Does the projected spin state of the d+id mean-field Hamiltonian on a triangular lattice has time-reversal(TR) symmetry?

To determine whether the projected spin state of the d + id mean-field Hamiltonian on a triangular lattice possesses time-reversal (TR) symmetry, we need to delve into the properties of the Hamiltonian and the nature of time-reversal symmetry itself.

Understanding Time-Reversal Symmetry

Time-reversal symmetry is a fundamental concept in quantum mechanics. It implies that the equations governing a physical system remain invariant when time is reversed. For a system to exhibit TR symmetry, the Hamiltonian must satisfy certain conditions under the transformation of time reversal, typically denoted by the operator T.

Characteristics of the d + id Hamiltonian

The d + id Hamiltonian is often associated with topological superconductors and is characterized by its pairing symmetry. In the context of a triangular lattice, this Hamiltonian can be expressed as:

• H = -t ∑ (c†_i c_j + h.c.) + Δ ∑ (c†_i c†_j + h.c.)

Here, t represents the hopping term, and Δ denotes the pairing amplitude, which in this case has a complex form (d + id). The d-wave pairing indicates that the pairing amplitude has angular dependence, which is crucial for understanding its symmetry properties.

Analyzing Time-Reversal Symmetry

For a system to exhibit TR symmetry, the following must hold:

• The Hamiltonian must remain unchanged under the time-reversal operation.
• The time-reversal operator typically involves complex conjugation and may also involve a spin flip.

In the case of the d + id Hamiltonian, when we apply the time-reversal operator, we need to consider how the complex pairing term transforms. The d + id pairing can be expressed in terms of its components, and under time-reversal, the imaginary part (associated with the i in the pairing) will change sign. This leads to:

• Δ* = -Δ

This transformation indicates that the pairing state is not invariant under time reversal, as the sign of the pairing amplitude flips. Therefore, the presence of the imaginary unit in the pairing function suggests that the system does not maintain TR symmetry.

Conclusion on TR Symmetry in the d + id State

In summary, the projected spin state of the d + id mean-field Hamiltonian on a triangular lattice does not exhibit time-reversal symmetry. The complex nature of the pairing function leads to a transformation that alters the Hamiltonian under time reversal, indicating a violation of TR symmetry. This characteristic is significant in the study of topological phases and their implications in condensed matter physics.