Chiral d-wave RVB state on honeycomb lattice as a generalized staggered flux phase

Abstract: We show the chiral d-wave RVB state on honeycomb lattice stands as a natural generalization of the staggered flux phase on square lattice. Although the state is generated from a time reversal symmetry broken mean field ansatz, it actually represents a fully symmetric spin liquid state with a positive definite wave function in the sense of Marshall sign rule for unfrustrated antiferromagnets. The evolution of the state with the parameter $Δ/χ$ follows exactly the same manner as that of the staggered flux phase on square lattice. The critical pairing strength corresponding to the $π$-flux phase is found to be $Δ/χ=\sqrt{2}$. As a result of the geometric frustration between neighboring plaquette on honeycomb lattice, a direct generalization of the U(1) staggered flux pattern on square lattice to honeycomb lattice is impossible. Replacing it is the chiral d-wave state with $Z_{2}$ gauge structure. However, this $Z_{2}$ gauge structure is found to be ineffective after Gutzwiller projection and the system does not support topological degeneracy. The chiral d-wave RVB state is also found to be a rather good variational state for the Heisenberg model on honeycomb lattice. The spin correlation of the chiral d-wave state is found to be greatly enhanced as compared to the mean field prediction.

• The paper demonstrates that the chiral d-wave RVB state on the honeycomb lattice acts as a generalized staggered flux phase with distinct gauge properties.
• It employs a Gutzwiller-projected BCS framework to analyze transitions among SU(2), U(1), and Z2 gauge structures influencing spin liquid behaviors.
• Variational studies reveal optimized Heisenberg model energies and enhanced short-range spin correlations, suggesting potential for unconventional superconducting states.

Chiral d-wave RVB State on Honeycomb Lattice

This paper explores the chiral d-wave Resonating Valence Bond (RVB) state on a honeycomb lattice, establishing it as a generalized staggered flux phase analogous to those observed on square lattices. It explores the structural and energetic properties of such states and examines their evolution under varying parameters, focusing on implications for spin liquid states and superconductivity phenomena.

RVB States and Gauge Structures

The RVB state is derived from a Gutzwiller-projected BCS mean field ground state characterized by hopping ($\chi_{i,j}$) and pairing ($\Delta_{i,j}$) order parameters. The central concept is the classification of states by gauge structure: $SU(2)$, $U(1)$, and $Z_{2}$, determined by loop operators constructed from the order parameters. This classification directly affects the possible topological degeneracies and stability of resulting quantum phases.

Staggered Flux Phase on Square Lattice

The staggered flux phase, with d-wave symmetry on the square lattice, serves as a reference, showing how varying the flux ($\Phi$) affects gauge structure. At critical points, where $\Delta/\chi$ equals 0 or $\infty$, the system stabilizes in $SU(2)$ gauge structure, while intermediate states reflect a $U(1)$ structure. These correlates to physical configurations that enhance antiferromagnetic correlations, pivotal to phenomena seen in high-$T_c$ cuprate superconductors.

Chiral d-wave RVB State on Honeycomb Lattice

Mean Field Description

On the honeycomb lattice, the chiral d-wave state features complex pairing orders differing by defining angles, leading to zero-energy nodes at specific momentum points. The spectrum's zero modes establish a close kinship with the Dirac-like dispersions fundamental to this geometry at both $\Delta/\chi=0$ and $=\sqrt{2}$, where transitions resemble the square lattice’s staggered phases.

Gauge Structure and Symmetry

The honeycomb lattice’s peculiar geometry prohibits direct extension of $U(1)$ flux observed in square lattices, resulting instead in a $Z_{2}$ gauge structure except at critical points, where $SU(2)$ emerges. Notably, the supposed chiral nature resolves under gauge transformation, harmonizing time reversal and lattice rotation symmetries unseen in mere mean field renderings.

Topological and Sign Considerations

Despite possessing a $Z_{2}$ gauge structure, the bipartite nature and Marshall sign rule negate topological degeneracy, an assertion validated through overlap calculations. This property infers that any superconducting states emerging from such configurations would remain exempt from conventional topological classifications, potentially impacting theoretical descriptions of correlated electron systems on honeycomb lattices.

Variational Study and Spin Correlations

In variational terms, the chiral d-wave state offers a substantially optimized energy basis for the Heisenberg model on the honeycomb lattice, closely aligning with known experimental dips in energy minimalism, albeit with modest improvements compared to square lattice counterparts. Spin structure factor assessments further reinforce shorter-range enhancements, albeit without establishing long-range magnetic order, implying limits to analogy with antiferromagnetic states robust on square lattices.

Conclusion

The chiral d-wave RVB state on the honeycomb lattice decisively aligns its physical properties closely with those seen on square lattices, yet underscores intrinsic geometric constraints and variations from its staggered flux phase cousin. Situated in this expanded RVB framework, the research highlights avenues for further empirical inquiry particularly surrounding unconventional superconductivity, amid acknowledging the protective confounds induced by geometric and symmetric peculiarities unique to honeycomb configurations. Future exploration could yield deeper insights into potential superconductive states in honeycomb-structured materials, inviting theoretical and experimental examinations beyond classical categorizations.