Detecting Higher Berry Phase via Boundary Scattering

Abstract

Higher Berry phase has recently been proposed to study the topology of the space of gapped many-body quantum systems. In this work, we develop a boundary-scattering approach to detect higher Berry phases in one-dimensional gapped free-fermion systems. By coupling a gapless lead to the gapped system, we demonstrate that the higher Berry invariant can be obtained by studying the higher winding number of the boundary reflection matrix. The resulting topological invariant is robust against perturbations such as disorder. Our approach establishes a connection between higher Berry invariants and transport properties, thereby providing a potentially experimentally accessible probe of parametrized topological phases.

Figures (8)

• Figure 1: Schematic illustration of detecting the topological invariant of a parametrized family of gapped systems via boundary scattering. In the limit where the gapped system is infinitely long, an incoming wavefunction $\psi_{\mathrm{in}}$ from the left lead is completely reflected into an outgoing wavefunction $\psi_{\mathrm{out}}$. The resulting reflection matrix $R(\lambda)$ as defined in \ref{['def:Rmatrix']} depends on the gapped-system parameters $\lambda$. As these parameters are varied, the evolution of $R$ in parameter space defines a quantized topological winding number.
• Figure 2: Exactly solvable free-fermion lattice models that exhibit a nontrivial higher Berry phase with parameter space $X=S^3$. Each super-site (black box) contains two sites, and the orange lines correspond to the hopping characterized by $m_0$. Each blue line corresponds to a hopping characterized by $\vec{m}=(m_1,m_2,m_3)^T$. These parameters satisfy the constraint in \ref{['Eq:S3_mass']}.
• Figure 3: Current of Chern number pump as a function of the pumping parameter $\alpha$ for different velocity $v_1$ in the gapped system and $v_0=1$ in the lead. Here the system size for the gapped system is chosen as $L=20$.
• Figure 4: Current of Chern number pump as a function of the pumping parameter $\alpha$ for different onsite-disorder strengths $\overline{d}$. We choose $v_0=1$, $v_1 = 0.5$, and the gapped-system size is $L=20$.
• Figure 5: $\frac{1}{2\pi}\mathrm{Arg}(R)$ for different values of $m_0$ and $m_1$ in the decoupled limit, with lead velocity $v_0=1$.