Quantum complexity and topological phases of matter

TL;DR

This paper introduces spread complexity, defined via the Krylov (Lanczos) basis, as a diagnostic for topological phases in quantum many-body systems. Analytic results for the SSH model show ground-state spread complexity is phase-sensitive and becomes constant in the topological regime, while quench dynamics reveal universal early-time growth and late-time plateaus that distinguish phases. Extending to the Kitaev chain and discussing relations to known topological order parameters, the work advocates spread complexity as a universal, computable tool bridging quantum information and condensed matter, with potential experimental relevance. Overall, the approach provides a new perspective on identifying and characterizing topological phases through dynamical, information-theoretic probes.

Abstract

In this work, we find that the complexity of quantum many-body states, defined as a spread in the Krylov basis, may serve as a new probe that distinguishes topological phases of matter. We illustrate this analytically in one of the representative examples, the Su-Schrieffer-Heeger model, finding that spread complexity becomes constant in the topological phase. Moreover, in the same setup, we analyze exactly solvable quench protocols where the evolution of the spread complexity shows distinct dynamical features depending on the topological vs non-topological phase of the initial state as well as the quench Hamiltonian.

Figures (5)

• Figure 1: Behaviour of the spread complexity of formation $\mathcal{C}(t_1, t_2)$ (equation \ref{['kcmpl1_rs']}) for the ground state of the SSH model $|\Omega \rangle$.
• Figure 2: Evolution of spread complexity starting from $H_i$ with $(t^i_1=1,t^i_2=0.2)$ and $(t^f_1=0.7,t^f_2=1.5)$ (in black) and the same parameters in $H_i$ and $H_f$ switched (in red). At late times complexity approaches constant \ref{['LTCon']} (in blue).
• Figure 3: Spread complexity constant at late times $\mathcal{C}(t \to \infty ; \alpha , \beta)$, shown in (23) in the main text.
• Figure 4: Complexity $\mathcal{C}(s=1; \mu , \Delta)$ as a function of $\mu$ for various $\Delta$s ($\Delta = -2,-1,-1/2,1/2,1,2$). Between the two vertical gridlines $\mu = \pm 1$, spread complexity is $\mu$-independent. The two dashed horizontal gridlines are the analytical results of $\mathcal{C}(|\mu|<1, \Delta = \pm 1)$, respectively. For $|\mu| \to \infty$, the spread complexities of various $\Delta$s approach the dotted horizontal gridline $\mathcal{C} =1/2$.
• Figure 5: Complexity $\mathcal{C}(s=1; |\mu|<1 , \Delta)$ as a function of $\Delta$. The two red dots are the analytical results of $\mathcal{C}(|\mu|<1, \Delta = \pm 1)$, respectively.