What a Lee-Yang zero actually measures
The paper works with the partition function Z of a finite square-lattice quantum magnet and asks where Z vanishes when an imaginary source field is coupled to a chosen order-parameter channel. Those complex values of the source are the Lee-Yang zeros. At finite linear size L and inverse temperature β the zeros are discrete and stay off the real axis; a thermodynamic singularity appears only if, as L grows, the zeros condense into a continuum that pinches the real axis at a real coupling.
The diagnostic power comes from how fast the lowest zero approaches that axis. If the transition is continuous, the leading zero should scale as a power law, λ1 ∼ L^(-yh), where yh is the renormalization-group eigenvalue of the source. If instead two phases coexist, finite-size rounding is controlled by the free-energy difference between them, which is extensive in the spacetime volume βN, so the leading zero should approach as 1/(βN). These two laws are parametrically different, which is what makes the zeros a sharp discriminator rather than another real-axis observable.
The authors compute the zeros inside stochastic series expansion quantum Monte Carlo. Configurations are sampled from a real, symmetry-preserving anchor Hamiltonian, then reweighted to complex source values through a ratio estimator, so the zeros are extracted from the same ensemble rather than from separate simulations at each complex coupling. Because the source is projected onto a definite momentum and operator support, the analysis is symmetry-resolved: Néel order is probed in the (π,π) sector, columnar valence-bond order in the (π,0) and (0,π) bond sectors.
Benchmarks recover the orders they should
Before applying the method to the disputed models, the paper tests it on two cases whose answers are already established. The columnar dimerized Heisenberg model yields stable O(3) critical scaling, the behavior expected for a continuous transition in that model. The checkerboard J-Q model instead shows clear spacetime-volume scaling of the zeros, the signature expected for a first-order transition.
This pairing matters because it shows the same estimator can land on either side of the discriminator. A method that only ever produced drift, or only ever produced power-law scaling, would be uninformative about the J-Q transition. Here the two reference models separate cleanly along the predicted lines, which is the basis for trusting the contrast when the uniform J-Q models behave differently from both.
In J-Q2 and J-Q3 the zeros drift with size
Applied to the uniform J-Q2 and J-Q3 models, the leading-zero scaling does not stabilize. The authors report a pronounced and systematic drift with increasing system size, which they describe as an extended pseudocritical regime: over the lattice sizes accessible here, the system looks critical in a way that keeps changing as L grows, rather than sitting at a fixed point.
The drift is not an artifact of one microscopic probe. The main text uses bond-supported valence-bond operators, and a supplementary check repeats the analysis with interaction-supported operators, using Q2-plaquette-supported operators in J-Q2 and Q3-rectangle-supported operators in J-Q3. Both choices lie in the same columnar symmetry sector and can mix under coarse graining, and both reproduce the same qualitative downward drift. The authors therefore argue the drift belongs to the columnar valence-bond channel itself rather than to a particular estimator.
A further check varies the coupling. At J/Q2 = 0.04500, 0.04505 and 0.04510, three closely spaced values near the transition, the leading-zero exponents agree within uncertainties across the three couplings and between the Néel and valence-bond channels, and the two-size estimators from the first two zeros decrease systematically with L in both channels. The VBS data at J/Q2 = 0.04510 show the largest finite-size corrections but keep the same trend.
The effective scaling dimension shrinks toward zero
The drift can be restated in field-theoretic language. The Lee-Yang scaling implies an effective scaling dimension Δφ for the SO(5) order-parameter field, and this effective value decreases with system size and is consistent with vanishing in the thermodynamic limit. SO(5) here refers to the symmetry that would relate the Néel and valence-bond order parameters at a deconfined critical point; Δφ is the dimension that would be fixed at such a critical point.
The authors then invoke a bound: a vanishing Δφ lies below the scalar unitarity bound of any unitary relativistic conformal field theory in 2+1 dimensions. If the effective dimension keeps falling below that bound, the zeros cannot be described by a stable unitary fixed point, and the scaling is pushed toward inverse spacetime-volume behavior, the hallmark of a first-order transition. This is the paper's central inferential step, and it is an interpretation built on the measured drift plus the unitarity bound, not a direct observation of two coexisting phases.
Why conventional real-axis diagnostics stall
The paper frames its contribution against the difficulty of reading the transition from ordinary observables. Binder-cumulant scaling in these models persists through L = 512, and an apparent power-law regime in the spin-correlation derivative persists to L = 1024 in J-Q2. The same large-scale study reports evidence for a very weak first-order transition, with correspondingly weak coexistence signatures and very small Néel and valence-bond order parameters.
Crossing points can therefore look almost size independent on moderate lattices while still drifting once larger sizes are included, and exponents from standard finite-size fits depend noticeably on the fitting window. Joint order-parameter distributions showing approximate emergent symmetry are suggestive but do not by themselves separate true criticality from a long-lived pseudocritical regime. The Lee-Yang zeros are proposed as a complement because they target the finite-size singularity scale directly instead of inferring it from a crossing.
The scale and scope of the simulation
All simulations use stochastic series expansion quantum Monte Carlo with loop updates on L×L square lattices with periodic boundary conditions, at fixed aspect ratio β = L/2, chosen to remain in the expected z = 1 scaling window for these square-lattice models. Each parameter point accumulates 10^8 Monte Carlo steps. System sizes extend up to L = 128 for the columnar dimerized Heisenberg model and up to L = 96 for the checkerboard J-Q, J-Q2 and J-Q3 models.
Those sizes are what make the drift visible, and they are also the boundary of the claim. The statement that Δφ is consistent with vanishing in the thermodynamic limit is an extrapolation from a finite window in which the drift has not yet terminated; the paper's evidence is the systematic direction and persistence of the drift, not a settled limit. The conclusion is likewise specific to the square-lattice J-Q family and its Néel and columnar valence-bond channels, not to deconfined criticality in general.
What the drift changes for reading J-Q results
The practical upshot is a shift in what counts as evidence. A J-Q study that finds apparently critical Binder cumulants or a power-law regime over a finite window is not thereby showing a stable fixed point, because this paper's zeros keep drifting over sizes up to L = 96 while the same kind of real-axis behavior looks critical to far larger sizes. The discriminating question becomes whether the leading-zero scaling converges to a fixed power law or continues toward inverse spacetime-volume behavior.
The paper's own priority follows from that comparison: the low-lying Lee-Yang zeros, resolved separately in the Néel and valence-bond channels, are presented as the sensitive diagnostic of pseudocriticality and transition order in this model family. The weakly first-order reading rests on the direction and persistence of the measured drift together with the unitarity bound, so the result to track is whether larger lattices let the effective Δφ settle above that bound or continue its decrease.
Sources used in this answer
Lee-Yang Zeros and Pseudocritical Drift in J-Q N\'eel-VBS Transitions
Leading Lee-Yang zeros in the uniform J-Q2 and J-Q3 models drift systematically with system size, implying an effective SO(5) order-parameter scaling dimension that decreases with size and is consistent with vanishing in the thermodynamic limit, below the 2+1D scalar unitarity bound, which the authors interpret as a weakly first-order Néel-VBS transition.