Krylov complexity of density matrix operators

We consider a perturbative expansion of the Lanczos coefficients and the Krylov complexity for two-dimensional conformal field theories under integrable deformations. Specifically, we explore the consequences of $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ , $$ \textrm{J}\overline{\textrm{T}} $$ J T ¯ , and $$ \textrm{J}\overline{\textrm{J}} $$ J J ¯ deformations, focusing on first-order corrections in the deformation parameter. Under $$ \textrm{T}\overline{\textrm{T}} $$ T T ¯ deformation, we demonstrate that the Lanczos coefficients bn exhibit unexpected behavior, deviating from linear growth within the valid perturbative regime. Notably, the Krylov exponent characterizing the rate of exponential growth of complexity surpasses that of the undeformed theory for positive value of deformation parameter, suggesting a potential violation of the conjectured operator growth bound within the realm of perturbative analysis. One may attribute this to the existence of logarithmic branch points along with higher order poles in the autocorrelation function compared to the undeformed case. In contrast to this, both $$ \textrm{J}\overline{\textrm{J}} $$ J J ¯ and $$ \textrm{J}\overline{\textrm{T}} $$ J T ¯ deformations induce no first order correction to either the linear growth of Lanczos coefficients at large-n or the Krylov exponent and hence the results for these two deformations align with those of the undeformed theory.

Krylov complexity of deformed conformal field theories

Chattopadhyay,

Malvimat,

Mitra

2024

Operator size growth in Lindbladian SYK

Liu,

Meyer,

Xian

2024

We investigate the growth of operator size in the Lindbladian Sachdev-Ye-Kitaev model with q-body interaction terms and linear jump terms at finite dissipation strength. We compute the operator size as well as its distribution numerically at finite q and analytically at large q. With dissipative (productive) jump terms, the size converges to a value smaller (larger) than half the number of Majorana fermions. At weak dissipation, the evolution of operator size displays a quadratic-exponential-plateau behavior. The plateau value is determined by the ratios between the coupling of the interaction and the linear jump term in the large q limit. The operator size distribution remains localized in the finite size region even at late times, contrasting with the unitary case. Moreover, we also derived the time-independent orthogonal basis for operator expansion which exhibits the operator size concentration at finite dissipation. Finally, we observe that the uncertainty relation for operator size growth is saturated at large q, leading to classical dynamics of the operator size growth with dissipation.

Spread and spectral complexity in quantum spin chains: from integrability to chaos

Camargo,

Huh,

Jahnke

et al. 2024

We explore spread and spectral complexity in quantum systems that exhibit a transition from integrability to chaos, namely the mixed-field Ising model and the next-to-nearest-neighbor deformation of the Heisenberg XXZ spin chain. We corroborate the observation that the presence of a peak in spread complexity before its saturation, is a characteristic feature in chaotic systems. We find that, in general, the saturation value of spread complexity post-peak depends not only on the spectral statistics of the Hamiltonian, but also on the specific state. However, there appears to be a maximal universal bound determined by the symmetries and dimension of the Hamiltonian, which is realized by the thermofield double state (TFD) at infinite temperature. We also find that the time scales at which the spread complexity and spectral form factor change their behaviour agree with each other and are independent of the chaotic properties of the systems. In the case of spectral complexity, we identify that the key factor determining its saturation value and timescale in chaotic systems is given by minimum energy difference in the theory’s spectrum. This explains observations made in the literature regarding its earlier saturation in chaotic systems compared to their integrable counterparts. We conclude by discussing the properties of the TFD which, we conjecture, make it suitable for probing signatures of chaos in quantum many-body systems.