Chiral Charge Density Waves

• Chiral CDWs are charge-ordered states characterized by asymmetric lattice distortions and non-superimposable left/right domains, as seen in multi-Q systems.
• Experimental identification relies on techniques like STM, optical polarimetry, and ARPES to reveal unequal intensity peaks and rotated lattice motifs in materials such as 1T-TiSe₂ and 1T-VSe₂.
• Microscopic models explain chirality through mechanisms including orbital ordering, interlayer phase shifts, and chiral phonon-induced instabilities, enabling dynamic control via external fields.

Searching arXiv for recent and foundational papers on chiral charge density waves to ground the article in published work. Chiral charge density waves (CDWs) are charge-ordered states in which the periodic modulation of charge and the accompanying lattice distortion possess handedness, so that one domain is not superimposable on its mirror image. In the literature, chirality in CDWs has been established and interpreted in several distinct but partially overlapping ways: by unequal weighting and handed ordering of multiple CDW wavevectors in triple-$q$ systems; by rotated superlattice motifs such as Star-of-David reconstructions; by layer-dependent phase offsets that generate helical or spiral stacking of distortions; by lattice- and orbital-polarized displacement patterns; and, in more recent work, by mixed-parity multipolar order or phase-topological winding of a single-$Q$ order parameter. The subject emerged prominently from work on $1T$-TiSe$_2$, where scanning tunnelling microscopy (STM) and optical measurements revealed two enantiomorphic CDW states and symmetry lowering beyond ordinary triple-$q$ order (Ishioka et al., 2010), and has since expanded across transition-metal dichalcogenides, kagome metals, quasi-one-dimensional systems, and topological semimetals (Sugawara et al., 2018, Singh et al., 2021, Song et al., 2022, Kim et al., 2024, Yang et al., 2024, Asano et al., 3 Jul 2026).

1. Definitions and symmetry criteria

The minimal content of chirality in a CDW is the existence of two mirror-related domain types, commonly denoted left-handed and right-handed, whose charge/lattice patterns cannot be related by proper rotation or translation alone. In triple-$q$ CDWs, one experimental signature is that the three symmetry-related CDW components are not equivalent: their amplitudes differ, and the sequence of intensities decreases either clockwise or anticlockwise in Fourier space. This criterion was central in the original identification of chirality in $1T$-TiSe$_2$, where the CDW intensity ratio was reported as

$$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$

with two states in which the three intensity peaks decrease clockwise or anticlockwise in the Fourier transform of STM images (Ishioka et al., 2010). Closely related behavior was later reported in $1T$-VSe$Q$0, where three CDW satellite peaks have different intensities and the intensity ordering defines the handedness (Sugawara et al., 2018).

A widely used geometric criterion for triple-$Q$1 chiral CDWs is

$Q$2

In this definition, $Q$3 when the three CDW wavevectors are coplanar and $Q$4 when they are not. The proposal in $Q$5-TiSe$Q$6 was that nonzero $Q$7 should induce a real-space chirality in CDW systems because the triple-$Q$8 vectors do not lie in an identical plane in reciprocal space (Ishioka et al., 2010). The same condition was explicitly invoked for $Q$9-VSe$1T$0, where both $1T$1-TiSe$1T$2 and $1T$3-VSe$1T$4 were argued to satisfy $1T$5 (Sugawara et al., 2018).

Later work broadened the notion of chirality beyond this triple-$1T$6 criterion. In single-layer $1T$7-NbSe$1T$8, the chiral CDW was identified through the $1T$9 Star-of-David superstructure, whose close-packed direction is rotated by approximately $_2$0 relative to the close-packed direction of the top Se lattice; counterclockwise rotation corresponds to L chirality and clockwise to R chirality (Song et al., 2022). In kagome metal KV$_2$1Sb$_2$2, the CDW was described as a mixed-parity, triple-$_2$3, chiral polar-nematic state in which inversion and both mirror symmetries in the kagome plane are broken electronically, while the atomic lattice remains centrosymmetric at the surface according to nc-AFM (Shi et al., 22 Apr 2026). In the recently proposed “winding CDW,” chirality no longer requires multiple ordering wavevectors at all: a single-$_2$4 CDW becomes macroscopically chiral when its order parameter acquires an azimuthal phase winding,

$_2$5

with integer winding number $_2$6 set by crystal angular momentum transfer in the electron-phonon coupling (Asano et al., 3 Jul 2026).

These formulations are not identical. Some are reciprocal-space criteria, some are structural descriptors, and some are order-parameter constructions. A plausible implication is that “chiral CDW” is best regarded as a family of symmetry-breaking patterns rather than a single microscopic mechanism.

2. Experimental identification across material classes

The experimental literature on chiral CDWs is dominated by local real-space probes, especially STM, but decisive claims typically combine STM with reciprocal-space or optical observables. In $_2$7-TiSe$_2$8, STM revealed a $_2$9 CDW modulation with unequal amplitudes along the three crystallographic directions, while optical pump-probe or ellipsometry-type polarization measurements showed a macroscopic two-fold symmetry rather than the three-fold symmetry expected from the trigonal lattice (Ishioka et al., 2010). This combination was important because STM is local and surface sensitive, whereas the optical response established that the symmetry lowering persists over the light penetration depth.

In $q$0-VSe$q$1, STM resolved a $q$2 superlattice whose Fourier peaks have unequal intensities, and the real-space maxima form a kagome-like lattice rather than a triangular lattice. Optical polarimetry again found a twofold polarization dependence of the transient reflectivity variation, consistent with the microscopic symmetry lowering (Sugawara et al., 2018). The same study further reported that Friedel oscillations around defects inherit both the chirality and the periodicity of the background CDW (Sugawara et al., 2018).

In the nearly-commensurate CDW state of $q$3-TaS$q$4, the phenomenology differs. Sub-angstrom STM showed that the CDW lattice evolves continuously from domain wall to domain center rather than forming rigid commensurate islands. Chirality appears here as an intradomain, vortex-like displacement texture of the “true CDW” maxima relative to an “average CDW” lattice. The average displacement was reported as $q$5, with a maximum of $q$6, about $q$7 of the average CDW lattice spacing (Singh et al., 2021). In this material, the chiral signature resides in the rotation of satellite peaks and in the real-space displacement field, not in a clockwise or anticlockwise intensity progression of the main CDW peaks (Singh et al., 2021).

Atomic-resolution visualization reached a different level in single-layer $q$8-NbSe$q$9. STM directly resolved the top Se lattice and the triangular Star-of-David protrusions, allowing the chirality to be read off from the $q$0 rotation of the CDW superlattice relative to the atomic lattice. Time-resolved current-time spectroscopy with $q$1 ms per point tracked dynamic switching at domain boundaries, identifying metastable statuses and a more stable boundary configuration (Song et al., 2022). The study further reported a typical low state near $q$2 pA and a high state near $q$3 pA lasting about $q$4 ms, and found a power-law rate $q$5 with $q$6, interpreted as a one-electron process (Song et al., 2022).

Bulk-sensitive reciprocal-space methods have also become important. In $q$7-TaS$q$8, bulk x-ray diffraction identified a 2D chiral $q$9 CDW in the $1T$0 layers, while ARPES characterized the coexisting CDW structures in the inequivalent layers of the heterostructure (Yang et al., 2024). More recently, micro-ARPES with a $1T$1 spot resolved coexisting left- and right-handed chiral domains in $1T$2-TaS$1T$3, as well as four distinct spectral patterns arising from the combination of chirality and $1T$4 rotational stacking of $1T$5 and $1T$6 terminations (Gofman et al., 22 Aug 2025).

In kagome materials, a multi-technique approach has been essential. In KV$1T$7Sb$1T$8, STM resolved the $1T$9 CDW and real-space inversion breaking, nc-AFM showed a perfect honeycomb lattice with equal sublattice intensity, and optical second-harmonic generation detected a weak but reproducible signal below about $_2$0 K, supporting bulk inversion symmetry breaking (Shi et al., 22 Apr 2026). In CsV$_2$1Sb$_2$2, STM with individual Co atoms used as local magnetic probes identified a localized in-gap state inside the CDW pseudogap, interpreted as the quasiparticle signature of a chiral loop-current order parameter (Zheng et al., 24 Mar 2025).

3. Microscopic mechanisms: orbital, lattice, interlayer, and phonon scenarios

One major line of interpretation views chiral CDWs as orbitally structured lattice distortions. In the Ginzburg–Landau theory proposed for $_2$3-TiSe$_2$4, the three CDW components connect the Se $_2$5 band maximum at $_2$6 to three Ti $_2$7 minima at the inequivalent $_2$8 points, each dominated by a different Ti $_2$9 orbital. Because the electron-phonon coupling is anisotropic, the ionic displacement wave need not be purely longitudinal; in TiSe$$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$0 the displacement waves are essentially transversely polarized, and phase shifts between the three components produce a helical rotation of the dominant displacement direction from layer to layer (Wezel, 2011). In this picture, the chiral CDW is simultaneously an orbital-ordered state.

A different interpretation, grounded in atomistic structure optimization, was advanced in the later TiSe$$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$1 work. There the chiral CDW was attributed to helical Se–Ti bond distortions together with a half-unit-cell translation between neighboring layers, yielding a $$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$2 structure with broken inversion and reflection symmetry and lower total energy than the previously accepted non-chiral $$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$3 model (Kim et al., 2024). The reported energy gains were

$$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$4

$$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$5

so the $$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$6 structure is lower by about $$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$7 meV per CDW unit cell (Kim et al., 2024). This suggests that, at least in TiSe$$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$8, orbital and structural descriptions are not mutually exclusive but emphasize different levels of description.

In $$I_{a_1}:I_{a_2}:I_{a_3} = 1:0.7 \pm 0.1:0.5 \pm 0.1,$$9-TaS$1T$0, the nearly-commensurate chiral texture was explicitly interpreted as lattice-driven rather than orbital-driven. The evidence was the absence of the FFT intensity progression characteristic of the orbital-driven chirality reported in related TMDs, together with the direct correspondence between CDW displacements and atomic positions, the continuous deformation from wall to center, and the interpretation of satellite peaks as signatures of domain formation and chiral distortion of the CDW lattice (Singh et al., 2021).

Interlayer phase structure has become a unifying motif. In the predictive first-principles framework developed for chiral CDWs, the central mechanism is a previously overlooked phase difference of the CDW $1T$1-vectors between layers, linked to opposite collective atomic displacements across different layers. Phase-mismatched stacking produces a spiral arrangement of the $1T$2-vectors and hence a helical real-space structure (Shao et al., 2024). This framework was used to predict chiral CDWs in AV$1T$3Sb$1T$4 and in triangular-lattice NbSe$1T$5, including a $1T$6 chiral CDW in $1T$7-NbSe$1T$8 with space group $1T$9 (Shao et al., 2024).

A complementary microscopic route is based on chiral phonons. In the monolayer $Q$00-TiSe$Q$01 case study, the CDW instability was attributed to strong electron-phonon coupling rather than simple Fermi-surface nesting, with three in-plane acoustic phonons at the three $Q$02 points as the dominant soft modes. The paper argued that chiral phonons are necessary ingredients for a chiral CDW but not sufficient by themselves: in the symmetric monolayer, the three soft modes combine achirally unless mirror or $Q$03 symmetry is externally broken so that the frozen soft mode acquires chiral elliptical or circular mass-center motion (Zhang et al., 2024). In this EPC framework, a chiral CDW is engineered rather than automatic.

The “winding CDW” proposal generalizes the phonon-based mechanism in a different symmetry setting. In screw-symmetric chiral crystals, chiral phonons drive a Peierls instability and, through the crystal-angular-momentum selection rule

$Q$04

endow the CDW with integer azimuthal phase winding (Asano et al., 3 Jul 2026). This extends the concept of chiral CDW from structural helicity in multi-$Q$05 systems to phase topology in a single-$Q$06 order parameter.

4. Domains, switching, and low-dimensional control

Chiral CDWs generically admit two enantiomorphic domains, and domain structure is central to both interpretation and control. In $Q$07-TiSe$Q$08, STM already showed both clockwise and anticlockwise domains, and the boundary between them was identified as a chiral domain wall (Ishioka et al., 2010). The later TiSe$Q$09 structural study described a “translative CDW domain wall” that simultaneously switches chirality and shifts the CDW phase by one atomic unit cell along the row direction (Kim et al., 2024). This coupling between handedness and phase translation indicates that domain walls need not be purely Ising-like interfaces.

In single-layer $Q$10-NbSe$Q$11, chirality switching was directly visualized. As a domain boundary moves, an L domain expands while an R domain shrinks, and the top-layer Se atomic lattice remains continuous and intact throughout the process (Song et al., 2022). Fourier analysis showed that before the transition two mirror-related sets of CDW spots coexist, whereas after the transition only one set remains, while the atomic lattice spots stay unchanged (Song et al., 2022). The kinetics of the boundary motion were captured by time-resolved STM, leading to the conclusion that the chiral transition is fast, dynamic, and driven by tunneling electrons (Song et al., 2022).

The same work established reversible switching with an external electric field. Voltage pulses of about $Q$12 V induced L $Q$13 R switching, and pulses of about $Q$14 V induced R $Q$15 L switching. The switching could be triggered at lateral distances of roughly $Q$16 nm from the tip and propagated domino-like across islands as large as $Q$17 nm (Song et al., 2022). Since the atomic lattice stayed intact, the process was identified as a reorientation of the CDW order rather than structural damage.

Control by external fields also appears in other settings. In the quasi-one-dimensional In/Si(111) system, first-principles calculations predicted that chiral stacking orders should show distinct circular dichroism responses, and suggested that cooling under circularly polarized light may favor one handedness over the other (Kim et al., 2021). In KV$Q$18Sb$Q$19, the mixed-parity polar-nematic CDW has a ferroelectric dipole and a quadrupolar nematic component, so an in-plane electric field couples linearly to the dipole and quadratically to the quadrupole, enabling manipulation of the chirality through their relative orientation (Shi et al., 22 Apr 2026). In the chiral-phonon framework for monolayer TiSe$Q$20, polarized light, directional electric fields, magnetic fields, and especially shear strain were proposed as symmetry-breaking stimuli that lift the degeneracy of the soft modes and switch between L-CDW and R-CDW domains (Zhang et al., 2024).

These results indicate that chirality is not only an order-parameter label but also a switchable degree of freedom in low-dimensional correlated matter.

5. Representative materials and phenomenology

The materials presently associated with chiral CDWs span several crystallographic and electronic settings.

Material/system	Chiral-CDW signature	Representative features
$Q$21-TiSe$Q$22	Triple-$Q$23 intensity hierarchy and CW/ACW domains	$Q$24; macroscopic two-fold optical symmetry (Ishioka et al., 2010)
$Q$25-VSe$Q$26	Incommensurate chiral triple-$Q$27 order	Unequal CDW peak intensities; kagome-like real-space lattice; two-fold optical symmetry (Sugawara et al., 2018)
$Q$28-TaS$Q$29 (NC-CDW)	Lattice-driven intradomain chirality	Vortex-like displacement texture; rotating satellite peaks; continuous wall-to-center evolution (Singh et al., 2021)
Single-layer $Q$30-NbSe$Q$31	$Q$32 chiral superlattice	$Q$33 rotation; reversible electric-field switching (Song et al., 2022)
$Q$34-TaS$Q$35	Chiral $Q$36 CDW in $Q$37 layers	2D chiral CDW coexisting with 2D $Q$38 CDW in $Q$39 layers (Yang et al., 2024)
CoSi	Chirality-locking unidirectional CDW	$Q$40 at $Q$41; orientation set by crystal enantiomer (Li et al., 2021)
AV$Q$42Sb$Q$43	Mixed-parity or loop-current-related chiral charge order	Inversion breaking, polar-nematic order, or TRSB loop-current signatures depending on study (Shi et al., 22 Apr 2026, Zheng et al., 24 Mar 2025)

The TiSe$Q$44 and VSe$Q$45 cases are foundational because they introduced the now-standard connection between chirality, unequal triple-$Q$46 components, and reduced rotational symmetry (Ishioka et al., 2010, Sugawara et al., 2018). The $Q$47-TaS$Q$48 and $Q$49-NbSe$Q$50 results show that chirality can instead be encoded in lattice distortions and rotated cluster motifs (Singh et al., 2021, Song et al., 2022). The $Q$51-TaS$Q$52 work is important because it places a chiral CDW inside a natural quantum heterostructure consisting of alternating $Q$53-TaS$Q$54 and $Q$55-TaS$Q$56 layers, thereby separating a chiral $Q$57-layer subsystem from a nonchiral but intra-unit-cell-coupled $Q$58 subsystem (Yang et al., 2024).

CoSi introduces a different notion: the CDW is not merely chiral in itself but chirality-locked to the handedness of the crystal. The wavevector and stripe orientation are set by the enantiomer, and the two CDW patterns are related by mirror reflection (Li et al., 2021). This is distinct from spontaneous left/right domain formation in an achiral host.

Kagome systems remain the most controversial subclass. In KV$Q$59Sb$Q$60, the CDW was described as a parity-violating chiral polar-nematic state with substantial odd-parity topographic and superconducting contributions of about $Q$61 and $Q$62, respectively (Shi et al., 22 Apr 2026). In CsV$Q$63Sb$Q$64, a separate STM study used Co magnetic atoms to argue for a loop-current, time-reversal-symmetry-breaking chiral CDW ground state (Zheng et al., 24 Mar 2025). By contrast, a variational study of the kagome extended Hubbard model at van Hove filling reported no signatures of chiral phases and found that the purely electronic $Q$65-driven CDWs are incompatible with the charge disproportionation observed in AV$Q$66Sb$Q$67 (Ferrari et al., 2022). This disagreement is substantive rather than terminological.

6. Relation to topology, optical response, and superconductivity

The interplay of chiral CDWs with topology and collective transport is a major reason for current interest. In Weyl semimetals, a CDW can couple the two Weyl nodes and gap them by spontaneously breaking translational symmetry. For generic node separation, the Weyl CDW order parameter is complex and supports vortex defects; in the commensurate period-two case $Q$68, the order becomes real, defects become domain walls, and each wall binds a 2D Dirac fermion with half-quantized Hall conductivity

$Q$69

Although this paper is not about structural chirality in the TMD sense, it establishes a distinct topological context in which CDW order, anomaly inflow, and parity anomaly are intertwined (Sehayek et al., 2020).

CoSi provides an experimentally realized topological-material context. The unidirectional incommensurate CDW observed on CoSi(001) was accompanied by a particle-hole asymmetric V-shaped gap around the Fermi level, with representative $Q$70 meV and variation between about $Q$71 and $Q$72 meV across regions and samples (Li et al., 2021). Because CoSi is a Weyl semimetal with unconventional chiral fermions, the authors connected the chirality-locking CDW to the possibility of an axion-insulating correlated state (Li et al., 2021).

Optical activity is another recurring consequence. In quasi-one-dimensional In nanowires on Si(111), density-functional calculations predicted that left- and right-chiral stacking orders show distinct circular dichroism responses, while a nonchiral stacking order does not. The origin was traced to the presence or absence of glide mirror symmetry in the stacking order (Kim et al., 2021). In TiSe$Q$73 and VSe$Q$74, the macroscopic two-fold optical anisotropy provided early evidence that chirality lowers the rotational symmetry of the electronic response (Ishioka et al., 2010, Sugawara et al., 2018).

A further proposed transport signature is a Hall effect in nonmagnetic metallic chiral CDWs. In the predictive framework for chiral CDWs, a “spin-anomalous dual-Hall effect” was predicted to occur without external magnetic fields or intrinsic magnetization, with the striking feature that the Hall conductivity reverses sign when the injected current is reversed (Shao et al., 2024). The same work reported experimental confirmation in CsV$Q$75Sb$Q$76, where a finite zero-field Hall resistivity emerged below $Q$77 K and was extracted by comparing opposite current directions (Shao et al., 2024). This suggests that chiral CDW transport signatures need not rely on magnetic order.

The relation to superconductivity is particularly prominent in layered TaS$Q$78 and kagome metals. In $Q$79-TaS$Q$80, the coexistence of a 2D chiral flat band pinned at $Q$81 in the $Q$82 layers and intra-unit-cell-coupled $Q$83 CDW in the $Q$84 layers led the authors to suggest time-reversal-symmetry-breaking superconducting pairing in the $Q$85 subsystem and an orbital FFLO state in the $Q$86-bilayer subsystem (Yang et al., 2024). In KV$Q$87Sb$Q$88, the superconducting state was found to inherit parity-violating pair-density modulations at both the original lattice and CDW wavevectors, consistent with a parity-broken normal state and suggestive of mixed-parity superconductivity (Shi et al., 22 Apr 2026).

7. Open issues, competing interpretations, and emerging generalizations

Several persistent controversies shape the field. The TiSe$Q$89 case remains the most historically debated. One line of work interprets chirality as orbital order with phase-shifted polarized displacement waves (Wezel, 2011); another attributes it to helical Se–Ti bond distortions plus interlayer translation in a lower-energy $Q$90 structure (Kim et al., 2024). These are not simple contradictions, but they emphasize different microscopic degrees of freedom. A plausible implication is that the distinction between “orbital-driven” and “structural” chirality may depend on the level at which the order parameter is coarse-grained.

Kagome metals present a sharper dispute. On the one hand, mixed-parity, inversion-breaking, polar-nematic chirality has been directly imaged in KV$Q$91Sb$Q$92 by STM and supported by SHG, with the atomic lattice remaining centrosymmetric at the surface (Shi et al., 22 Apr 2026). On the other hand, a variational Jastrow-Slater study of the kagome extended Hubbard model found no chiral charge order and no time-reversal-breaking phases, concluding that electron-phonon coupling rather than pure electronic $Q$93-$Q$94 interactions is required to obtain experimentally relevant $Q$95 distortions (Ferrari et al., 2022). Yet another study on CsV$Q$96Sb$Q$97 interpreted impurity-induced in-gap states as evidence for a loop-current, time-reversal-symmetry-breaking chiral flux phase (Zheng et al., 24 Mar 2025). These results show that the phrase “chiral charge order” in kagome systems can refer either to inversion-breaking mixed-parity order, to loop-current TRSB order, or to both.

The stacking problem in CsV$Q$98Sb$Q$99 remains unresolved. A real-space Ginzburg–Landau analysis of the $1T$00 CDW found that inversion-breaking, mirror-breaking chiral stackings such as AABC and distorted $1T$01-ABCD are possible, but occupy only small fractions of parameter space, suggesting that other mechanisms for inversion symmetry breaking may be at play (Juan et al., 21 Apr 2026). This is consistent with the broader lesson that structural chirality, inversion breaking, and time-reversal breaking need not coincide.

The most important generalization is conceptual. The predictive first-principles framework asserts that chiral CDWs can be systematically identified by enumerating interlayer phase combinations of CDW wavevectors and relaxing all candidate structures (Shao et al., 2024). The winding-CDW framework goes further, proposing that macroscopic chirality can arise even for a single ordering wavevector when the CDW phase winds azimuthally around a symmetry axis (Asano et al., 3 Jul 2026). Together with the chiral-phonon perspective (Zhang et al., 2024), these advances shift the field from post hoc identification toward a more unified classification based on symmetry, interlayer phase structure, and phase topology.

This suggests that future work will likely treat chiral CDWs not as exceptional variants of conventional density waves, but as a broader class of correlated orders in which handedness may arise from unequal multi-$1T$02 amplitudes, rotated cluster motifs, interlayer phase helicity, mixed-parity multipoles, or integer phase winding.