Compact localized currents in flat bands with broken time-reversal symmetry

TL;DR

The paper develops a systematic method to construct all-bands-flat lattices that explicitly break time-reversal symmetry by flux threading through polygonal plaquettes and applying local entangling unitary transformations. By mapping from semi-detangled to non-detangled representations while preserving flux, the authors generate 1D, 2D, and 3D ABF models in which compact localized states support flux-tunable circulatory currents. They extend the framework to lattices with coexisting flat and dispersive bands, exploring both orthogonal and non-orthogonal CLS and illustrating cases with chiral flat bands. The work provides a controlled route to dispersionless lattices with tunable local currents, offering insights for caging phenomena, edge states under boundary modifications, and potential experimental realizations.

Abstract

We develop a systematic framework for constructing all-bands-flat (ABF) lattice Hamiltonians that explicitly break time-reversal symmetry (TRS). By threading magnetic flux through disconnected polygonal plaquettes and applying local entangling unitary transformations, we map plaquettes onto families of ABF models in one, two, and three dimensions. This procedure preserves the flux configuration while converting semi-detangled geometries into ABF lattices with nontrivial hopping structure. The resulting flat bands admit compact localized states (CLSs) whose support includes both the flux-threaded plaquettes and auxiliary sites introduced by the unitary transformations. In these TRS-broken constructions, the CLSs host localized circulatory currents whose magnitude depends on the applied flux. We further extend the framework to lattices with coexisting flat and dispersive bands, illustrating cases with both orthogonal and non-orthogonal CLSs. Our results provide a controlled route for generating dispersionless lattices supporting flux-induced local currents.

Figures (8)

• Figure 1: (a) 1D lattice formed by stacking unit cells, that are triangles threaded by a flux $\phi$. Green shadings mark the sites to which local unitary operation is applied (b) Resulting lattice after the entangling rotations. (c,d) Same as (a,b) but for triangular plaquettes with auxiliary sites.
• Figure 2: (a) 2D ABF lattice with a different tiling of triangular unit cells (inset). In both cases hoppings induced by local unitary transformation are shown by green bonds. (b) 2D ABF lattice generated from hexagonal tiling of triangular unit cells threaded by a flux $\phi$ (inset). In both panels, $\Vec{a}_1$ and $\Vec{a}_2$ are the primitive translation vectors, with $\Vec{a}_3 = \Vec{a}_2 - \Vec{a}_1$.
• Figure 3: (a) 3D ABF lattice generated from pyramid-like tiling of tetrahedral unit cells with appendices (shown in the inset). (b) a different tiling producing a 3D ABF network with $16$-band and a tetrahedral unit cell with appendices (see inset). In both the panels, $\Vec{a}_1,\Vec{a}_2$ and $\Vec{a}_3$ are the primitive translation vectors and the sites undergoing local unitary transformations are shown in shaded yellow and purple ellipses.
• Figure 4: (a) Two dimensional semi-detangled ABF lattice obtained by tiling augmented square unit-cells. With the shaded areas we indicate the sites chosen for the local unitary operations $\bm{U}$. The green and the pink colors denote the $x$ and $y$ direction in which the two rotations $U_x$ and $U_y$ which form $\bm{U}$ are applied. (b) Resulting non-detangled ABF lattice. With the green circles we show the location of CLS. In both the panels, $\Vec{a}_1$ and $\Vec{a}_2$ are the primitive translation vectors.
• Figure 5: (a) CLS amplitude shown in bright colors, whose intensities are encoded in the right colorbar. The CLS sites are displayed as larger markers, and the signs of their amplitudes are explicitly indicated for clarity. The blue shaded region indicates where the current circulates. (b) Probability current as a function of the flux $\phi$ for the eight different CLSs. In the lower right corner, we indicate the plaquettes where the current distribution is evaluated. On the right, it is indicated the plaquette labeled I where the current distribution is evaluated.

Theorems & Definitions (6)

• Lemma 1
• proof
• Theorem 1
• proof
• Corollary 1
• proof