Physical Observable for One-Dimensional Competing Many-Body Interactions

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1. Introduction

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 [1] [2] [3] [4] [5] . Quantum fluctuations, instead of thermal fluctuations, drive such transitions in absolute zero temperature. On the other hand, experiments in quantum simulators, such as ultra-cold atoms in optical lattices and trapped ions can access the dynamical properties of closed quantum many-body systems far from equilibrium [6] [7] [8] [9] . The one-dimensional competing many-body interactions are an important model and had been researched in experiment, however, trapped ions can be used to study exotic quantum phases which do not have a counterpart in nature.

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 [10] [11] [12] [13] [14] in one spatial dimension and the projected entanglement-pair states [15] [16] [17] in two and higher spatial dimensions. The good approximation ground-state wave-function for transitionally invariant quantum systems on an infinite-size lattice can be obtained by the variational algorithm.

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.

2. The Hamiltonian and Ground-State Phase Diagram

The Hamiltonian for the one-dimensional competing many-body interactions [18] is given as the follow

$H={J}_{2}{\displaystyle \underset{j}{\sum}{\sigma}_{j}^{z}{\sigma}_{j+1}^{z}}+{J}_{3}{\displaystyle \underset{j}{\sum}{\sigma}_{j}^{z}{\sigma}_{j+1}^{z}{\sigma}_{j+2}^{z}}-h{\displaystyle \underset{j}{\sum}{\sigma}_{j}^{x}}.$ (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.

${\sigma}_{x}=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\sigma}_{z}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$

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 Figure 1. In this paper, we set h = 1 and J_{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 [19] . The value
$\left|{J}_{3}\right|/h=1$ means a four-fold ferrimagnetic state emerged, which is improved by the symmetry broken state density and shown in Figure 2.

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 Figure 2, we know that the chance for the 20 labeled states of the 80 wavefunctions arrives to the peak, the maximum appears for the 30 labeled states of the 120 wavefunctions.

3. Physical Observable for the Competing Many-Body Interactions

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

Figure 1. Ground state phase diagram [18] of the Hamiltonian (1). The phase AF, F and P are separated with the solid line in different color. The red point is the tri-critical point among the AF, F and P phase.

Figure 2. 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.

(1) are shown in Figure 3 with truncation dimension χ = 4, 8, 16, 32 in different label, respectively. With the truncation dimension is larger and larger, the phase transition point is moving to the point J_{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

$S=a\mathrm{log}\left(\chi \right)+b$ (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 Figure 4. The ground state fidelity is shown in Figure 5.

Figure 3. The local order parameter for the Hamiltonian (1) with h = 1, J_{2} = 0 and J_{3} as the control parameter, the truncation dimensions are shown with χ = 4, 8, 16, 32. The inset figure is the extrapolation for the phase transition point given by the truncation dimension χ = 4, 8, 16, 32. The solid lines are the guide for eye.

Figure 4. The scaling relationship for entanglement entropy of the Hamiltonian (1) with truncation dimension χ = 8, 16, 20, 24, 28, 32, 48, the parameter for the Equation (2) is a = 0.225, b = −0.898. The solid lines are the guide for eye.

4. Summary

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

Figure 5. The ground state fidelity of the Hamiltonian (1) with truncation dimension χ = 4, 8, 16, 32, 64. The bifurcation point given by the fidelity is same as the point given by local order parameter with the corresponding dimension χ. The dot lines are the guide for eye.

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.

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