Introduction

In recent years topological phases have been intensively studied in condensed-matter systems. Their understanding is relevant to topological quantum computation, which provides the paradigm to store and manipulate information in topologically nontrivial systems. The topological states are immune to local noise due to their nonlocal topological nature1,2. The progress was impeded by the fact that non-Abelian anyons are known to appear in p-wave superconductors3,4. Recently, it has been realized that the p-wave pairing can be emulated with s-wave pairing provided with spin-orbit coupling and Zeeman splitting5,6. Especially, this provide the possibility of direct simulation7,8 of Kitaev's 1D p-wave spinless quantum wire model4, which lead to elementary experimental evidence9 for the existence of Majorana fermions. Therefore, these progresses have greatly advanced the field. In addition, The analytic eigenspectra in the 1D Kitaev model have been obtained and it was shown that this model has one gapless topological phase. But the characters of the topological phase from quantum information approach have never been well studied previously.

Quantum phase transition (QPT), originates from the quantum fluctuations at zero temperature, is characterized by nonanalytical changes in the physical properties of the ground state of a many-body system governed by the variation of a parameter of the system's Hamiltonian10. Quantum discord (QD)11 is defined by the difference between two quantum analogues of two classically equivalent expressions of mutual information. The QD has been introduced to analyze the QPT of the spin XY model12 and much effort has been devoted to various quantum critical systems13,14. It is worth to note that other approaches to characterize QPT have also been put forward, such as entanglement15,16, quantum fidelity17, fidelity susceptibility18, geometric phase19,20 and Loschmidt Echo21. Since the calculation of quantum discord is based on numerical maximization procedure, it does not guarantee exact results and in the literature there are few analytical expressions including special cases. To avoid this difficulty, Dakic et al.22 introduced geometric measure of quantum discord (GMQD) which measures the quantum correlations through the minimum Hilbert-Schmidt distance between the given state and zero discord state. Although quantum discord around a QPT for a fermionic lattice hamiltonian has been studied, they only consider the magnetic phase transition in the transformed picture instead of the physical picture. The Kitaev model is related by the Jordan-Wigner transformation to the transverse field Ising model, however, the Jordan-Wigner transformation is fundamentally non local, the information is not robust to local perturbations in the experimental system. This is because the Jordan-Wigner transformed local perturbations become exactly the non-local sort of term that can mess up topologically protected states. The geometric quantum discord of the ground state is naturally expected to shed some light on the understanding of topological QPT.

Traditionally, QPT can be understood in the frame of Landau's continuous phase transitions paradigm with local order parameter and long range correlations. However, it has been found that the local order parameters cannot describe all possible orders23. Therefore, it is interesting to investigate whether this success, describe QPT with geometric quantum discord (GQD) DG, can be extended to the purely topological QPT cases. Here, we explore this question taking the topological QPT in Kitaev's quantum wire model4 as a example. The present work aims to provide a more general understanding of geometric quantum discord in topological phase transitions. The reason why geometric quantum discord can describe topological phases or determine their boundaries is due to the fact that they depend only on the properties of the ground state of the system. We analyze zero temperature and finite-temperature scaling parameters, extracted from the DG of ground state. First, it is found that, in the thermodynamic limit, the derivative of ground-state DG is nonanalytic at the critical point. A finite size scaling analysis is carried out for model with different lattice sites and the scaling behavior is confirmed numerically. This allows us to extract the correlation length critical exponent. To confirm the universality, we also perform a finite-temperature scaling analysis for this model, aiming at extracting the correlation length critical exponent from the scaling behaviors of thermal-state DG. The finite-temperature scaling is manifested when the temperature approaches zero, in analogy with finite-size scaling.

Results

The model and topological phase transition

We consider the typical lattice model of the 1D p-wave superconductor, which is described by the L-site Hamiltonian

where μ is the on-site chemical potential, t is the nearest-neighbor hopping amplitude, Δ is the p-wave superconducting gap and and ci are fermionic operators that satisfy the anticommutation relations . The model has two distinct phases, i.e., topologically trivial and nontrivial phases. Under open boundary condition, the two phases can be distinguished by the presence or absence of zero-energy Majorana bound states at the ends. It is well known that the topologically non-trivial phase of the model is best illustrated for the choice of parametersfor u/t < 2 for any Δ ≠ 0. Without loss of generality, we assume t and Δ are both real and set μ = 2. Therefore, our choice makes the model possess a critical point at t = 1 for any Δ ≠ 0.

GQD close to the phase transition

To illustrate the intrinsic relation between the geometric quantum discord of ground-state and quantum phase transition in this model. We plot GQD DG and its derivative dDG/dt as a function of the Hamiltonian parameters t. As shown in Fig. 1(a), given the value of Δ, the GQD increases with increasing the nearest-neighbor hopping strength t. Meanwhile, it is shown in Fig. 1(b) that the derivatives of GQD (dDG/dt) exhibit sharp peaks around the critical points tc = 1. First, these sharp peaks manifest the nonanalytic property of the GQD at the critical point. Second, this also verified the assertion made by kitaev4 that the critical point will not change with the changing of Δ. This means the GQD of the ground state can signature the criticality in this model, as in the conventional QPT cases.

Figure 1
figure 1

GQD DG of the ground state (a) and its derivatives dDG/dt (b) as a function of t around the critical point tc = 1 for different superconducting gap Δ.

The lattice size L = 1001.

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Finite size scaling

In quantum critical phenomena, the most important themes are scaling and universality. In a finite lattice model, it is expected that the anomalies will become clearer and clearer as the size of the lattice increases. The relevant study is the so-called finite-size scaling. The critical features are characterized in term of a critical exponent ν defined by ξ = (t − tc)−ν with ξ representing the correlation length. To further understand the relation between GQD and quantum criticality, we calculate the derivative of GQD dDG/dt for Δ = 1 and different lattice sizes around the critical point, as shown in Fig. 2(a). For simplicity, we first look at the case of Δ = 1 and we will discuss the properties of the family of Δ ≠ 1 later. In fact, there is no real divergence for finite L, but the curves exhibit marked anomalies and the height of which increases with lattice size. The position of the peak tm can be regarded as a pseudo-critical point which changes and tends as L−1.46 towards the critical point and clearly approaches tc as L → ∞. This scaling behavior of dDG/dt is also clearly shown in in Fig. 2(b).

Figure 2
figure 2

(a)The derivatives of GQD dDG/dt as a function of t near the critical point tc = 1 for different system sizes L = 51, 101, 201, 401, 801, ∞. With increasing the system sizes, the maximum becomes more pronounced. (b) The position of maximum approaches the critical point tc = 1 as tm ~ L−1.46.

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We now further going to deal with the critical exponent that governs divergence of the correlation length. As shown in Fig. 3(a), the singular behavior of dDG/dλ for the infinite chain can be analyzed in the vicinity of the quantum criticality and we find the asymptotic behavior as

where κ1 = −0.0638. On the other hand, As shown in Fig. 3(b), the value of dDG/dt at the point tm diverges logarithmically with increasing lattice size as:

where κ2 = 0.0601. According to the scaling ansatz, the ratio |κ1/κ2| gives the exponent ν. Therefore, ν ~ 1 is obtained in our numerical calculation for the 1D p-wave superconductor model. Furthermore, by proper scaling and taking into account the distance of the maximum of DG from the critical point, it is possible to make all the data for the value of as a function of L1/ν(t − tc) for different L collapse onto a single curve. The result for several typical lattice sizes is shown in Fig. 3(c), where we can also extract the critical exponent ν = 1, in the vicinity of the critical point.

Figure 3
figure 3

(a) Plot of dDG/dt against ln |t − tc| for evaluating the thermodynamic approaching to the critical point tc = 1. The line slope close to the critical point is −0.0638(−0.0716) for Δ = 1.0 (Δ = 0.8). (b) The maximum value of the derivative at the pseudocritical point tm as a function of lattice sizes. The line slope close to the critical point tc = 1 is 0.0601 (0.0699) for Δ = 1.0 (Δ = 0.8). The critical exponent ν for the correlation length is determined by the two slopes in (a) and (b) for a fixed Δ. (c) A finite size scaling analysis is carried out for the value of as a function of L(t − tm) for different lattice sizes. All the data from L = 51 up to L = 401 for a fixed parameter Δ collapse on a single curve. This shows that the system at the critical point is scaling invariant and that the correlation length critical exponent ν = 1.

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Universality

As is well known, 1D p-wave superconductor with nearest neighbor hopping belongs to the same quantum universality class for non-zero Δ, with the same critical exponents ν = 1. To confirm the universality principle in this model, we need to check the scaling behaviors for different values of the parameter Δ. For instance, from Fig. 3(a) and Fig. 3(b) we get κ1 = 0.0716 and κ2 = −0.0699 for Δ = 0.8. Moreover, we also verify that, by proper scaling, all data for different L but a specific Δ and a critical point tc = 1 can collapse onto a single curve. The data for Δ = 0.8 are show in Fig. 3(c). We can extract the same critical exponent ν = 1 from all the above results.

Extended to finite temperatures

We would like to further study the relation between the thermal-state DG and quantum phase transitions at a finite temperature. The derivative of GQD dDG/dt as a function of t at different temperatures T (including zero temperature) are presented in Fig. 4(a). At zero temperature the derivative of DG shows a singularity at tc = 1, but at nonzero temperature, there are no real divergence. Nevertheless, there are clear anomalies at low temperature and the height of which increases with the decrease of the temperature. This can be regarded as the precursors of the QPT. What is more, the position of the maximum derivative (pseudocritical point) tm changes and tends as T1.44 and clearly approaches tc when T → 0, as shown in Fig. 4(b). Meanwhile, the maximum value of dDG/dt at the pseudocritical point tm diverges logarithmically with the decrease of the temperature

Our numerical results, as shown in Fig. 4(c), give κ3 = 0.0611. The ratio of the two slopes (κ1/κ3) for a fixed parameter Δ is equal to the critical exponent ν = 1. Moreover, we also verify that by proper scaling, all data for different temperatures T but a specific Δ will collapse onto the same curve. The data for Δ = 0.8 and Δ = 1.0 near the critical points tc = 1 are shown in Fig. 4(d). The numerical results agree with the finite-size scaling ansatz and the universality of the above results.

Figure 4
figure 4

(a) The derivatives of GQD dDG/dt as a function of the parameter t. The curve corresponds to different temperatures T = 0, 0.01, 0.02 and 0.04. With the decrease of the temperature, the maximum gets pronounced and the pseudopoint tm changes and tends as T1.44 towards the critical point tc = 1 (b). (c)The maximum value of dDG/dt at the pseudocritical point tm of this model as a function of temperature T. The slope of the line is −0.0611 (−0.0717) for Δ = 1(Δ = 0.8). (d) The value of as a function of (t − tm)/T for different temperature.

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It has also been noted that unlike pairwise entanglement, which is typically short ranged, quantum discord does not vanish even for distant lattice pairs. Fig. 5 shows that quantum discord provides the expected long-range behavior of quantum correlations LDG, L = 15 for 1D Kitaev model exhibiting QPTs. In a topological nontrivial case, one can observe stronger quantum discord. The critical exponent ν = 1 for the correlation length is determined by the two slopes in Fig. 5(c) and Fig. 5(d) for a fixed Δ. Such a behavior close to topological QPT is remarkable, even though quantum discord has been shown to be non-vanishing. Concerning QPTs, long-range quantum discord has been considered as an indicator of the critical point of topological QPT in several systems, having succeeded in this task even in situations where entanglement fails, which may shed some light on the understanding of topological phase transition for the other more complex systems from the view of quantum discord. Finally further numerical calculation of the quantum discord shows that for 2D Kitaev model exhibiting topological QPTs in the range of tx + ty = 1, critical exponent ν also gives exactly the same results with the theoretical results z = 2, ν = 1/2.

Figure 5
figure 5

(a)(b) Long-range quantum discord LDG and its derivatives dLDG/dt as a function of the parameter t. (c) dLDG/dt against ln |t − tc| for evaluating the thermodynamic approaching to the critical point tc = 1. (d) The maximum value of the derivative at the pseudocritical point tm as a function of lattice sizes. The critical exponent ν for the correlation length is determined by the two slopes in (c) and (d) for a fixed Δ.

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Discussion

Topological states that are protected from local perturbations need to be supported by quantitative calculations, in particular working with quantum systems of finite sizes. To study the experimental feasibility, we add a random defect perturbation to each hopping amplitude to check the robustness of the topological QPT under the perturbation. Taking the system defects with parameters as given in Fig. 6, the numerical results indicated that the scaling and the critical point of topological QPT are robust against defects. We find topological QPT located in the boundary is robust even the defect amplitude takes δ = 0.1. In summary, we have performed a finite size scaling analysis, whose analytical expression has been extracted from the DG corresponding to different values of the control parameter near the critical point. This makes it possible to extract the correlation length critical exponent. Finally, our study establishes the connection between the DG and topological QPT at both zero temperature and nonzero temperature. All key features of the quantum criticality, such as scaling, critical exponent, the universality are presented. We also would like to point out that the results obtained in this paper does not depend on the model. An interesting question is that when added with next-neighbor tunneling, i.e., , new critical points will appear and this needs further study.

Figure 6
figure 6

(a)(b)A random defect to each hopping amplitude to check the robustness of the topological QPT under the perturbation in Ising model and 1D Kitaev model.

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Added note

In the revise manuscript we find for the same question, Luo etal24 also investigate the fidelity susceptibility (FS) and the topological phase transition in a two-dimensional spin-orbit coupled (SOC) Fermi superfluid.

Methods

Diagonalization of the model hamiltonian

Under the open boundary condition, the Hamiltonian can be immediately diagonalized

by introducing a Bogoliubov-de Gennes (BdG) transformation , where λn denotes the quasi-particle energy and un,i and vn,i can be obtained by solving the corresponding BdG equations. the Bogoliubovde Gennes matrix is written in blocks as with and . In order to obtain the DG in the following, we need to calculate the ground-state correlation functions , , and , written in terms of the amplitude of the BdG transformation as

Computation of DG

An arbitrary two-qubit state can be written in Bloch representation:

where , are components of the local Bloch vectors, σi, i ∈ {1, 2, 3} are the three Pauli matrices and are components of the correlation tensor. For two-qubit case, Then a analytic expression of the DG is given by22

where x = (x1, x2, x3)T and kmax is the largest eigenvalue of matrix K = xxT + RRT.