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Register a new userExact Solution of a Yang-Baxter Spin-1/2 Chain Model and Quantum Entanglement
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Entanglement is believed to be crucial in macroscopic physical systems for understanding the collective quantum phenomena such as quantum phase transitions. We start from and solve exactly a novel Yang-Baxter spin-1/2 chain model with inhomogeneous and anisotropic short-range interactions. For the ground state, we show the behavior of neighboring entanglement in the parameter space and find that the inhomogeneous coupling strengths affect entanglement in a distinctive way from the homogeneous case, but this would not affect the coincidence between entanglement and quantum criticality.
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Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any Yang-Baxter solution can be interpreted as a two-qudit quantum gate. Here we show that if this gate is non-entangling, then the resulting invariant of knots is trivial. We thus obtain a general connection between topological entanglement and quantum entanglement, as suggested by Kauffman et al.
An integrable Heisenberg spin chain with nearest-neighbour couplings, next-nearest-neighbour couplings and Dzyaloshinski-Moriya interacton is constructed. The integrability of the model is proven. Based on the Bethe Ansatz solutions, the ground state and the elementary excitations are studied. It is shown that the spinon excitation spectrum of the present system possesses a novel triple arched structure. The method provided in this paper is general to construct new integrable models with next-nearest-neighbour couplings.
Starting from the Kauffman-Lomonaco braiding matrix transforming the natural basis to Bell states, the spectral parameter describing the entanglement is introduced through Yang-Baxterization. It gives rise to a new type of solutions for Yang-Baxter equation, called the type-II that differs from the familiar solution called type-I of YBE associated with the usual chain models. The Majorana fermionic version of type-II yields the Kitaev Hamiltonian. The introduced $ell_1$ -norm leads to the maximum of the entanglement by taking the extreme value and shows that it is related to the Wigners D-function. Based on the Yang-Baxter equation the 3-body S-Matrix for type-II is explicitly given. Different from the type-I solution, the type-II solution of YBE should be considered in describing quantum information. The idea is further extended to $mathbb{Z}_3$ parafermion model based on $SU(3)$ principal representation. The type-II is in difference from the familiar type-I in many respects. For example, the quantities corresponding to velocity in the chain models obey the Lorentzian additivity $frac{u+v}{1+uv}$ rather than Galilean rule $(u+v)$. Most possibly, for the type-II solutions of YBE there may not exist RTT relation. Further more, for $mathbb{Z}_3$ parafermion model we only need the rational Yang-Baxterization, which seems like trigonometric. Similar discussions are also made in terms of generalized Yang-Baxter equation with three spin spaces ${1,frac{1}{2},frac{1}{2}}$.
Usually the $ell_2$-norm plays vital roles in quantum physics, acting as the probability of states. In this paper, we show the important roles of $ell_1$-norm in Yang-Baxter quantum system, in connection with both the braid matrix and quantum entanglements. Concretely, we choose the 2-body and 3-body S-matrices, constrained by Yang-Baxter equation. It has been shown that for 2-body case, the extreme values of $ell_1$-norm lead to two types of braid matrices and 2-qubit Bell states. Here we show that for the 3-body case, due to the constraint of YBE, the extreme values of $ell_1$-norm lead to both 3-qubit $|GHZrangle$ (local maximum) and $|Wrangle$ (local minimum) states, which cover all 3-qubit genuine entanglements for pure states under SLOCC. This is a more convincing proof for the roles of $ell_1$-norm in quantum mechanics.
An integrable anisotropic Heisenberg spin chain with nearest-neighbour couplings, next-nearest-neighbour couplings and scalar chirality terms is constructed. After proving the integrability, we obtain the exact solution of the system. The ground state and the elementary excitations are also studied. It is shown that the spinon excitation of the present model possesses a novel triple arched structure. The elementary excitation is gapless if the anisotropic parameter $eta$ is real while the elementary excitation has an enhanced gap by the next-nearest-neighbour and chiral three-spin interactions if the anisotropic parameter $eta$ is imaginary. The method of this paper provides a general way to construct new integrable models with next-nearest-neighbour interactions.
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