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arXiv:2303.03601 · Physical review. E

Lee-Yang zeros and quantum Fisher information matrix in a nonlinear system

Hong Ge Tao, Yuguo Su, Xingyu Zhang, Jing Liu, Xiaoguang Wang

Published Aug 3, 2023• 2 citations
arXivHTMLPDF

Abstract

The distribution of Lee-Yang zeros not only matters in thermodynamics and quantum mechanics, but also in mathematics. Hereby we propose a nonlinear quantum toy model and discuss the distribution of corresponding Lee-Yang zeros. Utilizing the coupling between a probe qubit and the nonlinear system, all Lee-Yang zeros can be detected in the dynamics of the probe qubit by tuning the coupling strength and linear coefficient of the nonlinear system. Moreover, the analytical expression of the quantum Fisher information matrix at the Lee-Yang zeros is provided and an interesting phenomenon is discovered. Both the coupling strength and temperature can simultaneously attain their precision limits at the Lee-Yang zeros. However, the probe qubit cannot work as a thermometer at a Lee-Yang zero if it sits on the unit circle.

DOI
10.1103/physreve.108.024104

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The distribution of Lee-Yang zeros not only matters in thermodynamics and quantum mechanics, but also in mathematics. Hereby we propose a nonlinear quantum toy model and discuss the distribution of corresponding Lee-Yang zeros. Utilizing the coupling between a probe qubit and the nonlinear system, all Lee-Yang zeros can be detected in the dynamics of the probe qubit by tuning the coupling strength and linear coefficient of the nonlinear system. Moreover, the analytical expression of the quantum Fisher information matrix at the Lee-Yang zeros is provided and an interesting phenomenon is discovered. Both the coupling strength and temperature can simultaneously attain their precision limits at the Lee-Yang zeros. However, the probe qubit cannot work as a thermometer at a Lee-Yang zero if it sits on the unit circle.
The distribution of Yang-Lee zeros in the ferromagnetic Ising model in both two and three dimensions is studied on the complex field plane directly in the thermodynamic limit via the tensor network methods. The partition function is represented as a contraction of a tensor network and is efficiently evaluated with an iterative tensor renormalization scheme. The free-energy density and the magnetization are computed on the complex field plane. Via the discontinuity of the magnetization, the density of the Yang-Lee zeros is obtained to lie on the unit circle, consistent with the Lee-Yang circle theorem. Distinct features are observed at different temperatures---below, above, and at the critical temperature. Application of the tensor network approach is also made to the qqq-state Potts models in both two and three dimensions and a previous debate on whether, in the thermodynamic limit, the Yang-Lee zeros lie on a unit circle for q>2q>2q>2 is resolved: they clearly do not lie on a unit circle except at the zero temperature. For the Potts models (q=3,4,5,6)(q=3,4,5,6)(q=3,4,5,6) investigated in two dimensions, as the temperature is lowered the radius of the zeros at a fixed angle from the real axis shrinks exponentially towards unity with the inverse temperature.
Abstract To understand the distribution of the Yang–Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the 2D Yang–Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature β , can be computed in terms of the thermodynamics Bethe Ansatz based on its exact S -matrix. We extract the Yang–Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories. There is though an important difference between the interactive theory and the free theories, for the radius of the zeros in the interactive theory goes continuously to zero in the high-temperature limit β → 0 while in the free theories it remains close to 1 even for small values of β , jumping to 0 only at β = 0 .
Lee-Yang zeros are points on the complex plane of physical parameters where the partition function of a system vanishes and hence the free energy diverges. Lee-Yang zeros are ubiquitous in many-body systems and fully characterize their thermodynamics. Notwithstanding their fundamental importance, Lee-Yang zeros have never been observed in experiments, due to the intrinsic difficulty that they would occur only at complex values of physical parameters, which are generally regarded as unphysical. Here we report the first observation of Lee-Yang zeros, by measuring quantum coherence of a probe spin coupled to an Ising-type spin bath. The quantum evolution of the probe spin introduces a complex phase factor and therefore effectively realizes an imaginary magnetic field. From the measured Lee-Yang zeros, we reconstructed the free energy of the spin bath and determined its phase transition temperature. This experiment opens up new opportunities of studying thermodynamics in the complex plane.
Quantum Fisher Information and Lee‐Yang Zeros with Quantum Coherence of the Isotropic XYXYXYXY Model and the Energy Scale
Abstract Lee‐Yang zeros lie on the complex plane of the control parameters for many‐body systems with finite size, such as the transverse magnetic field and the inverse temperature. Here, an anisotropic spin system with nearest‐neighbor interactions for finite‐size parity space is considered. The Lee‐Yang zeros are obtained in anisotropic model for the complex plane of the transverse magnetic field and in the isotropic model for the complex fugacity plane of the quantum coherence. A characteristic function to represent the probe spin coherence for the isotropic model, and study on Lee‐Yang zeros, quantum coherence, and quantum Fisher information at the complex fugacity plane is presented. The relationship between quantum Fisher information and quantum coherence in decoherence channels is demonstrated and it is illustrated that information cannot be acquired at high temperatures. Meanwhile, the ground‐state energy on the complex plane for the anisotropic model is studied and the free energy cumulants with the partition function of the anisotropic model for parity space is calculated.
There exists the famous circle theorem on the Yang-Lee zeros of the ferromagnetic Ising model. However, the Yang-Lee zeros of the antiferromagnetic Ising model are much less well understood than those of the ferromagnetic model. The precise distribution of the Yang-Lee zeros of the antiferromagnetic Ising model only with nearest-neighbor interaction J on LxL square lattices is determined as a function of temperature a=e(2betaJ) (J<0), and its relation to the phase transitions is investigated. In the thermodynamic limit (L-->infinity), the distribution of the Yang-Lee zeros of the antiferromagnetic Ising model cuts the positive real axis in the complex x=e(-2betaH) plane, resulting in the critical magnetic field +/-H(c)(a), where H(c)>0 below the critical temperature a(c)=square root of 2-1. The results suggest that the value of the scaling exponent y(h) is 1 along the critical line for a<a(c).
Abstract We study the Lee–Yang zeros of the quantum ferromagnetic Ising bath via the interaction with the two probe spins. Similarly, as in an earlier paper (Wei and Liu 2012 Phys. Rev. Lett. 109 , 185701), the problem of detecting the zeros is reduced to the exploration of the time evolution of the probe spins. As a result, the relation between the Lee–Yang zeros of the bath and correlation functions of the probe system is obtained. Also, we obtain the relation between the Lee–Yang zeros and values of the entanglement of the probe spins. We apply these results to the 1D Ising spin model with nearest-neighbor interaction, which can be prepared on trapped atoms.