Multipartite quantum discord and quantum coherence in Heisenberg–Ising bond alternating chain

Multipartite quantum discord and quantum coherence in Heisenberg–Ising bond alternating chain

WAJID JOYIA

Department of Physics, Quaid-i-Azam University, Islamabad, Pakistan E-mail: wajidjoya@yahoo.com

MS received 21 February 2019; revised 25 July 2019; accepted 29 July 2019

Abstract. The behaviour of multipartite quantum discord and quantum coherence in Heisenberg–Ising bond alternating spin-1/2 chain are computed by exploiting the method of quantum renormalisation group (QRG). At larger number of iterations, both quantum coherence and quantum discord have found to exhibit quantum phase transition (QPT) between the spin fluid phase and the Neel phase. In addition, at the critical point, the first derivative of both quantum coherence and quantum discord have shown a non-analytical behaviour. Finally, the scaling behaviour of both quantum coherence and quantum discord has been investigated.

Keywords. Quantum coherence; tripartite quantum discord; quantum phase transition.

PACS Nos 03.65.Ud; 03.65.Ta; 75.10.Jm; 03.67.Mn

1. Introduction

Quantum discord and entanglement (quantum corre- lations) play crucial roles in enacting and employing quantum computation and quantum information [1–3].

In literature, these quantum correlations have been quan- tified with many methods [4–10]. Apart from these quantum correlations, quantum coherence [11–13] has a very vital role in the field of quantum information.

In spite of its importance, it was not addressed prop- erly because there is no method, which is universally accepted, for its measurement. Baumgratzet al[11] and Streltsovet al[12] have diligently overcome these bar- riers and put forward a method to quantify the quantum coherence in any given system.

Quantum phase transition (QPT) [14], a very signif- icant topic in the area of many-body systems, attracted much attention over the last decade. In literature, it is well established that QPT can be probed with non- analytical behaviour of quantum correlations [15–17].

Therefore, numerous attempts have been made to inves- tigate quantum criticality in many-body systems with the help of quantum correlations [18–23]. To study the many-body systems, the method of quantum renormali- sation group (QRG) is observed to be an efficient method [24,25]. This method has been applied successfully to study quantum criticality in well-known quantum spin chains [16,17,26–32]. The Heisenberg–Ising quantum

spin chain is among these quantum spin chains and was the first studied in ref. [33] and then reviewed in refs [34,35] via Jordan–Wigner and Bogoliubov transforma- tions. Recently, this model is solved by QRG iteration technique to study bipartite quantum correlations and quantum phase transition [36,37].

In QRG iteration technique, a block of three spin sites (Kadanoff’s block approach) is used and hence corre- sponding to the block, each degenerate ground state has three spin states. Then, to study quantum correlation between any two spin sites (bipartite quantum correla- tion), one has to partially trace any of the spin site. In fact, we may lose some important information by partial tracing. Therefore, the measurement of bipartite quan- tum correlation for a three-spin site block is not a good option. This motivated us to study the quantum correla- tion in a block of three spin sites without partial tracing.

In the current study, this is achieved by tripartite quan- tum coherence and tripartite quantum discord. Although tripartite quantum discord has several advantages com- pared to bipartite quantum discord, because of the very difficult optimisation procedure, the tripartite quantum discord is very less studied [38,39]. In this work we solved tripartite quantum discord analytically [40] and tripartite quantum coherence byl₁norm.

This work is organised as follows: In §2, we briefly introduce the QRG method to investigate the model.

Section 3 reports the formalism of both quantum

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coherence and tripartite quantum discord. Section4 is devoted to results and discussions. Finally, we conclude in §5.

2. Renormalisation of the model

A Hamiltonian of spin-1/2 Heisenberg–Ising bond alternating chain is given by

H =

N

i=1

Jσ_2i^x₋₁σ_2i^x +Jσ_2i^y₋₁σ_2i^y

+J_zσ_2i^z₋₁σ_2i^z +λσ_2i^zσ_2i^z₊₁, (1) whereNrepresents a finite number of sites correspond- ing to thermodynamics limits of the system andσ^x, σ^y andσ^zare Pauli matrices. The parametersJ, Jzandλin the above Hamiltonian are coupling constants. The first two parameters represent the Heisenberg interaction and the last parameter corresponds to the Ising interaction.

The degenerate ground state of the Hamiltonian are

|0 = 1

√1+a²(a|↑↓↑ + |↓↑↑), (2) ₀

= 1

√1+b²(|↓↑↓ +b|↑↓↓), (3) with

a = −λ+√

4J²+λ²

2J , b= λ−√

4J²+λ²

2J ,

and the corresponding energy is E0 = −Jz−

4J²+λ².

The effective Hamiltonian of the renormalised chain is [37]

H^eff =

N/3

m=1

Jσ_2m−1^x σ_2m^x +Jσ_2m^y ₋₁σ_2m^y

+J_zσ_2m^z ₋₁σ_2m^z +λσ_2m^z σ_2m^z ₊₁, (4) where the renormalised couplings are

J = 4J³

4J²+λ², J_z= Jz, λ= λ³

4J²+λ². (5) To study both Heisenberg and Ising interactions together, we set the ratio of both interaction as

g= λ

J, (6)

wheregis the ratio of the Ising interaction to the Heisen- berg interaction. In renormalised coupling form, the parametergis

g = g³

4 . (7)

The most important advantage of QRG method is that it can provide stable and unstable fixed points. The stable and unstable points define the stable phases and crit- ical point respectively. By solving g = g, we can obtain unstable point at g = 2, and stable point at g =0,∞.

3. Quantum coherence and quantum correlations Here, we discuss the measure of quantum coherence and tripartite quantum discord for the density matrix ρ=ρklm = |00|

ρ= 1

1+a²(a²|↑↓↑ ↑↓↑| +a|↓↑↑

↑↓↑| +a|↑↓↑ ↓↑↑| + |↓↑↑ ↓↑↑|). (8) In agreement with the usual definition of the two-qubit system [3,8], the tripartite quantum discord D³(ρ)can be written as [40]

D³(ρklm)=T³(ρklm)−J³(ρklm). (9) Giorgiet al[40] have showed that the tripartite quantum discord for the pure state is reduced to

D³(ρklm)=J³(ρklm)= T³(ρklm)

=min 2

n [S(ρn)], (10)

wheren = k,l,m,ρn is the reduced density matrix of ρklm and S(ρn) is the Von Neumann entropy. Density matrix of our system (eq. (8)) is pure, and therefore it takes less effort to evaluate the analytical result of the tripartite quantum discord

D³(ρ)= −log

1/(1+a²)

−a²log

a²/(1+a²) log 2+a²log 2 .

(11) The quantum coherence is defined as the absolute sum of the off-diagonal elements of a density matrix [11]

C(ρ)=

i =j

|ρi j|. (12)

For our system, the quantum coherence can be simply derived as

C(ρ)= 2|a|

1+a². (13)

4. Results and discussion

To investigate the quantum phase transition with reference to quantum coherence, eq. (13) is plotted againstg for different QRG iterations in figure1. For zero and second QRG steps, quantum coherence (C(ρ)) decreases gradually and both QRG steps join each other at the critical point g_c = 2. However, with increase in QRG steps, the gradually decreasing behaviour turned to non-analytic one. At the sixth iteration, this gradually decreasing behaviour completely becomes non-analytic withC(ρ)→0 forgc>2 andC(ρ)→1 for 0≤ gc<

2. Here,gc = 2 is the critical point that is exactly the same as we found earlier in renormalised couplings of QRG in §2.

The phases, separated by the critical point, may be called the Neel (zero quantum coherence) phase and the spin fluid (saturated quantum coherence) phase. The non-analytic behaviour of quantum coherence can also be studied from the divergence of its first derivative. For this purpose,(d/dg)(C(ρ))vs.gis plotted in figure2.

Figure 1. The behaviour of quantum coherenceC(ρ)against gin terms of QRG iterations.

Figure 2. The behaviour of(d/dg)(C(ρ))againstgin terms of QRG iterations.

Figure 3. Logarithm of the absolute value of the minimum of ln(|(d/dg)(C(ρ))|min)against ln(|N|).

The figure illustrates that the minimum of the first derivative of quantum coherence attained more promi- nent peak at the critical point with the number of iterations. This emerging singularity is linked to the second-order QPT. Further, the link between the quan- tum coherence and quantum phase transition has been investigated by the finite-size scaling behaviours of quantum coherence. Therefore, in figure 3the scaling behaviour of ln(|(d/dg)(C(ρ))|min) against ln(|N|)is plotted. The plot shows a linear behaviour of ln(|(d/dg) (C(ρ))|min)with system size ln(N).This linear beha- viour is certainly the characteristics of quantum crit- icality of spin systems in thermodynamic limits. The exponent|(d/dg)(C(ρ))|min ∼ N^0.99 depicts the cor- relation length close to the critical point. The value of critical exponent(0.99)obtained with quantum coher- ence is in agreement with the results already existing in [36,37]. This definitely assured the quantum criti- cal behaviour of Heisenberg–Ising model captured with quantum coherence.

Next, we investigate the quantum phase transition with tripartite quantum discord, as an order parameter.

In figure4, the analytically calculated tripartite quantum discord againstgfor different number of QRG iterated steps is plotted. It can be seen that like quantum coher- ence, all plots of tripartite quantum discordD³(ρ)also intersect each other atg_c=2.With increase in the QRG steps, tripartite quantum discord developed two differ- ent behaviour separated atgc, which are linked with the Neel and the spin fluid phases.

The quantum phase transition can also be investigated by diverging the derivative of tripartite quantum discord atg_c.For this purpose, we plot(d/dg)(D³(ρ))against different values ofgfor different numbers of iterations in figure5. The plot shows that in the thermodynamic limit, the minima of (d/dg)(D³(ρ)) becomes more promi- nent atgc= 2.Finally, in figure6, we plot the scaling behaviours of tripartite quantum discord against system

Figure 4. The behaviour ofD³(ρ)againstgin terms of QRG iterations.

Figure 5. The behaviour of (d/dg)(D³(ρ)) against g in terms of QRG iterations.

Figure 6. Logarithm of the absolute value of the minimum of ln(|(d/dg)(D³(ρ))|min)against ln(|N|).

size. The plot shows the linear behaviour of ln(|(d/dg) (D³(ρ))|min)vs. ln(|N|). This shows that the tripartite quantum discord also captured the critical behaviour of the system. Again the value of the critical exponent (0.98) is consistent with the critical exponent of quantum coherence and entanglement for the Heisenberg–Ising model.

5. Conclusion

In the current work, we explored the quantum coherence and tripartite quantum discord in the Heisenberg–Ising bond alternating spin-1/2 chain by using the QRG tech- nique. The result has shown that after sufficient iteration steps, quantum coherence developed two phases, one with saturated coherence (spin fluid phase) and the other without coherence (Neel phase). Similarly, quantum phase transition between these two phases have been acquired with the tripartite quantum discord. In addition, the first derivative of both quantum coherence and tripar- tite quantum discord have shown non-analytic behaviour at the critical point. Finally, our results show that in thermodynamic limit the scaling behaviour of quantum coherence and tripartite quantum discord accompanied the quantum criticality with identical critical exponent.

In the end, we deduced that quantum coherence is not only equivalently as good as the other quantum corre- lations for the study of QPT in many-body system, but also have advantage in its easy computing.

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