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arXiv:2609.35940

Discrete Scale Invariance of Ising Mesons

Denis V. Vasilyev, Abbas Ali Saberi

cond-mat.dis-nncond-mat.quant-gascond-mat.stat-mechquant-phSubmitted Sep 28, 2026• Announced Sep 29, 2026• 0 citations
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Abstract

Discrete scale invariance often arises from resonant few-body or singular scale-invariant dynamics. We show that it can instead emerge in the internal motion of a composite excitation in an ordered quantum magnet. In the marginal $1/r^4$ transverse-field Ising chain, long-range bonds generate an asymptotic inverse-square attraction between dressed domain walls, with strength fixed by the spontaneous magnetization. The dressed two-kink threshold curvature sets the relative kinetic scale. Together, these independently accessible quantities determine the scale-anomaly exponent and hence the meson hierarchy. In the supercritical regime, the same exponent governs geometric binding-energy and size ratios, logarithmic level accumulation, and log-periodic threshold scattering. Full-spin exact diagonalization and a fourth-order weak-field Hamiltonian support the coupled energy--size scaling in the finite window between core and ring effects. This provides a microscopic many-body realization of quantum limit-cycle physics, with its universal scaling fixed by magnetic order and kink mobility.

Comments
26 pages, 8 figures

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Discrete scale invariance often arises from resonant few-body or singular scale-invariant dynamics. We show that it can instead emerge in the internal motion of a composite excitation in an ordered quantum magnet. In the marginal 1/r41/r^41/r4 transverse-field Ising chain, long-range bonds generate an asymptotic inverse-square attraction between dressed domain walls, with strength fixed by the spontaneous magnetization. The dressed two-kink threshold curvature sets the relative kinetic scale. Together, these independently accessible quantities determine the scale-anomaly exponent and hence the meson hierarchy. In the supercritical regime, the same exponent governs geometric binding-energy and size ratios, logarithmic level accumulation, and log-periodic threshold scattering. Full-spin exact diagonalization and a fourth-order weak-field Hamiltonian support the coupled energy--size scaling in the finite window between core and ring effects. This provides a microscopic many-body realization of quantum limit-cycle physics, with its universal scaling fixed by magnetic order and kink mobility.
Universal meson spectra near (1+1)(1+1)(1+1)-dimensional Ising criticality
Near (1+1)(1+1)(1+1)-dimensional [(1+1)(1+1)(1+1)D] Ising criticality, a magnetic perturbation induces confinement and produces a cascade of bound-state excitations known as mesons. Here we show these mesons share a universal mass scaling after independently rescaling the model-dependent microscopic couplings. The number of stable mesons is controlled by the lightest two-meson threshold, while the lightest-meson mass follows a continuous trajectory characterized by a single scaling parameter. Using Hamiltonian truncation method, we obtain the trajectory numerically in both Ising field theory and the near-critical mixed-field Ising chain (MFIC). Under the rescaling, the trajectory and stable-meson-count crossover windows of MFIC both collapse onto the field-theory results. To further demonstrate the above universal organization of the meson spectra, we consider a class of four-periodic spin-1/21/21/2 Heisenberg-Ising chains under transverse fields, whose parameter space contains a family of quantum Ising critical points. The Hamiltonian family includes effective spin models for the quasi-one-dimensional antiferromagnets Ba(Sr)Co2_22​V2_22​O8_88​. Using tensor-network calculations, we obtain the corresponding lightest-meson mass trajectory for BaCo2_22​V2_22​O8_88​ and find that it also collapse onto the same universal curve given by field-theory result. Our results suggest that the universal scaling structure of quantum Ising criticality extends into the nearby confining regime, governing the organization of the meson spectrum. They thereby provide a practical criterion for interpreting excitations of quasi-1D Ising-like magnets in mixed fields beyond E8E_8E8​ integrability.
We investigate the interplay between confinement and Krylov complexity in the long-range transverse-field Ising chain, where power-law interactions ∼1/rα\sim 1/r^α∼1/rα generate an effective confining potential between domain walls without an explicit symmetry-breaking field. Using the time-averaged Krylov state complexity after a quench as a diagnostic, we find that the complexity is strongly suppressed and only weakly system size-dependent at small ααα and small transverse field hxh_xhx​, where confinement is strongest, and crosses over to rapid growth at larger ααα, where it increases with system size. Within the confined regime, the oscillation frequencies of the complexity agree with the meson masses of an effective two-kink model, with the dominant frequency growing linearly with the lightest meson mass and the time-averaged complexity decreasing approximately as its inverse square. We further study operator Krylov complexity at finite temperature using the Wightman inner product. At low temperature, it exhibits bounded oscillations that directly encode the meson spectrum, whereas this structure is progressively washed out as the temperature increases. These results identify Krylov complexity as a spectroscopic probe of interaction-induced confinement and reproduce, in a microscopic lattice model, the hallmarks of Krylov complexity identified in confining field theories.
Fisher's [Phys. Rev. B 51,51,51, 6411 (1995)] real-space renormalization-group (RG) treatment of random transverse-field Ising spin chains is extended into the strongly ordered and strongly disordered Griffiths phases, and asymptotically exact results are obtained. In the noncritical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a nonlinear quantum control parameter \ensuremath{\Delta}, which is a natural measure of the distance from the quantum critical point. \ensuremath{\Delta}, which is found to stay invariant along the RG trajectories, and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc.), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities do not depend on the form of the disorder, provided the nonlinear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated, and the critical point magnetization singularity is determined in a simple, direct way.
Shannon--Rényi entropies probe the full measurement distribution of a quantum many-body state, but their microscopic evaluation remains difficult even in free-fermion systems. We show that the Born distribution of the critical transverse-field Ising chain is exactly the parity boundary of a weak inverse-square classical Ising model, a structure revealed by the Cauchy form of its probability kernel. The construction is exact at finite size for every real n>0n>0n>0 and isolates the long-distance sectors controlling the n>1n>1n>1 thermodynamics. Cutting a finite interval generates the universal c n/[8(n−1)]c\,n/[8(n-1)]cn/[8(n−1)] logarithm directly from the missing 1/r21/r^21/r2 interactions, with c=1/2c=1/2c=1/2 the Ising central charge. On the ring, the explicit finite-size factor has no constant term, while any additional noninteger singularity is confined to a residual cycle-only free energy. All explicit long-distance contributions become marginal only at n=1n=1n=1.
We investigate the scaling relation between the physical entanglement time τent and the modular time τmod in the dynamics of the integrable Heisenberg XXX spin chain. Contrary to the hypothesis of a stable universal exponent γ ≈ 2, we find no evidence for a time-independent power-law scaling. Instead, the effective exponent γ(t) exhibits a systematic drift toward smaller values at longer evolution times, following a robust power-law relaxation γ(t) −γ∞ ∼ t−α with α ≈ 0.65, supported by an excellent data collapse with relative spread below 6%. Remarkably, the amplitude of this relaxation is independent of system size N within the accessible window N ∈ {14,16,18,20}. We interpret this behavior as an effective renormalization group flow of the scaling exponent in time, suggesting that γ is not a fixed scaling dimension but a running quantity controlled by a marginally irrelevant operator. Our results demonstrate that previously reported values γ ≈ 2 originate from preasymptotic regimes rather than a true scaling law, and reveal a dynamical mechanism by which simple scaling breaks down in quantum entanglement evolution.
By means of free fermionic techniques combined with multiple precision arithmetic we study the time evolution of the average magnetization, , of the random transverse-field Ising chain after global quenches. We observe different relaxation behaviors for quenches starting from different initial states to the critical point. Starting from a fully ordered initial state, the relaxation is logarithmically slow described by , and in a finite sample of length L the average magnetization saturates at a size-dependent plateau here the two exponents satisfy the relation . Starting from a fully disordered initial state, the magnetization stays at zero for a period of time until with and then starts to increase until it saturates to an asymptotic value , with . For both quenching protocols, finite-size scaling is satisfied in terms of the scaled variable . Furthermore, the distribution of long-time limiting values of the magnetization shows that the typical and the average values scale differently and the average is governed by rare events. The non-equilibrium dynamical behavior of the magnetization is explained through semi-classical theory.