Magnetic-flux-driven topological quantum phase transition and manipulation of perfect edge states in graphene tube

We study the tight-binding model for a graphene tube with perimeter N threaded by a magnetic field. We show exactly that this model has different nontrivial topological phases as the flux changes. The winding number, as an indicator of topological quantum phase transition (QPT) fixes at N/3 if N/3 equals to its integer part [N/3], otherwise it jumps between [N/3] and [N/3] + 1 periodically as the flux varies a flux quantum. For an open tube with zigzag boundary condition, exact edge states are obtained. There exist two perfect midgap edge states, in which the particle is completely located at the boundary, even for a tube with finite length. The threading flux can be employed to control the quantum states: transferring the perfect edge state from one end to the other, or generating maximal entanglement between them.


Results
Parameter Δ is the hopping integral with zero magnetic field (we take Δ = 1 hereafter for simplification) and φ = πΦ /(NΦ 0 ), where Φ is the flux threading the tube, Φ 0 is flux quantum. In Fig. 1, the geometry of the model is illustrated schematically. We are interested in the effect of φ on the property of the system with small N and large M. To this end, a proper form of solution is crucial. We employ the Fourier transformation , ( 4 1, 4 2) ,   by defining κ = J(1 + δ). The topologically nontrivial phase for the Hamiltonian H K with even M is realized for |κ| < 1. It can only be transformed into a topological trivial phase by either breaking the symmetries which protect it or by closing the excitation gap. We note that κ is the function of φ, which indicates the flux can drive a topological QPT. The present SSH ring described by H K could lead to two transition points at κ = ± 1, respectively, which will be seen from another point of view in the next section.
Topological characterizations. Now we investigate the QPT from another perspective. For each SSH ring, we can represent H K in a simple form , , which obey the Lie algebra relations , . Here the fermion operators , and the components of the field  r K k ( , ) in the rectangular coordinates are . When x 0 = ± 1, the circle crosses the origin, leading to the switch of the winding number of the loops around the origin. Since a given honeycomb tube consists of a set of SSH rings, the topology of the corresponding circles reflects the feature of the system. A straightforward analysis shows that the winding number ν depends on N and flux φ in a simple form: where [x] stands for the integer part of x and φ Λ = N [ 2 sin( )/ 3 ]. It indicates that the topology of the tube state is unchanged if N is a multiple of 3, or jumps between two cases, as φ varies. In Fig. 2, we demonstrate the relationship between ν and the geometry of the loops for several typical N. We have seen that the quantum phases can be characterized by the topology of the loops in the parameter space. It seems that the obtained result is based on the Fourier transformation. However, the theory of topological insulator claims that the topological character is independent of the representation and can be observed in experiment. To demonstrate this point, we consider another representation, which maps the original Hamiltonian to a single ring but with long-range hoppings and is shown in detail in Methods section. The geometrical representation is clear, still consisting of N cycles with unit radius. The centers of these circles locate at another unit-radius circle with equal distance, called circle-center circle. As flux varies, the origin of the parameter space moves along the circle-center circle, resulting in various values of winding numbers. A straightforward analysis shows that although the pattern is different, it presents the same topology with the first representation and Fig. 2 illustrates the patterns for small N.
Control of perfect edge states. The above results indicate that the quantum phase of the model H exhibits topological characterization. Another way to unveil the hidden topology behind the model is exploring the zero modes of the system with open boundary condition. Consider the graphene system with zigzag boundary and its Hamiltonian could be rewritten as Performing the Fourier transformation, H open can be decomposed into N independent SSH chains. The number of zero modes is determined by the sign of |κ| − 1, which leads to the same conclusion as that from the above two geometrical representations. In Fig. 3 we plots the band structures for the systems demonstrated in Fig. 2. It clearly demonstrates the processes of the emergence and disappearance of zero modes. According to the Bethe ansaz results (see Methods section.), the exact wave functions of edge states for a finite N but infinite M tube can be expressed as denotes the position state and represents the edge state at left or right. The features of the edge states are obvious: (i) Nonzero probability is only located at the same one triangular sublattice. Then they have no chirality, without current for any flux, which is different from the square lattice [41][42][43]  can be regarded as the qubit states ↑ and ↓ of a nanoscale qubit protected by the gap or topology. The qubit states can be controlled by varying the flux adiabatically. We take ψ K L c as an initial state, for example. As flux changes adiabatically from φ c , state ψ K L c is separated as two instantaneous eigenstates at the edges of two bands. When φ (τ) = φ c + nπ with n = 0, 1, 2, ..., the evolved state becomes an edge state again. However, it may be the superposition of two edge states, i.e., ψ ψ where the coefficients c L and c R arise from the dynamical phase, c L = cosα, c R = isinα, and , depending on the passage of the adiabatical process. Here E + (t) is the eigenenergy at the edge of positive band. Proper passage allows to obtain the maximal entangled edge Here we demonstrate this process by a simplest case. We consider a graphene tube with N = 3 and M = 4, which can be mapped to three 4-site SSH chains. Initially, we have K c = 2π/3 and φ c = π/6. Two perfect edge states are  When we vary the flux adiabatically from φ (0) = π/6 to φ (τ) = 7π/6, state |L〉 can evolve to |R〉 if For large N, the Zener tunneling occurs, reducing the fidelity of quantum control schemes. In order to demonstrate the scheme, we perform the numerical simulations to compute the evolved wave function We employ the fidelities T to demonstrate the two schemes. Under the assumption of adiabatical process, F E (t) and F T (t) reach unit when σ takes some discrete values, meeting the restrictions for α (τ) and φ (τ). However, Zener tunneling will influence the fidelity as σ increases. We plot the fidelities in Fig. 4 as functions of time with several typical values of σ for N = 3, M = 24. We see that this scheme is achievable with high fidelity even for the quasi-adiabatic process.
To conclude our analysis, we briefly comment on the experimental prospects of detecting topological phase transition, edge states, and quantum state transfer. Artificial honeycomb lattices have been designed and fabricated in semiconductors [44][45][46] , molecule-by-molecule assembly 47 , optical lattices [48][49][50] , and photonic crystals [51][52][53][54] . These offer a tunable platform for studying their topological and correlated phases. According to our analysis above, the topological phase transition in honeycomb lattice is equivalent to that in the SSH model. In fact, the Zak phase which is a phase degree of freedom for the SSH model is measured in reciprocal space by using spin-echo interferometry with ultracold atoms 55 . As for the detection of the perfect edge states found in our work, we propose the following scheme. We note that a perfect zero-mode state is independent of M, i.e., it can appear in a small sized system.
We consider a graphene tube with N = 4 and M = 4, as an example to illustrate the main point. In the absence of flux, there are two perfect edge states which have zero energy and are well separated from other levels. When a small flux φ is switched on, the two degenerate levels are splitted as ± 4φ 2 . In the half-filled case, a small graphene tube acts as a two-level system. This artificial atom can be detected by the absorption and the emission of photons with frequency resonating to 8φ 2 .
For the experimental realization of quantum state transfer, the varying flux affects the fidelity in the following two aspects in practice. (i) The deviation of a magnetic flux from the optimal form in the adiabatic limit. Here we consider a pulse flux as the form where δ F is introduced as a quantity to express the strength of deviation. Figure 5 is the plots of the fidelities under the control of Gaussian pulse fluxes φ (δ F , t), which are obtained by numerical simulations for several typical values of δ F . It indicates that the deviation of the flux would reduce the fidelity for the process of adiabatic transfer. And we believe that a similar conclusion could also be achieved from the process of generating the maximal entanglement. (ii) The speed of varying flux. The adiabatic process requires a sufficiently slow speed of the flux. A fast pulse would induce to Zener tunneling, reducing the fidelity. From Fig. 4, we can see the influence of the speed on the fidelity, which provides a theoretical estimation for experiments.

Discussion
We have proposed two ways to rewrite the Hamiltonian of a flux-threaded graphene tube by the sum of several independent sub-Hamiltonians. Each of them represents a system that may have nontrivial topology or not, which can be determined by the threading flux. For an open tube, such topology emerges as zero-mode edge states according to the bulk-boundary correspondence, which is still controllable by the flux. In addition, we have shown the existence of the perfect edge state even for a small length tube. Such kind of states can be transferred and entangled by adiabatically varying the flux, which has the potential application to design a nanoscale quantum device. There are three advantages as a quantum device: (i) The perfect edge states of two ends are well distinguishable, pointer states, even for a small length tube, (ii) Midgap states are well protected by the gap, (iii) Owing to the finite size effect, the gap between the controlled levels and others is finite for any flux, suppressing the Zener tunneling and allowing the realization of an adiabatic process.  In order to demonstrate the obtained winding numbers in Eq. (7) are topological invariant, we consider another way to exhibit the topological feature of the honeycomb tube. We index the lattice in an alternative way, which is illustrated in Fig. 6  can be infinite, leading to a thermodynamic limit system, which shares the same property as that mentioned in the former sections. Such an arrangement allows us to rewrite the Hamiltonian in the form    The field → = We are interested in the geometry of the curve represented by the above parameter equations. We note that π π and π π which allow us to rewrite the Hamiltonian as k k where the field and pseudo-spin operator are redefined as The equivalent Hamiltonian (18) provides another platform to study the geometry of the band. We find that  r k ( ) is not smooth as k increases continuously from 0 to 4π, which indicates that the curve  r k { ( )} consists of several independent loops. Actually, considering discrete k, we can decompose the set of k into N groups by dividing l into l n = mN + n with m ∈ [0, M − 1], n ∈ [1, N]. Then we have n n n n which could be rewritten as x y Solving these equations, we have While, in the case of κ = 0, from the Hamiltonian (37) it is easy to check that there exist a perfect edge state 1 and two types of highly degenerate eigenstates ± + ( ) m m 2 2 1 / 2 with their eigenenergies 1 (here m ≥ 1). To demonstrate the processes of perfect transfer and entanglement generation for a perfect edge state, we plot the profile of probability distributions at several typical time in Fig. 7. The results are obtained by exact diagonalization.