This paper reinterprets the thermodynamic limit of statistical mechanics as aprojective dynamics process driven by the accumulation of informational potentialdifferences. By rigorously constructing the condensation operator as a sequenceof orthogonal projections on finite-dimensional Hilbert spaces, it is proved thatwhen the cumulative informational potential difference crosses the ln2 threshold,the projection sequence converges in the strong operator topology on the inductive limit Hilbert space to a limit projection, i.e., the system undergoes projectiveclosure. The commutativity between the regularised limit and the thermodynamiclimit is shown to be equivalent to the joint uniform convergence of the family ofprojection operators with respect to the regularisation parameter and the hierarchical index. Within this framework, the locking of Lee-Yang zeros on the imaginaryaxis of the complex magnetic field is shown not to be a consequence of finite-sizeextrapolation or reflection positivity, but rather a necessary consequence, in thethermodynamic limit, of the gap opening of the effective Hamiltonian when its realpart is non-zero and of the extremal condition of the dissipative cohomology. Thisconclusion relies on four premises: the axiom of information conservation, the axiom of computability, the ferromagnetic condition, and the Hermitian structure ofthe connection.
This paper reinterprets the thermodynamic limit of statistical mechanics as aprojective dynamics process driven by the accumulation of informational potentialdifferences. By rigorously constructing the condensation operator as a sequenceof orthogonal projections on finite-dimensional Hilbert spaces, it is proved thatwhen the cumulative informational potential difference crosses the ln2 threshold,the projection sequence converges in the strong operator topology on the inductive limit Hilbert space to a limit projection, i.e., the system undergoes projectiveclosure. The commutativity between the regularised limit and the thermodynamiclimit is shown to be equivalent to the joint uniform convergence of the family ofprojection operators with respect to the regularisation parameter and the hierarchical index. Within this framework, the locking of Lee-Yang zeros on the imaginaryaxis of the complex magnetic field is shown not to be a consequence of finite-sizeextrapolation or reflection positivity, but rather a necessary consequence, in thethermodynamic limit, of the gap opening of the effective Hamiltonian when its realpart is non-zero and of the extremal condition of the dissipative cohomology. Thisconclusion relies on four premises: the axiom of information conservation, the axiom of computability, the ferromagnetic condition, and the Hermitian structure ofthe connection.
Yang-Lee edge singularities are the branch point of free energy on the complex plane of physical parameters and have been shown to be the simplest universality class of phase transitions.However, Yang-Lee edge singularities have not been regarded as experimentally observable since they occur at complex physical parameters which are unphysical.A recent discovery regarding the relation between partition functions and probe spin coherence makes it experimentally feasible to access the complex plane of physical parameters.However, how to extract the critical point and the critical exponent of Yang-Lee edge singularities in many-body systems, which occurs only at the thermodynamic limit, has still been elusive.Here we show that the quantum coherence of a probe spin coupled to finite-size Ising-type spin systems presents universal scaling behavior near the Yang-Lee edge singularity.The finite-size scaling behavior of the quantum coherence of the probe spin predicts that one can extract the critical point and the critical exponent of the Yang-Lee edge singularity of an Ising-type spin system in the thermodynamic limit from the spin coherence measurement of the probe spin coupled to finite-size Ising-type spin systems.This finding provides a practical approach to studying the nature of Yang-Lee edge singularities of many-body systems.In 1952, Yang and Lee [1, 2] initiated the examination of phase transitions by studying the partition function zeros, termed Lee-Yang zeros, on the complex plane of a magnetic field h (or fugacity z for fluids).They found that for a system above its critical temperature T c , the partition function must be nonzero throughout some neighborhood of the real axis of the magnetic field within a gap determined by two edges at ( ) h T YL .Therefore the free energy is an analytic function of the real magnetic field.On the other hand, below the critical temperature T c , the Lee-Yang zeros will come arbitrarily to the real axis as the thermodynamic limit is taken, destroying the analyticity of the free energy in h for those real fields at which the Lee-Yang zeros accumulate.Kortman and Griffiths [3] pointed out that the edges of Lee-Yang zeros of a many-body system in the thermodynamic limit are singularity points, termed Yang-Lee edge singularities [4].Fisher [4] proposed that the Yang-Lee edge singularity could be considered as a new second order phase transition point with associated critical exponents.Renormalization group analyses [4] show that the Yang-Lee edge singularity is associated with an j i 3 theory and the crossover dimensionality of the Yang-Lee edge singularity is d c = 6.Later, the investigation of the Yang-Lee edge singularity has been extended to many other models, such as the classical nvector model [5], the quantum Heisenberg model [5], the spherical model [6], the quantum one-dimensional (1D) transverse Ising model [7], the hierarchical model [8], branched polymers [9], the directed-site animalsenumeration problem [10], Ising models on fractal lattices [11], Ising systems with correlated disorder [12], fluid models with repulsive-core interactions [13], the antiferromagnetic Ising model [14] etc.A comprehensive review of the study of Yang-Lee edge singularities can be found in [15, 16].Recently, Kibble-Zurek scaling was shown to appear at the Yang-Lee edge singularity [17], which supports that the Yang-Lee edge singularity is similar to conventional second order phase transitions.Yang-Lee edge singularity occurs in the ferromagnetic Ising models above its critical temperatures in a purely imaginary magnetic field h i (figure 1).For > ( ) h h T YL , the partition function acquires zeros, which becomes dense on the line with > ( ) Ih h T YL .The density of Lee-Yang zeros ( ) g T h , in the thermodynamic limit behaves as ~ -s ( ) | ( )| g T h h h T , YL when ( ) h h T YL from above, with σ being the universal critical
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.
This paper constructs a four-tier axiomatic system of the stability ladder toreveal the underlying logic of phase transitions in statistical mechanics. Startingfrom information conservation (L0) and short-range interactions (L1), we rigorously prove, using DLR measure theory and large deviation principles in the thermodynamic limit, that the coexistence of macroscopically distinguishable states insystems of spatial dimension not less than two necessarily requires a strictly positiveenergy barrier in the free energy landscape (L2). In one-dimensional systems, information protection is instead realized via topological boundary conditions. Takingthe two-dimensional Ising model as a paradigm, we reconstruct the ontological basisof the existence of a phase transition in Onsager’s exact solution from the axioms,and extend the framework to general lattice systems and quantum many-bodymodels. The derivation does not rely on mean-field approximations or specific calculations of the renormalization group, thereby providing a necessary explanationfor the universality of phase transitions based on the logic of information conservation. In critical dynamics and quantum phase transitions, the barrier conceptis generalized to a dynamical barrier or quantum gap, demonstrating the broaderapplicability of the stability ladder.
We study Yang-Lee zeros as they approach the thermodynamic limit of the two-dimensional (2D) nearest-neighbor antiferromagnetic Ising model on square and triangular lattices. We employ the numerical linked cluster expansion (NLCE) equipped with exact enumeration (EE) of the partition function to estimate the Laplacian of the free energy, which is proportional to the zeros density. Using a modified NLCE, where the expansion can be carried directly on the Yang-Lee zeros of the involved clusters, we estimate the density of Yang-Lee zeros in the thermodynamic limit. NLCE gives significantly more zeros than EE in the complex field plane providing more insights on how the root curves look in the thermodynamic limit. For the square lattice at $T\ensuremath{\ll}{T}_{c}$, the results suggest that two vertical lines at $\ifmmode\pm\else\textpm\fi{}{h}_{c}(T)$ in the complex field plane (i.e., two concentric circles in the complex fugacity plane) are the thermodynamic root curves. A similar picture is found for the triangular lattice for phase transitions at large values of magnetic field while further study is needed for phase transitions at smaller values of magnetic field. The convergence of the NLCE and (EE) calculations of the partition function to the thermodynamic limit is studied in both lattices and the temperature-field phase diagram is obtained from Yang-Lee zeros using both methods. This NLCE-based approach will facilitate the study of different types of phase transitions using Yang-Lee zeros in future research.
Non-Hermitian criticality is often treated as a single phenomenon, yet two distinct objects travel under that name. An exceptional point is a finite-N spectral object—the coalescence of eigenvectors of a single operator; a Lee–Yang edge is a thermodynamic-limit c-number object—the accumulation of partition-function zeros. We show that which of these an imaginary coupling produces is set by the direction the coupling occupies. When the coupling multiplies a conserved charge, [Q, H] = 0, the deformation H − iµQ is normal for every complex µ: its spectrum is entire, its eigenvectors are µ-independent, and no exceptional point exists at any coupling strength or volume, while Lee–Yang zeros persist and can carry an edge. When the coupling lies on a symmetry-invariant but non-conserved direction, exceptional points appear, but the associated universality is the analytic continuation of the Hermitian thermal transition, not the Yang–Lee edge. Only on the order-parameter (symmetry-odd) direction do the two coincide, at the M(2, 5) edge. The direction of an imaginary coupling therefore sets how much genuine non-unitarity its spectrum acquires. We give the theorem, a three-arm criterion, exactly solvable and exact-diagonalization checks, and falsifiable kill conditions.
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.