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International Journal of Modern Physics B

LEE-YANG ZEROS AND QUANTUM PHASE TRANSITIONS OF NONUNIFORM ANISOTROPIC CHAINS IN A TRANSVERSE FIELD

Ming Zhong, Xiaoxian Liu, Peiqing Tong

Published Sep 30, 2007• 1 citation
Publisher

Abstract

The quantum phase transitions of four kinds of nonuniform anisotropic XY quantum spin chains in a transverse field are studied by the Lee-Yang zeros method. It is found that the closed curves of the partition function zeros cut the positive real axis at the quantum phase transition points. The results are checked by the explicit calculations of the magnetization.

DOI
10.1142/s021797920704544x

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The quantum phase transitions of four kinds of nonuniform anisotropic XY quantum spin chains in a transverse field are studied by the Lee-Yang zeros method. It is found that the closed curves of the partition function zeros cut the positive real axis at the quantum phase transition points. The results are checked by the explicit calculations of the magnetization.
The partition function zeros of the anisotropic XY chain in a complex transverse field are studied analytically and numerically. It is found that the partition function zeros of the periodic and quasiperiodic quantum Ising chain lie on the circle at zero temperature and the radius equal to the values of the critical field. For the periodic and quasiperiodic anisotropic XY chains, the closed curves are split to two or three closed curves as the anisotropic parameter gamma decreases at a given ratio of two kinds of exchange interactions. For the isotropic XX case, the partition function zeros lie on the straight segments which are parallel to the real axis and the segments move towards the real axis as the temperature goes to zero.
The zero-temperature phase diagram of a quasiperiodic anisotropic XY\mathrm{XY}XY quantum spin chain in a transverse magnetic field is studied. In the isotropic limit, the system exhibits an infinite number of quantum phase transitions with a devil's-staircase-like magnetization curve. Anisotropy decreases the number of transitions. A mechanism is proposed for the successive transitions based on the ordering of finite clusters of spins along the chain.
We investigate the Lee--Yang zeros (LYZ) of inhomogeneous anisotropic XY spin chains with Dzyaloshinskii--Moriya (DM) interactions in the complex transverse-field plane, focusing on their fundamental connection to quantum phase transitions. We systematically study uniform chains, period-2 and period-3 modulated chains, and Fibonacci quasiperiodic chains of lengths 5 and 8. As the DM coupling strength DDD increases, the LYZ exhibit qualitatively distinct topological evolutions on the complex plane: the complex zeros of the uniform chain collapse toward the real axis; periodic chains feature either bifurcation of closed zero contours before all zeros become real or a single re-emergence of complex zeros; quasiperiodic chains exhibit repeated annihilation and revival of complex zeros. Analytical derivations demonstrate that this diverse behavior originates from DM-induced shifts of folded bands and the modulation of zero positions by anisotropic pairing at particle--hole band crossings. Our results establish a direct correspondence between LYZ topology and band deformation: isolated contact points of zeros with the real axis correspond to discrete quantum critical fields, while continuous real-zero intervals directly identify gapless chiral phases. Accordingly, beyond locating phase boundaries, LYZ can distinguish characteristic phases and serve as an intuitive probe for band folding and phase diagram restructuring.
We study the Yang-Lee theory in quantum phase transitions from the perspective of quantum entanglement in one-dimensional many-body systems. We primarily focus on the distribution of Yang-Lee zeros and its associated Yang-Lee edge singularity of two prototypical models: the Su-Schrieffer-Heeger model and the XXZ spin chain. By taking the zero-temperature limit, we show how the Yang-Lee zeros approach the quantum phase transition points on the complex plane of parameters. To characterize the edge singularity induced by Yang-Lee zeros in a quantum phase transition, we introduce the entanglement entropy of the ground state to show that the edges of Yang-Lee zeros lead to the ground-state entanglement transition. We further show that our results are also applicable to the general noninteracting parity-time-symmetric Hamiltonians.
The spin-1/2 quantum anisotropic XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.
The spin-1/2 quantum anisotropic XY\mathrm{XY}XY spin chain in a transverse random magnetic field parallel to the z axis is numerically studied by means of the density-matrix renormalization group. The dependence of the spontaneous magnetization and the energy gap on both the strength of the random magnetic field and the factor of anisotropy is determined. The critical line for the order-disorder phase transition is obtained and the resulting phase diagram is drawn. Our results are compatible with the fact that models with different factors of anisotropy fall within the same universality class as the quantum Ising model in a transverse random field.