Chiral Flux Phase in Topological Materials

• Chiral flux phase is a state characterized by complex phases in microscopic interactions that break time-reversal symmetry and induce topologically nontrivial effects.
• It models phenomena in RVB systems, multiflavor electrons, Kagome lattices, and chiral fluids, influencing superconductivity, quantum Hall responses, and active matter dynamics.
• Experimental efforts employ SP-STM, muon spin resonance, and other high-sensitivity probes to detect chiral loop currents and anomalous Hall signals linked to these phases.

A chiral flux phase is a broadly applicable concept in condensed matter and statistical physics, denoting many-body quantum or classical states where the emergent fluxes—manifested via complex phases in microscopic degrees of freedom—simultaneously break time-reversal symmetry and often generate topologically nontrivial or dynamically distinctive phenomena. Originally introduced in the context of resonating valence bond (RVB) physics, the chiral flux phase now encompasses a range of lattice Hamiltonians (from correlated electrons to spin systems and chiral fluids), is relevant for both equilibrium and nonequilibrium systems, and underlies various experimentally observed or hypothesized phases, including chiral superconductors, Chern (quantum anomalous Hall) insulators, spin liquids, topological superconductors, and multifluid chiral active matter.

1. Chiral d-wave RVB and the Generalized Staggered Flux Phase

The paradigmatic chiral flux phase in quantum magnets is exemplified by the chiral d-wave RVB state on the honeycomb lattice, which generalizes the U(1) staggered flux (orbital current) state established for the square lattice. The mean-field ansatz consists of complex BCS pairing terms,

$$\mathcal{H}_{\text{MF}} = \sum_{\langle ij \rangle} (t\, c^\dagger_{i\sigma} c_{j\sigma} + \Delta_{ij} c_{i\uparrow} c_{j\downarrow} + \text{h.c.}),$$

where, for the honeycomb geometry, the pairing amplitudes $\Delta_{ij}$ pick up phases differing by $\pm 2\pi/3$ on each bond emanating from a site. Although this construction breaks time-reversal at the mean-field level, after Gutzwiller projection the state is fully symmetric and positive definite according to the Marshall sign rule for bipartite unfrustrated antiferromagnets, precluding sign frustration and topological degeneracy (Li, 2011).

Tuning the ratio $\Delta/\chi$ (pairing over hopping strength) interpolates between a uniform RVB state (Dirac semimetal of spinons, $\Delta/\chi=0$), a $\pi$-flux phase at $\Delta/\chi = \sqrt{2}$, and back to a uniform limit for large $\Delta/\chi$. Unlike the square lattice case—with its U(1) flux pattern and protected symmetric points—the honeycomb's geometric frustration necessitates a $Z_2$ gauge structure in the intermediate regime, evidenced by noncommuting loop operators across elementary hexagons. However, the $Z_2$ structure is "ineffective" after projection, as all projected states remain Marshall-sign positive, again forbidding topological degeneracy.

The chiral d-wave RVB state optimizes energy for the Heisenberg model on the honeycomb lattice and demonstrates enhanced spin correlations relative to the uniform RVB state. The evolution of the spin structure factor, as calculated before and after projection, reveals nontrivial dependence on $\Delta/\chi$; the distinction between the uniform and the $\pi$-flux point is sharply manifested in both the algebraic decay exponents of spin correlations and the amplitude of the correlations at long distances.

2. Chiral Flux Phases in Multiflavor Electron Systems and Moiré Heterostructures

In multiflavor (SU($N_f$)) correlated electron models on hexagonal and square lattices, analysis via weak-coupling parquet renormalization group (RG) reveals the robust emergence of chiral flux phases as leading instabilities. For hexagonal lattices near Van Hove filling, the divergent density of states and strongly nested Fermi surface trigger either chiral $d+id$ superconductivity or a triply degenerate imaginary charge density wave (iCDW) instability, the latter associated with loop current order (circulating currents on bonds with zero net flux per unit cell) (Lin et al., 2019).

In $N_f\geq4$ systems, the dominant weak-coupling instability at exact Van Hove filling is the fully gapped, chiral quantum anomalous Hall insulator (QAHI): a Chern insulator with quantized Hall conductivity $\sigma_{xy} = \pm N_f e^2/h$, arising purely from interaction-induced spontaneous time-reversal breaking and loop current ordering. Upon doping away from the Van Hove point, the leading instability shifts to chiral $d+id$ superconductivity, with an order parameter winding by $4\pi$ around the Fermi surface:

$$\Delta^\pm(\mathbf{k}) = \Delta \big(1, e^{\pm 2\pi i/3}, e^{\mp 2\pi i/3}\big).$$

A comparable analysis on the square lattice uncovers a nodal staggered flux phase (with loop currents at a single nesting momentum, giving rise to gapless points in the spectrum) for $N_f>2$ at perfect nesting, giving way to $d$-wave superconductivity upon doping.

3. Classification and Symmetry Structure on the Kagome Lattice

The chiral flux phase is central to the phenomenology of Kagome-lattice systems, especially AV$_3$Sb$_5$ (A=K, Rb, Cs) compounds. The low-energy effective theory for the Kagome lattice focuses on the three van Hove points; order parameters are constructed from the SU(3) Gell-Mann matrix degrees of freedom:

$\Delta_\alpha = \sum_i \Delta_{\alpha,i} \Gamma_i,$

split into real bond (Δb), imaginary flux (Δφ), and on-site charge (Δ_s) orders (Feng et al., 2021). The CFP order parameter resides in the A$_{1g}$ representation of the magnetic group D$_{6h}^*$, respects all point group symmetries except time reversal, and corresponds to

$$\Delta_{\mathrm{CFP}} = i\lambda\,(\cos(\mathbf{Q}_a\cdot\mathbf{R}), \cos(\mathbf{Q}_b\cdot\mathbf{R}), \cos(\mathbf{Q}_c\cdot\mathbf{R})).$$

A brute-force classification of flux phases within 2×2 unit cells, enforcing charge conservation and equal-magnitude complex hopping, yields 183 distinct classes, with the CFP the highest-symmetry 2×2 phase. Experimental signatures include anomalous Hall conductivity associated with nonzero Chern numbers in the occupied bands and local current patterns seen in STM studies—a direct consequence of the imaginary hopping structure inherent in these states (Feng et al., 2021).

4. Experimental Detection, Constraints, and Controversies

Despite robust theoretical predictions, direct detection of chiral flux currents remains experimentally nontrivial. Spin-polarized scanning tunneling microscopy (SP-STM) and spectroscopy (STS) experiments in CsV$_3$Sb$_5$ (Li et al., 2021) establish clear 2$a_0\times 2a_0$ CDW and $4a_0$ stripe orders but do not resolve local magnetic moments or feature modulation attributable to chiral loop currents, placing an upper bound of $0.2\,\mu_B$ per site on possible moments. This lack of spin contrast suggests either that predicted chiral orbital moments are below the experimental detection threshold or that the chiral flux phase is not realized (or is masked) in some material contexts, motivating alternative experimental approaches (e.g., muon spin resonance, higher-sensitivity probes) and more detailed theoretical modeling. The measured anomalous Hall effects in AV$_3$Sb$_5$ compounds, nevertheless, are consistent with the presence of a chiral flux phase, emphasizing the complexity of reconciling direct and indirect signatures.

5. Chiral Flux Phases Beyond Quantum Solids: Active and Chiral Fluids

The chiral flux phase paradigm extends well beyond quantum solids into soft active matter and chiral fluids:

• In chiral active particle (CAP) systems—self-propelled particles with fixed angular velocity (chirality) and alignment interactions—flocking transitions and rotating macro- or micro-droplet phases emerge (Jian et al., 24 Mar 2025). By mapping the nonequilibrium potential landscape (in density-alignment phase space) and analyzing the probability flux field, it is shown that transitions are driven by a dynamical origin (mean probability flux) and a thermodynamic origin (entropy production rate). In the macro-droplet (high-order) regime, a unified rotating "chiral flux phase" is observed. Flux vector fields in phase space destabilize potential landscape basins, underpinning both continuous and discontinuous phase transitions.
• In classical chiral fluids, such as assemblies of Lennard–Jones particles with nonconservative transverse forces modeling spinning colloids (Caporusso et al., 2023), the chiral drive increases interfacial surface tension and generates robust edge currents at the liquid-gas interface. The rotational (odd) viscosity, measured via both mechanical stress and hydrodynamic edge current profiles, provides a direct link between microscopic chiral dynamics and macroscopic transport properties, fulfilling the statistical mechanical signature of the "chiral flux phase". At high density, solid phases melt into mosaics of rotating hexatic domains due to persistent chiral stresses, highlighting the broad manifestation of chiral flux phases in both equilibrium and nonequilibrium systems.

6. Implications for Topological Quantum Matter and Future Directions

Chiral flux phases constitute a unifying structural ingredient in a range of topological and correlated states:

• In chiral spin liquids (CSLs) realized in Hubbard models with synthetic gauge fields (e.g., via optical Raman lattices (Yang et al., 2023) or SU($N$) alkaline-earth atomic systems (Chen et al., 2015)), the phase diagram reveals the CSL stabilized by flux, separating quantum Hall and magnetically ordered states. The essential stabilization arises from chiral three-spin (scalar chirality) interactions generated by effective gauge fluxes, leading to Chern-number-protected edge modes, fractionalization, and enhanced robustness against thermal fluctuations.
• In higher-order topological insulators, Floquet engineering of a π-flux square lattice enables extrinsic HOTI phases protected by emergent chiral symmetry. The generation of edge and robust corner states, even in the absence of bulk inversion, demonstrates the utility of periodic driving in producing chiral-symmetry-enforced flux phases and their associated boundary modes (Bhat et al., 2020).
• In heavy-fermion systems near Kondo breakdown, the emergence of orbital antiferromagnetic phases with checkerboard modulated internal flux—accompanied by spontaneous translation and time-reversal symmetry breaking—is predicted (Drechsler et al., 2023). The chiral flux structure in these "hidden order" phases is detectable via anomalous Hall signatures and modulated current arrangements, and its continuous onset is understood via nonanalytic terms in a Landau-type effective free energy.

These insights collectively illustrate the foundational role of chiral flux phases—as encoded in complex order parameters, emergent gauge field structures, and edge phenomena—across a spectrum of quantum and classical systems, making them essential for understanding the emergence, classification, and detection of topological and symmetry-broken phases in contemporary condensed matter and materials physics.