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International Journal of Modern Physics B

Exact out-of-time-ordered correlators in free Luttinger liquids: Theory and perturbative corrections

Prafulla S. Oak

Published May 29, 2026• 0 citations
Publisher

Abstract

In a free Luttinger liquid, we calculate the exact out-of-time-ordered correlator (OTOC) for vertex operators, which is valid for arbitrary operator charges without small-charge approximations. The OTOC grows purely as a power law [Formula: see text] inside the light cone without exponential spreading, characteristic of integrable systems. We develop a systematic perturbative expansion in the interaction strength [Formula: see text] and construct the full structure of first- and second-order corrections via Wick’s theorem. At finite temperature [Formula: see text], thermal effects modify the late-time scaling from power-law to exponential growth [Formula: see text]; however, this reflects thermal statistics rather than quantum chaos. These results lay a solid foundation for studies of information scrambling and quantum chaos in one-dimensional integrable systems.

DOI
10.1142/s0217979226501869

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In a free Luttinger liquid, we calculate the exact out-of-time-ordered correlator (OTOC) for vertex operators, which is valid for arbitrary operator charges without small-charge approximations. The OTOC grows purely as a power law [Formula: see text] inside the light cone without exponential spreading, characteristic of integrable systems. We develop a systematic perturbative expansion in the interaction strength [Formula: see text] and construct the full structure of first- and second-order corrections via Wick’s theorem. At finite temperature [Formula: see text], thermal effects modify the late-time scaling from power-law to exponential growth [Formula: see text]; however, this reflects thermal statistics rather than quantum chaos. These results lay a solid foundation for studies of information scrambling and quantum chaos in one-dimensional integrable systems.
Mimicry of chaos and kkk-design in higher order OTOCs of Luttinger liquids
Out-of-time-order correlators (OTOCs) provide a fundamental metric for quantum chaos, but capturing the fine structure of information scrambling requires exploring their higher-order generalizations. Here, we systematically investigate the sequence of higher-order OTOCs in a Luttinger liquid and its lattice realization, the XXZ Heisenberg chain. Using bosonization and numerics, we extract the full temporal dynamics of the first three OTOCs, revealing that they rapidly erase memory of the initial state, and quickly saturate to their steady state values. Strikingly, we show that calculating the late time saturation values for the entire sequence of higher-order OTOCs maps exactly onto determining the partition function of a non-Hermitian Harper model. Through this mapping, we demonstrate that for moderately strong interactions, the steady-state OTOCs become parametrically small up to the seventh order, mimicking higher kkk-design. Our results reveal that Luttinger liquids exhibit an unexpectedly profound degree of apparent scrambling when viewed through the lens of higher-order OTOCs.
Information scrambling and the butterfly effect in chaotic quantum systems can be diagnosed by out-of-time-ordered (OTO) commutators through an exponential growth and large late time value. We show that the latter feature shows up in a strongly correlated many-body system, a Luttinger liquid, whose density fluctuations we study at long and short wavelengths, both in equilibrium and after a quantum quench. We find rich behavior combining robustly universal and nonuniversal features. The OTO commutators display temperature- and initial-state-independent behavior and grow as t^{2} for short times. For the short-wavelength density operator, they reach a sizable value after the light cone only in an interacting Luttinger liquid, where the bare excitations break up into collective modes. This challenges the common interpretation of the OTO commutator in chaotic systems. We benchmark our findings numerically on an interacting spinless fermion model in 1D and find persistence of central features even in the nonintegrable case. As a nonuniversal feature, the short-time growth exhibits a distance-dependent power.
We study out-of-time-ordered correlators (OTOCs) in hard-core boson models with short-range and long-range hopping and compare the results to the OTOCs in the Luttinger-liquid model. For density-density correlations, a related expectation value of the squared commutator starts at zero and decays back to zero after the passage of the wavefront in all three models, while the wavefront broadens as t1/3{t}^{1/3}t1/3 in the short-range model and shows no broadening in the long-range model and the Luttinger-liquid model. For the boson creation operator, the corresponding commutator function shows saturation inside the light cone in all three models, with similar wavefront behavior as in the density-density commutator function, despite the presence of a nonlocal string in terms of Jordan-Wigner fermions. For the long-range model and the Luttinger-liquid model, the commutator function decays as a power law outside the light cone in the long-time regime when following different fixed-velocity rays. In all cases, the OTOCs approach their long-time values in a power-law fashion, with different exponents for different observables and short-range versus long-range cases. Our long-range model appears to capture exponents in the Luttinger-liquid model (which are found to be independent of the Luttinger parameter in the model). This conclusion also comes to bear on the OTOC calculations in conformal field theories, which we propose correspond to long-ranged models.
We present exact results for the classical version of the out-of-time-order commutator (OTOC) for a family of power-law models consisting of N particles in one dimension and confined by an external harmonic potential. These particles are interacting via power-law interaction of the form ∝∑_{i,j=1(i≠j)}^{N}|x_{i}-x_{j}|^{-k}∀k>1 where x_{i} is the position of the ith particle. We present numerical results for the OTOC for finite N at low temperatures and short enough times so that the system is well approximated by the linearized dynamics around the many-body ground state. In the large-N limit, we compute the ground-state dispersion relation in the absence of external harmonic potential exactly and use it to arrive at analytical results for OTOC. We find excellent agreement between our analytical results and the numerics. We further obtain analytical results in the limit where only linear and leading nonlinear (in momentum) terms in the dispersion relation are included. The resulting OTOC is in agreement with numerics in the vicinity of the edge of the "light cone." We find remarkably distinct features in OTOC below and above k=3 in terms of going from non-Airy behavior (1<k<3) to an Airy universality class (k>3). We present certain additional rich features for the case k=2 that stem from the underlying integrability of the Calogero-Moser model. We present a field theory approach that also assists in understanding certain aspects of OTOC such as the sound speed. Our findings are a step forward towards a more general understanding of the spatiotemporal spread of perturbations in long-range interacting systems.
Out-of-time-ordered correlators (OTOCs) are useful tools for investigating foundational questions such as thermalization in closed quantum systems because they can potentially distinguish between integrable and nonintegrable dynamics. Here, we discuss the properties of wavefronts of OTOCs by focusing on the region around the main wavefront at x=vBtx={v}_{B}tx=vB​t, where vB{v}_{B}vB​ is the butterfly velocity. Using a Heisenberg spin model as an example, we find that the leading edge of a propagating Gaussian with the argument $\ensuremath{-}m(x){(x\ensuremath{-}{v}_{B}t)}^{2}+b(x)t$ gives an excellent fit to the region around x=vBtx={v}_{B}tx=vB​t for both the free and chaotic cases. However, the scaling in these two regimes is very different: in the free case the coefficients m(x)m(x)m(x) and b(x)b(x)b(x) have an inverse power law dependence on xxx, whereas in the chaotic case they decay exponentially. We conjecture that this result is universal by using catastrophe theory to show that, on the one hand, the wavefront in the free case has to take the form of an Airy function and its local expansion shows that the power law scaling seen in the numerics holds rigorously and, on the other hand, an exponential scaling of the OTOC wavefront must be a signature of nonintegrable dynamics. We find that the crossover between the two regimes is smooth and characterized by an S\mathsf{S}S-shaped curve giving the lifting of Airy nodes as a function of a chaos parameter. This shows that the Airy form is qualitatively stable against weak chaos and consistent with the concept of a quantum Kolmogorov-Arnold-Moser theory.
Out-of-time-ordered correlators (OTOCs) have emerged as a diagnostic of information scrambling and quantum chaos in many-body systems. We investigate the imprints of chaos in the dynamics of OTOCs in the Heisenberg spin-1/21/21/2 chain with random fields. The system is parameterized to exhibit a crossover from integrable to chaotic dynamics. We demonstrate numerically that the approach to saturation of the OTOC can distinguish between integrable and chaotic regimes, with a power-law (1/t)(1/t)(1/t) relaxation for integrable systems and a higher-degree power-law decay (1/tα;α≥1)(1/t^α; α\ge 1)(1/tα;α≥1) followed by an exponential relaxation for the chaotic regime. We further show that long-range spectral statistics, such as the number variance, are more effective in characterizing quantum chaos in the regime near saturation of OTOC. We also demonstrate that the relaxation and initial scrambling regimes exhibit distinct and universal features, with the former being sensitive and the latter being robust against different realizations of random-fields. The long-time saturation of OTOC also fluctuates with different realizations, and its exact expression is derived through the Eigenstate Thermalization Hypothesis.