Effect of thermal fluctuations on topological crossover in the chiral d+id superconducting phase

The effect of thermal fluctuations on the temperature dependence of the topological index C1 of the chiral d+id superconducting phase of a two-dimensional single-band model on a triangular lattice is investigated. Thermal fluctuations are taken into account within the framework of the self-consistent functional-integral theory. It is established that when the nodal points are located far inside (outside) the Fermi contour of the normal phase, thermal fluctuations expand the relative temperature ranges in which the values of the topological index are close to integer values C1=4(-2). This expansion depends both on the value of the topological index and on the magnitude of the effective attraction between the electrons. However, as the nodal points approach the Fermi contour, topological crossovers to new C1 values are observed, which can persist over a wide temperature range. The nature and degree of influence of thermal fluctuations on these crossovers are established. It is assumed that the observed effects may also manifest in the edge state behavior of a similar system with open boundaries.

💡 Research Summary

This paper presents a detailed theoretical investigation into how thermal fluctuations influence the topological properties of a chiral d+id superconducting phase on a two-dimensional triangular lattice at finite temperatures. Moving beyond the standard zero-temperature or mean-field approaches, the study employs the self-consistent functional-integral theory to incorporate thermal fluctuations of the superconducting order parameter. This leads to an effective non-Hermitian Hamiltonian with a complex self-energy, which accounts for quasiparticle lifetime effects.

The core of the methodological advance is the derivation of a generalized formula for the topological index (C1) at finite temperatures (Eq. 2). This formula, valid in the presence of thermal fluctuations, reduces to the familiar Berry phase/Chern number expression when the self-energy vanishes. The model Hamiltonian includes electron hopping between nearest neighbors and an effective attraction (V) between electrons on second-nearest-neighbor sites. By varying the chemical potential (μ), the system can be tuned to phases where the gap function’s nodal points lie inside (leading to C1=4), outside (C1=-2), or near the Fermi contour of the normal phase.

The key findings are multifaceted. First, when nodal points are located far from the Fermi contour, thermal fluctuations expand the relative temperature range (T/T_c) over which the topological index remains close to its integer quantized values (C1=4 or -2). This stabilization effect is more pronounced for the C1=4 phase and for larger values of the attraction parameter V. The physical mechanism is linked to the temperature dependence of the root-mean-square phase fluctuations (RMSF, σ) of the order parameter. A larger V results in a more gradual increase of σ with temperature, delaying the loss of topological coherence to a higher relative temperature.

Second, and most significantly, the study examines the “topological crossover” regime, where nodal points are near the Fermi contour. In this regime, C1 values close to 1 are observed over a wide temperature range. The influence of thermal fluctuations here is highly dependent on the strength of Fermi contour renormalization. For a moderate V=2, renormalization is weak, and the temperature dependence of C1 with fluctuations closely follows the mean-field (Hartree-Fock) results. However, for a strong V=4, thermal fluctuations cause a substantial temperature-dependent shift in the renormalized chemical potential (μ*). This strong renormalization can cause the nodal points to intersect the shifted Fermi contour at a much lower temperature than predicted by mean-field theory, thereby altering the onset, trajectory, and temperature extent of the topological crossover in a qualitative way.

In conclusion, this work demonstrates that thermal fluctuations are not merely a perturbative effect but can actively reshape the topological phase diagram of a superconductor. The impact is asymmetric between different topological phases, contingent on the proximity of nodal points to the Fermi surface, and heavily modulated by the strength of electron interaction. The authors suggest that these bulk topological crossovers should have observable consequences for the edge state behavior in finite systems. This research underscores the critical importance of going beyond static mean-field approximations to accurately predict and understand the finite-temperature behavior of topological quantum materials for future applications.