Fidelity zeros and Lee-Yang theory of quantum phase transitions

Abstract

Lee-Yang theory is central to the analysis of thermal phase transitions. However, the underlying mechanism of the theory and the nature of Lee-Yang zeros in quantum many-body systems remains elusive. Here, we develop a unified framework for understanding quantum phase transitions from fidelity zeros induced by symmetry breaking. These zeros, arising from transitions between symmetry sectors, obey the Lee-Yang theorem and give rise to fidelity edges near critical points. Quantum criticality is further characterized through the finite-size scaling of fidelity zeros. As concrete examples, we analytically and numerically investigate fidelity zeros in one- and two-dimensional ferromagnetic Ising models under a complex magnetic field. Our results provide new insights into the mechanism of Lee-Yang theory and open avenues for exploring unexplored landscapes of phase transitions in quantum many-body systems.

Figures (5)

• Figure 1: Geometric structures of one dimensional (1D) and two dimensional (2D) transverse-field quantum Ising models. (a) Quantum Ising model in a chain, (b) Quantum Ising model in a square lattice.
• Figure 2: Non-Hermitian parity-symmetry breaking. (a) Real-part energy difference $\text{Re}(E_{0}^{\text{odd}})-\text{Re}(E_{0}^{\text{even}})$ between the odd- and even-parity symmetric ground state as the function of $\text{Re}(h)$ for $L=10$ at $\text{Im}(h) =0.5$, (b) Fidelity $F(h,h^{\prime})$ with $\delta h=0.01$ as a function of $\text{Re}(h)$ under the same parameters as in (a). Here, the solid symbols represent the numerical results, and the solid and dashed lines act as eye guides.
• Figure 3: Fidelity zeros and finite-size scaling of 1D transverse-field quantum Ising model. (a) Distribution of fidelity zeros in the complex plane $h$ with $L=10$ and $\delta h=10^{-4}$. The scatter red points mark the complex field values at which the fidelity vanishes. (b) Finite-size scaling of the fidelity-zero positions $h_{L}$ as a function of system size $N$ ranging from $10$ to $32$. Here, The grey-red dots denote the complex field values possessing the largest real and smallest imaginary parts. The black dashed line represents the fitted curve. The pink solid dot on the real axis marks the critical value $h_{c1} = 1.0045$, consistent with the theoretical prediction.
• Figure 4: Fidelity zeros and fidelity edges of 1D quantum Ising model. (a) and (b) Distributions of fidelity zeros in the complex field plane for $L=10$ at $g=0.5$ and $g=1.5$. (c) and (d) Distributions of fidelity zeros in the complex field plane for $L=32$ at $g=0.5$ and $g=1.5$. All distributions of fidelity zeros are located at the unit circle $Z = e^{i\varphi}$ with $\varphi = [0, 2\pi)$. For $g=1.5$, branch points dubbed fidelity edges exist. Here $\delta \varphi = 2\pi/400$ is used for computation of fidelity.
• Figure 5: Fidelity zeros and fidelity edges of 2D quantum Ising model. (a) Distribution of fidelity zeros in the complex plane $h$ with $N=3 \times 3$ and $\delta h = 10^{-4}$. The scatter blue points mark the complex field values at which the fidelity vanishes. The critical point is $\text{Re}(h_{L}) \approx 2.57$ for $N=3 \times 3$. (b) Finite-size scaling of the fidelity-zero positions as a function of system size $L$ ranging from $2$ to $5$. Here, The grey-blue dots denote the complex field values with the largest real and smallest imaginary parts. The black dashed line represents the fitted curve. The pink solid dot on the real axis marks the critical value $h_{c2} = 3.05$, consistent with previous results. (c) and (d) Distributions of fidelity zeros in the complex field plane for $g=1.5$ and $g=3.5$ with $N=4 \times 4$, where the distributions of fidelity zeros are located at the unit circle $Z = e^{i \varphi}$ with $\varphi = [0, 2\pi)$. For $g=3.5$, fidelity edges exist. Here $\delta \varphi = 2\pi/400$ is used for computation of fidelity.