Spotlighting quantum phase transition in spin -1/2 Ising–Heisenberg diamond chain employing Measurement-Induced Nonlocality

doi:10.1016/j.physa.2021.125932

Physica A: Statistical Mechanics and its Applications

Volume 573, 1 July 2021, 125932

https://doi.org/10.1016/j.physa.2021.125932 Get rights and content

Highlights

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Quantum correlations in a spin-1/2 Ising–Heisenberg diamond chain are studied.
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A simple relationship between Hilbert–Schmidt Measurement-Induced Nonlocality (MIN) and trace distance based MIN is identified.
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Phase transitions between different magnetic phases are identified using MIN.
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The role of DM interaction on quantum correlations and phase transition is studied.

Abstract

We examine thermal quantum correlations characterized by Measurement-Induced Nonlocality (MIN) in an infinite spin-1/2 Ising–Heisenberg spin chain with Dzyaloshinskii–Moriya (DM) interaction. We evaluate MIN analytically in the thermodynamic limit using the transfer matrix approach and show that the MIN and its first-order derivative may spotlight the quantum criticality and quantum phase transition (QPT). We observe that the DM interaction reduces the role of anisotropy parameter in initiating QPT. Further, the DM interaction also induces the nonlocality in the system if the spins are unentangled and greatly enhances the quantum correlations if the spins are correlated. The impact of the magnetic field and temperature on quantum correlations is also brought out at a critical point.

Introduction

Nonlocality, a unique and fundamental characteristic feature of composite quantum systems [1], [2], [3], has been proven to be a useful resource for real-life applications, notably in the domain of secure communication, cryptography, etc. In the realm of Bell nonlocality [4], the violation of any Bell inequality can manifest itself as nonlocality and entanglement can be recognized as one of the most important signatures of nonlocal aspects of quantum states. Any pure entangled two-qubit state violates the Clauser, Horne, Shimony, and Holt (CHSH) inequality, while it is not true for mixed states [5]. Werner showed that the entanglement is not a complete picture of nonlocality [6]. Further, the deterministic quantum computation with one qubit (DQC1) [7] protocol is demonstrated through the separable state (zero entanglement), implying that there exists a special kind of property beyond entanglement, which is also efficient in solving some classically intractable problems. In recent times, the detailed investigation on these domains reveal that the entanglement does not measure the quantum correlations present in the quantum system completely [7]. To address this issue, various measures have been identified to measure the quantum correlations which cannot be grasped by the entanglement [8], [9], [10], [11], [12].

In particular, Luo and Fu introduced a new measure called Measurement-Induced Nonlocality (MIN) [11], which is based on the fact that local disturbance due to von Neumann projective measurements on marginal state can influence globally. Under these local measurements, the marginal states are invariant, and due to the local invariance, MIN is considered to be a secure resource of quantum communications and cryptography. Originally, this quantity is defined in terms of maximal Hilbert–Schmidt distance between the pre- and post-measurement states and is considered to be more general than that of Bell’s version of nonlocality. It is well-known that the correlation quantifiers based on the Hilbert–Schmidt distance are not a faithful measure [13]. In order to resolve the local ancilla problem, different forms of MIN have been introduced [14], [15], [16], [17], [18], [19]

Quantum phase transition (QPT) is defined as a transition between distinct ground states of quantum many-body systems when a controlled or tuning parameter in the Hamiltonian crosses a critical point. In general, QPT can be observed at low temperatures when quantum fluctuations dominate the thermal effects [20]. The condensed matter systems can be considered as a natural playground for manipulating quantum information. The natural mineral azurite Cu₃(CO₃)₂(OH)₂ is an interesting quantum antiferromagnetic system described by the Heisenberg model on a generalized diamond chain [21]. Due to its experimental realization and spectacular magnetic properties, the diamond chain has caught the attention of physicists working on quantum information theory. In recent times, a lot of attention has been focused on understanding the connection between quantum information theory and condensed matter systems. Entanglement not only captures quantum correlations, but also detects phase transition in physical systems [22], [23], [24], [25], [26], [27], [28], [29]. It is found that quantum discord is also helpful in the identification of phase transition [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41].

In the present paper, we consider the spin-1/2 Ising-XXZ diamond chain and construct transfer matrix. We then identify the trace distance based measurement-induced nonlocality as an indicator of QPT in Ising-XXZ diamond chain. We evaluate MIN analytically and demonstrate how this measure could be useful in detecting QPT in Ising-XXZ diamond spin chain. Further, the role of DM interaction and magnetic field on quantum correlations and QPT is also highlighted.

Section snippets

Measurement-induced nonlocality

In recent times, Measurement-Induced Nonlocality (MIN) has been employed as quantum correlation quantifiers. MIN is defined as the maximal nonlocal and global effect due to locally invariant, originally captured via the maximal Hilbert–Schmidt distance between the quantum state of a bipartite system and the corresponding state after performing a local measurement on one subsystem which does not change the state of this subsystem. Mathematically, it is defined as [11] $N_{2} (ρ) =_{Π^{a}}^{max} {‖ ρ - Π^{a} (ρ) ‖}_{2}^{2}$ where

The model

In this section, we introduce the Hamiltonian of the spin-1/2 Ising-XXZ model on a diamond chain with Dzyaloshinskii–Moriya (DM) interaction $D$ in the presence of an external magnetic field $h$ . The model consists of the interstitial Heisenberg spins ${(S_{i}^{a}, S_{i}^{b})}_{Δ}$ and Ising spins located in the nodal site. Fig. 1 exemplifies the schematic representation of Ising-XXZ diamond spin chain. The total Hamiltonian of the model can be written as $H = \sum_{i}^{N} H_{i}$ with $i$ th block Hamiltonian being $H_{i} = J {(S_{i}^{a}, S_{i}^{b})}_{Δ} + J_{1} (S_{z, i}^{a} + S_{z, i}^{b})$

Transfer-matrix (TM) approach

In order to study the quantum correlations and phase transition, we first obtain a partition function for a diamond chain. This model can be solved exactly using the transfer-matrix (TM) approach [42]. In order to introduce TM approach, we will define the following operator as a function of Ising spin particles $μ_{i}$ and $μ_{i + 1}$ : $ϱ (μ_{i}, μ_{i + 1}) = e^{- β H_{i} (μ_{i}, μ_{i + 1})}$ where $H_{i}$ corresponds to the $i^{th}$ block Hamiltonian which depends on the neighboring Ising spins $μ_{i}$ and $μ_{i + 1}$ , $β = 1 ∕ k_{B} T$ , wherein $k_{B}$ is the Boltzmann’s

Results and discussion

It is quite well-known that the system under our investigation exhibits three different magnetic phases such as a frustrated (FRU) state, a ferrimagnetic (FIM) state, and a ferromagnetic (FM) state [43], [44]. Further, in terms of entanglement, there are two phases namely, entangled and unentangled states [42]. It is believed that entanglement is an incomplete manifestation of nonlocality. To capture complete picture of nonlocal aspects of the thermal system given in Eq. (17), we employ trace

Conclusion

To summarize, we have studied the behaviors of thermal quantum correlations captured by measurement-induced nonlocality (MIN) in a spin-1/2 Ising–Heisenberg diamond infinite spin chain. It is shown that MIN has an advantage in spotlighting the quantum critical phenomena in the system. At low temperatures, depending on the external magnetic field, MIN identifies the phase transition from an unentangled state in the ferrimagnetic phase to an entangled state in the ferrimagnetic phase $(h \geq 2)$ , from

CRediT authorship contribution statement

S. Bhuvaneswari: Formulated with necessary theoretical background, Written the paper. R. Muthuganesan: Formulated with necessary theoretical background, Written the paper. R. Radha: Identifined the problem, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Authors thank the reviewers for their critical comments to improve the contents of the paper. SB and RR thank the Council of Scientific and Industrial Research (CSIR), Government of India for the financial support under Grant No. 03(1456)/19/EMR-II. RM acknowledge the financial support from the Council of Scientific and Industrial Research (CSIR), Government of India, under Grant No. 03(1444)/18/EMR-II. RR wishes to acknowledge the DAE-NBHM for the financial support under the scheme

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Spotlighting quantum phase transition in spin -1/2 Ising–Heisenberg diamond chain employing Measurement-Induced Nonlocality

Highlights

Abstract

Introduction

Section snippets

Measurement-induced nonlocality

The model

Transfer-matrix (TM) approach

Results and discussion

Conclusion

CRediT authorship contribution statement

Declaration of Competing Interest

Acknowledgments

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