Circuit Complexity across a Topological Phase Transition

TL;DR

This work uses Nielsen's geometric circuit-complexity framework to quantify state-to-state transformations in the Kitaev chain across topological transitions. By reducing to momentum sectors and deriving a simple pairwise complexity $\mathcal{C}_k = |\Delta\theta_k|^2$, the authors obtain a total complexity $\mathcal{C} = \sum_k |\Delta\theta_k|^2$ that scales as $\mathcal{C} \propto L$ and exhibits non-analytic behavior at equilibrium critical points. They analytically show divergences in derivatives of $\mathcal{C}$ with respect to target parameters via contour-integral methods and reveal a dichotomy in real-space locality: same-phase evolutions are effectively local, while across phases require long-range circuits. Dynamically, the long-time steady-state circuit complexity after quenches shows nonanalytic features signaling dynamical topological transitions, and the results generalize to long-range 1D Kitaev chains and to higher dimensions, including $p+ip$ superconductors in 2D. Together, these findings position circuit complexity as a powerful tool for diagnosing both equilibrium and dynamical topological phenomena in quantum many-body systems.

Abstract

We use Nielsen's geometric approach to quantify the circuit complexity in a one-dimensional Kitaev chain across a topological phase transition. We find that the circuit complexities of both the ground states and non-equilibrium steady states of the Kitaev model exhibit non-analytical behaviors at the critical points, and thus can be used to detect both {\it equilibrium} and {\it dynamical} topological phase transitions. Moreover, we show that the locality property of the real-space optimal Hamiltonian connecting two different ground states depends crucially on whether the two states belong to the same or different phases. This provides a concrete example of classifying different gapped phases using Nielsen's circuit complexity. We further generalize our results to a Kitaev chain with long-range pairing, and discuss generalizations to higher dimensions. Our result opens up a new avenue for using circuit complexity as a novel tool to understand quantum many-body systems.

Figures (7)

• Figure 1: (a) Phase diagram of the Kitaev chain, with $W$ denoting the winding number. (b) Ground state circuit complexity and (c) its derivative versus target state chemical potential ($\mu_T$) for several reference states, each with a different chemical potential $\mu_R$. (d) Bogoliubov angle difference, $\Delta \theta_{k_n}$, for different target ground states, with $\mu_R= 0$. $\Delta_{R}=\Delta_{T}=1$ for (b)--(d), and $L=1000$ for (b) and (c).
• Figure 2: Derivative of circuit complexity as a function of $\mu_T$ and $\Delta_T$. Panel (a) plots the derivative with respect to $\mu_T$ (in units of $1/\Delta_T$), and panel (b) plots the derivative with respect to $\Delta_T$. The reference state is chosen as the ground state of Eq. \ref{['kitaev']} with $\mu_R=0$ and $\Delta_R=-1$, and $L=1000$.
• Figure 3: (a) Circuit complexity growth for various post-quench chemical potentials, $\mu_f$. The initial state (serves as the reference state) is the ground state of Eq. \ref{['kitaev']} with $\mu_i=0$. (b) Steady-state values of complexity versus $\mu_f$. The different lines denote different initial/reference states. $\Delta_i=\Delta_f=1$ and $L=1000$ in both plots.
• Figure 4: (a) Derivative of circuit complexity with respect to $\mu_T$ for three different reference ground states of the long-range Kitaev chain, with $\Delta_R=\Delta_T=1.3$. (b) Steady-state value of circuit complexity versus $\mu_f$ for three different initial ground states, with $\Delta_i=\Delta_f=1$. $L=1000$ and $\alpha=0$ in both plots.
• Figure S1: The phase diagram of the Kitaev chain, where in each phase we list which of the two branch points given in Eq. \ref{['eq:zia']} lie inside the contour integrals in Eq. \ref{['eq:divC2']}. The integrals can only diverge at the phase transitions, where the branch points cross the contour,