Abstract Here we employ trace distance and bipartite correlations as the detectors for quantum phase transitions (QPTs). Their performances behaving in Ising spin chains with Dzyaloshinskii–Moriya (DM) interaction and an external transverse field are studied, respectively. Exploiting the quantum renormalization group method, all the detectors can effectively characterize the QPTs at the critical points. Moreover, the block–block correlations existing in the multipartite spin systems never violate the Bell inequality with the DM interaction, whereas they are enough to violate this inequality under the transverse field. Finally, we detail the nonanalytic and scaling behavior of the trace distance, which is tightly connected to the critical features of spin systems.
Abstract With the aid of the quantum renormalization group (QRG) method, we study the multipartite entanglement and quantum phase transitions (QPTs) in the XXZ model with Dzyaloshinskii–Moriya (DM) interaction. The measures employed here are genuine multipartite entanglement (GME-concurrence) and global entanglement. The results show that both multipartite entanglement measures can accurately detect QPTs at the critical points after sufficient iteration steps. By the difference analysis, moreover, we derive an inequality such that the global entanglement is exactly an upper bound of the square GME-concurrence. Actually, this inequality is established whether or not the QRG method is carried out, which discloses some intrinsic features of long-scale multipartite entanglement. Based on the first derivative of GME-concurrence, the non-analytic phenomenon and scaling behaviors with different theoretical exponents are analyzed in detail, showing that the ground states of the XXZ model with DM interaction are asymmetrical.
The ground state of interacting spin chains in external magnetic fields can undergo a quantum phase transition (QPT) characterized by dramatic changes at a critical value of the magnetic field. In this paper, we use Bell-type inequalities to study the multipartite correlations (including multipartite entanglement and multipartite nonlocality in an
n-spin subsystem) in the QPT of an infinite
XY chain. An efficient numerical optimization procedure is proposed to figure out the violation measure
Mn of the inequalities. For $n\ensuremath{\le}7$, the magnetic-field ($\ensuremath{\lambda}$) dependence of
Mn is studied. We find the derivative of
Mn is divergent exactly at the QPT point ${\ensuremath{\lambda}}_{c}=1$ for any
n. In addition, with the increase of
n,
Mn converges quickly for $\ensuremath{\lambda}<{\ensuremath{\lambda}}_{c}$ and converges very slowly for $\ensuremath{\lambda}>{\ensuremath{\lambda}}_{c}$, which can be regarded as another signal for the QPT. Furthermore, in the vicinity of ${\ensuremath{\lambda}}_{c}$, high-order Bell-type inequalities will be violated as long as
n is large enough. This indicates that high-level multipartite correlation will be present when the system is in the vicinity of the QPT point. Nevertheless, genuine
n-partite entanglement or genuine
n-partite nonlocality is not observed in the QPT.
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.
Abstract We investigate quantum phase transitions (QPTs) in various spin chain systems using the multipartite entanglement measure τ S E F based on the monogamy inequality of squared entanglement of formation (EOF). Our results demonstrate that τ S E F is more effective and reliable than bipartite entanglement or bipartite correlation measures such as EOF, von Neumann entropy, and quantum discord in characterizing QPTs. τ S E F not only detects critical points that may go unnoticed by other detectors but also avoids the issue of singularity at non-critical points encountered by other measures. Furthermore, by applying τ S E F , we have obtained the phase diagram for the XY spin chain with three and four interactions and discovered a new quantum phase.
We study the quantum entanglement and quantum phase transition (QPT) of the anisotropic spin-
1/2 XY model with staggered Dzyaloshinskii-Moriya (DM) interaction by means of the quantum renormalization group method. The scaling of coupling constants and the critical points of the system are obtained. It is found that when the number of renormalization group iterations tends to infinity, the system exhibit a QPT between the spin-fluid and N\'eel phases which correspond with two saturated values of the concurrence for a given value of the strength of DM interaction. The DM interaction can enhance the entanglement and influence the QPT of the system. To gain further insight, the first derivative of the entanglement exhibit a nonanalytic behavior at the critical point and it directly associates with the divergence of the correlation length. This shows that the correlation length exponent is closely related to the critical exponent, i.e., the scaling behaviors of the system.
We investigate the quantum phase transitions of spin systems in one and two dimensions by employing trace distance and multipartite entanglement along with the real-space quantum renormalization group method. As illustration examples, a one-dimensional and a two-dimensional XY models are considered. It is shown that the quantum phase transitions of these spin-chain systems can be revealed by the singular behaviors of the first derivatives of renormalized trace distance and multipartite entanglement in the thermodynamics limit. Moreover, we find that the renormalized trace distance and multipartite entanglement obey certain universal exponential-type scaling laws in the vicinity of the quantum critical points.