Chiral d-wave RVB State

• Chiral d-wave RVB state is a quantum many-body phase characterized by a complex d-wave order parameter that breaks time-reversal symmetry, yielding a fully gapped and topologically nontrivial structure.
• Variational Monte Carlo and mean-field theories reveal close energetic competition with nodal d-wave states, highlighting the role of geometric frustration in stabilizing the chiral phase.
• Real-space bond patterns, SU(2) gauge properties, and quantized topological features such as chiral edge modes provide clear experimental fingerprints for identifying these states on triangular and honeycomb lattices.

A chiral d-wave RVB (Resonating Valence Bond) state is a quantum many-body state characterized by a complex d-wave order parameter that spontaneously breaks time-reversal symmetry. This state arises naturally in frustrated spin systems and in correlated lattice models where the pairing channel belongs to a two-dimensional irreducible representation of the lattice point group, most prominently on triangular and honeycomb lattices. The chiral d-wave RVB state is fully gapped and can host topological order, with potential for topologically protected chiral edge modes. Its stability, energetics, gauge structure, real-space bond patterns, and experimental fingerprints are the subject of extensive theoretical investigations using variational Monte Carlo (VMC), mean-field theory (MFT), Gutzwiller projection methods, cluster extensions of dynamical mean-field theory, and tensor network formalisms.

1. Order Parameter Structure and Symmetry Classification

In systems with hexagonal or triangular symmetry (point group $C_{6v}$), the nearest-neighbor singlet-pairing (d-wave) manifold forms a two-dimensional irreducible representation. The generic d-wave gap function on the triangular lattice is given in momentum space as: $$\Delta_{\mathbf{k}} = \Delta \left[\eta_1 \Phi_1(\mathbf{k}) + \eta_2 \Phi_2(\mathbf{k})\right], \quad |\eta_1|^2 + |\eta_2|^2 = 1,$$ with $$\Phi_1(\mathbf{k}) = \cos k_x - \cos(\frac{1}{2}k_x)\cos(\frac{\sqrt{3}}{2}k_y)$$ and $$\Phi_2(\mathbf{k}) = \sqrt{3} \sin(\frac{1}{2}k_x)\sin(\frac{\sqrt{3}}{2}k_y)$$. The chiral combination is realized for $\eta_2 = i\eta_1$, yielding a fully gapped $d_{x^2-y^2} + i\,d_{xy}$ state that breaks time-reversal symmetry. Any linear combination $\Delta_{\mathbf{k}}(\theta, \phi)$ with real or complex coefficients parametrizes a two-component order parameter.

On the honeycomb lattice, the chiral d-wave (often $d \pm id$) takes the form: $$\Delta_{j} = \Delta_s \exp[i(j-1)\varphi_{\mathrm{sc}}],\; j=1,2,3;\;\; \varphi_{\mathrm{sc}} = \pm 2\pi/3,$$ assigning distinct phases to pairings along the three nearest-neighbor bonds.

A key property is that the chiral d-wave state belongs to symmetry class D (or C with spin conservation) and supports a topological Chern number $|C|=2$ per spin sector on two-dimensional lattices, guaranteeing robust chiral edge states.

2. Variational Energetics and Stability

Mean-Field Landscape

BCS and slave-boson mean-field theory predict that the fully gapped chiral $d_{x^2-y^2} + i d_{xy}$ combination is the unique minimum of the variational energy within the two-dimensional d-wave space (e.g., for the Heisenberg or $t$-$J$ model on the triangular or honeycomb lattice). The Landau-type expansion of free energy includes a quartic term $$F[\Delta_1, \Delta_2] \sim \gamma(\Delta_1^2\Delta_2^{*2} + \text{c.c.})$$ with positive $\gamma$, selecting the chiral solution.

Variational Monte Carlo and Beyond-Mean-Field

VMC calculations, which enforce the no-double-occupancy constraint exactly, reveal a significant reversal of mean-field predictions. On the triangular lattice Heisenberg model, the nodal nonchiral $d_{xy}$ state (real order parameter) achieves the global energy minimum, while the fully gapped chiral state becomes a local maximum. The energy anisotropy in the d-wave manifold is less than $5\%$ of the condensation energy ($\Delta E\lesssim0.003$ per site for $L=14$), confirming the two-component nature but indicating the decisive role of geometric frustration and projection effects (Li, 2010, Yamada, 30 Sep 2024).

In the half-filled Hubbard model on isotropic triangular lattice (VCA approach), the $d_{xy}$ ground state is found when $U/t \lesssim 6$, with the chiral $d+id$ solution only $0.01t$–$0.03t$ higher in energy for $U/t \simeq 5$, and both are below the normal state by $0.02t$–$0.06t$. For $U/t \gtrsim 6$ the system undergoes a Mott transition to a spin-liquid phase (Yamada, 30 Sep 2024).

3. Real-Space Structure and Gauge Properties

Bond Patterns and Flux

The chiral d-wave state imposes complex pairing amplitudes on the bonds, generating a nontrivial SU(2) flux when traversing lattice loops. On the honeycomb lattice, the state can be characterized by the product of RVB matrices around a hexagon, resulting in

$$\mathcal{P}_k = -\rho^6 \exp\left[i\,\Phi\,\vec{n}_k \cdot \vec{\tau}\right];\;\; \rho=\sqrt{\chi^2+\Delta^2},$$

with a staggered SU(2) flux direction for each hexagon. At the special point $\Delta/\chi=\sqrt{2}$ the flux is $\pi$, corresponding to a $\pi$-flux Dirac state (Li, 2011).

Gauge Structure and Topological Degeneracy

The mean-field IGG (invariant gauge group) of the chiral d-wave ansatz is generically $\mathbb{Z}_2$, indicating potential for topological order and vison excitations. However, after Gutzwiller projection, the wave function on bipartite lattices obeys the Marshall sign rule, rendering the projected state fully symmetric and erasing topological degeneracy: vison-sector overlaps remain finite in the thermodynamic limit (Li, 2011). On nonbipartite lattices (e.g., triangular), this cancellation may not occur, leaving possible room for stable $\mathbb{Z}_2$ spin liquids.

In tensor network PEPS constructions with $d+id$ symmetry, a genuine fourfold topological ground-state degeneracy is observed, with orthogonal sectors corresponding to spinon parity and $\mathbb{Z}_2$ flux threading. Transfer matrix analysis indicates critical algebraic correlations for singlets, finite correlation length for triplets, and chiral edge modes (Poilblanc et al., 2015).

4. Improved Gutzwiller Approximations and Frustration Effects

The conventional single-site Gutzwiller approximation, which works well for unfrustrated square lattices, drastically fails on triangular lattices due to the strong modulation of local singlet formation by the third site of a triangle. This leads to overestimated condensation energies and incorrect chiral/nonchiral energy ordering.

Two-site ("Hsu's") and three-site Gutzwiller renormalization schemes introduce short-range bond or triangle correlations: $$g_s^{(2)} = \frac{4}{1 + |\chi_{ij}|^2 + |\Delta_{ij}|^2},\qquad g_s^{(3)} = \frac{1}{\sum_\alpha W^0_\alpha},$$ where $W^0_\alpha$ is the unprojected triangle configuration weight. These modifications reduce the mean-field errors (from $\sim$50% to ~10% in the energy), yet only exact (VMC) projections recover the correct nonchiral $d_{xy}$ ground state (Li, 2010).

5. Topological Features and Edge Phenomena

The $d+id$ chiral state is a prototype of a two-dimensional topological superconductor with nonzero Chern number ($|C|=2$), ensuring chiral Majorana edge modes and quantized thermal Hall conductance $\kappa_{xy}/T = (\pi/12)(k_B^2/\hbar)C$. On cylinders, PEPS analyses reveal four orthogonal sectors, chiral edge branches in entanglement spectra, and long-range entanglement Hamiltonian structure (Poilblanc et al., 2015).

In models of twisted bilayer graphene, self-consistent mean-field analysis of the effective $t$-$J$-$D$ model demonstrates a chiral $d+id$ pairing instability, a Chern number $C=2$, and explicit chiral Majorana edge modes in finite geometries (Gu et al., 2019). For the honeycomb $t$-$J$ model, the chiral $d\pm id$ RVB superconductor remains topological ($C=2$) in the superconducting phase, with physical consequences for edge transport and valley/spin-selective ARPES (Ho et al., 2022).

6. Experimental Signatures and Implications

The chiral d-wave state, if realized, is associated with spontaneous time-reversal symmetry breaking, which can manifest in internal fields detectable by $\mu$SR or Kerr rotation, quantized thermal Hall effects, and chiral edge contributions observable in transport experiments or tunneling spectroscopy. Proximate near-degeneracy of nodal and chiral d-wave solutions suggests that modest perturbations (disorder, magnetic field, strain) may enable local or mesoscopic domains of chiral superconductivity, even when the true ground state is nodal (Yamada, 30 Sep 2024). Such phenomena may explain transient time-reversal breaking observed in organic triangular-lattice superconductors and moiré van der Waals materials.

A plausible implication is that in the spin-liquid regime near the Mott transition, quantum fluctuations can sample both chiral and nodal components, leading to rich crossover and phase coexistence phenomena upon doping or applying external fields.

7. Comparative Landscape and Physical Significance

The chiral d-wave RVB state is a natural extension of the square-lattice staggered flux (d-wave) phase to frustrated geometries. On the honeycomb lattice, the mean-field chiral d-wave represents the $Z_2$-gauge analog of the $\pi$-flux state, though topological order is destroyed by Gutzwiller projection due to the Marshall sign structure. On the triangular lattice, the RVB variational landscape critically depends on geometric frustration and higher-order bond correlations, with the nodal $d_{xy}$ phase stabilizing at half filling and the chiral state remaining close by in energy.

Tensor network studies (chiral RVB PEPS) demonstrate that constructing local tensors with $d+id$ symmetry can produce bona fide chiral spin liquids with topological degeneracy and chiral conformal edge modes, providing a route to realizing these states in artificial quantum matter platforms (Poilblanc et al., 2015).

In summary, the chiral d-wave RVB state constitutes a fundamentally important and nearly-realized class of topological quantum liquids in correlated frustrated systems, with implications for unconventional superconductivity, spin liquid physics, and quantum topology in strongly interacting electron systems.