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.
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.
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.
Yang-Lee edge singularities (YLES) are the edges of the partition function zeros of an interacting spin model in the space of complex control parameters. They play an important role in understanding non-Hermitian phase transitions in many-body physics, as well as characterizing the corresponding nonunitary criticality. Even though such partition function zeroes have been measured in dynamical experiments where time acts as the imaginary control field, experimentally demonstrating such YLES criticality with a physical imaginary field has remained elusive due to the difficulty of physically realizing non-Hermitian many-body models. We provide a protocol for observing the YLES by detecting kinked dynamical magnetization responses due to broken PT symmetry, thus enabling the physical probing of nonunitary phase transitions in nonequilibrium settings. In particular, scaling analyses based on our nonunitary time evolution circuit with matrix product states accurately recover the exponents uniquely associated with the corresponding nonunitary CFT. We provide an explicit proposal for observing YLES criticality in Floquet quenched Rydberg atomic arrays with laser-induced loss, which paves the way towards a universal platform for simulating non-Hermitian many-body dynamical phenomena.
We study the driven dynamics across the critical points of the Yang-Lee edge singularities (YLESs) in a finite-size quantum Ising chain with an imaginary symmetry-breaking field. In contrast to the conventional classical or quantum phase transitions, these phase transitions are induced by tuning the strength of the dissipation in a non-Hermitian system and can occur even at finite size. For conventional phase transitions, universal behaviors in driven dynamics across critical points are usually described by the Kibble-Zurek mechanism, which states that the scaling in dynamics is dictated by the critical exponents associated with one critical point and topological defects will emerge after the quench. While the mechanism leading to topological defects breaks down in the YLES, we find that for small lattice size, the driven dynamics can still be described by the Kibble-Zurek scaling with the exponents determined by the (0+1)-dimensional YLES. For medium finite size, however, the driven dynamics can be described by the Kibble-Zurek scaling with two sets of critical exponents determined by both the (0+1)-dimensional and the (1+1)-dimensional YLESs.
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
A Kibble–Zurek rate scan gives one slope per observable, and three exponentsare often read from it: the field eigenvalue yg, the dynamic exponent z and theoperator dimension xO. We prove that the rate map carries an exact one-parameterredundancy: within this scaling family the response, freeze-out-field and freeze-outtime rate exponents are invariant under (yg, z, xO) → λ(yg, z, xO), so no number offurther ramp shapes or further observables of unknown dimension separates them,and rate-only data determine ratios alone. Within the homogeneous informationfamily considered here, we give a necessary-and-sufficient criterion for removing theredundancy: the information must contain a non-vanishing quantity homogeneousof definite, non-zero scaling weight, reaching the inversion either as an externallyknown exponent or as one further measured exponent whose scaling dimension isfixed by construction. A measured length does this; the freeze-out time, despiteits microscopic units, does not. Formal invertibility does not imply measurableidentifiability. Against exact diagonalisation of a quantum Ising chain in an imaginaryfield, with a free-fermion control, a freeze-out scale rebuilt from the response returnsthe normalisation it was given: a planted z = 2 comes back as z = 1.0000. Anunconstrained finite-size collapse does remove it, recovering all three exponents withnone pinned; but one invariant sits ten per cent from unity without improving withsize, and replacing response-conditioned sampling by a fixed grid of durations movesthem at the ten-per-cent level and above. The thermodynamic branch is left open,the criterion set in advance to test it having no resolving power at these sizes; at theYang–Lee edge the ratios come back as they should, −yg/xO = 2 and −z/xO = 1 forthe biorthogonal order parameter—though a two-level model with no lattice returnsthe same exponents, so what they establish is a square-root exceptional point.