Sequential Adiabatic Generation of Chiral Topological States

TL;DR

The paper advances a method to generate chiral topological states from simple product states by sequentially evolving a gapped Hamiltonian rather than applying a sequence of local unitaries. It proves and demonstrates, for free-fermion chiral phases such as the Chern insulator and the p+ip superconductor, that a gap can be preserved while building the topological state one subregion at a time, including a coupled-wire intuition and numerical validation. It then shows how coupling to a discrete gauge group can be implemented via a sequential circuit, enabling the construction of interacting chiral states from free-fermion ones. The work discusses the equivalence and tradeoffs between sequential adiabatic evolution and sequential circuits, highlighting locality properties, nonlocal tails, and the potential applicability to gauging and more general symmetries, with open questions about tail truncation and extension to more complex chiral phases.

Abstract

In previous work, it was shown that non-trivial gapped states can be generated from a product state using a sequential quantum circuit. Explicit circuit constructions were given for a variety of gapped states at exactly solvable fixed points. In this paper, we show that a similar generation procedure can be established for chiral topological states as well, despite the fact that they lack a zero-correlation-length exactly solvable form. Instead of sequentially applying local unitary gates, we sequentially evolve the Hamiltonian by changing local terms in one subregion and then the next. The Hamiltonian remains gapped throughout the process, giving rise to an adiabatic evolution mapping the ground state from a product state to a chiral topological state. We demonstrate such a sequential adiabatic generation process for free fermion chiral states like the Chern Insulator and the $p+ip$ superconductor. Moreover, we show that coupling a quantum state to a discrete gauge group can be achieved through a sequential quantum circuit, thereby generating interacting chiral topological states from the free fermion ones.

Figures (10)

• Figure 1: Sequential patterns in a sequential unitary transformation. A sequential unitary transformation can be a sequential quantum circuit or a sequential adiabatic evolution.
• Figure 2: The coupled wire picture of the sequential adiabatic evolution process that generates chiral states. Starting from gapped decoupled wires (a), 1. first tune the intra-wire coupling in wires 1 and 2 (red in (a)), then turn down this coupling while turning up inter-wire coupling between the two wires (green in (b)); 2. exchange wire 2 and 3; 3. first tune the couplings shown in red in (c), then turn down these couplings while increasing those shown in green in (d). Upward (downward) arrows indicate right-moving (left-moving) modes. See the text for more details.
• Figure 3: Intermediate step in the Sequential Adiabatic Evolution process for generating the Chern Insulator state from the atomic insulator state. The left half of the system is in the Chern insulator state while the right half of the system is in the atomic insulator state. Labels on arrows indicate coupling strength between neighboring fermion modes. The system is translation invariant in the $y$ direction but not in the $x$ direction.
• Figure 4: Spectrum of the Chern insulator state with respective to $k_y$ for $N_x=50$, $m=-1$, $\epsilon=0.3$. Left: open boundary condition in $x$ direction; Right: closed boundary condition in $x$ direction.
• Figure 5: The adiabatic evolution of the spectrum as one extra wire is added to the Chern insulator state with width $N_x=20$. The spectrum remains gapped as the wire merges into the bulk. In all of the plots, the $x$ axis represents momentum and the $y$ axis represents energy.