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Structural and practical identifiability in multi-protocol Kibble–Zurek scaling

Rongzhen Dai

Published Sep 17, 2026• 0 citations

Abstract

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.

DOI: 10.5281/zenodo.22810023

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Near a critical point, the equilibrium relaxation time of a system diverges and any change of control/thermodynamic parameters leads to nonequilibrium behavior. The Kibble-Zurek problem is to determine the dynamical evolution of the system parametrically close to its critical point when the change is parametrically slow. The nonequilibrium behavior in this limit is controlled entirely by the critical point and the details of the trajectory of the system in parameter space (the protocol) close to the critical point. Together, they define a universality class consisting of critical exponents, discussed in the seminal work by Kibble and Zurek, and scaling functions for physical quantities, which have not been discussed hitherto. In this article, we give an extended and pedagogical discussion of the universal content in the Kibble-Zurek problem. We formally define a scaling limit for physical quantities near classical and quantum transitions for different sets of protocols. We report computations of a few scaling functions in model Gaussian and large-

N

problems and prove their universality with respect to protocol choice. We also introduce a protocol in which the critical point is approached asymptotically at late times with the system marginally out of equilibrium, wherein logarithmic violations to scaling and anomalous dimensions occur even in the simple Gaussian problem.

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.

We revisit the problem of \textit{real-time} quantum dynamics of the paradigmatic two dimensional transverse-field Ising model using the recently developed fuzzy sphere regularization scheme. By linearly ramping the transverse field from the paramagnetic phase to criticality, we study the finite-time scaling behavior of the squared order parameter

\langle m_z^2 \rangle

, the excitation energy density

Q

, and the two-point correlation function of

m_z

. We establish numerically that, at intermediate quench rate,

\langle m_z^2 \rangle

follows the conventional Kibble-Zurek prediction set by the critical exponents of the

3

D Ising universality class, and the correlation function exhibits the expected exponential decay whose correlation length can be used to estimate the non-universal scaling coefficient in the freeze-out time/length. In contrast, the excitation energy density

Q

does not reach the same scaling regime at available system sizes due to large effective finite-size gap from symmetry-enforced level sparsity in the energy spectrum. At slow quench rates the universal quasi-adiabatic scaling for both

\langle m_z^2 \rangle

and

Q

is recovered. Since the fuzzy sphere construction can realize not only the Ising conformal field theory (CFT), but a broad family of

(2+1)d

CFTs, our results establish a route to the real-time critical dynamics of strongly coupled CFTs that are otherwise computationally challenging to study.

We investigate driven dynamics across critical and exceptional points in the one- and two-dimensional imbalanced-pairing Kitaev models using both the wave-function normalization approach and the biorthogonal framework. For a positive pairing imbalance parameter, the quasiparticle spectrum remains real, and a pairing imbalance neither shifts the equilibrium phase boundaries nor generates imaginary eigenenergies. In this regime, the defect density follows the conventional Kibble--Zurek scaling in one dimension and the extended Kibble--Zurek scaling, arising from a gapless manifold, in two dimensions within both frameworks. The corresponding scaling exponents are therefore governed by those of the Hermitian transition. For a negative pairing imbalance parameter, time-reversal symmetry is broken, the quasiparticle spectrum develops complex eigenvalues, and the gap closes at exceptional points. For ramps ending at an exceptional point, the defect density follows the modified Kibble--Zurek scaling in the wave-function normalization approach, whereas it obeys the conventional Kibble--Zurek scaling in the biorthogonal framework. When the ramp traverses the time-reversal-symmetry-broken region, a finite density of defects remains even in the adiabatic limit, leading to defect freezing in both frameworks. Although this frozen background indicates a breakdown of adiabaticity, the excess defects generated on top of this background continue to obey the conventional Kibble--Zurek scaling in one dimension and the extended Kibble--Zurek scaling in two dimensions.