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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.
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We determine a previously unknown universal quantity, the location of the Yang-Lee edge singularity for the O($N$) theories in a wide range of $N$ and various dimensions. At large $N$, we reproduce the $Ntoinfty$ analytical result on the location of the singularity and, additionally, we obtain the mean-field result for the location in $d=4$ dimensions. In order to capture the nonperturbative physics for arbitrary $N$, $d$ and complex-valued external fields, we use the functional renormalization group approach.
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.
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.
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.
Li and Haldane conjectured and numerically substantiated that the entanglement spectrum of the reduced density matrix of ground-states of time-reversal breaking topological phases (fractional quantum Hall states) contains information about the counting of their edge modes when the ground-state is cut in two spatially distinct regions and one of the regions is traced out. We analytically substantiate this conjecture for a series of FQH states defined as unique zero modes of pseudopotential Hamiltonians by finding a one to one map between the thermodynamic limit counting of two different entanglement spectra: the particle entanglement spectrum, whose counting of eigenvalues for each good quantum number is identical (up to accidental degeneracies) to the counting of bulk quasiholes, and the orbital entanglement spectrum (the Li-Haldane spectrum). As the particle entanglement spectrum is related to bulk quasihole physics and the orbital entanglement spectrum is related to edge physics, our map can be thought of as a mathematically sound microscopic description of bulk-edge correspondence in entanglement spectra. By using a set of clustering operators which have their origin in conformal field theory (CFT) operator expansions, we show that the counting of the orbital entanglement spectrum eigenvalues in the thermodynamic limit must be identical to the counting of quasiholes in the bulk. The latter equals the counting of edge modes at a hard-wall boundary placed on the sample. Moreover, we show this to be true even for CFT states which are likely bulk gapless, such as the Gaffnian wavefunction.
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