Sequential Topological Superconductivity in a Square Lattice with Chiral Charge Density Waves

Technical Summary: Sequential Topological Superconductivity in a Square Lattice with Chiral Charge Density Waves

Problem Statement
The interplay between charge order and superconductivity is a central theme in condensed matter physics, particularly in unconventional superconductors where multiple order parameters coexist. While recent studies on kagome lattices have demonstrated that chiral flux phases (CFP) alone can drive topological phase transitions, the specific role of lattice geometry in determining the topological fate of such modulations remains an open question. Specifically, there is a gap in understanding the simultaneous presence of real-part hopping modulations (Charge Bond Order, CBO) and imaginary-part hopping modulations (Chiral Flux Phase, CFP) within a superconducting square lattice. Previous works on square lattices typically addressed these modulations in isolation or focused on their interplay with superconductivity separately, leaving the systematic study of the full composite charge density wave (CDW) in a superconducting context unexplored.

Methodology
The authors theoretically investigate a square-lattice superconductor coexisting with a composite CDW. The system is modeled using a tight-binding Hamiltonian comprising three terms:

1. Tight-binding term ($H_{TB}$): Describes nearest-neighbor hopping and chemical potential ($\mu$).
2. CDW term ($H_{CDW}$): Incorporates both CBO (real modulation of hopping strengths, $\lambda_{CBO}$) and CFP (imaginary modulation breaking time-reversal symmetry, $\lambda_{CFP}$). These orders form a $2\times2$ modulation pattern on the bipartite lattice.
3. Superconducting term ($H_{SC}$): Assumes an isotropic $s$-wave pairing amplitude ($\Delta$).

The system belongs to Altland-Zirnbauer class D (broken time-reversal symmetry, preserved particle-hole symmetry), where topological properties are characterized by the Chern number ($C$). The authors employ the following computational approaches:

• Band Structure Analysis: Diagonalization of the Bogoliubov-de Gennes (BdG) Hamiltonian in momentum space to determine the energy spectrum and identify Dirac cones and band-touching points.
• Edge State Visualization: Imposition of periodic boundary conditions in the $x$-direction and open boundary conditions in the $y$-direction to visualize chiral Majorana edge modes.
• Thermal Hall Conductivity: Calculation of the thermal Hall conductivity ($\kappa_{xy}$) using the linear-response Kubo formula to serve as an experimental fingerprint for the topological invariant.
• Symmetry Analysis: Utilization of $k \cdot p$ perturbation theory and symmetry operators (specifically parity $P$) to analyze the nature of gap-closing points at high-symmetry points ($\Gamma$, $M$) and lines ($X-M$).

Key Contributions and Results
The study reveals a distinctive phenomenon specific to the bipartite square lattice: the coexistence of CBO and CFP drives the system through sequential topological phase transitions as the CFP strength ($\lambda_{CFP}$) increases.

1. Sequential Topological Transitions:

   • The system transitions from a trivial phase ($C=0$) to a topological phase with $C=+2$, and subsequently to a phase with $C=-2$.
   • This sequential evolution ($0 \to +2 \to -2$) has not been previously reported in square-lattice systems.
   • The phase diagram is mapped in the parameter spaces of chemical potential ($\mu$), CBO strength ($\lambda_{CBO}$), and CFP strength ($\lambda_{CFP}$).

2. Mechanism of Transitions:

   • $C=0 \to C=2$ Transition: Occurs via a quadratic band touching at the $\Gamma$ point. This transition is protected by a symmetry operator $P$, rendering the critical condition independent of $\lambda_{CBO}$ (resulting in a horizontal phase boundary in the $\lambda_{CBO}$-$\lambda_{CFP}$ plane).
   • $C=2 \to C=-2$ Transition: Occurs via linear Dirac crossings along the $X-M$ lines of the Brillouin zone boundary. Here, the symmetry constraint is lifted, allowing linear momentum-dependent terms. This transition's critical $\lambda_{CFP}$ decreases monotonically as $\lambda_{CBO}$ increases.

3. Role of CBO and CFP:

   • CBO alone induces anisotropic band reconstruction, splitting high-energy bands.
   • CFP alone cannot lift the symmetry-protected degeneracy at the $M$ point, leaving bands degenerate.
   • In the coexistence regime, CBO and CFP act cooperatively to reshape the low-energy effective Hamiltonian. Strong CFP renormalization suppresses superconducting pairing, leading to remarkably narrow bulk gaps in the $C=-2$ phase.

4. Experimental Signatures:

   • Edge Modes: The $C=2$ and $C=-2$ phases exhibit two pairs of chiral Majorana edge modes localized at the boundaries, consistent with the bulk-edge correspondence.
   • Thermal Hall Conductivity: The zero-temperature thermal Hall conductivity is quantized as $\kappa_{xy} = C \kappa_0$, where $\kappa_0$ is the thermal conductance quantum. The authors demonstrate that $\kappa_{xy}$ exhibits quantized plateaus proportional to the Chern number at low temperatures, providing a clear experimental fingerprint.
   • Spectral Features: The zero-energy density of states (DOS) shows distinct features: a sharp peak for CBO-dominated reconstruction, a V-shape for CFP order, and peak splitting in the coexistence regime.

Significance
The paper establishes the square lattice as a "pristine platform" for engineering topological superconductivity through the synergy of real and imaginary bond modulations. By isolating these interactions from the geometric frustration and multi-orbital complexities inherent in kagome systems, the authors uncover universal physics governing the interplay of intertwined orders. The work provides a concrete theoretical guide for identifying topological quantum states in square-lattice materials and heterostructures, specifically proposing thermal Hall conductivity measurements as a viable tool for detecting these exotic phases and mapping their phase diagrams.