-plane spin polarization with spin- and angular-resolved photoemission spectroscopy (SARPE) techniques [5]. The RSO effect is manifested through the anisotropic spin-splitting of the bands at the K (K ) points [7,8] and split the original Dirac point into four gapless points [8–10]. One of them remains at the K (K ) point and other three link up a equilateral triangle around the K (K ) point. It causes a topological change of the low energy band structure at the K (K ) points of the graphene’s Brillouin zone (BZ), known as the trigonal warping [8,11–14]. This topological change of the energy dispersion directly affects the linear [15,16] and nonlinear [18] optical conductivity. The types of gapless band structures are topologically nontrivial and had been studied early in the pioneering work of Volovik [17]. Recently many theoretical works have been devoted to the studies on the electronic structure resulting from the RSO [9,10,19,20]. In Ref. [9], band split accompanied by the appearance of additional gapless points is confirmed by exactly solving the tight-binding model Hamiltonian. Meanwhile, it is also found that the eigenstates of spin–orbit coupled graphene are polarized in-plane and are perpendicular to the crystal momentum k. The magnitude of spin polarization S at the Bloch eigenstate |k vanishes when k → 0 [19]. Moreover, it was proven that photoconversion of a quasiparticle inside a crystal into a
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Corresponding author. E-mail addresses: [email protected] (C. Zhang), [email protected] (J.C. Cao).
https://doi.org/10.1016/j.physleta.2018.10.049 0375-9601/© 2018 Published by Elsevier B.V.
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ˆ RSO = i H
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l∈ T
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δ
can be written as
⎛
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ϕ0∗
25
where |ϕk (r ) = √1
N
ν
λR
i ϕ− 0
∗ − i ϕ+
√
lν ∈ T ν
1 + 2e
eigenvalues of Eq. (2) is given by
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E η1 ,η2 (k) = η1 th
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where W = |ϕ0 |2 + 2
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|ψ± = √
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ϕ+ = e
iak y
√
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1 − 2e
W 2 − |ϕ + 2 ϕ− ϕ+ |2 , 2 0
2
and
|ϕk B ⊗ |θ B , φ B ± |ϕk A ⊗ |θ A , φ A
bA =
c3 , |a A |2 1+|c 2 |2
√ − 12 i 3ak y
+ |b A | = 1. The pertinent parameters are following 2
c1 = −
sin
akx 2
+ π6
,
ϕ− = e
iak y
√
3
2
c3 =
) ∗)
(2)
c4 = 1
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1 + 2e
√ − 12 i 3ak y
sin
akx 2
− π6
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. The
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(3)
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(4) c2 1+|c 2 |2
, bB =
c4 1+|c 2 |2
c1 , 1+|c 2 |2
, |a B |2 +|b B |2 = 1 and a A =
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∗
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− ϕ0 (ϕ+ 2 + ϕ− ϕ0∗ 2 (ϕ+ ) ∗ 2 |ϕ− | 2 − E − E − (ϕ− (ϕ0
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(ϕ+ ) ∗ 2 |ϕ− | 2 − E − + ϕ− ϕ0 2 ∗ 2 i 2 ϕ0 (ϕ+ ϕ− ) ∗ + |ϕ0 | 2 − E − ϕ0 c2 = 2 ∗ 2 2 (ϕ+ ) |ϕ− | − E − + ϕ− ϕ0∗ 2 )∗
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i E − −2 |ϕ− | 2 − |ϕ0 | 2 + E −
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(5)
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Note that the defined spin quantum state |θ A ( B ) (k), φ A ( B ) (k) is the pure spin state and the average of the spin vector σˆ (also known as the spin polarization) in the two states are the unit vectors on the surface of the bloch shere (as shown in Fig. 3), which point to the some specific directions. The directions can be determined by calculating that θ A ( B ) , φ A ( B ) |σˆ z |θ A ( B ) , φ A ( B ) ≡ σˆ z A ( B ) ≡ cos θ A ( B ) and σˆ y A ( B ) /σˆ x A ( B ) ≡ tan φ A ( B ) . Using the relations |c 1 |2 = |c 4 |2 = 1, and |c 2 |2 = |c 3 |2 , it is found that cos θ B = − cos θ A and tan φ A = tan φ B ,
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which means that θ B = π − θ A ≡ π − θ(k), φ B = φ A ≡ φ(k). The normalized eigenstate Eq. (4) can be explicitly rewritten as
|ϕk B ⊗ |π − θ, φ ± |ϕk A ⊗ |θ, φ ,
1
|ψ± = √
2
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where |θ B , φ B ≡ a B (k)| ↑ + b B (k)| ↓, |θ A , φ A ≡ a A (k)| ↑ + b A (k)| ↓ with a B =
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|ϕ+ |2 +|ϕ− |2
cos
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ϕ0
normalized eigenvector for the ± E − branches
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η1 , η2 = ±1 for the different energy branches. In the work we are only interested in the low energy branch where η2 = −1, η1 = ±1. Neglecting the subscript η1 and denote | E ±,− | as E − (k) and after some algebras we get the
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W + η2
∗ − i ϕ−
√ − 12 i 3ak y
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,
i ϕ+ 0 ⎟ ⎟
=
34
(1)
e ik·lν φ2p z (r − lν ) (ν = A , B ) with φ2p z (r ) usually chosen as the 2p z orbit wave function of the carbon atom and
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th
ϕ0 = e
iak y
⎞
0
0
0
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ϕ0
0
⎜ ˆ = th ⎜ H ⎝
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l +δ
l
where the subscript σ represents the spin index, σˆ are the Pauli matrices δ is the three nearest-neighbor vectors in real space [9, λR λR δ = − d zˆ × δ = − d −δ y , δx , 0 , u δ · σˆ = λdR (δy σˆ x − δx σˆ y ), d0 = √a and λ R is the RSO coupling parameter. In the non-coupling 21], u 0 0 0 3
ˆ representation of the true and pseudo spin where the basis is |ϕk ⊗ | ↑, |ϕk ⊗ | ↑, |ϕk ⊗ | ↓, |ϕk ⊗ | ↓ , the full Hamiltonian H B B A A
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photoelectron can be accompanied with a dramatic change of its spin polarization, up to a total spin flip with the model of the spin–orbit coupled graphene [20]. In this paper we demonstrate a rare phenomenon of true spin-pseudo spin entanglement in graphene with RSC. We emphatically analyze the quantum states based on the tight-binding model with RSO and show that these states can be explicitly expressed as the superposition of two nonorthogonal, but mirror symmetrical spin states entangled with the pseudo spin states. Under the linear approximation of the relative crystal momentum the intrinsic interconnection between the geometry of the equi-energy contour and the orbit curve in the spin-polarization parameter space is revealed. Furthermore, the effect on the entangled states due to the coupling strength is investigated. • Schmidt decomposition of entangled eigenstates – For the monolayer graphene with RSO the Hamiltonian is given as Hˆ = Hˆ 0 + ˆ 0 is the Hamiltonian of a normal monolayer graphene Hˆ 0 = i , j th Cˆ † Cˆ j σ with the nearest-neighbor hopping energy ˆ H R S O , where H iσ th ≈ 2.7 eV [21], and H R S O is the Hamiltonian of Rashba SO interaction [9,22–24]
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where |θ(k), φ(k) =
− cos θ2 e
sin
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e
iφ 2
iφ 2
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2
2 x
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cos β − sin β sin β cos β
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ity 2 2/3 and also implies that the electronic excitation around the satellite Dirac points behaves like the photon in the anisotropic
√
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ˆ (β) = where L = K A , K B , K C and δk˜ L = (k − k L )/a, E˜ = E /th are dimensionless. D
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triangle in Fig. 1) relative to the point K is independent of the coupling strength . In Fig. 1 it is noted that the equi-energy contours around the satellite Dirac points K A , K B , K C are not strictly circular. To explore the properties of the equi-energy contours around these points we expand the square of the energy around these points with respect to the parameters δk x , δk y , respectively. Up to the second order of these parameters the square of the energy around the degeneracy points is written uniformly as
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pseudo spin states (orbit states). The renormalized states in Eq. (6) has exactly the same form of the Bell states which is extensively used in the study of the quantum information [25–28]. • Energy dispersion relation and the spin states around the Dirac points – RSO causes the topological change of the low energy band structure and gives rise to the trigonal warping shown in Fig. 1. The coordinates of the satellite Dirac points are k K A = (2γ , 0) a−1 , 2 2π 2π −3γ k K B ( K C ) = 2π − γ , ± √ a−1 , where the angle γ () is defined as γ () ≡ cos−1 22 −1 with lim γ = , which implies that 2 +1 3 →0 3 k K ( B ,C ) → k K = ( 4π , 0)a−1 when → 0, recovering the initial topological configuration. The side length of the equilateral triangle comA 3
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decomposition of the entangled energy eigenvector of the spin–orbit-coupled system Eq. (2) and it implies that for each fixed k point in the 2D Brillouin Zone (in x–y plane) the entangled eigenvector of the system H (k) can be decomposed as the superposition of the two direct product states between the pure true spin state the polarization orientation of which is in the direction θ(k), φ(k) and the
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Here the angle θ is also the angle between the spin polarization vector σˆ θ,φ and the polar axis which points the direction of the spin polarization in the state | ↑. In this paper the polarization orientation of the basis state | ↑ is taken as the direction of the z axis that is perpendicular to the plane of the graphene. It is worth pointing out that |θ, φ, |π − θ, φ are not orthogonal since π − θ, φ|θ, φ = sin θ . However the direct product vectors |ϕk ⊗ |π − θ, φ, |ϕk ⊗ |θ, φ may be normalized and orthogonal if it is taken that ϕk |ϕk ≈ δν ,ν . For now we obtained the Schmidt
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∗ −ϕ i (ϕ+ − ϕ− ) ϕ0∗ + ϕ0 ϕ+ c 1 c ∗ − c 1∗ c 3 a A b∗A − a∗A b A σˆ y A − ∗ tan φ = =i = i 3∗ = ∗ ∗ ∗ ∗ ∗ σˆ x A aAbA + aAbA c1 c3 + c1 c3 (ϕ− + ϕ+ ) ϕ0 − ϕ0 ϕ− + ϕ+ ∗ 2 2 2 ∗ − 2E 2 ϕ 4 + ϕ0 4 |ϕ− | 4 + |ϕ0 | 2 2 |ϕ− | 2 − 2 |ϕ+ | 2 − 2E − ϕ− ϕ+ + ϕ02 + 2 (ϕ− ) ∗ ϕ02 ϕ+ − − + E− cos θ = 2 ∗ ϕ 2 ϕ ∗ + 2E 2 ϕ 4 + ϕ0∗ 2 ϕ02 − 2 ϕ− ϕ+ − 2 ϕ− 4 |ϕ− | 4 + |ϕ0 | 2 32 |ϕ− | 2 + 2 |ϕ+ | 2 − 2E − − − + E− 0 +
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Fig. 1. The equi-energy contours around the Dirac points at = 0.2 and the red arrow denote the spin polarization. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.)
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2 k2 k homogeneous-medium with the dispersion relation ωc = x + y . The denominators in Eq. (8) play the role of the effective permittivity y x 2
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parameter it is a remarkable fact that the direction φ(k) of the spin polarization at each k is independent of the coupling strength and the amplitude of the polarization n T (k, ) is tuned by the factor sin θ(k, ). To find the distribution of the orientation of the T (k, ) in the 2D crystal momentum space we expand the quantity n T · δk˜ around K point where spin polarization (spin texture) n δk˜ 3 −3δk˜ 2 δk˜ y δk˜ = (k − k K )/a and find that n T · δk˜ ≈ y 4 x + o(δk˜ 5 ). For the small δk˜ and under the linear approximation it can be taken as that T · δk˜ ≈ 0, i.e. the spin polarization at wavevector δk relative to K point is perpendicular to δk, which is in accordance with the result in n ˜ Ref. [19], where the Hamiltonian around the Dirac point is linearized about the parameter δk. For the pure spin state |θ, φ we can find (k). At Dirac points K , K A , K B , K C the eigen the dependence of the spin state |θ, φ on the crystal momentum k by exploring the vector n energy is zero and the degenerate eigenstates can be expressed as the same form in Eq. (6) with φ| K A = 32π , φ| K B = 56π , φ| K C = π6 , θ| K = 0,
θ| K A = θ| K B = θ| K C ≡ α , where cos α
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numerically calculated and shown in Fig. 4. It can be found that the geometry of the two kinds of curve are very similar, indicating the fine linear relationship between the intrinsic spin polarization and the relative crystal momentum which results from the spin-momentum locking caused by the Rashba SOC. For the calculation of macroscopic physical quantities such as conductivity, susceptibility, the free electron gas model or mean field approximation may be valid when the electron–electron interaction can be ignored or averaged. In this physical picture for the system where the RSO is involved, the quantum state of the many-body system is characterized by the reduced density matrix, a situation may be encountered in the field of quantum information and computing. To explore the possible application of the entangled quantum system with true-pseudo spin and to further understand the physical meaning of the angle θ we calculate the reduced density matrix of the eigenstates in Eq. (6) for the true spin part by taking the partial trace over the pseudo spin space. It is found,
ρˆtrue (k, ) = Trpseudo (|ψ+ ψ+ |) = Trpseudo (|ψ− ψ− |) = Moreover, we obtain
Tr
Tr
1 + sin2 θ
2
2 ρˆtrue =
ρˆtrue σˆ = n T (k, )
1 2
1 i φ(k) e sin θ(k, ) 2
1 −i φ(k) e sin θ(k, ) 2 1 2
(10)
.
ρˆ (t ) =
2π 2
f (k, t )ρˆtrue dk2 ,
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(11)
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(12)
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where f (k, t ) is the distribution function in the non-equilibrium state. Considering that the parameter θ reflects the coherence between
the two true spin states in the mixed state Eq. (10), we expand n z (δk˜ L ) around the Dirac points to the linear order of δk˜ L and the second order of to obtain the following results,
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which corresponds to the pure spin state on the equator of the Bloch sphere (see Fig. 3). The off-diagonal matrix element 12 e ±i φ sin θ describe the degree of coherence between the two true spin states and the magnitude of spin polarization in the mixed states, which is exactly the physical meaning of the angle θ . The density matrix of the whole ensemble under the RSO interaction may be expressed as,
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2 2 The trace 1/2 ≤ Tr ρˆtrue ≤ 1 implies that the reduced quantum true spin states are mostly the mixed states except θ = π /2, Tr ρˆtrue = 1,
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T (δk˜ L ) = Dˆ −1 n T − ζL is The equi-energy contours of δk˜ L around the Dirac points K , K B and the corresponding spin-polarization curve n
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T (δk L ) and the crystal momentum δk L relative to the Dirac Points k L implies that there may exist a similarity in-plane spin polarization n between the geometry of the equi-energy contour and the one of the curves of the spin polarization on the same equi-energy contour.
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1−2 sin
where L = K , K A , K B , K C , ζL () is the spin offset vector. Moreover, in the linear it is found that at each Dirac point, √ approximation ˜ ˜ 2 2 ζL = ψ(k L )|σ |ψ(k L ) T are given as ζ K = 0, ζ K A = (0, −2) 1 − 2 , ζ K B (C ) = ∓ 3, 1 1 − 2 . The linear relationship between the
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ˆ (π /2)δk˜ L + ζL () T (δk˜ L ) √ n D 3
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92 − 2 +1
T (k) around the Dirac points K , K A , K B , K C with respect to the parameters δk˜ x , δk˜ y , respectively. By keeping δk˜ and −1 to the linear n order we obtain,
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γ + π6 2 = 2 2 2 2 . For the spin states in Eq. (3) we expand the in-plane spin polarization 9 + +1 1−2 sin γ + π6
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the one around the K point and it may be called the squeezing effect in analogy with the trigonal wrapping effect. |ψ± = θ, φ|σ |θ, φT , where ˆI s is The spin polarization of the electron at the electronic eigenstate |ψ± (k) defined as ψ± | ˆI s ⊗ σ the unit operator in the pseudo space and the subscript T denote the in-plane component of the vector. The mean value of the Pauli |θ, φ = sin θ(cos φe x + sin φe y ) + cos θ(k)e z ≡ n(k) ≡ n T + n z (k)e z . Since the factor tan φ in Eq. (7) does not include the spin θ, φ|σ
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eff,x =
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. The geometry of dispersion relation around the satellite Dirac points seems to be created by squeezing
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n z δk˜ K
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For a typical value δk˜ x = 0.01, δk˜ y = 0 the corresponding n z δk˜ L = cos θ δk˜ L for the L = K A and L = K with respect to parameter are compared in Fig. 2 It is found that at the k˜ = k˜ L , cos α and cos θ δk˜ K A decrease increases and cos θ δk˜ K is nearly constant with θ δk˜ K ≈ 0. It shows that the coherence between the two true spin states for a fixed δk˜ L around points K A , K B , K C can be enhanced by
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increasing the coupling strength. • Summary and conclusions – Using the tight-binding model we obtained the Schmidt decomposition of the entangled eigenstate massless Dirac Fermions in the presence of the Rashba spin–orbit (RSO) interaction. It has the standard form of the Bell state. The results show that the entangled eigenstates can be exactly decomposed as the superposition of the pseudo spin states entangled respectively with the two mirror symmetrical pure spin states the direction of which is dependent on the crystal momentum and the coupling strength. The spin texture around the Dirac points in the 2D momentum space has been explored in detail and it is rather remarkable that the orientation of the spin polarization of the eigenstate at each crystal momentum k is independent of the coupling strength , which seems to be a little counterintuitive. The stability of the orientation can be used to resist the perturbation from the coupling strength. We guess that the stability may has something to do with some kind of topological invariant in the momentum space. Moreover, the amplitude of the spin polarization is tuned by the coupling strength . Because of the nonzero coupling strength , which leads to the topologically change of the local geometry of the dispersion relation, the new equi-energy contours around the new emergent Dirac points can be obtained by squeezing the one around the original Dirac point. The locking between the relative crystal momentum and the spin polarization is also analyzed quantitatively and an exact relation between them has been obtained. The similarity between the shape of
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Fig. 3. The spin states at the Dirac Points K , K A , K B , K C in the Bloch sphere, where φ| K A =
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Fig. 2. Different n z − curves at a typical value δk˜ x = 0.01, δk˜ y = 0.
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T (red line): (a) around the K point Fig. 4. The equi-energy contours of δk˜ L (blue line) around the Dirac points at = 0.2 and the corresponding spin-polarization curve n with the energy E ( K ) = 0.01th , (b) around the K B point with energy E ( K ) = 0.004th .
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the euqi-energy contour and the shape of the corresponding curve of spin polarization geometrically demonstrate the close locking of spin-momentum. The -independence of the orientation of the equilateral triangle composed of Dirac points and the eccentricity of the contour ellipse implies that the nonzero coupling strength may lead to a geometrically transforming effect in the energy–momentum space that is similar to the conformal transform. By taking the partial trace we obtain the reduced density matrix for the true spin part and the off-diagonal elements reveal a possible physical meaning of the zenith angle θ in the Bloch sphere that characterizes the coherence between the true spin states.
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Acknowledgements
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This work was supported by the 973 Program of China (Grant No. 2014CB339803), the National Natural Science Foundation of China (Grant No. 61404150), and the Shanghai Municipal Science and Technology Commission (Project No. 15JC1403800).
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