^{1}

^{2}

^{*}

The one-dimensional 3-spin couplings and 2-spin competing many-body interactions with magnetic fields are an interesting field in quantum system, which is theoretically analysed in an experimental configuration. In this paper, we study the physical observable information for one-dimensional competing many-body interactions. The figure for symmetry broken state density, local order parameter, scaling of entanglement entropy is shown. The results agree with the phase diagram.

Quantum phase transition occurs as a result of a sudden change in the ground state as a system’s parameter (such as external field) is slowly changed [

The numerical simulation algorithm is made much progress in time and space, which is used to understand the collective behavior of quantum many-body systems. More is different. The collective behavior is a formidable challenge due to the exponential growth of Hilbert space dimension with system size. N. G. Vidal and his colleagues had introduced the matrix product states [

In this paper, the figure for symmetry broken state density, local order parameter, scaling of entanglement entropy and ground state fidelity per latter site is obtained by the approximation ground-state wave-function.

The Hamiltonian for the one-dimensional competing many-body interactions [

H = J 2 ∑ j σ j z σ j + 1 z + J 3 ∑ j σ j z σ j + 1 z σ j + 2 z − h ∑ j σ j x . (1)

where J_{2} and J_{3} are the exchange couplings for two-spins and three-spins, respectively, h is the external magnetic field, and σ is the spin-1/2 operator on the site j.

σ x = ( 0 1 1 0 ) , σ z = ( 1 0 0 − 1 )

The competition between the different terms of Hamiltonian determined the ground state. With the exchange couplings J_{2} = 0 or J_{3} = 0, Equation (1) is deduced into one-dimensional quantum Ising model. When exchange coupling J_{2}(>0) induces antiferromagnetic (AF) order, J_{3}(<0) will be shown to induce a novel ferrimagnetic (F) phase, and h(>0) encourages the system to lie in a disordered paramagnetic (P) regime, the phase diagram is shown in _{2} = 0, that is to say, we study the line along the x-direction. The 3-spin interactions induce a novel quantum phase, which is no longer exactly solvable, but shows self-duality properties [

The symmetry broken state density of the Hamiltonian (1) with h = 1, J_{2} = 0 and J_{3} = 2. 8000 approximation ground state wave-functions are obtained with the truncation dimension χ = 4. We pick up 80 wavefunctions and label one state from the four-fold ferrimagnetic state. The times we pick up are 100,000. The line for green-ball is yield. The same method is for 120 wavefunctions with the triangle line. From

The local order parameter is an important observable in phase transition field, which is obtained by order parameter and the good approximation ground state wavefunction. The simulation results of the local order parameter for Equation

(1) are shown in _{3} = 1, which is the phase transition point in theory. The inset figure is the extrapolation for the phase transition point given by the truncation dimension χ = 4, 8, 16, 32, the result is well agree with the J_{3} = 1.

The amount of entanglement can be quantified in terms of the von Neumann entropy, which is known to obey scaling properties for an infinite chain. We may try to find the exact amount of entanglement which is captured in different truncation dimension by using matrix product states. The maximum entanglement entropy for each truncation dimension meets the scaling relationship

S = a log ( χ ) + b (2)

The results are a = 0.225, b = −0.898 with the truncation dimension χ = 8, 16, 20, 24, 28, 32, 48. The scaling relationship for (1) is shown in

The one-dimensional competing many-body interactions are investigated by using matrix product states. Although the amount of entanglement supported by the matrix product states approximation is limited by the size χ of the matrices, we obtained interesting results. The physical observable parameters: local order

parameter and scaling relationship of the entanglement entropy are shown in this paper. Besides, we also give the symmetry broken state density. All the results well agree with the known phase diagram. The numerical work we have performed is qualitatively valid for other models.

The authors declare no conflicts of interest regarding the publication of this paper.

Xiang, C.H. and Wang, H.L. (2019) Physical Observable for One-Dimensional Competing Many-Body Interactions. Open Journal of Applied Sciences, 9, 595-600. https://doi.org/10.4236/ojapps.2019.97047