Quantum Renormalization of Spin Squeezing in Spin Chains

By employing quantum renormalization group (QRG) method, we investigate quantum phase transitions (QPT) in the Ising transverse field (ITF) model and in the XXZ Heisenberg model, with and without Dzyaloshinskii Moriya (DM) interaction, on a periodic chain of N lattice sites. We adopt a new approach called spin squeezing as an indicator of QPT. Spin squeezing, through analytical expression of a spin squeezing parameter, is calculated after each step of QRG. As the scale of the system becomes larger, (after many QRG steps), the ground state (GS) spin squeezing parameters show an abrupt change at a quantum critical point (QCP). Moreover, in all of the studied models, the first derivative of the spin squeezing parameter with respect to the control parameter is discontinuous, which is a signature of QPT. The spin squeezing parameters develop their saturated values after enough QRG iterations. The divergence exponent of the first derivative of the spin squeezing parameter in the near vicinity of the QCP is associated with the critical exponent of the correlation length.

correlations may help to clarify the QPTs in many-body systems. Therefore, it is helpful to investigate the renormalization of various quantum correlation measures as indicators of QPT in every system. Spin squeezing, connected to the quantum correlations between the spins, is one of the most successful approaches to detect the multipartite entanglement in many-spin systems [40][41][42][43][44][45] . Because of important applications, spin squeezing has attracted significant attention as a subject of theoretical and experimental investigations [46][47][48] . The notion of spin squeezing has many advantages where the most important one is the simplicity in generating and measuring it, for instance, in atomic interferometry, weak magnetic fields 49,50 , spin noise in quantum fluids 51,52 , magnetometry with a spinor Bose-Einstein condensate 53 , Ramsey spectroscopy, atom clocks and in quantum computing [54][55][56][57][58][59] . Specially, it improves the precision of experimental measurements [60][61][62] . Another advantage is that the spin squeezing is a multipartite entanglement witness 14,42,63,64 . Moreover, the sensitivity of a state with respect to SU(2) rotations can be characterized by the spin squeezing parameters 65 . Measuring spin squeezing in a many-spin system needs no bipartition or reduction process, unlike some other measures of entanglement, e.g. concurrence or negativity. So, it is easy to be used for every spin model, independent of size, dimension and geometry of the system. Among various definitions of the spin squeezing parameter 40,42 , here we use the most widely studied one, defined by Kitagawa and Ueda 40 .
To the best of our knowledge, there is no report on the renormalization of a spin squeezing parameter in detection of QPT in spin chains, till now. Therefore, our main purpose in this work is to use the QRG method to study the scaling behaviour of the spin squeezing parameter in the vicinity of QCP in some spin-1/2 models.
The paper is organized as follows; We apply the QRG method on a spin-1/2 ITF chain model to investigate the scaling behaviour of the spin squeezing parameter. Then, we briefly review the renormalization of the one dimensional anisotropic XXZ model with DM interaction. We aim to investigate the behaviour of the spin squeezing parameter in detecting QPTs. We also study the renormalization of the spin squeezing parameter in the one-dimensional anisotropic XXZ-Heisenberg model.

Renormalization of the ITF Model
The Hamiltonian of the ITF model 66,67 on a periodic chain of N spin-1/2 sites is where i is the number of the site, J denotes the nearest-neighbor coupling that scales the energy, λ is the strength of transverse magnetic field and σ α i (α = x, y, z) are the usual Pauli matrices of the i-th spin. From the exact solution 68,69 it can be seen that the value of the order parameter, i.e. the magnetization, in the ground state of this model changes from the non-null value for λ < 1, i.e. the ferromagnetic phase, to the null value for λ > 1, in the paramagnetic phase. In other words, system exhibits QPT with the QCP λ c = 1. In order to employ the idea of QRG approach, we use the Kadanoff 's block method for one dimensional spin systems and divide the N-spin chain into N/2 two-spin blocks as shown in Fig. 1.
Then, the total Hamiltonian is rewritten in two parts as The second part is the interblock interactions and is equal to     . We can collect all these relations together and obtain the full renormalized Hamiltonian after one step of renormalization, It is clear that, our choice of QRG transformation preserves the form of the original Hamiltonian, so J (1) and λ (1) are the new renormalized coupling constant and the strength of the transverse magnetic field. One obtains the following iterative relations (1) 2 (1) 2 After setting λ (1) = λ, the resulting fixed point equation is λ* = λ* 2 with a nontrivial fixed point which is the critical point of the ITF model, i.e. λ c = 1. These results were deduced before in several studyings 2,23,30,31 . The n-fold renormalization of the coupling constants can be obtained for a chain with = + N n B n 1 spins, where n B is the number of spins in a block, in the Kadanoff 's block method. For the ITF model, n B = 2.

Renormalization of the Spin Squeezing Parameter in the ITF Model
In this section, we study the spin-1/2 squeezing parameter in the GS of the ITF model. First, we briefly review the definition of the spin squeezing parameter ξ S 2 according to ref. 40 , where N is the number of spins on the chain and  ⊥ n refers to the direction perpendicular to the mean spin direction. Minimization will be done over all directions.
x y z denotes the angular momentum operator of an ensemble of spin-1/2 particles. If ξ < 1 S 2 , the state is squeezed while for the coherent spin state (CSS), ξ S 2 is equal to 1. A spin squeezed state, i.e. ξ < 1 S 2 , is pairwise entangled, while a pairwise entangled state may not be a spin-squeezed state, according to the squeezing parameter ξ S 242 . The components of → J for a N spin-1/2 chain is given by i N i 1 We assume that  = 1 for the sake of simplicity. In continue, we choose one of the degenerated GSs, i.e. |e 1 〉, and calculate the ξ S 2 for this state by substituting N = 2 in the relations. The first step to calculate the parameter ξ S 2 is to calculate the mean spin direction which is obtained as e J e e J e ( , , ) (0, 0, 1) where, ϕ is an arbitrary angle and the following variance is minimized over ϕ. After calculations, we obtain it can be seen that ξ S 2 is a function of the strength of the transverse magnetic field. Consequently, the spin squeezing parameter in the n-th step of the QRG can be written as By substituting λ c = 1 into the Eq. (12), we get ξ = .
In other words, λ c = 1 is the fixed point of the spin squeezing parameter of all different QRG steps.
Spin squeezing parameter versus transverse magnetic field is plotted in Fig. 2 for different QRG steps. The cross point appearing in Fig. 2, represents the QCP, λ c = 1. Spin squeezing parameter develops two different saturated values; ξ = 0 S 2 for λ < 1 and ξ = 1 S 2 for λ > 1. In Fig. 2, the QCP is detected by iterative renormalization steps which justifies that the spin squeezing parameter can be used as an indicator of QPT. We have plotted the first derivative of the spin squeezing parameter with respect to the transverse magnetic field, i.e.  peak versus the size of the system is plotted. It can be seen from the scaling behaviour λ max = λ c + N −0.95 , that λ max goes to λ c at the ther- . Near the critical point, In Fig. 6, we have plotted thermal spin squeezing parameter, ξ λ J T ( , , ) S 2 , for two cases of low and high temperatures. Clearly, at low temperatures, the spin squeezing parameter does not depend on parameter J, while it depends on J at high temperatures.

Renormalization of the XXZ Model with DM Interaction
The Hamiltonian of the one-dimensional anisotropic XXZ model with DM interaction in the z-direction on a periodic chain of N spin-1/2 is where J is the exchange coupling constant, Δ is the anisotropy parameter and D is the strength of the z-component of the DM interaction. To obtain a self-similar Hamiltonian after each QRG step, the chain is divided into three-site blocks, as shown in Fig. 7. The inter-block Hamiltonian for the three site block and the corresponding eigenstates and eigenenergies are given in Appendix A of ref. 70 . It has two degenerate ground states as   where J′, Δ′ and D′ are the new scaled coupling constants given by the following recursion relations

Renormalization of the Spin Squeezing in the XXZ Model with DM Interaction
In this subsection, we calculate the spin squeezing parameter of the XXZ model with DM interaction by considering one of the degenerated GSs. For |ψ 0 〉, one obtains The results are the same if one uses ψ | ′ 〉 0 . In Fig. 8, the evolution of the spin squeezing parameter, ξ S 2 , with QRG steps is plotted as a function of the strength of the DM interaction at Δ = 2 . All plots cross at D = 1, that is in correspondence with the fixed point   Fig. 8, gives the QCP. By increasing the scale of the system, i.e. some QRG iterations, ξ S 2 drops suddenly from the value 1 for D < 1 to the value 0 for D > 1. In Fig. 9, the spin squeezing parameter at ninth QRG step is plotted as a function of D for different values of the anisotropy parameter, Δ = 2 , 5 , 10 , 17. The sudden change of graphs at points D = 1, 2, 3, 4 correspond-   , at Δ = 2 in Fig. 12 for some QRG steps. It can be seen from Fig. 12, there is a peak for each QRG step with position D max that tends to D = 1 at the thermodynamic limit.
In Fig. 13, the absolute value of D max − D c is plotted versus the size of the system. Figure 13 illustrates that the position of the maximum of

Renormalization of the XXZ Model
The Hamiltonian of the one dimensional anisotropic XXZ model on a periodic N spin-1/2 chain is The recursion relations of the QRG flow are where J′ and Δ′ are the renormalized couplings. By solving Δ′ = Δ ≡ Δ c , the stable and unstable fixed points of the QRG equations can be obtained. The critical point of this model is located at Δ c = 1

Renormalization of the Spin Squeezing in the XXZ Model
We can calculate the spin squeezing parameter of the XXZ model for the GS by substituting D = 0 into Eq. (24). Then, one obtains for ξ S In Fig. 16, the evolution of the spin squeezing parameter after n-th QRG steps is plotted as a function of the anisotropy parameter, Δ. All plots cross at Δ = 1, that is in correspondence with a previous study 71 . The scale-free point of Fig. 16, gives the QCP. By increasing the scale of the system, i.e. after QRG iterations, the spin squeezing parameter changes suddenly from a value between zero and one for Δ < 1 to one for Δ > 1.
For more details of the critical behaviour, we study the evolution of the first derivative of the spin squeezing parameter with QRG steps as a function of the anisotropy parameter, i.e.; , as is plotted in Fig. 17. As it can be seen from the figure, there is a peak for each QRG step with position Δ max that tends to Δ = 1 at the thermodynamic limit.
In Fig. 18, the absolute value of Δ max − Δ c is plotted versus the size of the system. Figure 18 illustrates that Δ max scales as |Δ The numerical results that is plotted in Fig. 19, show the behaviour of the absolute value of the maximum of The simple XXZ model has a critical line that can not be detected without imposing the effect of boundary conditions on each block Hamiltonian. When we add the DM interaction to the XXZ model, its critical behaviors become detectable by studying the spin squeezing parameter, which are in agreement with the other analytical or numerical results.
The XXZ model on a periodic chain of N spin-1/2, has a critical line for Δ ⩽ ⩽ 0 1 , which is not detected by our approach. Here, we use another QRG method which uses the concept of quantum groups, proposed by Delgado, et al. 68 . The renormalization of entanglement in this method used by Kargarian et al. 24 , leads to the detection of the critical line of the XXZ model. This approach is suggested by the fact that quantum groups describe symmetries in the presence of appropriate boundary conditions, e.g. boundary magnetic fields. The main where q is an arbitrary complex parameter. To get a self-similar Hamiltonian, the system is divided to three spin blocks as shown in Fig. 8. The inter-block Hamiltonian and the intrablock Hamiltonian are as follows:  Figure 18. The scaling behaviour of |Δ − Δ | c max versus the size of the system N, where Δ max is the position of the peak in Fig. 16. The effective Hamiltonian with the renormalized coupling constants is , for q being a complex number and 0 < Δ < 1), the coupling constant q can be written as a pure phase. We can calculate the spin squeezing parameter of the GS, |ψ 0 〉. The result is ξ = . Because of the fixed value of the spin squeezing parameter in the region 0 < Δ < 1, it does not evolve under QRG transformation which is a signature of the critical line of the XXZ model in this region.
It is also noticable that the size scaling of the in Figs 13, 14, 18 and 19 is not logarithmic as may be expected based on the fact that the corresponding QPT is Kosterlitz-Thouless type. The power scaling behaviour was also seen in the maximum of the fidelity susceptibility, which is in close correspondence with ξ Δ d d S 2 in our study, while the maximum in the entanglement entropy itself shows logarithmic scaling 72 . This suggests that the scaling behaviour is measure and method dependent.
The main requirement of the QRG method is that after renormalization the effective Hamiltonian must be similar to the original Hamiltonian, i.e. the system is renormalizable. It is not clear that if renormalization can yield the physical properties of QPTs precisely for every model system, thought some models are not even renormalizable. The models we have selected here are particular ones where the method works confidently. Counter example are spin-1 XYZ Heisenberg model with DM interaction or spin-1 isotropic bilinear biquadratic Heisenberg model in the presence of magnetic field. For summary, we have gathered some examples of renormalizable spin models and QPT indicators in QRG approach in Table 1. Investigation of critical behaviours of a non trivial example such as XXZ Heisenberg model with single ion anisotropy in a one dimensional spin 1 chain, is under debate by authors of this paper.

Conclusion
In this work the relation between the spin squeezing and QPT is addressed based on the QRG procedure. We have used the idea of QRG to study the quantum information properties of the ITF, the XXZ with and without DM interaction. Spin squeezing is used as the witness of quantum entanglement. In order to explore the critical behaviour of the spin models, the evolution of the spin squeezing parameter with renormalization of the model is examined. As the number of QRG iterations increases, the spin squeezing parameter develops its saturated values in both sides of the QCP. The first derivative of the spin squeezing parameter diverges close to the QCP as the scale of the system becomes larger. In the near vicinity of the QCP, the critical exponent of the spin squeezing parameter is in correspondence with the critical exponent of the correlation length. 23