Introduction

Quantum phase transitions (QPTs) signify that the ground state of the many-body system dramatically changes by varying a physical parameter—such as pressure or magnetic field. The one-dimensional Ising model in a transverse field (ITF)1,2,3,4,5,6 can be used to explain the phenomena of ferromagnetic, ferroelectric and order-disorder transformations. It has been obtained comprehensive study as the simplest exactly solvable model to demonstrate QPTs.

Traditionally, the way of studying the phase transitions is the mean field theory. But researchers have found that the mean field theory results are not in agreement with the experiments because the mean field theory ignores the effect of fluctuation. It is the quantum fluctuation that induces QPTs. One of the most important progresses happened in 1970 when Wilson introduced the concept of renormalization in the quantum field theory to quantum statistical physics7. He used the renormalization theory to investigate the Ising model and derived the universal law of the second order phase transitions. The result is the most revolutionary breakthrough to find the nature of QPTs8.

Besides the direct investigation on the relations between entanglement and QPTs in different systems9,10,11, combining the quantum renormalization group (QRG) method and the quantum entanglement theory to study quantum critical phenomena also has attracted great attention12,13,14,15,16,17,18. Many interesting and meaningful results have been got by studying the low dimensional spin system. M. Kargarian12,13 have found that the derivative of the concurrence between two blocks each containing half of the system size diverges at the critical point. The behavior of the entanglement near the critical point is directly connected to the quantum critical properties. The divergent behavior of the first derivative of the concurrence was accompanied by the scaling behavior in the vicinity of the critical point14.

But things have changed dramatically as some new developments took place that the quantum entanglement cannot be viewed as the whole quantum correlation and the only useful resource in quantum information processing19. One of the important concepts came up in 2001 when Henderson et al. and Ollivier et al. have concluded that entanglement does not account for all nonclassical correlations and even those separable states contain nonclassical correlations that can be demonstrated by quantum discord19,20,21. Inspired by the meaningful results about quantum discord, many similar quantum correlation measures based on information-theoretic have been proposed, such as measurement-induced disturbance, geometric discord, measurement-induced nonlocality, quantum deficit and so on19,22,23,24,25. The investigations on these methods also have obtained much attention26,27,28,29,30,31,32,33,34,35,36,37. Among them one of the topics deserves more attention that is the relations between these quantum correlation measures and QPTs. Some of the questions still remain to be solved: are there some general and special properties when we use these quantum correlation measures to study the QPTs if the QRG theory is adopted? How does the scaling behavior or the critical phenomena change as we take different measures? If the answer can give some universal result, this will be very important and meaningful to know the relations between QPTs and quantum information theory because QPTs can be used to recover the qubit in quantum information processing. This paper is our attempt to solve these questions.

Results

Renormalization of the Hamiltonian

The Hamiltonian of ITF on a periodic chain of N qubits can be written as12

where J is the exchange interaction, are Pauli operators at site i and g is the transverse field strength. In order to implement QRG, one needs to divide the Hamiltonian into two-site blocks. The Hamiltonian then can be resolved into the block Hamiltonian and interacting Hamiltonian , here are the Pauli matrices at site j of the block labelled by I.

The two lowest eigenstates of the corresponding Ith block Hamiltonian is given by

where , and . Now we can establish the relations between the original Hamiltonian and the renormalized one that is . The projection operator can be constructed by using the two lowest eigenstates |ψ0〉 and |ψ1〉, where and are renamed states of each block to represent the effective site degrees of freedom12. The effective Hamiltonian of the renormalized chain is again an ITF model which is similar to the original Hamiltonian H

where the renormalized couplings are

Accordingly, the density matrix of the ground state is given by

In the following, after briefly introducing the definitions of different measures, we will investigate the characteristic of them.

Negativity

Firstly, we use the negativity to calculate the entanglement in this model. According to the Peres-Horodecki criterion, a non-entangled state has necessarily a positive partial transpose (PPT)38. The Peres-Horodecki criterion gives a qualitative way to judge whether the state is entangled or not. Negativity, firstly introduced by Vidal and Werner, is defined by39

where denotes the trace norm of the partial transpose ,

It is easy to compute the negativity of eq. (5)

The properties of entanglement Ne, the first derivative of Ne and the scaling behavior of have been plotted in Fig. 1. It is shown that the negativity develops two different saturated values with the increasing of the size of the system in Fig. 1(a). The saturated value of negativity is zero corresponding to the paramagnetic phase when the magnetic field parameter is in 1 < g ≤ 2.5, while the saturated value of negativity is 0.5 corresponding to the long-ranged ordered Ising phase for 0 ≤ g < 1. Therefore, the two phases are separated by critical point gc = 1. The nonanalytic feature of the first derivative of negativity at the critical point is given in Fig. 1(b). The system exhibits singular property as the number of QRG iterations increases. In order to give a more detailed analysis, the values of the minimum as a function of the system size are depicted in Fig. 1(c) after enough times of iteration. It can be seen that the shows a linear characteristic with ln (N) and gmin gradually approaches to the critical point gc. The relation is . So the critical exponent θ is 1. Referenc 12 have demonstrated that the exponent θ is directly related to the correlation length exponent v close to the critical point. So we can easily compute the correlation length through QRG theory.

Figure 1
figure 1

The negativity (a) and the first derivative of negativity (b) of the model versus g at different quantum renormalization group steps. The logarithm of the absolute value of minimum (c) in terms of system size ln (N).

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Quantum discord and measurement-induced disturbance

Quantum discord (QD)21,29 is defined by the following expression

For an arbitrary bipartite state ρAB, the total correlations are expressed by quantum mutual information

where the mutual information measures the total correlation, including both classical and quantum, for a bipartite state ρAB. Here denotes the von Neumann entropy, with ρA and ρB being the reduced density matrix of ρAB obtained by tracing out A and B, respectively. The classical correlation CC(ρAB) is defined as

where is quantum conditional entropy. The maximum is achieved from a complete set of projective measurements on subsystem B locally.

Measurement-induced disturbance (MID) is defined as the difference of two quantum mutual information respectively of a given state ρAB shared by two parties (A and B) and the corresponding post-measurement state Π(ρAB)25,29

where the mutual information is the same as defined in eq. (10). I(Π(ρ)) quantifies the classical correlation in ρAB with , where the measurement is induced by the spectral resolutions of the reduced states and .

After some standard algebra, we can get QD and MID as

Here , , x = β2 − α2, t1 = 2αβ, t2 = −2αβ, t3 = β2 + α2. The function h(z) can be expressed as

Since the eq. (2) is a pure state and the quantum discord reduces to entanglement entropy (E) in such case. So eq. (13) can also be expressed as . We find that eq. (13) is identical with eq. (14), therefore QD, MID and entanglement entropy are equal in this case. The characteristic of QD & MID, the first derivative of QD & MID and the scaling behavior of have been displayed in Fig. 2. The QD & MID also can be used to discover the critical point gc correlated with QPTs through enough steps of QRG. The difference is that the saturated value of QD & MID is 1 for 0 ≤ g < 1. In addition, the singular behavior of QD & MID is more pronounced than negativity at the thermodynamic limit from Fig. 2(b). The scaling behavior between the minimum value of dQD&MID/dg and the size of the system also can be found in Fig. 2(c). The critical exponent θ of QD & MID are equal to negativity, i.e. with θ = 1.

Figure 2
figure 2

The QD & MID (a) and the first derivative of QD & MID (b) versus g at different quantum renormalization group steps. The logarithm of the absolute value of minimum (c) in terms of system size ln (N).

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Measurement-induced nonlocality and geometric discord

Luo et al. introduced the measurement-induced nonlocality (MIN) in 2011 to quantify the quantum correlation24,29

here is the von Neumann measurements and .

Dakie et al. introduced the geometric measure of quantum discord as22,23,29

where Ω means the set of zero-discord states and is the square of the Hilbert-Schmidt norm.

Applying these formulas to the eq. (5), one gets

where , , , , .

The results of MIN&GQD, the first derivative of MIN&GQD and the scaling behavior of are given in Fig. 3. The MIN&GQD also can detect the critical point. The singular behavior at the vicinity of the critical point can be seen in the Fig. 3(b) and the scaling behavior of exist too. The critical exponent θ of is 1 and have no change.

Figure 3
figure 3

The MIN&GQD (a) and the first derivative of MIN&GQD (b) of the model versus g at different quantum renormalization group steps. The logarithm of the absolute value of minimum (c) in terms of system size ln (N).

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Quantum deficit

For any bipartite state, the quantum deficit (QDe) is defined as the relative entropy of the state ρAB with respect to its classically decohered counterpart as below28

The quantum deficit QDeAB determines the quantum excess of correlations in the state ρAB, with reference to its classical counterpart . The classical state has the same marginal states ρA, ρB as that of ρAB. It is diagonal in the eigenbasis{|a〉, |b〉}of ρA, ρB and the expression is

where stands for the diagonal terms of ρAB and .

So, it is easy to see that , which leads to

where λi signify the eigenvalues of the state ρAB. After some algebra, we can get QDe

The performances of QDe, the first derivative of QDe and the scaling behavior of are plotted in Fig. 4. Just like before, all curves in Fig. 4(a) cross each other at the critical point gc = 1. The saturated values QDe = 0.6931 for 0 ≤ g < 1and 0 for g > 1 after enough steps of renormalization. The singular behavior at the critical point and the scaling behavior of also can be observed if we use QDe to quantify the quantum correlation. The exponent θ of still is 1. It is found that the critical exponent θ does not change with the variation of different measures.

Figure 4
figure 4

The QDe (a) and the first derivative of QDe (b) of the model versus g at different quantum renormalization group steps. The logarithm of the absolute value of minimum (c) in terms of system size ln (N).

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Bell violation

The Bell violation can be adopted to prove the existence of quantum nonlocality. The following expression is the Bell operator corresponding to Clauser-Horne-Shimony-Holt (CHSH) inequality29,40,41,42,43

where a, a′, b, b′ are the unit vectors in and σ = (σx, σy, σz). The CHSH inequality can be expressed as

The maximum violation of CHSH inequality is defined by

So, we can get the analytical result for this model as

where the parameters are exactly the same as eq. (17).

The features of BCHSH, the first derivative of BCHSH and the scaling behavior of are shown in Fig. 5. In Fig. 5(a), it is found that the block-block correlations will violate the CHSH inequality and also exhibit QPTs at the critical point. The saturated values are different from before: one is BCHSH = 2.828 for 0 ≤ g < 1 and the other is 2 for g > 1. The scaling behaviors of convince us that the Bell violation also catches the critical behavior of the ITF due to the nonanalytic behavior of the Bell nonlocality44. The exponents relation of this property is . The values of the critical exponents are identical with before.

Figure 5
figure 5

The BCHSH (a) and the first derivative of BCHSH (b) of the model versus g at different quantum renormalization group steps. The logarithm of the absolute value of minimum (c) in terms of system size ln (N).

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Discussions

In this study, we have combined the methods of quantum correlation and QRG theory to analyze the critical behavior of ITF model. Our results indicate that the critical behavior of the system can be described by quantum correlation or Bell violation. These quantum-information theoretic measures share the same singularity and the same finite-size scaling. The critical exponent which relates with the correlation length exponent will remain the value 1 even with the variation of different quantum correlation measures. Based on numerical computation, we have conjectured that the correlation length can be easily gotten by applying the QRG theory. The similarities and differences between each quantum correlation measures also are given. Furthermore, by applying QRG method the block-block correlations in ITF will violate the CHSH inequality.

Additional Information

How to cite this article: Qin, M. et al. Universal quantum correlation close to quantum critical phenomena. Sci. Rep. 6, 26042; doi: 10.1038/srep26042 (2016).