While the zeros of complex partition functions, such as Lee-Yang zeros and Fisher zeros, have been pivotal in characterizing temperature-driven phase transitions, extending this concept to zero temperature remains an open question. In this work, we propose a solution to this issue by calculating the imaginary-temperature zeros (ITZs), which are defined as the roots of the imaginary-temperature partition function. We illustrate the analytical properties of ITZs in the transverse-field Ising chain, showing that the ITZs' distribution can distinguish between various phases and signify the critical exponents. Universal singular behaviors manifest in such quantities as the edge density of ITZs and the magnetization, with the scaling exponents remarkably differing from those in Lee-Yang theory. We further illuminate the consistency between ITZs and the zeros of the spectral form factor, which offers a practical path for the experimental detection of ITZs.
The densities of Yang-Lee zeros for the Ising ferromagnet on the L x L square lattice are evaluated from the exact grand partition functions (L=3 approximately 16). The properties of the density of Yang-Lee zeros are discussed as a function of temperature T and system size L. The three different classes of phase transitions for the Ising ferromagnet--first-order phase transition, second-order phase transition, and Yang-Lee edge singularity--are clearly distinguished by estimating the magnetic scaling exponent yh from the densities of zeros for finite-size systems. The divergence of the density of zeros at Yang-Lee edge in high temperatures (Yang-Lee edge singularity), which has been detected only by the series expansion until now for the square-lattice Ising ferromagnet, is obtained from the finite-size data. The identification of the orders of phase transitions in small systems is also discussed using the density of Yang-Lee zeros.
Lee-Yang theory, based on the study of zeros of the partition function, is widely regarded as a powerful and complimentary approach to the study of critical phenomena and forms a foundational part of the theory of phase transitions. Its widespread use, however, is complicated by the fact that it requires introducing complex-valued fields that create an obstacle for many numerical methods, especially in the quantum case where very limited studies exist beyond one dimension. Here we present a simple and statistically exact method to compute partition function zeros with general complex-valued external fields in the context of large-scale quantum Monte Carlo simulations. We demonstrate the power of this approach by extracting critical exponents from the leading Lee-Yang zeros of 2D quantum antiferromagnets with a complex staggered field, focusing on the Heisenberg bilayer and square-lattice
J-
Q models. The method also allows us to introduce a complex field that couples to valence bond solid order, where we observe extended rings of zeros in the
J-
Q model with purely imaginary staggered and valence bond solid fields.
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.
Concepts of the complex partition functions and the Fisher zeros provide intrinsic statistical mechanisms for finite temperature and real time dynamical phase transitions. We extend the utility of these complexifications to quantum phase transitions. We exactly identify different Fisher zeros on lines or closed curves and elucidate their correspondence with domain-wall excitations or confined mesons for the one-dimensional transverse field Ising model. The crossover behavior of the Fisher zeros provides a fascinating picture for criticality near the quantum phase transition, where the excitation energy scales are quantitatively determined. We further confirm our results by tensor network calculations and demonstrate a clear signal of deconfined meson excitations from the disruption of the closed zero curves. Our results unambiguously show significant features of Fisher zeros for a quantum phase transition and open up a new route to explore quantum criticality.
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.
We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable
u=e−4K, where
K=J/(kBT), for nonzero external magnetic field,
H. Exact results are given for the phase diagram in the
u plane for the model in one dimension and on infinite-length quasi-one-dimensional strips. In the case of real
h=H/(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex
h=H/(kBT) and
μ=e−2h. Calculations of complex-
u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary
h, i.e.,
μ=eiθ, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the
u plane. We find that in this case, the phase boundary
Bu contains a real line segment extending through part of the physical ferromagnetic interval
0≤u≤1, with a right-hand endpoint
urhe at the temperature for which the Yang-Lee edge singularity occurs at
μ=e±iθ. Conformal field theory arguments are used to relate the singularities at
urhe and the Yang-Lee edge.