Spread complexity and topological transitions in the Kitaev chain

Abstract

A number of recent works have argued that quantum complexity, a well-known concept in computer science that has re-emerged recently in the context of the physics of black holes, may be used as an efficient probe of novel phenomena such as quantum chaos and even quantum phase transitions. In this article, we provide further support for the latter, using a 1-dimensional p-wave superconductor — the Kitaev chain — as a prototype of a system displaying a topological phase transition. The Hamiltonian of the Kitaev chain manifests two gapped phases of matter with fermion parity symmetry; a trivial strongly-coupled phase and a topologically non-trivial, weakly-coupled phase with Majorana zero-modes. We show that Krylov-complexity (or, more precisely, the associated spread-complexity) is able to distinguish between the two and provides a diagnostic of the quantum critical point that separates them. We also comment on some possible ambiguity in the existing literature on the sensitivity of different measures of complexity to topological phase transitions.

Time evolution of spread complexity in quenched Lipkin–Meshkov–Glick model

“…Consequently, in some sense, the SC can be thought to provide a robust characterization of QPTs using information theoretic quantities. Furthermore, in [59,60], the SC was used to study topological phase transition in SSH and Kitaev models, respectively, in both in-and out-of-equilibrium scenarios, and the utility of SC as a marker of phase transition was established. In our work, though, we have mainly focused on the zero-temperature QPTs in an infinite range interacting LMG model, we briefly consider quenches in the SSH model showing the topological phase transition in the appendix below.…”

“…In this case, the complete set of Lanczos coefficients can be obtained from the 'survival amplitude', which is the overlap between the initial reference state and the unitary evolved target state. This notion of SC was explored further in [59] to distinguish different topological phases of the Su-Schrieffer-Heeger (SSH) model and in the Kitaev chain [60]. It was concluded that the SC shows markedly different behavior in the two phases, in contrast to that of the NC.…”

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

“…This orthonormal basis is known as the Krylov basis, and in a sense is an optimal choice of basis in Krylov space. In the context of quantum state complexity, the Krylov basis has been shown to minimize the spread complexity [29,33,77].…”

Quantum state complexity meets many-body scars

“…Often called as 'spread complexity,' in its simplest avatar it is associated to the optimal spread of a particular state under timeevolution [31]. Recently, this has found applications in the context of quantum many-body theory [31][32][33][34][35][36][37][38][39]. The notion of quantum complexity comes from the theory of quantum computation.…”