# Factorization and Criticality in the Anisotropic XY Chain via Correlations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Spin-1/2 Anisotropic XY Chain in a Transverse Field

_{c}= 1 that separates a paramagnetic (disordered) phase from a ferromagnetic (ordered) phase. There is another QPT in the region λ > 1 at γ

_{c}= 0, which is of the Berezinskii–Kosterlitz–Thouless type and separates a ferromagnet ordered phase in the x-direction from a ferromagnet ordered phase in the y-direction. However, this transition will not be deeply explored in this review.

^{2}+ λ

^{−2}= 1. Therefore, the ground state becomes completely factorized at any field point satisfying the equation,

_{2}symmetry. Taking this symmetry into account together with the translational invariance of the system, the reduced density matrix of two spins have an X-shaped form, and in terms of the magnetization and two-spin correlation functions, it can be written as:

_{r}is given as follows:

^{z}⟩ is the transverse magnetization, and the density matrix is written on the basis of the eigenvectors of σ

_{z}. Note that due to the translational invariance of our system, it is not important which spin we trace out. Indeed, all of the single-spin reduced density matrices are the same in this case.

_{c}= 1 in the 0 < γ ≤ 1 region has important consequences regarding the ground state of the system. At the CPof the QPT, Z

_{2}symmetry is broken with the expectation value of the magnetization in the x-direction, ⟨S

_{x}⟩, being the order parameter for this transition with a non-zero value for λ > 1, and the ground state of the system becomes two-fold degenerate. In real physical settings, the system chooses one of the degenerate ground states as the real ground state due to some small external perturbation, which is called the mechanism of spontaneous symmetry breaking (SSB). However, in this case, the ground state of the system does not possess the symmetries of the Hamiltonian, for example the Z

_{2}symmetry in our case, and the reduced density matrix given in Equation (2) no longer has the X-shaped form. In the vast majority of the studies present in the literature, the effect of the broken symmetry is not taken into account, apart from a few exceptions [69,70,75–78], due to the complications it introduces into the calculations. We should note that, throughout this paper, we also neglect the effects of SSB and consider the so-called thermal ground state of the system. The thermal ground state is an equal mixture of these two degenerate states. Indeed, it is nothing but the limit β → ∞ of the canonical ensemble,

## 3. Correlations, Coherence and Uncertainty

#### 3.1. Geometric Measure of Quantum Discord

^{AB}is the density matrix of the total system, ρ

^{a}and ρ

^{b}are the reduced density matrices of the subsystems and S(ρ) = −tr ρ log

_{2}ρ is the von Neumann entropy. On the other hand, it is possible to quantify the classical correlations contained in a quantum system as follows [7]:

^{2}denotes the square of the Hilbert–Schmidt norm.

_{i}: i = 0, 1,⋯, m

^{2}− 1} and {Y

_{j}: j = 0, 1,⋯, n

^{2}− 1}, satisfying tr(X

_{k}X

_{l}) = tr(Y

_{k}Y

_{l}) = δ

_{kl}, define an orthonormal Hermitian operator basis associated with the subsystems a and b, respectively. The components of the local Bloch vectors $\overrightarrow{x}=\{{x}_{i}\},\overrightarrow{y}=\{{y}_{j}\}$ and the correlation matrix T = t

_{ij}can be obtained as:

_{i}} = {0, 2π, 4π} and $\theta =\mathrm{arccos}\{(2\mathrm{tr}{S}^{3}-9\mathrm{tr}S\mathrm{tr}{S}^{2}+9\mathrm{tr}{S}^{3})\sqrt{2/{(3\mathrm{tr}{S}^{2}-{(\mathrm{tr}S)}^{2})}^{3}}\}$. Furthermore, observing that $\mathrm{cos}\left(\frac{\theta +{\alpha}_{i}}{3}\right)$ reaches its maximum for α

_{i}= 0 and choosing θ to be zero, a very tight lower bound to the GMQD can be obtained, and it is given by:

#### 3.2. Measurement Induced Nonlocality

^{a}locally, meaning $\sum}_{k}{\mathrm{\Pi}}_{k}^{a}{\rho}^{a}{\mathrm{\Pi}}_{k}^{a}={\rho}^{a$, and ║.║

^{2}denotes the square of the Hilbert–Schmidt norm. Although MIN, as given by Equation (14), has no closed form formula for an arbitrary bipartite state, it can be calculated analytically for pure states of arbitrary dimension and for 2 × N dimensional mixed states. MIN for a 2 × 2 dimensional system (two-spin system), which will be the focus of this work, can be analytically evaluated as:

^{t}is a 3 × 3 dimensional matrix with λ

_{3}being its minimum eigenvalue, and ${\Vert \overrightarrow{x}\Vert}^{2}={\displaystyle {\sum}_{i}{x}_{i}^{2}}$ with $\overrightarrow{x}={({x}_{1},{x}_{2},{x}_{3})}^{t}$. Due to the special form of the density matrix considered in this work, which is given in Equation (2), and since the local Bloch vector $\overrightarrow{x}$ is never zero in our investigation, MIN takes the simple form:

#### 3.3. Wigner–Yanase Skew Information

- It is upper bounded by the variance with respect to the considered observable, I(ρ, K) ≤ V (ρ, K) = TrρK
^{2}− (TrρK)^{2}, with the equality reached for pure states. - I(ρ, K) is convex, such that:$$I\left({\displaystyle \sum _{i}{\lambda}_{i}{\rho}_{i},K}\right)\le {\displaystyle \sum _{i}{\lambda}_{i}I({\rho}_{i},K),}$$
_{i}is an arbitrary quantum state and λ_{i}is a probability distribution satisfying the constraint ∑_{i}λ_{i}= 1. - For a given bipartite state ρ
^{ab}and an observable K acting on the subsystem a, one has:$$I({\rho}^{ab},K\otimes I)\ge I({\rho}^{a},K).$$

_{AB}, K

_{A}⊗ I

_{B}), which quantifies the coherence with respect to a local observable acting on the first subsystem.

_{max}is the maximum eigenvalue of the 3×3 symmetric matrix W

_{AB}, whose elements are given by:

_{i}} is a family of observables, which constitutes an orthonormal basis. It is possible to use Q(ρ) to capture the total information content of a bipartite quantum system ρ

^{ab}with respect to the local observables of the subsystem a as follows:

_{i}}. As a result, the difference between the information content of the composite system ρ

^{ab}and the Kronecker product of the local subsystems ρ

^{a}⊗ ρ

^{b}with respect to the local observables of the subsystem a can be introduced as a measure for total correlations for ρ

^{ab},

#### 3.4. Concurrence

^{y}is the Pauli spin operator and ρ* is obtained from ρ via complex conjugation. Then, concurrence reads:

_{i}} are the eigenvalues of the product matrix $\rho \tilde{\rho}$ in decreasing order. For the simple form of the reduced density matrix given in Equation (2), concurrence reduces to:

## 4. Behavior of Correlations

_{2}symmetry, a general spin flip symmetry exhibited by most of the spin chain Hamiltonians, but not necessarily exhibited by the ground state density matrix in the ordered phase, as we have mentioned. In order to illustrate the findings, the author has investigated the behavior of correlations in XXZ, Ising and LMG models, both in the thermodynamic limit and in the finite size cases. He has shown that the effects of first-, second- and infinite-order QPTs in these systems can be found in the classical correlations and quantum discord, either via the measure’s or its derivatives behaviors near criticality. Furthermore, the scaling law obeyed by the derivative of the quantum discord was shown to differ compared to that of the entanglement. A comprehensive study of the two-site scaling of quantum discord in the XXZ, XY and Ising models in a transverse magnetic field has been done in [59]. It is important to note that the geometric version of the quantum discord has also been explored in the anisotropic XY model [57]. It has been shown that the derivative of the geometric quantum discord is also singular at the CP, together showing a universal finite-sized scaling behavior.

_{f}, the energy levels of the ground and the first excited states cross, which are of opposite parity. What happens at this point is actually a transition between the different parity ground states. They have also examined the factorization phenomena at finite temperatures and seen that as the temperature increases, the factorization point widens and becomes a region of separability.

#### 4.1. Correlation Measures

_{c}= 1, both measures make a finite jump, resulting in a divergence in their first derivatives, which points to the existence of the second-order QPT of the considered system.

_{f}≈ 1.15, which can be calculated from Equation (1). More interestingly, in all of the remaining three correlation measures considered in this work, only WYSIM is able to detect the existence of the factorized ground state by a non-analytic behavior in its derivative. This is a rather peculiar property of WYSIM, since, as mentioned, we do not take into account the effects of SSB. No other correlation measure in the literature is capable of revealing this phenomena without explicitly considering the effects of SSB with a single evaluation of the measure. The intersection point of the QD calculated using the thermal (symmetry protected) state for different spin distances can detect factorization in this model; however, for WYSIM, a single calculation is sufficient.

#### 4.2. Local Quantum Coherence

_{x}coherence (coherence contained in ρ when measuring σ

_{x}), I(ρ

_{0}, σ

_{x}) and its experimentally-friendly lower bound, I

^{L}(ρ

_{0}, σ

_{x}) in our system, which are presented in Figure 5. We plot these quantities for γ = 0.5 and γ = 1, which is the Ising model in transverse field.

_{x}coherence as a representative.

_{x}coherence and its lower bound decreases as the inverse field λ increases, meaning that in the ordered phase, a randomly chosen spin contains less coherence than it contains the disordered phase. It can be seen from Figure 5b,d that the derivatives of the measures are capable of detecting the presence and the order of the CP by displaying a divergent behavior at λ

_{c}= 1. On the other hand, we see no non-trivial behavior at the factorization field λ

_{f}≈ 1.1547 for γ = 0.5, i.e., the coherence of a single spin behaves smoothly at this point. We do not expect to see any effect of factorization for γ = 1, since in this case, the factorizing field tends to infinity, λ → ∞. It is remarkable and important that even the experimentally-friendly, simplified version of LQC, I

^{L}(ρ

_{0}, σ

_{x}), which does not require the full tomography of the state, can spotlight the CP of the QPT.

_{x}LQC are presented in Figure 6, again for two different values of γ, γ = 0.5 and γ = 1. Recall that in the two-spin case, we calculate the coherence contained in one of the subsystems locally. Therefore, the measurement operator only acts on the chosen subsystem, and the mathematical expression for this is given as I(ρ

_{AB}, σ

_{x}⊗I

_{B}). We have considered nearest neighbor spins in our discussion; however, similar results can also be obtained considering next-nearest neighbor spins. In the left panel, we can see the behavior of the measures and conclude that they seem very similar to that of the single-spin coherence. Both the original definition and its lower bound follow a decreasing trend with increasing inverse field. The derivatives of the measures, presented in the right panel, again signal the existence of the CP via a divergence in their derivatives, thus also giving the correct information about the order of the phase transition. The important difference from the single-spin case is the appearance of a small discontinuity in the original measure for γ = 0.5 at λ = 1.1547. As we have mentioned, this is the critical field in order to observe the factorized ground state of the system for the considered value of γ. While the WYSI can detect the occurrence of this phenomenon, the lower bound for WYSI does not get affected by it. It is rather striking that WYSI itself can signal the factorized ground state, since neither is it an entanglement measure for mixed states, nor have we considered the effects of symmetry breaking on the ground state of the system.

_{f}. First of all, it is important to note that these discontinuities are present for all values of γ, not just for the γ = 0.5 case. The explanation of such a behavior stems from the presence of the square root of the density matrix, $\sqrt{{\rho}^{0r}}$, in the definition of WYSI. The elements of $\sqrt{{\rho}^{0r}}$ are themselves discontinuous at the factorization point, resulting in a physical quantity depending on them to be also discontinuous at the same point. Only correlation measures other than WYSI that can identify the factorization phenomenon in a single evaluation are entanglement measures, such as concurrence and the entanglement of formation, which is itself a function of concurrence. Interestingly, looking at the definition of concurrence, we see that it also involves $\sqrt{{\rho}^{0r}}$ and vanishes at λ

_{f}as a result of the fact that even the thermal ground state is factorized at this point. Furthermore, as soon as we drop the square root of the density matrix, for example as in the case of I

^{L}, we suddenly lose the information on the ground state factorization. We stress that the detection mechanism for the factorization point is not the same as the detection of the CP, since no thermodynamic quantity is discontinuous at this point. To be clear, we do not present an explicitly constructed correspondence between the discontinuities of the elements of $\sqrt{{\rho}^{0r}}$ and the emergence of the factorization point; instead, we point to a possible explanation of the results that we have obtained.

_{x}LQC and its simplified version for γ = 0.5 as a function of temperature. We see that the experimentally-friendly lower bound remains quite accurate up to a temperature kT = 0.1. This is particularly important, since it proves that I

^{L}can be a strong candidate to detect the CP in experimental applications. Figure 7b shows the estimated factorization point calculated from the non-trivial behavior of σ

_{x}, σ

_{y}and σ

_{z}coherence for γ = 0.5 as a function of temperature. All three measures behave very similarly as the temperature increases. They stay extremely accurate until kT = 0.015 and start to deviate significantly from the actual point from kT = 0.02

#### 4.3. Local Quantum Uncertainty

_{c}= 1 and finite discontinuity at λ

_{f}= 1.1547, in its derivative, we can spot the presence of the CP and the factorization point, respectively. However, in Figure 8a,c, there are two extremum points, resulting in discontinuities in the derivatives, at the Hamiltonian parameter values where the system exhibits no non-trivial behavior. The reason behind these non-analyticities in the derivative of LQU is, in fact, due to the optimization procedure involved in the calculation of LQU. An abrupt change in the optimizing observable causes such extremum points in the measure itself. Specifically, in this case, two extremum points of LQU exactly correspond to the values of λ, where the optimizing observable changes from σ

_{z}to σ

_{x}. Correlation measures involving optimization procedures might sometimes cause such ambiguities and display finite discontinuities that are not rooted in the elements of the reduced density matrix. Therefore, when dealing with such measures, one should be careful not to confuse a non-analyticity resulting from the optimization with a QPT.

## 5. Conclusions

_{x}coherence for a single spin, we can identify the QPT by the divergence in the first derivative. Furthermore, the experimentally-friendly simplified version of the coherence measure can also detect the QPT, which is an important result for possible experimental applications. On the other hand, both of them fail to recognize the factorized ground state. Moving on to the two-spin σ

_{x}local coherence, we again observe that the location of the CP can be spotlighted by the divergence in the measure. However, even though the simplified measure again does not feel the presence of the FP, LQC is able to pinpoint the position of the FP. Therefore, we have concluded that the discontinuous behavior of a correlation measure at the FP stems from the non-trivial behavior of the elements of the square root of the density matrix. It is important to note that using different observables from σ

_{x}does not provide any additional insight about the system under consideration. We have then turned our attention to the LQC, which is the optimized version of the LQC over all sets of observables. The divergence of the derivative of LQU is again present at the CP pinpointing the QPT. However, we see sharp extrema points, which do not correspond to any kind of non-triviality of the XY model. A detailed analysis of these points revealed that the extrema points are in fact caused by the change in the optimizing observable in the definition of LQC. Thus, it is important to be careful about the origin of non-analyticities showing-up in the behavior of quantum correlations before reaching conclusions about criticality and factorization.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**(

**a**) The thermal total correlations quantified by measurement-induced nonlocality (MIN) and the Wigner–Yanase skew information-based measure (WYSIM) as a function of λ for γ = 0.001, 0.5, 1 at kT = 0 (solid line), kT = 0.1 (dashed line) and kT = 0.5 (dotted line). (

**b**) The first derivatives of MIN and WYSIM as a function of λ for γ = 0.001, 0.5, 1 at kT = 0 (solid line), kT = 0.1 (dashed line) and kT = 0.5 (dotted line). The graphs are for the first nearest neighbors.

**Figure 2.**(

**a**) The thermal quantum correlations quantified by observable measure of quantum correlations (OMQC) and concurrence as a function of λ for γ = 0.001, 0.5, 1 at kT = 0 (solid line), kT = 0.1 (dashed line) and kT = 0.5 (dotted line). (

**b**) The first derivatives of OMQC and concurrence as a function of λ for γ =0.001, 0.5, 1 at kT = 0 (solid line), kT = 0.1 (dashed line) and kT = 0.5 (dotted line). The graphs are for first nearest neighbors.

**Figure 3.**The estimated values of the CPas a function of kT for three different values of the anisotropy parameter γ = 0.001, 0.5, 1. The CPs in the graphs are estimated by OMQC (denoted by o), WYSIM (denoted by +), MIN (denoted by *) and concurrence (denoted by x). Concurrence is not included for γ = 1 and r = 2, since it vanishes at even very low temperatures.

**Figure 4.**Long-range behavior of the thermal total and quantum correlations for γ = 0.001 and γ = 1 at kT = 0.1, 0.5. The circles, squares, diamonds and triangles correspond to λ = 0.75, λ = 0.95, λ = 1.05 and λ = 1.5, respectively.

**Figure 5.**Single-spin σ

_{x}-coherence for γ = 0.5 (

**a**) and γ = 1 (

**c**), along with its first derivative (with respect to λ) for γ = 0.5 (

**b**) and γ = 1 (

**d**), as a function of λ. As the red solid line denotes the measure, the dashed blue line corresponds to its simplified version.

**Figure 6.**Two-spin local σ

_{z}-coherence for γ = 0.5 (

**a**) and γ = 1 (

**c**), along with its first derivative (with respect to λ) for γ = 0.5 (

**b**) and γ = 1 (

**d**), as a function of λ. As the red solid line denotes the measure, the dashed blue line corresponds to its simplified version.

**Figure 7.**(

**a**) The critical point estimated by single-spin σ

_{x}-coherence (red line) and its simplified version (blue line) as a function of the temperature for γ = 0.5. (

**b**) The factorization field estimated by local two-spin σ

_{x}-coherence (red line), σ

_{y}-coherence (blue line) and σ

_{z}-coherence (green line) as a function of time for γ = 0.5.

**Figure 8.**Two-spin local quantum uncertainty for γ = 0.5 (

**a**) and γ = 1 (

**c**), along with its first derivative (with respect to λ) for γ = 0.5 (

**b**) and γ = 1 (

**d**), as a function of λ.

© 2015 by the authors; licensee MDPI, Basel, Switzerland This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Çakmak, B.; Karpat, G.; Fanchini, F.F. Factorization and Criticality in the Anisotropic XY Chain via Correlations. *Entropy* **2015**, *17*, 790-817.
https://doi.org/10.3390/e17020790

**AMA Style**

Çakmak B, Karpat G, Fanchini FF. Factorization and Criticality in the Anisotropic XY Chain via Correlations. *Entropy*. 2015; 17(2):790-817.
https://doi.org/10.3390/e17020790

**Chicago/Turabian Style**

Çakmak, Barış, Göktuğ Karpat, and Felipe F. Fanchini. 2015. "Factorization and Criticality in the Anisotropic XY Chain via Correlations" *Entropy* 17, no. 2: 790-817.
https://doi.org/10.3390/e17020790