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.
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.
A purely imaginary coupling added to a spin chain can produce two qualitatively different critical objects — a finite-size exceptional point (eigenvector coalescence) ora thermodynamic-limit Lee–Yang edge (partition-function-zero accumulation) — andwhich one appears is set by the direction the coupling occupies relative to the symmetry. We propose a protocol with two independent knobs: (i) the direction of theimaginary coupling (conserved-charge / symmetry-invariant / order-parameter), whichselects one of three arms, and (ii) the linear-ramp rate, which reads the frozen singularresponse density. Two on-chip observables discriminate the arms: the eigenvector nonorthogonality (Petermann factor), bounded on the conserved arm and divergent on theother two; and the sign of the Kibble–Zurek frozen-response exponent, which is positive (response grows as the quench slows) only on the order-parameter arm, wherethe edge is the non-unitary 𝑀(2, 5) point. Each arm carries a falsifiable kill clause.The ingredients — imaginary fields, per-site imaging, tunable interactions — exist onnon-Hermitian Rydberg / PXP and open-system photonic platforms.
The Yang–Lee edge singularity is the critical point of a scalar order parameter coupled to a purely imaginary symmetry-breaking field; in two dimensions it is the non-unitary minimal model , with the negative order-parameter dimension and edge exponent . We study a linear ramp through this edge. Three results are exact and mutually independent. (i) A reduction theorem: any real-analytic Landau functional that is even under the order-parameter reflection, perturbed by an imaginary linear source, flows term by term near its edge to the Fisher action, hence in two dimensions to ; the cubic coupling is automatically imaginary because the edge saddle lies on the imaginary axis. An accompanying all-order phase lemma shows the odd and even sectors are locked exactly apart, with no tunable phase. (ii) Kibble–Zurek exponents that are rational with the unusual denominator : the freeze-out scale, correlation length, and frozen singular-response density scale as , , and . Because , the frozen response grows as the quench slows—the reverse of ordinary Kibble–Zurek defect suppression—giving a sharp, falsifiable signature. (iii) The exact operator-product coefficient , negative as required by non-unitarity. The exponents are generic: they require only a nonzero quadratic dispersion, the fine-tuned quartic-dominated (Lifshitz) alternative belonging to a distinct class. The sign-reversed response is within reach of non-Hermitian Rydberg, PXP, and open-system Yang–Lee platforms.
The Yang-Lee edge singularity is an intriguing critical phenomenon characterized by nonunitary field theory. However, its experimental realization for interacting many-body systems remains elusive. We show that Yang-Lee edge singularities, regarded as many-body exceptional points, can be observed using both the self-normal and the associated-biorthogonal Loschmidt echoes, leveraging the advantages of nonunitary dynamics in non-Hermitian systems. The Loschmidt echoes are demonstrated to display unitary dynamics in the PT-symmetric regime but exhibit nonunitary dynamics in the PT-symmetry-broken regime, leading to a sharp change near an exceptional point. We hereby identify exceptional points in both the non-Hermitian transverse-field Ising model and the Yang-Lee model, and determine the critical exponent that is consistent with nonunitary conformal field theory. This paper provides a direct observation of Yang-Lee edge singularities in non-Hermitian many-body systems arising from nonunitary dynamics.
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
In a compression-based account of charge, five distinct objects share the single name “charge”: a global conserved charge , an einselected reduced record , a -odd Yang–Lee ordering operator , a decaying Gaussian occupation , and a trajectory charge-transfer count . We separate them inside an instrumented, charge-conserving open quantum process and ask which relations are invariants of the full process and which are only diagnostics of a monitoring scheme or of the no-jump conditional slice. In minimal completely-positive models we establish: (i) [THM] if every jump commutes with and , charge-diagonal monitoring makes the reduced record an exact fixed point, and [CALC] its survivor -algebra is the two-dimensional diagonal (charge-sector) algebra; a charge-violating perturbation makes it a finite-lifetime metastable record whose lifetime rises with the monitoring rate in the strong-monitoring (Zeno) branch. (ii) [THM] -covariance does not imply charge conservation — amplitude damping is covariant yet lowers the occupation. (iii) [CALC] the asymptotic () stationary fixed-point -algebra obeys a three-way classification: for a bare record; with unchanged charge centre when a non-abelian seed survives (intra-sector protected structure); and a refined centre when a second abelian label survives. Charge labels all asymptotic survivors iff the stationary and there are no nonzero-frequency peripheral modes; the weaker excludes neither intra-sector survivors nor rotating (nonzero-frequency) peripheral modes, which are counted separately from . (iv) [CALC] a genuine finite-time trajectory Lee–Yang zero of a two-site transport counting statistics exists (complex Newton root, residual , winding number one); we do not call it an edge. The two strongest results are negative: [NEG] the no-jump effective Hamiltonian is not a representation invariant of the physical generator — at an exceptional point the eigenvalue splitting opens from to under a redundant Lindblad representation that leaves fixed to , so the exceptional point itself is not a process invariant; and [NEG] in the stated model, at a no-jump exceptional point no non-analyticity of the physical scaled cumulant generating function is detected in the scanned range. One minimal model simultaneously instantiates three charge objects () and two derived dynamical objects (); and remain comparison entries supplied by other minimal models / companions. In that model the conserved record and the zero-generating transferred charge are different objects: the ledger and language are unified, but a direct, representation-invariant bridge from the record to the dynamical Lee–Yang zeros is not established. We do not derive , a dynamic exponent , or any universality class.
We study the scaling behavior of the Berry phase in the Yang-Lee edge singularity (YLES) of the non-Hermitian quantum system. A representative model, the one-dimensional quantum Ising model in an imaginary longitudinal field, is selected. For this model, the dissipative phase transition (DPT), accompanying a parity-time (PT) symmetry-breaking phase transition, occurs when the imaginary field changes through the YLES. We find that the real and imaginary parts of the complex Berry phase show anomalies around the critical points of YLES. In the overlapping critical regions constituted by the (0 + 1)D YLES and (1 + 1)D ferromagnetic-paramagnetic phase transition (FPPT), we find that the real and imaginary parts of the Berry phase can be described by both the (0 + 1)D YLES and (1 + 1)D FPPT scaling theory. Our results demonstrate that the complex Berry phase can be used as a universal order parameter for the description of the critical behavior and the phase transition in the non-Hermitian systems.