On Chaos in QFT

TL;DR

The paper probes quantum chaos in 1+1D QFTs by contrasting integrable SG with non-integrable DSG using a Truncated Hamiltonian Space Approach to generate large spectra and then mapping spectral statistics to GOE predictions from Random Matrix Theory. It surveys multiple chaos diagnostics—NN spacings, spacing ratios, higher-order spacings, pair correlations, spectral form factor, and spectral rigidity—to distinguish chaotic (GOE-like) from integrable (Poisson-like) behavior, and demonstrates GOE-like statistics in DSG for appropriate parameters while highlighting truncation-induced deviations in SG. It additionally reveals linear growth in mean higher-order spacings ⟨s_k⟩ ≈ k and linear behavior of ⟨r_k⟩ with a notable symmetry around k = N/2, along with a scale-dependent GOE regime in DSG evidenced by a finite Thouless time t_{Th} ≈ 4.45×10^4. These results advance the use of RMT-based chaos diagnostics in quantum field theories and motivate applying similar analyses to other truncation-based QFT models, OTOCs, and Krylov complexity to broaden the chaoticity landscape.

Abstract

In this note we explore the chaotic behavior of non-integrable QFTs and compare them to integrable ones. We choose as prototypes the double sine-Gordon and the sine-Gordon models. We analyze their discrete spectrum determined by a truncation method. We examine the map of the corresponding energy eigenvalues to the eigenvalues of the random matrix theory (RMT) Gaussian orthogonal ensemble (GOE). This is done by computing the following properties: (a) The distribution of the adjacent spacings and their ratios (b) Higher order spacings and ratios (c) Pair correlations (d) Spectral form factors and (e) Spectral rigidity. For these properties we determine the differences between the integrable and non-integrable theories and verify that the former admits a Poisson behavior and the latter GOE (apart from the spectral rigidity).

Figures (16)

• Figure 1: Histogram of the spectrum taken from GUE, $N=1000$. The blue curve in (\ref{['fig:GUE-N=1000-hist']}) is the average level density of GUE, eq. (\ref{['eq:rho_0-GUE']}) and it is used for the unfolding. (\ref{['fig:GUE-N=1000-hist-unfolded']}) shows the spectrum after the unfolding.
• Figure 2: The high range spectrum of DSG with $mL=1$. The orange bars in (\ref{['fig:DSG-spectrum-before-unfolding']}) is the binned density and the blue curve on top is the interpolated $\rho_0$. (\ref{['fig:DSG-spectrum-before-unfolding']}) shows the same spectrum after unfolding, using the numerically evaluated $\rho_0$. The fluctuations are still visible in (\ref{['fig:DSG-spectrum-before-unfolding']}) as variations of the bins heights around the constant average density.
• Figure 3: RMT expressions for the connected parts of the SFF, after unfolding and normalization. The black curve corresponds to the GOE (\ref{['eq:SFF-connected-GOE']}) and the orange curve to the GUE (\ref{['eq:SFF-connected-GUE']}).
• Figure 4: (\ref{['fig:DSG-NN-spacing']}) The unfolded DSG nearest neighbor spacing (orange bars) with the GOE Wigner-surmise (blue curve. (\ref{['fig:DSG-NN-spacing-ratio']}) The unfolded DSG nearest neighbor spacing ratio (orange bars) with the expression for nearest neighbor spacing ratio distribution (\ref{['eq:NN-spacing-ratio-distribution']}), with $\beta=1$ (blue curve).
• Figure 5: Level spacing and spacing ratios statistics from the SG data. Fig. \ref{['fig:SG-NN-spacing']} shows the consecutive level spacing distribution, where the black dashed line is the RMT prediction for a Poisson distribution (\ref{['eq:Poisson-distribution']}) and the grey dashed line is the GOE WD distribution (\ref{['eq:WD-distribution']}). Fig. \ref{['fig:SG-NN-spacing-ratio']} shows the consecutive level spacing ratio distribution, with the black dashed line being the Poisson ensemble prediction for level spacing ratios (\ref{['eq:NN-spacing-ratio-distribution-Poisson']}) and the grey dashed line being the GOE level spacing ratio from RMT, (\ref{['eq:P(r,beta,k)']}).