Spread Complexity and Topological Transitions in the Kitaev Chain

TL;DR

We study spread (Krylov) complexity as a diagnostic for topological phase transitions in the 1D Kitaev chain with Hamiltonian H = sum_j [ -J/2 (c_j^† c_{j+1} + c_{j+1}^† c_j) - μ (c_j^† c_j - 1/2) + Δ/2 (c_j^† c_{j+1}^† + c_{j+1} c_j) ]. By decomposing into momentum modes and using SU(2) coherent states, we compute the spread complexity for several circuit choices connecting simple reference states to the Kitaev ground state. We find that spread complexity can distinguish the topological phase |μ|<|J| from the trivial phase, with distinct behaviors across the three circuits: a μ-independent plateau in the topological phase for a μ,Δ-independent reference, and a monotonic μ-curve without plateau for a vacuum reference; derivatives with respect to Δ or μ exhibit cusps or divergences at the phase boundary. These results support spread complexity as an efficient, less ambiguous probe of topological quantum phase transitions and motivate extensions to twisted boundaries and other topological models.

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.

Figures (10)

• Figure 1: The spread complexity in the continuum limit for the circuit connecting the free fermion ground state to the Kitaev chain ground state. We have chosen $J=1$. When $|\mu| < 1$ the system is in the topological phase and the spread complexity is a $\Delta$-dependent constant.
• Figure 2: a) The spread complexity for the circuit connecting the free fermion ground state and Kitaev chain ground state as a function of the chemical potential, with $J =1$, and in the continuum limit. The complexity is constant in the region $|\mu| <1$. b) The spread complexity for the circuit connecting the free fermion ground state and Kitaev chain ground state as a function of the chemical potential, with $J =1$ and $\Delta=2$. The spread complexity has been divided by $\sqrt{L}$ for display purposes. The top curve corresponds to $L=40$, the middle $L=20$ and the bottom $L=10$.
• Figure 3: The derivative of spread complexity with respect to $\Delta$ in the continuum limit for the circuit connecting the free fermion ground state and Kitaev chain ground state. We have set $J=1$. The blue line has $\mu = 0.98$, the orange $\mu = 1.02$ and the green $\mu = 1.1$. When crossing the topological phase transition points at $|\mu| =1$ the derivative develops a discontinuity.
• Figure 4: Complexity $\mathcal{C}(s=1; \mu , \Delta)$ as a function of $\mu$ for various choices of $\Delta$ ($\Delta = -2,-1,-1/2,1/2,1,2$) for the circuit connecting the Cooper-pair ground state and Kitaev chain ground state. 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 $\Delta$ for the circuit connecting the Cooper-pair ground state and Kitaev chain ground state. The two red dots are the analytical results of $\mathcal{C}(|\mu|<1, \Delta = \pm 1)$, respectively.