Signatures of Chaos in Nonintegrable Models of Quantum Field Theories

Abstract

We study signatures of quantum chaos in (1+1)D Quantum Field Theory (QFT) models. Our analysis is based on the method of Hamiltonian truncation, a numerical approach for the construction of low-energy spectra and eigenstates of QFTs that can be considered as perturbations of exactly solvable models. We focus on the double sine-Gordon, also studying the massive sine-Gordon and φ 4 model, all of which are non-integrable and can be studied by this method with sufficiently high precision from small to intermediate perturbation strength. We analyse the statistics of level spacings and of eigenvector components, which are expected to follow Random Matrix Theory predictions. While level spacing statistics are close to the Gaussian Orthogonal Ensemble as expected, on the contrary, the eigenvector components follow a distribution markedly different from the expected Gaussian. Unlike in the typical quantum chaos scenario, the transition of level spacing statistics to chaotic behaviour takes place already in the perturbative regime. Moreover, the distribution of eigenvector components does not appear to change or approach Gaussian behaviour, even for relatively large perturbations. Our results suggest that these features are independent of the choice of model and basis.

Weak integrability breaking and level spacing distribution

“…involve only two levels. It is interesting to note that in quantum field theories obtained as perturbation of conformal field theories the crossover to the Wigner-Dyson behaviour takes place already in the perturbative regime [38]. The essential difference with the case considered here is that for the conformal spectrum the levels are generically multiply degenerate.…”

Snowmass White Paper: Hamiltonian Truncation

“…For instance, using the distribution of eigenvalue spacings, one can observe the chaotic nature of 2d φ 4 theory at both weak and strong couplings (see Fig. 5, left plot), and more generally in deformations of rational theories [40,41]; one can also study the emergence of chaos through the statistics of eigenvector components. From the density of eigenvalues as a function of energy, one can extract the entropy and other thermodynamic quantities.…”

Chaos in the three-site Bose-Hubbard model -- classical vs quantum

“…To compare the Bose-Hubbard system with the GOE case, we will compare the coefficients of Bose-Hubbard eigenstates with the Gaussian distribution with mean 0 and variance 1/D. It has been observed that mid-spectrum eigenstates of chaotic/ergodic many-body systems generally have coefficients with a near-Gaussian distribution [54,[74][75][76][77][78][79][80][81][82], in accord with Berry's conjecture [83]. On the other hand, eigenstates of integrable or many-body-localized systems, as well as eigenstates at the spectral edges of nominally chaotic systems, typically have markedly non-Gaussian distributions [74,75,78,84].…”