Krylov complexity in free and interacting scalar field theories with bounded power spectrum

Abstract

We study a notion of operator growth known as Krylov complexity in free and interacting massive scalar quantum field theories in d-dimensions at finite temperature. We consider the effects of mass, one-loop self-energy due to perturbative interactions, and finite ultraviolet cutoffs in continuous momentum space. These deformations change the behavior of Lanczos coefficients and Krylov complexity and induce effects such as the “staggering” of the former into two families, a decrease in the exponential growth rate of the latter, and transitions in their asymptotic behavior. We also discuss the relation between the existence of a mass gap and the property of staggering, and the relation between our ultraviolet cutoffs in continuous theories and lattice theories.

A bulk manifestation of Krylov complexity

“…This has to be assumed, since analytic continuation cannot be invoked because for it to apply one needs to have a function defined over at least an interval, not a discrete domain. Our Lanczos coefficients for DSSYK (3.10) certainly satisfy this assumption, but there are examples in the literature of Lanczos sequences that do not fulfill it: the works [47][48][49] have studied systems where the Lanczos sequence shows staggering, the even and odd coefficients following different profiles. For such systems, the Lanczos sequence does not have a well-defined continuum limit.…”

Krylov complexity in a natural basis for the Schrödinger algebra

“…Krylov complexity is conjectured to grow at a maximal rate for chaotic systems, although there are subtleties that need to be taken into account, especially for quantum field theories due to their infinite degrees of freedom in contrast to ordinary quantum mechanical systems, see discussions in [3][4][5][6][7][8][9][10]. When one considers quantum many body systems such as spin chains it is more straightforward to produce evidence that Krylov complexity can for example distinguish between integrable and chaotic dynamics as was argued in [11][12][13].…”

“…We believe that there are two main reasons. First, Krylov complexity can be applied to any quantum system making it computationally available, at least in principle, for a plethora of different cases including but not limited to condensed matter and many-body systems [13][14][15][16][17], quantum and conformal field theories [3][4][5][18][19][20], open systems [21][22][23][24][25], topological phases of matter [26,27] and many other topics related to aspects of the above and not only [28][29][30][31]. Second, it is related by its construction to inherent properties and characteristic parameters of the system, namely the Hamiltonian and the Hilbert space that it defines.…”