The massive scalar field in 1+1 dimensions perturbed by ϕ4\phi^4 or ϕ6\phi^6 is the canonical continuum laboratory for the kinetic-theory classification of weak versus strong integrability breaking. Fitos and Takács (SciPost Phys. 15, 137 (2023)) confirmed the classification spectrally: in level-spacing statistics computed with the renormalization-group-improved truncated Hamiltonian approach, the crossover coupling to Wigner-Dyson statistics scales as a power of the free-boson particle number NN with well-separated exponents for the two perturbations (γ4∈[−3.7,−3.0]\gamma_4 \in [-3.7,-3.0] versus γ6∈[−4.8,−4.2]\gamma_6 \in [-4.8,-4.2]), while its volume scaling is weak and inconclusive. They left open the comparison of the full quantum time evolution of the two theories after quenches — the direct dynamical test of the weak/strong distinction — because the truncated Hamiltonian approach requires further methodological development (higher-order counterterms, controlled truncation errors at long times) to yield reliable real-time dynamics; the supporting particle-number-scaling evidence covers only moderate NN, and the interpretation of particle number, not volume, as the finite-size control parameter is acknowledged by the authors to be heuristic. The dynamical comparison of the two theories after quenches therefore remains open.

Background

An interacting quantum field theory (QFT) is called integrable when it possesses infinitely many independent local conserved charges, which force all particle scattering to be elastic and factorized (any multiparticle process decomposes into independent two-particle collisions). The free massive scalar field in 1+1 space-time dimensions is the simplest relativistic integrable QFT, and adding a polynomial self-interaction ϕ4\phi^4 or ϕ6\phi^6 to it destroys the higher charges, making these the standard continuum models for studying how integrability breaks. A standard diagnostic of quantum chaos is the statistics of energy level spacings: after 'unfolding' the spectrum (rescaling by the smooth local level density so the mean spacing is one), generic quantum systems show Wigner-Dyson statistics of random matrix theory — for a real symmetric Hamiltonian those of the Gaussian orthogonal ensemble (GOE), with level repulsion — while integrable systems show Poissonian statistics with frequent near-degeneracies (the Berry-Bohigas-Giannoni-Schmit and Berry-Tabor conjectures). Fitos and Takács quantified the crossover with two standard measures: the mean consecutive level-spacing ratio r~n=min⁡(sn/sn−1, sn−1/sn)\tilde r_n=\min(s_n/s_{n-1},\,s_{n-1}/s_n) mapped affinely onto [0,1][0,1] between its Poisson and GOE values, and the Brody parameter β\beta of the interpolating Brody distribution Pβ(s)∝sβe−csβ+1P_\beta(s)\propto s^\beta e^{-c s^{\beta+1}} (β=0\beta=0 Poisson, β=1\beta=1 GOE). In a system restricted to a finite Hilbert space the crossover is continuous in the coupling, and a crossover coupling g~∗\tilde g^* is defined by fitting the measure to 1−exp⁡(−g~/g~∗)1-\exp(-\tilde g/\tilde g^*); in spin chains, the scaling of g~∗\tilde g^* with system size characterizes how strongly a perturbation breaks integrability and is a standard probe of the onset of many-body quantum chaos. Separately, Durnin, Bhaseen and Doyon introduced a kinetic-theory classification of integrability-breaking perturbations based on the Boltzmann equation for the quasiparticle distribution: a perturbation is 'weak' when the leading-order collision processes preserve a complete set of quasiparticle occupations, so true relaxation begins only at higher perturbative order, and 'strong' when the leading-order processes already degrade them. Their kinetic theory places the ϕ4\phi^4 deformation of the massive scalar in the weak class and ϕ6\phi^6 in the strong class, predicting parametrically different relaxation after a quantum quench (a sudden change of a Hamiltonian parameter followed by unitary time evolution). Fitos and Takács addressed the spectral side with the truncated Hamiltonian approach (THA), pioneered by Yurov and Zamolodchikov and extended to ϕ4\phi^4 by Bajnok and Lájer: the Hamiltonian H=HKG+gn∫0Ldx :ϕn(x):H = H_{KG} + g_n \int_0^L dx\, :\phi^n(x): (free massive Klein-Gordon theory plus a normal-ordered interaction on a circle of circumference LL, with dimensionless volume mLmL and coupling g~n\tilde g_n) is diagonalized in the free Fock basis truncated at a total free-energy cutoff Λ\Lambda, with leading-order renormalization-group (RG) counterterms cancelling the cutoff dependence (the ϕ6\phi^6 counterterms were derived in that work). Working in the zero-momentum sector even under parity and under ϕ→−ϕ\phi\to-\phi, in an energy window well above the low-lying quasiparticle levels and well below Λ\Lambda, they found: (i) the volume scaling of the crossover coupling is weak and strongly dependent on the chosen energy window and cutoff, giving no reliable separation between the two perturbations; (ii) restricting the statistics to states of fixed free-boson particle number NN and analyzing each NN-sector separately yields a robust power law g~∗(N)∼Nγ\tilde g^*(N)\sim N^{\gamma} with γ4∈[−3.7,−3.0]\gamma_4\in[-3.7,-3.0] versus γ6∈[−4.8,−4.2]\gamma_6\in[-4.8,-4.2] over mL=4mL=4-88 and even particle numbers N=6N=6-1616 (the N=18N=18 data were discarded as too close to the energy-window edge), cleanly separating the perturbations and supporting the kinetic classification of ϕ6\phi^6 as the stronger integrability breaking. Their interpretation — that particle number, not volume, is the correct finite-size control parameter for the chaos crossover in a continuum QFT — rests on heuristic arguments: the dominant level-splitting processes are particle-number-preserving 3→33\to3 scatterings, while 2→42\to4 processes are kinematically suppressed by a threshold energy and 4→24\to2 processes need a simultaneous four-particle encounter; this is supplemented by an analogy to the Kolmogorov-Arnold-Moser (KAM) theorem of classical Hamiltonian dynamics, under which weakly perturbed finite-dimensional integrable systems retain most invariant tori, with the torus-breakup threshold vanishing only as the number of degrees of freedom diverges. What remains missing is the dynamical side of the story: a direct comparison of the full quantum time evolution of the two theories, which the authors explicitly left open because the available THA methodology does not yet give reliable long-time dynamics in the regimes of interest.

Problem Statement

The problem is to complete the quantitative characterization of weak versus strong integrability breaking in the ϕ4\phi^4- and ϕ6\phi^6-perturbed massive scalar QFT in 1+1 dimensions, by establishing whether the distinction inferred from kinetic theory and level statistics is realized in the actual full quantum dynamics of the two theories. Its components serve this single objective and are scientifically linked, but only the central deliverable (b) is required; the methodological route (a) is advisory, and any controlled real-time method that yields (b) is acceptable: (a) Methodological (advisory enabling route): develop the real-time truncated Hamiltonian approach for these non-integrable theories to controlled reliability — extending the renormalization-group improvement beyond leading-order counterterms, providing systematic truncation-error estimates, and demonstrating stable long-time evolution in the parameter regimes of interest (the source paper's preliminary attempts showed the present leading-order machinery is insufficient). (b) Central deliverable: compute and compare the full quantum time evolution after quenches in the ϕ4\phi^4 and ϕ6\phi^6 theories at matched volumes, energy windows and couplings, e.g. following the earlier full-quantum ϕ4\phi^4 quench study of Szász-Schagrin, Lovas and Takács, and test the kinetic-theory predictions of Durnin, Bhaseen and Doyon in dynamical observables (for example relaxation timescales, prethermal plateaus, and the approach of occupation numbers or one-point functions toward equilibrium), deciding whether the weak (ϕ4\phi^4) versus strong (ϕ6\phi^6) distinction manifests dynamically as predicted. An answer that only improves the method, without confronting the dynamical comparison (b), does not meet the objective. The dynamical comparison (b), enabled by such a controlled real-time method, is the decisive deliverable: a completion is a controlled comparison that confirms, refutes, or delimits the weak-versus-strong classification in the dynamics.

References

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