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We describe all linear operators on spaces of multivariate polynomials preserving the property of being non-vanishing in open circular domains. This completes the multivariate generalization of the classification program initiated by Polya-Schur for univariate real polynomials and provides a natural framework for dealing in a uniform way with Lee-Yang type problems in statistical mechanics, combinatorics, and geometric function theory. This is an announcement with some of the main results in arXiv:0809.0401 and arXiv:0809.3087.
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Lee-Yang zeros are points on the complex plane of magnetic field where the partition function of a spin system is zero and therefore the free energy diverges. Lee-Yang zeros and their generalizations are ubiquitous in many-body systems and they fully characterize the analytic properties of the free energy and hence thermodynamics of the systems. Determining the Lee-Yang zeros is not only fundamentally important for conceptual completeness of thermodynamics and statistical physics but also technically useful for studying many-body systems. However, Lee-Yang zeros have never been observed in experiments, due to the intrinsic difficulty that Lee-Yang zeros would occur only at complex values of magnetic field, which are 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. As recently proposed, the quantum evolution of the probe spin introduces a complex phase factor, and therefore effectively realizes an imaginary magnetic field on the bath. From the measured Lee-Yang zeros, we reconstructed the free energy of the spin bath and determined its phase transition temperature. This experiment demonstrates quantum coherence probe as a useful approach to studying thermodynamics in the complex plane, which may reveal a broad range of new phenomena that would otherwise be inaccessible if physical parameters are restricted to be real numbers.
We show that a class of $mathcal{PT}$ symmetric non-Hermitian Hamiltonians realizing the Yang-Lee edge singularity exhibits an entanglement transition in the long-time steady state evolved under the Hamiltonian. Such a transition is induced by a level crossing triggered by the critical point associated with the Yang-Lee singularity and hence is first-order in nature. At the transition, the entanglement entropy of the steady state jumps discontinuously from a volume-law to an area-law scaling. We exemplify this mechanism using a one-dimensional transverse field Ising model with additional imaginary fields, as well as the spin-1 Blume-Capel model and the three-state Potts model. We further make a connection to the forced-measurement induced entanglement transition in a Floquet non-unitary circuit subject to continuous measurements followed by post-selections. Our results demonstrate a new mechanism for entanglement transitions in non-Hermitian systems harboring a critical point.
Statistical physics provides the concepts and methods to explain the phase behavior of interacting many-body systems. Investigations of Lee-Yang zeros --- complex singularities of the free energy in systems of finite size --- have led to a unified understanding of equilibrium phase transitions. The ideas of Lee and Yang, however, are not restricted to equilibrium phenomena. Recently, Lee-Yang zeros have been used to characterize non-equilibrium processes such as dynamical phase transitions in quantum systems after a quench or dynamic order-disorder transitions in glasses. Here, we experimentally realize a scheme for determining Lee-Yang zeros in such non-equilibrium settings. We extract the dynamical Lee-Yang zeros of a stochastic process involving Andreev tunneling between a normal-state island and two superconducting leads from measurements of the dynamical activity along a trajectory. From the short-time behavior of the Lee-Yang zeros, we predict the large-deviation statistics of the activity which is typically difficult to measure. Our method paves the way for further experiments on the statistical mechanics of many-body systems out of equilibrium.
Lee-Yang zeros are points in the complex plane of an external control parameter at which the partition function vanishes for a many-body system of finite size. In the thermodynamic limit, the Lee-Yang zeros approach the critical value on the real-axis, where a phase transition occurs. Partition function zeros have for many years been considered a purely theoretical concept, however, the situation is changing now as Lee-Yang zeros have been determined in several recent experiments. Motivated by these developments, we here devise a direct pathway from measurements of partition function zeros to the determination of critical points and universal critical exponents of continuous phase transitions. To illustrate the feasibility of our approach, we extract the critical exponents of the Ising model in two and three dimensions from the fluctuations of the total energy and the magnetization in lattices of finite size. Importantly, the critical exponents can be determined even if the system is away from the phase transition. Moreover, in contrast to standard methods based on Binder cumulants, it is not necessary to drive the system across the phase transition. As such, our method provides an intriguing perspective for investigations of phase transitions that may be hard to reach experimentally, for instance at very low temperatures or at very high pressures.
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 $q$-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 $q>2$ 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) 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.
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