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Visualizing topological orders through Zitterbewegung in ultracold atoms
Physics Letters A 381 (2017) 252–256
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Physics Letters A www.elsevier.com/locate/pla
Visualizing topological orders through Zitterbewegung in ultracold atoms Xin Shen, Feng Mei, Zhi Li ∗ National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China
a r t i c l e
i n f o
Article history: Received 19 July 2016 Received in revised form 19 September 2016 Accepted 16 October 2016 Available online 20 October 2016 Communicated by V.A. Markel Keywords: Zitterbewegung Topological order Haldane model Ultracold atoms
a b s t r a c t Topological Haldane model recently has been experimentally realized with ultracold atoms. The low energy Hamiltonian in the model can be formally described by the Dirac equation. In this paper, we explore the Zitterbewegung effect of the Dirac quasiparticles in the vicinity of the two nonequivalent Dirac points in the Haldane model. Through analytical and numerical calculation of the evolution of Gaussian wave packets, we find that the Zitterbewegung patterns could be used to characterize the topological phases of the Haldane model. Since the patterns can be detected through standard timeof-flight measurements, our proposal provides a promising scheme to directly visualize the topological phases in ultracold atoms. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Within the frame of relativistic theory, the motion of a free electron is described by the Dirac equation. In 1930 [1], Schrödinger found that such free electron would not only move with a uniform velocity but also oscillate, which he called the Zitterbewegung (ZB). The ZB turns out to be a relativistic effect because it is explained by the interference between the positive and negative energy states. In this sense, the ZB is of fundamental significance in understanding the quantum field theory. It is believed that ZB cannot be observed since both the characteristic amplitude and frequency are beyond the scope that one can detect [2,3] with current technologies. The quantum simulation [4,5], however, provides a new way to approach ZB, which would have the possibility to observe the nontrivial phenomenon. Theoretical predictions have been made in a wide range of systems, including the semiconductor quantum well [6], the ultracold atom system [7,8] and the photonic crystals [9] and a further study of ZB has been extended to the time-modulated Dirac equation [10,11]. A general theory of ZB has been further developed in the context of a multiband system [12], which shows that the couplings between different energy eigenstates are responsible for the appearance of ZB. Experimental observations of ZB phenomenon have also been achieved in trapped ions [13], photons [14] as well as ultracold atoms [15–17].
*
Corresponding author. E-mail address: [email protected] (Z. Li).
http://dx.doi.org/10.1016/j.physleta.2016.10.029 0375-9601/© 2016 Elsevier B.V. All rights reserved.
Topological insulators recently have attracted intense interests in condensed matter physics [18,19]. The classification of such states are beyond the traditional method using symmetry-breaking order parameters and Landau–Ginzburg theory. Nevertheless, a topologically invariant index [20] can be defined to characterize the states and thus leads to concept of topological quantum phase transition (TQPT). Historically, the first discovered topological state is the integer quantum Hall state [21,22], which occurs when electrons are confined to two dimensions and subjected to a strong magnetic field perpendicular to the plane. In 1988, Haldane constructed quantum Hall states with a honeycomb lattice model [23]. Such model does not require an external magnetic field while a complex next nearest neighbor hopping term is introduced in. The low energy Hamiltonian in this model could be written in the form of Dirac equation which lies at the heart of the underling topological Dirac insulator. This discovery directly promoted the birth of topological insulators [24,25]. In the context of ultracold atoms, tremendous theoretical progresses have been made using such system to mimic different classes of topological insulators [26–37], which finally leads to the experimental measurement of Chern number [38] and demonstration of Haldane model [23]. Note that cold atoms are charge neutral and the methods devised in electronic system for measuring topological insulator will not yet be applied here. Some methods recently have been proposed to measure topological phases, including probing atomic density and momentum distribution to further extract topological invariant [28,39–45], or using Bloch oscillation to directly derive Berry phase [46,47]. However, it is still desirable to explore new phys-
X. Shen et al. / Physics Letters A 381 (2017) 252–256
ical features to characterize topological orders in ultracold atoms system. Motivated by the recent experimental demonstration of Haldane model with ultracold atoms trapped in optical lattice, in this paper, we study in detail the relation between ZB and topological phases in the Haldane model. It is well known that the low energy states of the Haldane model should be described by the Dirac equation. In the first Brillouin Zone (BZ), there are two different Dirac points K and K . The physical features near these two Dirac points directly relate to the topological features of the Haldane model, which provides another fashion to find out the topological properties of the system by investigating the Dirac equation whose dynamics can be presented through the ZB. Thus in this sense the two kinds of nontrivial phenomena (ZB and TQPT) are related and we can show that for a proper choice of initial wave packet different phases can be distinguished by the time-evolved distribution of atoms in an optical lattice. In Ref. [48], the authors considered a spin-decoupled topological model and pointed out that at a single valley the ZB of spin-up and spin-down behave differently in different topological phases. The topological Haldane model we are considering here is equivalent to the spin-up (or spin-down) part in Ref. [48] and essentially the topological information is encoded in the mass terms at the two different valleys (K, K ), so our method could be generalized to visualize different classes of topological Dirac insulators. Our study not only connect ZB with topological insulators, despite the fact [49] that the ZB can be expressed in terms of the nondiagonal elements while the Chern number is connected with the diagonal elements, but also may provide an alternative scheme to probe topological orders in ultracold atoms system. The paper is organized as follows. We will first review the Haldane’s model and it’s experimental realization in the ultracold atom system in Sec. 2. Then the ZB of the effective Hamiltonian will be examined analytically in Sec. 3 and we can see that the phases of the system and ZB can be related through the coefficients of the effective Hamiltonian for a specific initial state. The numerical results shall manifest the relationship, which can be experimentally detected through time-of-flight imaging in ultracold atom system. In Sec. 4, we will discuss the proper choice of initial states which is necessary to ensure the relation and a brief summary of our results is given in the end. 2. The experimental realization of Haldane model In the original paper [23], Haldane constructed a honeycomb lattice with the next-nearest-neighbor hopping term being complex which amounts to be two staggered magnetic fluxes while without net flux. Such a scheme breaks the time reversal symmetry that eventually leads to the possibility of the nontrivial phase transition. For a long time, the strange magnetic field has been a challenge in the realistic material in condensed matter systems. In a recent work [50], it is reported that the model has been realized experimentally in the ultra-cold atom system through periodically time-modulated optical lattice. The lattice sites were moved on the elliptical trajectories, which can be effectively seen as the application of an electric field so that the time reversal symmetry breaks. When the frequency is high enough, according to the Floquet theory, the system can well be approximated by a time-independent Hamiltonian
H=
i j
†
t1 ci c j +
i j
†
t 2 e i i j c i c j + A B
†
ci ci .
(1)
i∈ A
The summations in the first and second term correspond to the nearest and next-nearest-neighbors respectively. The hopping terms t 1 and t 2 are real numbers. A B is the energy offset between A and B sublattice. i j is the additional phase when an
253
atom hops from i site to j site, which is a constant and experimentally tunable. By Fourier transformation, the Chern number can then be calculated by the integral of the Berry curvature over the Brillouin zone [18,19]. Theoretically, the topological property can be manifested by the expansion of the Hamiltonian in the reciprocal space at the degenerate points which are the (K, K ) usually labeled for the graphene system. At half-filling, the low-energy excitations are described by the Dirac Hamiltonian
H ± = ∓ v F σ1 p 2 − v F σ2 p 1 + m ± σ3 .
(2)
The minus (plus) sign √ represents the expansion at K (K ). v F
=
3t 1 /2, m± = A B /2 ± 3 3t 2 sin i j . The σi is the Pauli matrix and the lattice constant has been taken to be 1. For the Haldane model, the topological transition can be tracked to the comparison of the signs of the effective masses at (K, K ). If the signs of the two effective masses are contrary, which signals a complete winding of skyrmion in the Brillouin zone [51], the system is topologically nontrivial. Based on this, we then can uncover the topological information of the system by examining the masses. The next section we will show that this can be done through the evolution of the wave packet, which is the ZB in the current context. 3. Zitterbewegung and the topological transition To observe the effect of the ZB with wave packets, we need to calculate expectation value of the coordinate operators. The timedependent position operator in the Heisenberg picture is written as (¯h = 1)
r(t ) = e i Ht r(0)e −i Ht .
(3)
The effective Hamiltonian at K (K ) can be represented by the Hamiltonian
H = ασ1 p 2 + β σ2 p 1 + mσ3 ,
(4)
where the α , β , m represent the corresponding coefficients in Eq. (2). The explicit results of the two coordinate components for both K and K can then be obtained typically as
x(t ) = x(0) + v 1 t + Z 11 (cos(2Et ) − 1) + Z 12 sin(2Et ), y (t ) = y (0) + v 2 t + Z 21 (cos(2Et ) − 1) + Z 22 sin(2Et ), with E = α 2 p 22 + β 2 p 21 + m2 , and v1 = Z 12 =
β2 E2 1
Hp 1 t , Z 11 =
2E 3
1 2E 2
(5)
(α β p 2 σ3 − β mσ1 ),
[(β α 2 p 22 + β m2 )σ2 −
(6)
α β 2 p 1 p 2 σ1 − β 2mp 2 σ3 ], for the x component
v2 = Z 22 =
α2 E2
Hp 2 t , Z 21 =
1 2E 3
1 2E 2
(αmσ2 − α β p 1 σ3 ),
[(α β 2 p 21 + αm2 )σ1 −
(7)
α 2 β p 1 p 2 σ2 − α 2mp 2 σ3 ], for the y component. The first two terms in x(t )[ y (t )] are similar with the counterpart classical kinematics, as expected for a free particle, whereas the last two terms induce ZB. Consider the wave packet initially prepared in the spinor state
1
|ψ(r, t = 0) = √
πd
e
2 + y2
−x
2d2
|,
(8)
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X. Shen et al. / Physics Letters A 381 (2017) 252–256
with the width of the √ wave packet characterized by d. Currently we set | = (1, 1) T / 2 and the general case will be discussed in Sec. 4. Given the above coordinate operators and the initial state, the calculation of expectation value is direct. r¯i (t ) = ψ|r i (t )|ψ can be obtained as
x¯ (t ) = −β m
dp 1 dp 2 g (p)
y¯ (t ) = α 3 t
dp 1 dp 2 g (p)
α
dp 1 dp 2 g (p)
cos(2Et ) − 1
p 22 E2
(9)
2E 2
+
(β 2 p 21 + m2 ) sin(2Et ) 2E 3
(10)
,
with g (p) = d2 /π e −( p 1 + p 2 )d . It can be seen clearly that there is a drift term in y¯ (t ) and ZB terms in both x¯ (t ) and y¯ (t ). Beyond that, we can also obtain x¯ (t ), y¯ (t ) by directly solving the Schrödinger’s equation i ∂t |ψ = H |ψ. Both analytical and numerical results are plotted in Fig. 1. The ZB oscillation can be visualized through the evolution of the wave packet. The numerical results are showed in Fig. 2. For numerical convenience, we have set the parameters α , β , m are in units of v F throughout the paper. From the expressions, we can estimate that amplitude of the oscillation is in order of |β/m| (|α /m|) for x¯ ( y¯ ), which equals to the lattice constant in our numerical simulation, by considering only p 1 = 0, p 2 = 0 in the integral. The oscillations are periodically and can be analyzed by Fourier transformation. Simply we can also estimate that the period equals to |π /m| approximately. Since both the amplitude and period are proportional to 1/m, experimentally it is favorable to observe the oscillations by adjusting m. Considering the graphene system is topologically trivial and the wave packets are centered on the (K, K ), which corresponds to the two figures of Fig. 1(a) and Fig. 2(a). In Fig. 1(a), the y¯ oscillates in 2
2
2
Table 1 The signs of the coefficients in Eq. (4) at K (K ). The the topologically nontrivial (trivial) case. ch
K K
ν are the same (opposite) for
+1 (−1)
0
α
βm
ν
α
βm
ν
− +
+ (−) + (−)
− (+) + (−)
− +
+ (−) − (+)
− (+) − (+)
the contrary direction while the x¯ oscillates in the same direction. Thus from Fig. 2(a) we can see that either of the two distributions can be viewed as the image of the other in a mirror. When the system turns topologically nontrivial, Fig. 1(b) shows that both x¯ and y¯ oscillate in the contrary directions. Thus the two density profiles at (K, K ) can be viewed as space reflection partners, which can be seen from Fig. 2(b). Here we have set the magnitude of effective masses at (K, K ) to be equal to make the comparison clearer, which is not true ( M = 0) for the model considered here. Nevertheless, it does not matter since we distinguish different phases by observing the way that the wave packets rotate, which only depends on the signs of (α , β, m) as we will see later. The absolute value of the mass contributes to the amplitude and the period of the oscillation. The above conclusion can be understood by examining the expressions in Eq. (9) and Eq. (10). The sign of x¯ ( y¯ ) is determined by β m (α ), as the integral part consists of the squared terms. All possible cases are listed in Table 1. From the table we can see that the sign of β m (m more specifically) at (K, K ) in the considered model is crucial for us to distinguish the trivial and nontrivial phase. The distinctions of ZB in different phases can be related to the handness of Cartesian coordinate system, as the handness of a two dimensional Cartesian system can be changed by reversing one of the axes. In this sense, a pair of ν = sign(α β m) for (K, K ) can be defined to characterize the ZB for a specific phase. From Table 1, we can see that topological trivial (nontrivial) case corresponds to the same (opposite) ν at (K, K ). Back to Fig. 1, we can also read the information by checking the direction where the center of mass of the wave packet oscillates at the initial time. For example, in Fig. 1(b) both x¯ and y¯ move in the positive direction while the negative direction in the cases of K and K , respectively, which corresponds to the same ν . Thus we can speculate that the system is topologically nontrivial and the Chern number ch = −1 according to the above analysis. 4. Discussion and summary The above results are dependent on the choice of the spinor
Fig. 1. (Color online.) The expectation values of the coordinates for different Dirac points. The horizontal scale is in units of a/ v F . The lines (symbols) are calculated numerically (analytically). (a) The topological trivial case. (b) The topological nontrivial case of ch = −1.
|√. Consider the general √ case that | = a|φ1 + b|φ2 with |φ1 = 1/ 2(1, 1) T , |φ2 = 1/ 2(1, −1) T and a, b being the c-number satisfying |a|2 + |b|2 = 1. We calculate the expectation values of the oscillatory parts in Eq. (5)–Eq. (7). We arrive at
Fig. 2. (Color online.) The snapshots of numerically calculated probability distributions, | (r, t )|2 . The width of the wave packet is d = 2. The probability value is rescaled from 0 to 1. The wave packets at (K, K ) oscillates (a) clockwise and anti-clockwise (ch = 0), (b) both clockwise (ch = −1), (c) both anti-clockwise (ch = +1).
X. Shen et al. / Physics Letters A 381 (2017) 252–256
255
x¯ o (t ) = −β m(|a|2 − |b|2 )
β(ia∗ b − ib∗ a)
dp 1 dp 2 g (p) A 1 (p, t ) +
(11) dp 1 dp 2 g (p) B 1 (p, t ),
y¯ o (t ) = α (|a|2 − |b|2 )
dp 1 dp 2 g (p) A 2 (p, t ) +
αm(ia∗ b − ib∗a)
(12) dp 1 dp 2 g (p) B 2 (p, t ),
with
A 1 (p, t ) = B 1 (p, t ) = A 2 (p, t ) = B 2 (p, t ) =
1 2E 2
sin(2Et ),
(α 2 p 22 + m2 ) 2E 3 (β 2 p 21 + m2 ) 2E 3 1 2E 2
(cos(2Et ) − 1), (13)
(cos(2Et ) − 1),
sin(2Et ).
From the above expressions, we can see that in order to make the sign of the mass term determining the direction of the ZB oscillation explicitly, we need to drop either the (ia∗ b − iba∗ ) terms or (|a|2 − |b|2 ) terms. So we obtain the conditions for the spinor that could manifest the sign change of the mass term, which are
|a| = |b|, Im(b∗ a) = 0
(14)
or
|a| = |b|, Im(b∗ a) = 0.
(15)
Given the above conditions, we shall note that the two differ in a global minus sign. Experimentally it is convenient to choose the a = 1, b = 0, i.e., | = (1, 1) T , which is the case in our paper. If the condition in Eq. (14) is fulfilled, a global minus sign should be considered in the analysis. The ν defined in Table 1 shall be modified by adding the global minus sign. In summary, we have calculated the ZB in the Haldane’s model. In the topologically trivial phase, ZB at (K, K ) rotate in the opposite direction, while the same direction in the nontrivial case. The two different nontrivial phases can be discriminated by different signatures of the oscillations and they can be visualized in ultracold atom system. The results can be seen as the property of the Dirac equation. The topological transition of the system is determined by the effective masses which result in ZB with different kinds of oscillations, hence in general the results are applicable to the system in which the Dirac points characterize the topological property of the system. Acknowledgements We thank Shi-Liang Zhu and D. W. Zhang for useful discussions. This work was supported by the NKRDP of China (Grant No. 2016YFA0301803), the NSFC (Grants No. 11474153 and 91636218), and the PCSIRT (Grant No. IRT1243). References [1] E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. 24 (1930) 418. [2] A.O. Barut, W. Thacker, Covariant generalization of the Zitterbewegung of the electron and its so(4, 2) and so(3, 2) internal algebras, Phys. Rev. D 31 (1985) 1386–1392, http://dx.doi.org/10.1103/PhysRevD.31.1386, http://link.aps.org/doi/ 10.1103/PhysRevD.31.1386. [3] H. Feshbach, F. Villars, Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles, Rev. Mod. Phys. 30 (1958) 24–45, http://dx.doi.org/10.1103/ RevModPhys.30.24, http://link.aps.org/doi/10.1103/RevModPhys.30.24.
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Physics Letters A www.elsevier.com/locate/pla
Visualizing topological orders through Zitterbewegung in ultracold atoms Xin Shen, Feng Mei, Zhi Li ∗ National Laboratory of Solid State Microstructures and School of Physics, Nanjing University, Nanjing 210093, China
a r t i c l e
i n f o
Article history: Received 19 July 2016 Received in revised form 19 September 2016 Accepted 16 October 2016 Available online 20 October 2016 Communicated by V.A. Markel Keywords: Zitterbewegung Topological order Haldane model Ultracold atoms
a b s t r a c t Topological Haldane model recently has been experimentally realized with ultracold atoms. The low energy Hamiltonian in the model can be formally described by the Dirac equation. In this paper, we explore the Zitterbewegung effect of the Dirac quasiparticles in the vicinity of the two nonequivalent Dirac points in the Haldane model. Through analytical and numerical calculation of the evolution of Gaussian wave packets, we find that the Zitterbewegung patterns could be used to characterize the topological phases of the Haldane model. Since the patterns can be detected through standard timeof-flight measurements, our proposal provides a promising scheme to directly visualize the topological phases in ultracold atoms. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Within the frame of relativistic theory, the motion of a free electron is described by the Dirac equation. In 1930 [1], Schrödinger found that such free electron would not only move with a uniform velocity but also oscillate, which he called the Zitterbewegung (ZB). The ZB turns out to be a relativistic effect because it is explained by the interference between the positive and negative energy states. In this sense, the ZB is of fundamental significance in understanding the quantum field theory. It is believed that ZB cannot be observed since both the characteristic amplitude and frequency are beyond the scope that one can detect [2,3] with current technologies. The quantum simulation [4,5], however, provides a new way to approach ZB, which would have the possibility to observe the nontrivial phenomenon. Theoretical predictions have been made in a wide range of systems, including the semiconductor quantum well [6], the ultracold atom system [7,8] and the photonic crystals [9] and a further study of ZB has been extended to the time-modulated Dirac equation [10,11]. A general theory of ZB has been further developed in the context of a multiband system [12], which shows that the couplings between different energy eigenstates are responsible for the appearance of ZB. Experimental observations of ZB phenomenon have also been achieved in trapped ions [13], photons [14] as well as ultracold atoms [15–17].
*
Corresponding author. E-mail address: [email protected] (Z. Li).
http://dx.doi.org/10.1016/j.physleta.2016.10.029 0375-9601/© 2016 Elsevier B.V. All rights reserved.
Topological insulators recently have attracted intense interests in condensed matter physics [18,19]. The classification of such states are beyond the traditional method using symmetry-breaking order parameters and Landau–Ginzburg theory. Nevertheless, a topologically invariant index [20] can be defined to characterize the states and thus leads to concept of topological quantum phase transition (TQPT). Historically, the first discovered topological state is the integer quantum Hall state [21,22], which occurs when electrons are confined to two dimensions and subjected to a strong magnetic field perpendicular to the plane. In 1988, Haldane constructed quantum Hall states with a honeycomb lattice model [23]. Such model does not require an external magnetic field while a complex next nearest neighbor hopping term is introduced in. The low energy Hamiltonian in this model could be written in the form of Dirac equation which lies at the heart of the underling topological Dirac insulator. This discovery directly promoted the birth of topological insulators [24,25]. In the context of ultracold atoms, tremendous theoretical progresses have been made using such system to mimic different classes of topological insulators [26–37], which finally leads to the experimental measurement of Chern number [38] and demonstration of Haldane model [23]. Note that cold atoms are charge neutral and the methods devised in electronic system for measuring topological insulator will not yet be applied here. Some methods recently have been proposed to measure topological phases, including probing atomic density and momentum distribution to further extract topological invariant [28,39–45], or using Bloch oscillation to directly derive Berry phase [46,47]. However, it is still desirable to explore new phys-
X. Shen et al. / Physics Letters A 381 (2017) 252–256
ical features to characterize topological orders in ultracold atoms system. Motivated by the recent experimental demonstration of Haldane model with ultracold atoms trapped in optical lattice, in this paper, we study in detail the relation between ZB and topological phases in the Haldane model. It is well known that the low energy states of the Haldane model should be described by the Dirac equation. In the first Brillouin Zone (BZ), there are two different Dirac points K and K . The physical features near these two Dirac points directly relate to the topological features of the Haldane model, which provides another fashion to find out the topological properties of the system by investigating the Dirac equation whose dynamics can be presented through the ZB. Thus in this sense the two kinds of nontrivial phenomena (ZB and TQPT) are related and we can show that for a proper choice of initial wave packet different phases can be distinguished by the time-evolved distribution of atoms in an optical lattice. In Ref. [48], the authors considered a spin-decoupled topological model and pointed out that at a single valley the ZB of spin-up and spin-down behave differently in different topological phases. The topological Haldane model we are considering here is equivalent to the spin-up (or spin-down) part in Ref. [48] and essentially the topological information is encoded in the mass terms at the two different valleys (K, K ), so our method could be generalized to visualize different classes of topological Dirac insulators. Our study not only connect ZB with topological insulators, despite the fact [49] that the ZB can be expressed in terms of the nondiagonal elements while the Chern number is connected with the diagonal elements, but also may provide an alternative scheme to probe topological orders in ultracold atoms system. The paper is organized as follows. We will first review the Haldane’s model and it’s experimental realization in the ultracold atom system in Sec. 2. Then the ZB of the effective Hamiltonian will be examined analytically in Sec. 3 and we can see that the phases of the system and ZB can be related through the coefficients of the effective Hamiltonian for a specific initial state. The numerical results shall manifest the relationship, which can be experimentally detected through time-of-flight imaging in ultracold atom system. In Sec. 4, we will discuss the proper choice of initial states which is necessary to ensure the relation and a brief summary of our results is given in the end. 2. The experimental realization of Haldane model In the original paper [23], Haldane constructed a honeycomb lattice with the next-nearest-neighbor hopping term being complex which amounts to be two staggered magnetic fluxes while without net flux. Such a scheme breaks the time reversal symmetry that eventually leads to the possibility of the nontrivial phase transition. For a long time, the strange magnetic field has been a challenge in the realistic material in condensed matter systems. In a recent work [50], it is reported that the model has been realized experimentally in the ultra-cold atom system through periodically time-modulated optical lattice. The lattice sites were moved on the elliptical trajectories, which can be effectively seen as the application of an electric field so that the time reversal symmetry breaks. When the frequency is high enough, according to the Floquet theory, the system can well be approximated by a time-independent Hamiltonian
H=
i j
†
t1 ci c j +
i j
†
t 2 e i i j c i c j + A B
†
ci ci .
(1)
i∈ A
The summations in the first and second term correspond to the nearest and next-nearest-neighbors respectively. The hopping terms t 1 and t 2 are real numbers. A B is the energy offset between A and B sublattice. i j is the additional phase when an
253
atom hops from i site to j site, which is a constant and experimentally tunable. By Fourier transformation, the Chern number can then be calculated by the integral of the Berry curvature over the Brillouin zone [18,19]. Theoretically, the topological property can be manifested by the expansion of the Hamiltonian in the reciprocal space at the degenerate points which are the (K, K ) usually labeled for the graphene system. At half-filling, the low-energy excitations are described by the Dirac Hamiltonian
H ± = ∓ v F σ1 p 2 − v F σ2 p 1 + m ± σ3 .
(2)
The minus (plus) sign √ represents the expansion at K (K ). v F
=
3t 1 /2, m± = A B /2 ± 3 3t 2 sin i j . The σi is the Pauli matrix and the lattice constant has been taken to be 1. For the Haldane model, the topological transition can be tracked to the comparison of the signs of the effective masses at (K, K ). If the signs of the two effective masses are contrary, which signals a complete winding of skyrmion in the Brillouin zone [51], the system is topologically nontrivial. Based on this, we then can uncover the topological information of the system by examining the masses. The next section we will show that this can be done through the evolution of the wave packet, which is the ZB in the current context. 3. Zitterbewegung and the topological transition To observe the effect of the ZB with wave packets, we need to calculate expectation value of the coordinate operators. The timedependent position operator in the Heisenberg picture is written as (¯h = 1)
r(t ) = e i Ht r(0)e −i Ht .
(3)
The effective Hamiltonian at K (K ) can be represented by the Hamiltonian
H = ασ1 p 2 + β σ2 p 1 + mσ3 ,
(4)
where the α , β , m represent the corresponding coefficients in Eq. (2). The explicit results of the two coordinate components for both K and K can then be obtained typically as
x(t ) = x(0) + v 1 t + Z 11 (cos(2Et ) − 1) + Z 12 sin(2Et ), y (t ) = y (0) + v 2 t + Z 21 (cos(2Et ) − 1) + Z 22 sin(2Et ), with E = α 2 p 22 + β 2 p 21 + m2 , and v1 = Z 12 =
β2 E2 1
Hp 1 t , Z 11 =
2E 3
1 2E 2
(5)
(α β p 2 σ3 − β mσ1 ),
[(β α 2 p 22 + β m2 )σ2 −
(6)
α β 2 p 1 p 2 σ1 − β 2mp 2 σ3 ], for the x component
v2 = Z 22 =
α2 E2
Hp 2 t , Z 21 =
1 2E 3
1 2E 2
(αmσ2 − α β p 1 σ3 ),
[(α β 2 p 21 + αm2 )σ1 −
(7)
α 2 β p 1 p 2 σ2 − α 2mp 2 σ3 ], for the y component. The first two terms in x(t )[ y (t )] are similar with the counterpart classical kinematics, as expected for a free particle, whereas the last two terms induce ZB. Consider the wave packet initially prepared in the spinor state
1
|ψ(r, t = 0) = √
πd
e
2 + y2
−x
2d2
|,
(8)
254
X. Shen et al. / Physics Letters A 381 (2017) 252–256
with the width of the √ wave packet characterized by d. Currently we set | = (1, 1) T / 2 and the general case will be discussed in Sec. 4. Given the above coordinate operators and the initial state, the calculation of expectation value is direct. r¯i (t ) = ψ|r i (t )|ψ can be obtained as
x¯ (t ) = −β m
dp 1 dp 2 g (p)
y¯ (t ) = α 3 t
dp 1 dp 2 g (p)
α
dp 1 dp 2 g (p)
cos(2Et ) − 1
p 22 E2
(9)
2E 2
+
(β 2 p 21 + m2 ) sin(2Et ) 2E 3
(10)
,
with g (p) = d2 /π e −( p 1 + p 2 )d . It can be seen clearly that there is a drift term in y¯ (t ) and ZB terms in both x¯ (t ) and y¯ (t ). Beyond that, we can also obtain x¯ (t ), y¯ (t ) by directly solving the Schrödinger’s equation i ∂t |ψ = H |ψ. Both analytical and numerical results are plotted in Fig. 1. The ZB oscillation can be visualized through the evolution of the wave packet. The numerical results are showed in Fig. 2. For numerical convenience, we have set the parameters α , β , m are in units of v F throughout the paper. From the expressions, we can estimate that amplitude of the oscillation is in order of |β/m| (|α /m|) for x¯ ( y¯ ), which equals to the lattice constant in our numerical simulation, by considering only p 1 = 0, p 2 = 0 in the integral. The oscillations are periodically and can be analyzed by Fourier transformation. Simply we can also estimate that the period equals to |π /m| approximately. Since both the amplitude and period are proportional to 1/m, experimentally it is favorable to observe the oscillations by adjusting m. Considering the graphene system is topologically trivial and the wave packets are centered on the (K, K ), which corresponds to the two figures of Fig. 1(a) and Fig. 2(a). In Fig. 1(a), the y¯ oscillates in 2
2
2
Table 1 The signs of the coefficients in Eq. (4) at K (K ). The the topologically nontrivial (trivial) case. ch
K K
ν are the same (opposite) for
+1 (−1)
0
α
βm
ν
α
βm
ν
− +
+ (−) + (−)
− (+) + (−)
− +
+ (−) − (+)
− (+) − (+)
the contrary direction while the x¯ oscillates in the same direction. Thus from Fig. 2(a) we can see that either of the two distributions can be viewed as the image of the other in a mirror. When the system turns topologically nontrivial, Fig. 1(b) shows that both x¯ and y¯ oscillate in the contrary directions. Thus the two density profiles at (K, K ) can be viewed as space reflection partners, which can be seen from Fig. 2(b). Here we have set the magnitude of effective masses at (K, K ) to be equal to make the comparison clearer, which is not true ( M = 0) for the model considered here. Nevertheless, it does not matter since we distinguish different phases by observing the way that the wave packets rotate, which only depends on the signs of (α , β, m) as we will see later. The absolute value of the mass contributes to the amplitude and the period of the oscillation. The above conclusion can be understood by examining the expressions in Eq. (9) and Eq. (10). The sign of x¯ ( y¯ ) is determined by β m (α ), as the integral part consists of the squared terms. All possible cases are listed in Table 1. From the table we can see that the sign of β m (m more specifically) at (K, K ) in the considered model is crucial for us to distinguish the trivial and nontrivial phase. The distinctions of ZB in different phases can be related to the handness of Cartesian coordinate system, as the handness of a two dimensional Cartesian system can be changed by reversing one of the axes. In this sense, a pair of ν = sign(α β m) for (K, K ) can be defined to characterize the ZB for a specific phase. From Table 1, we can see that topological trivial (nontrivial) case corresponds to the same (opposite) ν at (K, K ). Back to Fig. 1, we can also read the information by checking the direction where the center of mass of the wave packet oscillates at the initial time. For example, in Fig. 1(b) both x¯ and y¯ move in the positive direction while the negative direction in the cases of K and K , respectively, which corresponds to the same ν . Thus we can speculate that the system is topologically nontrivial and the Chern number ch = −1 according to the above analysis. 4. Discussion and summary The above results are dependent on the choice of the spinor
Fig. 1. (Color online.) The expectation values of the coordinates for different Dirac points. The horizontal scale is in units of a/ v F . The lines (symbols) are calculated numerically (analytically). (a) The topological trivial case. (b) The topological nontrivial case of ch = −1.
|√. Consider the general √ case that | = a|φ1 + b|φ2 with |φ1 = 1/ 2(1, 1) T , |φ2 = 1/ 2(1, −1) T and a, b being the c-number satisfying |a|2 + |b|2 = 1. We calculate the expectation values of the oscillatory parts in Eq. (5)–Eq. (7). We arrive at
Fig. 2. (Color online.) The snapshots of numerically calculated probability distributions, | (r, t )|2 . The width of the wave packet is d = 2. The probability value is rescaled from 0 to 1. The wave packets at (K, K ) oscillates (a) clockwise and anti-clockwise (ch = 0), (b) both clockwise (ch = −1), (c) both anti-clockwise (ch = +1).
X. Shen et al. / Physics Letters A 381 (2017) 252–256
255
x¯ o (t ) = −β m(|a|2 − |b|2 )
β(ia∗ b − ib∗ a)
dp 1 dp 2 g (p) A 1 (p, t ) +
(11) dp 1 dp 2 g (p) B 1 (p, t ),
y¯ o (t ) = α (|a|2 − |b|2 )
dp 1 dp 2 g (p) A 2 (p, t ) +
αm(ia∗ b − ib∗a)
(12) dp 1 dp 2 g (p) B 2 (p, t ),
with
A 1 (p, t ) = B 1 (p, t ) = A 2 (p, t ) = B 2 (p, t ) =
1 2E 2
sin(2Et ),
(α 2 p 22 + m2 ) 2E 3 (β 2 p 21 + m2 ) 2E 3 1 2E 2
(cos(2Et ) − 1), (13)
(cos(2Et ) − 1),
sin(2Et ).
From the above expressions, we can see that in order to make the sign of the mass term determining the direction of the ZB oscillation explicitly, we need to drop either the (ia∗ b − iba∗ ) terms or (|a|2 − |b|2 ) terms. So we obtain the conditions for the spinor that could manifest the sign change of the mass term, which are
|a| = |b|, Im(b∗ a) = 0
(14)
or
|a| = |b|, Im(b∗ a) = 0.
(15)
Given the above conditions, we shall note that the two differ in a global minus sign. Experimentally it is convenient to choose the a = 1, b = 0, i.e., | = (1, 1) T , which is the case in our paper. If the condition in Eq. (14) is fulfilled, a global minus sign should be considered in the analysis. The ν defined in Table 1 shall be modified by adding the global minus sign. In summary, we have calculated the ZB in the Haldane’s model. In the topologically trivial phase, ZB at (K, K ) rotate in the opposite direction, while the same direction in the nontrivial case. The two different nontrivial phases can be discriminated by different signatures of the oscillations and they can be visualized in ultracold atom system. The results can be seen as the property of the Dirac equation. The topological transition of the system is determined by the effective masses which result in ZB with different kinds of oscillations, hence in general the results are applicable to the system in which the Dirac points characterize the topological property of the system. Acknowledgements We thank Shi-Liang Zhu and D. W. Zhang for useful discussions. This work was supported by the NKRDP of China (Grant No. 2016YFA0301803), the NSFC (Grants No. 11474153 and 91636218), and the PCSIRT (Grant No. IRT1243). References [1] E. Schrödinger, Sitzungsber. Preuss. Akad. Wiss. Phys. Math. Kl. 24 (1930) 418. [2] A.O. Barut, W. Thacker, Covariant generalization of the Zitterbewegung of the electron and its so(4, 2) and so(3, 2) internal algebras, Phys. Rev. D 31 (1985) 1386–1392, http://dx.doi.org/10.1103/PhysRevD.31.1386, http://link.aps.org/doi/ 10.1103/PhysRevD.31.1386. [3] H. Feshbach, F. Villars, Elementary relativistic wave mechanics of spin 0 and spin 1/2 particles, Rev. Mod. Phys. 30 (1958) 24–45, http://dx.doi.org/10.1103/ RevModPhys.30.24, http://link.aps.org/doi/10.1103/RevModPhys.30.24.
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