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Chinese Physics B

Steered coherence and entanglement in the Heisenberg XX chain under twisted boundary conditions*

Yuhang Sun, Yu-Xia Xie

Published Feb 1, 2021• 1 citation
Publisher

Abstract

We study steered coherence (SC) and entanglement in a three-spin Heisenberg XX model under twisted boundary conditions and show that their strengths can be significantly enhanced by tuning the twist angle. The optimal twist angle θ opt for achieving the maximum l 1 norm of SC is π in the region of weak field B and decreases gradually from π to 0 when B increases after a critical value, while for the relative entropy of SC, θ opt equals π in the weak field region and 0 otherwise. The entanglement and the critical temperature above which the entanglement vanishes can also be significantly enhanced by tuning the twist angle from 0 to π.

DOI
10.1088/1674-1056/abe1a9

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We study steered coherence (SC) and entanglement in a three-spin Heisenberg XX model under twisted boundary conditions and show that their strengths can be significantly enhanced by tuning the twist angle. The optimal twist angle θ opt for achieving the maximum l 1 norm of SC is π in the region of weak field B and decreases gradually from π to 0 when B increases after a critical value, while for the relative entropy of SC, θ opt equals π in the weak field region and 0 otherwise. The entanglement and the critical temperature above which the entanglement vanishes can also be significantly enhanced by tuning the twist angle from 0 to π.
The finite XXZ Heisenberg spin chain with twisted boundary conditions is considered. For the case of an even number of sites N , anisotropy parameter -1/2 and twisting angle 2π/3 the Hamiltonian of the system possesses an eigenvalue -3 N /2. The explicit form of the corresponding eigenvector was found for N ⩽12. Conjecturing that this vector is the ground state of the system we made and verified several conjectures related to the norm of the ground state vector, its component with maximal absolute value and some correlation functions, which have combinatorial nature. In particular, we conjecture that the squared norm of the ground state vector coincides with the number of half-turn symmetric alternating sign N × N matrices.
We consider an open isotropic Heisenberg quantum spin chain, coupled at the ends to boundary reservoirs polarized in different directions, which sets up a twisting gradient across the chain. Using a matrix product ansatz, we calculate the exact magnetization profiles and magnetization currents in the nonequilibrium steady state of a chain with N sites. The magnetization profiles are harmonic functions with a frequency proportional to the twisting angle θ. The currents of the magnetization components lying in the twisting plane and in the orthogonal direction behave qualitatively differently: In-plane steady-state currents scale as 1/N^{2} for fixed and sufficiently large boundary coupling, and vanish as the coupling increases, while the transversal current increases with the coupling and saturates to 2θ/N.
We investigate the entanglement of the three-qubit Heisenberg XXX chain in the presence of impurity and obtain the analytical expressions of the concurrence C.It is found that for impurity entanglement,C appears only when J1 J for J 0,and J1 0 for J 0,and in these two regions C increases with the increase of J1,so is the critical temperature Tc.When J1 |J|,C reaches its maximum value 0.5 and Tc reaches the asymptotic value Tc = 3.41448J1.For entanglement between the normal lattices,C appears only when J 0 and 2J J1 J,and initially increases with the increase of J1 and arrives at the maximum value Cmax =(e4JIT-3)/(e4JIT+3)before it decays to zero gradually,so is the critical temperature Tc with,however,the maximum value Tcmax = 4J/ln3.
In the present thesis, we report several general properties of the enhancement of the entanglement by external fields. First, we investigate the thermal entanglement of interacting two qubits. We maximize it by tuning a local Hamiltonian under a given interaction Hamiltonian. We prove that the optimizing local Hamiltonian takes a simple form which dose not depend on the temperature and that the corresponding optimized thermal entanglement decays as 1/(T log T ) at high temperatures. We also find that at low temperatures the thermal entanglement is maximum without any local Hamiltonians and that the second derivative of the maximized thermal entanglement changes discontinuously at the boundary between the highand low-temperature phases. Second, we investigate the maximized entanglement of indirectly interacting two spins, that is, through other spins. We present a necessary condition for the indirect interaction to give a non-zero maximized entanglement between the focused spins. We also prove that if the focused spins are separated by two spins, there is a critical temperature above which the maximized entanglement between the focused spins vanishes. Then, we numerically calculate the maximized entanglement between the end spins of three-spin chains and four-spin chains. We discover that the maximizing local fields on the end spins have asymmetric forms. In the three-spin chains, we attribute the entanglement enhancement to the asymmetry of the local fields qualitatively and quantitatively in terms of the magnons. In XX and XY four-spin chains, we find that the critical temperature shows qualitatively different behavior depending on the conservation of the angular momentum in the z direction.
Abstract Steered coherence characterizes the ability of one party to steer the coherence of another when they share a bipartite state. It is crucial to control its strength so as to use it as a resource for practical applications. We investigate average steered coherence (ASC) in the Heisenberg XXZ model. It is found that for the thermal state at finite temperature, the ASC can always be noticeably enhanced by adjusting the Dzyaloshinsky–Moriya and Kaplan–Shekhtman–Entin-Wohlman–Aharony interactions. There are also parameter regions in which the ASC approaches to its maximum. This result suggests an alternative for manipulating ASC in a spin system.
The entanglement in an anisotropic spin-1 Heisenberg chain with a uniform magnetic field is investigated. The ground-state entanglement will undergo two different kinds of transitions when the anisotropy Δ and the amplitude of the magnetic field B are varied. The thermal entanglement of the nearest neighbour always declines when B increases no matter what the value of the anisotropy is. It is very interesting to note that the entanglement of the next-nearest neighbour can increase to a maximum at a certain magnetic field. Regardless of the boundary condition, the nearest-neighbour entanglement always decreases and approaches to a constant value when the size of the system is very large. The constant value of open boundary condition is much larger than that of periodic boundary condition.