The 2025 work of Sun, Yu, Wang and Zhai (arXiv:2506.00919, published in Chinese Physics B in 2026) introduced and numerically verified a hybrid scaling mechanism for the static critical behavior of the transverse-field Ising chain in an imaginary longitudinal field, whose (0+0)D and (1+0)D classical and (0+1)D and (1+1)D quantum Yang-Lee edge singularity (YLES) critical regions overlap at low temperature and finite system size: in a pairwise overlap, the scaling functions of both critical regions remain simultaneously valid and are linked by constraints mixing the critical exponents of both regions. The authors explicitly leave open three directions that jointly complete this framework: (i) the static hybrid scaling of regions where three or four YLES critical regions coexist simultaneously, which was not analyzed; (ii) a unified non-equilibrium theory - a hybridized Kibble-Zurek mechanism - for overlapping regions combining one classical and one quantum critical region, since existing hybrid Kibble-Zurek theory covers only quantum-quantum overlaps; and (iii) quantitative experimental validation of the hybrid scaling constraints at finite temperature, finite system size, and complex field. This problem asks for that completion.

Background

In 1952, Lee and Yang showed that the zeros of the partition function of an Ising ferromagnet, as a function of magnetic field, lie on the unit circle in the complex-field plane; these are called Lee-Yang zeros. In the thermodynamic limit their approach to the real axis produces the ordinary critical point, while the edge of the zero distribution is itself singular: the Yang-Lee edge singularity (YLES), which Fisher showed to be governed by a non-unitary φ3\varphi^3 field theory (non-unitary because the cubic self-interaction is tied to the imaginary magnetic field, so the theory is not Hermitian) with its own universality class - a set of critical points that share the same critical exponents. Although the YLES occurs at complex field and was long considered unphysical, non-Hermitian quantum physics has made it accessible: an imaginary longitudinal field ihih applied to a quantum spin chain acts as a complex Boltzmann weight, and YLES quantum criticality has been observed experimentally in open and non-Hermitian quantum systems, with further proposals for Rydberg atomic arrays.

The model at the center of this problem is the transverse-field Ising chain in an imaginary longitudinal field,

H=−∑n=1Lσnzσn+1z−λ∑n=1Lσnx−ih∑n=1Lσnz,\mathcal{H}=-\sum_{n=1}^{L}\sigma_n^z\sigma_{n+1}^z-\lambda\sum_{n=1}^{L}\sigma_n^x-ih\sum_{n=1}^{L}\sigma_n^z,

where σnx,z\sigma_n^{x,z} are Pauli matrices, λ\lambda is the transverse field, hh is the imaginary longitudinal field, and LL is the chain length. The Hamiltonian is non-Hermitian - not equal to its own complex-conjugate transpose - so its eigenvalues are generally complex. Its ordinary ferromagnetic-paramagnetic phase transition (FPPT) occurs at (λ,h)=(1,0)(\lambda,h)=(1,0). Its YLES critical points occur at nonzero imaginary longitudinal field, i.e., at real parameter pairs (λ,h)(\lambda,h) with h≠0h\neq 0 in the parametrization above (so the physical field ihih is genuinely complex), and, unusually, exist already at finite system size. Following the source paper, the YLES critical regions are labeled by effective dimensionality as '(space + time)': at zero temperature, finite chains show a (0+1)(0+1)-dimensional (D) quantum YLES (zero spatial dimensions - the finite chain has no thermodynamic spatial extent - plus one time dimension) at finite-size critical points (λQL,hQL)(\lambda_Q^L,h_Q^L) with λ>1\lambda>1, flowing to a (1+1)(1+1)D quantum YLES at the thermodynamic critical point (λQ∞,hQ∞)(\lambda_Q^\infty,h_Q^\infty); at finite temperature, classical YLES appear - a (0+0)(0+0)D classical YLES for finite LL at (λCL,hCL)(\lambda_C^L,h_C^L) and a (1+0)(1+0)D classical YLES at (λC∞,hC∞)(\lambda_C^\infty,h_C^\infty). The critical exponents used throughout are, as established in the literature cited by the source: for (0+0)(0+0)D, ν=−0.62\nu=-0.62, β=1\beta=1, δ=−1\delta=-1, z=1.633z=1.633; for the (0+1)(0+1)D and (1+0)(1+0)D universality class, ν=−1\nu=-1, β=1\beta=1, δ=−2\delta=-2, z=1z=1; for (1+1)(1+1)D, ν=−5/2\nu=-5/2, β=1\beta=1, δ=−6\delta=-6, z=1z=1 - where ν\nu is the correlation-length exponent, β\beta and δ\delta govern the order parameter (the complex magnetization MM along the field direction) as a function of the distance gg to a YLES critical point (the field distance h−hch-h_{\rm c} to the critical point hch_{\rm c} in question), and zz is the dynamical exponent.

Because the third law of thermodynamics forbids reaching T=0T=0 in a finite number of steps, any real experiment sits at finite temperature, where thermal and quantum fluctuations coexist; near a quantum critical point the finite-temperature classical critical region and the zero-temperature quantum critical region overlap. Sun, Yu, Wang and Zhai analyzed how the four YLES critical regions of the model above overlap and formulated a hybrid scaling mechanism for the overlap of two critical regions: (i) in the overlapping region the scaling functions of both critical regions remain simultaneously valid, and (ii) the two scaling functions obey a constraint that mixes the critical exponents of both regions. For example, in the overlap of the (0+0)(0+0)D classical and (0+1)(0+1)D quantum critical regions the order-parameter scaling implies

M=Tξ−η (gCL)1/δ00,ξ=β01ν01z01,  η=β01δ01ν01z01δ00,M = T^{\xi-\eta}\,(g_C^L)^{1/\delta_{00}},\qquad \xi=\frac{\beta_{01}}{\nu_{01}z_{01}},\ \ \eta=\frac{\beta_{01}\delta_{01}}{\nu_{01}z_{01}\delta_{00}},

where gCL=h−hCLg_C^L=h-h_C^L is the distance in field from the finite-size classical YLES critical point and the subscripts label the universality classes as above, and in the overlap of the (1+0)(1+0)D and (1+1)(1+1)D regions an analogous constraint relates the two scaling functions through the combination LT1/z11LT^{1/z_{11}}. Since such constraints contain the exponents of both the classical and the quantum YLES, a finite-temperature (classical-regime) measurement can in principle reveal quantum critical information. The authors verified this static hybrid scaling numerically for the pairwise overlaps they analyze - quantum-quantum, classical-classical, and classical-quantum.

On the non-equilibrium side, the Kibble-Zurek mechanism (KZM) describes a system driven through a critical point at a finite rate: when the driving timescale becomes shorter than the relaxation time, the system falls out of equilibrium and observables scale universally with the driving rate. Kibble-Zurek scaling has been verified for a single YLES critical region, and a hybridized Kibble-Zurek mechanism (HKZM) exists for driven dynamics across an overlapping critical region of two quantum transitions - the (0+1)(0+1)D YLES and the (1+1)(1+1)D FPPT of the same model. What does not exist is the corresponding theory for overlaps that mix a classical (finite-temperature) and a quantum critical region, a static analysis of regions where more than two YLES critical regions coexist, and any experimental test of the hybrid scaling constraints. These are precisely the directions the source work leaves open.

Problem Statement

Complete the hybrid scaling theory of overlapping classical and quantum Yang-Lee edge singularities in the transverse-field Ising chain in an imaginary longitudinal field, H=−∑n=1Lσnzσn+1z−λ∑n=1Lσnx−ih∑n=1Lσnz\mathcal{H}=-\sum_{n=1}^{L}\sigma_n^z\sigma_{n+1}^z-\lambda\sum_{n=1}^{L}\sigma_n^x-ih\sum_{n=1}^{L}\sigma_n^z, by resolving the three directions the source work (Sun, Yu, Wang, Zhai, 2025) explicitly leaves open:

(1) Static hybrid scaling with multi-region coexistence. In medium-sized systems (in the regime where the finite-size YLES critical points (λQL,hQL)(\lambda_Q^L,h_Q^L) and (λCL,hCL)(\lambda_C^L,h_C^L) are still distinct from their thermodynamic limits), parameter regions may exist where three or even all four of the (0+0)(0+0)D, (0+1)(0+1)D, (1+0)(1+0)D and (1+1)(1+1)D YLES critical regions overlap simultaneously. Derive the hybrid scaling description of such regions: the scaling form of the order parameter when three or four critical regions are simultaneously relevant, and the constraint(s) among the scaling functions of all coexisting regions, and verify them in the model (the source notes this requires more careful parameter tuning than the pairwise case).

(2) Unified non-equilibrium theory for classical-quantum overlaps. Construct a hybridized Kibble-Zurek mechanism for finite-rate driving across an overlapping critical region composed of one classical and one quantum YLES critical region - for example the (0+0)(0+0)D-(0+1)(0+1)D or the (1+0)(1+0)D-(1+1)(1+1)D overlap: derive the scaling forms of the observables (order parameter and related quantities) as functions of driving rate, temperature and system size, together with the constraints linking them, generalizing the existing hybridized Kibble-Zurek mechanism that covers only quantum-quantum overlaps. Establish the theory at least for the model Hamiltonian above, with verification of the predicted scaling.

(3) Experimental validation. Realize an overlapping critical region of a classical and a quantum YLES in a laboratory system - necessarily at finite temperature and finite system size, with controlled complex (imaginary) field, for instance in a non-Hermitian or open quantum simulator - and quantitatively test the hybrid scaling constraints, thereby validating the proposed route to extract quantum critical information from finite-temperature measurements. Component (3) is resolved by a concrete experimental proposal on a named platform, together with a simulated demonstration of the predicted hybrid-scaling collapse; an actual laboratory measurement is not required.

These three components jointly complete one objective: a scaling theory of overlapping classical and quantum YLES criticality covering statics (arbitrary numbers of coexisting critical regions), dynamics (driven transitions through mixed classical-quantum overlaps), and experimental reality.

References

  1. Yue-Mei Sun, Wen-Jing Yu, Xin-Yu Wang, Liang-Jun Zhai, 'Hybrid scaling mechanism of critical behavior in the overlapping critical regions of classical and quantum Yang-Lee edge singularities', arXiv:2506.00919 (2025); published in Chinese Physics B (2026), DOI 10.1088/1674-1056/ae6176, https://arxiv.org/abs/2506.00919, https://doi.org/10.1088/1674-1056/ae6176 (source paper)
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