Intervalley Coherent (IVC) Phase

• IVC phase is a coherent electronic state where K and K' valley states form a macroscopic relative phase, breaking the valley U(1) symmetry without net imbalance.
• Experimental detection employs techniques like STM, compressibility, and quantum oscillations to reveal momentum-space hybridization gaps and real-space modulations.
• IVC is pivotal in driving correlated regimes and unconventional superconductivity in graphene and moiré systems by mediating critical intervalley fluctuations.

The intervalley coherent (IVC) phase is a broken-symmetry electronic state in which quasiparticles from inequivalent valleys, typically $K$ and $K'$, develop a macroscopically coherent relative phase, so that valley is no longer a good quantum number. Its defining order parameter is a valley-off-diagonal bilinear, commonly written as $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$, and in valley-pseudospin language it corresponds to an in-plane polarization $\langle \tau_x\rangle$ or $\langle \tau_y\rangle$ rather than a valley imbalance $\langle \tau_z\rangle$ (Arp et al., 2023). In contemporary condensed-matter settings, IVC has become a central organizing principle for correlated phases in rhombohedral graphene, twisted graphene, twisted transition-metal dichalcogenides, and, more recently, three-valley moiré systems, where it appears in metallic, insulating, and superconductivity-adjacent regimes (You et al., 2021).

1. Definition and symmetry structure

In the simplest two-valley setting, IVC is the spontaneous coherent superposition of electronic states from $K$ and $K'$. A standard order parameter is

$$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$

with $\phi$ the valley $K'$0 phase. This breaks the continuous valley $K'$1 symmetry associated with independent phase rotations of the two valleys, while leaving the total charge $K'$2 intact (Tanaka et al., 11 Dec 2025). In rhombohedral trilayer graphene (r-TLG), the low-energy spin-valley manifold is nearly $K'$3 symmetric, but inter- versus intravalley scattering reduces the Hamiltonian symmetry to $K'$4; IVC then spontaneously breaks the independent $K'$5 valley charge conservation by establishing a coherent relative phase between $K'$6 and $K'$7 wave functions (Arp et al., 2023).

This symmetry characterization immediately distinguishes IVC from valley-polarized states. In a valley imbalanced (VI) or valley-polarized phase, $K'$8, one valley is preferentially occupied, and the state carries orbital magnetization. In an IVC phase, by contrast, the occupied states are coherent superpositions of $K'$9 and $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$0, so the net valley imbalance can vanish even though the state is strongly symmetry broken (Arp et al., 2023). A recurrent misconception is therefore to identify all valley order with valley polarization; the central distinction is whether the order is diagonal ($$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$1) or off-diagonal ($$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$2) in valley space.

The same language extends beyond two-valley systems, but with qualitatively richer possibilities. In three-valley $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$3-point moiré systems, the order parameter becomes a Hermitian $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$4 matrix $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$5, and IVC can couple all three valleys identically, with a non-trivial sign structure, or with different magnitudes. The resulting phases include $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$6, $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$7, $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$8, and $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle$$9, with one or two neutral phase modes depending on how many independent valley $\langle \tau_x\rangle$0 symmetries are broken (Park et al., 15 Jun 2026).

2. Order-parameter textures and real-space manifestations

A useful microscopic representation of IVC is a valley-basis mean-field Hamiltonian

$\langle \tau_x\rangle$1

with quasiparticle spectrum

$\langle \tau_x\rangle$2

Here the IVC order appears as the in-plane components of $\langle \tau_x\rangle$3, while $\langle \tau_x\rangle$4 acts as a valley-diagonal component (Chau et al., 2024). In this language, IVC opens hybridization gaps on nested parts of the $\langle \tau_x\rangle$5 and $\langle \tau_x\rangle$6 Fermi surfaces.

Real-space structure depends on the ordering wavevector. A commensurate intervalley coupling at wavevectors connecting $\langle \tau_x\rangle$7 to $\langle \tau_x\rangle$8 produces a Kekulé-type enlargement of the unit cell. In graphene, the local density acquires an interference term at $\langle \tau_x\rangle$9, giving a $\langle \tau_y\rangle$0 reconstruction on the atomic lattice (Fan et al., 3 Jan 2025). By contrast, an incommensurate IVC couples $\langle \tau_y\rangle$1 and $\langle \tau_y\rangle$2 at $\langle \tau_y\rangle$3, so the reconstructed bands become periodic only in an emergent supercell; in ABC graphene this can take the form of an IVC crystal with three symmetry-related wavevectors or an IVC spiral with a single wavevector (Vituri et al., 2024).

IVC also need not have a unique relation to time-reversal symmetry. In magic-angle twisted bilayer graphene, “Kramers IVC” and “time-reversal-symmetric IVC” are distinct many-body states. A notable result is that coupling the two valleys does not always imply a Kekulé distortion in scanning tunneling microscopy: K-IVC states and their nonchiral $\langle \tau_y\rangle$4 rotations do not exhibit Kekulé distortion, whereas time-reversal-symmetric IVC states do (Călugăru et al., 2021). More generally, whether IVC preserves or breaks time reversal depends on the spin structure, momentum dependence, and the phase texture of the condensate.

3. Identification in graphene multilayers

In rhombohedral graphene, IVC has been identified through a combination of compressibility, local magnetometry, quantum oscillations, anomalous Hall measurements, and direct STM/STS imaging. In r-TLG quarter metals, the decisive contrast is between an IVC phase with vanishing net orbital moment and a VI phase with finite orbital magnetization (Arp et al., 2023).

Phase	Valley structure	Key signatures
IVC	$\langle \tau_y\rangle$5, no net valley imbalance	no $\langle \tau_y\rangle$6 contrast within phase; no anomalous Hall hysteresis; boundary fixed vs $\langle \tau_y\rangle$7
VI	finite $\langle \tau_y\rangle$8	$\langle \tau_y\rangle$9 contrast; anomalous Hall hysteresis; cusp in $\langle \tau_z\rangle$0
Hybrid IVC–VI	single Fermi surface, mixed character	intermediate susceptibilities; grows with $\langle \tau_z\rangle$1; suppressed by $\langle \tau_z\rangle$2

The compressibility and nanoSQUID study of r-TLG established an IVC quarter metal with a single-flavor Fermi surface $\langle \tau_z\rangle$3, no anomalous Hall hysteresis, and no orbital magnetization inside the phase. By contrast, the VI quarter metal exhibits orbital ferromagnetism, $\langle \tau_z\rangle$4 contrast at first-order boundaries, and a strong cusp-like shift of the VI–IVC boundary with $\langle \tau_z\rangle$5. The same work used the curvature of the VI–IVC boundary versus $\langle \tau_z\rangle$6 to extract an intrinsic spin-orbit scale $\langle \tau_z\rangle$7, consistent with a general estimate $\langle \tau_z\rangle$8 for hBN-encapsulated graphene (Arp et al., 2023).

Direct real-space imaging later showed that IVC in rhombohedral graphene is not restricted to the commensurate Kekulé limit. STM/STS on rhombohedral trilayer graphene visualized an incommensurate, $\langle \tau_z\rangle$9-symmetric IVC crystal near both boundaries of a half-metal phase. At high hole density, the IVC Bragg peaks were measured at $K$0, larger than the commensurate expectation $K$1, implying an outward shift $K$2 (Liu et al., 2024). In unrestricted Hartree–Fock, this is the regime where a half metal is followed by an IVC crystal and then by an IVC spiral as density is tuned (Vituri et al., 2024).

A complementary route is substrate engineering. In rhombohedral tetralayer graphene on MoS$K$3, STM at $K$4 resolved a robust intervalley-coherent reconstruction at $K$5 and $K$6 fillings of the flat band, together with a spectroscopic splitting of $K$7. The observed $K$8-related pattern was absent in hBN-based tetralayers under the same conditions, which the authors associate with spin-orbit proximity and screening from MoS$K$9 (Liao et al., 2024). Earlier STM on PtSe$K'$0/HOPG had likewise visualized a $K'$1 modulation and a small $K'$2 gap around the Fermi level, with anti-phase LDOS at the two gap edges, consistent with intervalley mixing magnified by a higher-order moiré superlattice (Fan et al., 3 Jan 2025).

4. Collective modes and anomalous transport

Because IVC breaks a continuous valley $K'$3, it supports neutral collective modes. In an easy-plane pseudospin description, the low-energy hydrodynamics are

$K'$4

which imply

$K'$5

This is the Goldstone mode of the IVC phase, directly analogous to the phase mode of a superfluid (Xiong et al., 24 Jul 2025).

That mode has now been imaged. In twisted WSe$K'$6 moiré superlattices, ultrafast space-and-time-resolved transport revealed a fast neutral mode with velocity $K'$7, consistent with an IVC Goldstone mode, and a slow gapped mode identified as an amplitude mode. The fast mode disappears above $K'$8, the slow mode above $K'$9, and both occur only near the van Hove singularity (Xiong et al., 24 Jul 2025).

Several works have emphasized that IVC supports anomalous transport responses with a direct superfluid analogy. A phase-only theory yields a London-like valley current,

$$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$0

and, in rhombohedral graphene, surface acoustic waves can drive a dc valley current whose nonreciprocal pseudo-superfluid contribution diverges as $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$1 at low frequency. For an IVC gap $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$2, the estimated current is of order $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$3 for representative SAW parameters (Tanaka et al., 11 Dec 2025).

A different phase-dynamical response appears in multilayer graphene under magnetic field. A phase–number functional

$$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$4

leads to a momentum-space AC Josephson effect, with oscillating intervalley current and orbital magnetization. For rhombohedral trilayer graphene, microscopic estimates give $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$5, $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$6, $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$7, and an oscillation frequency of approximately $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$8 at $$\Delta_{\mathrm{IVC}}=\langle c^\dagger_K c_{K'}\rangle = |\Delta|e^{i\phi},$$9 (Das et al., 27 Mar 2025).

Transport in the diffusive regime can also diagnose time-reversal-invariant IVC. In graphene multilayers, IVC gaps one of the two competing Cooperons that otherwise cancel in the valley-conserving normal state, so low-field magnetoconductance becomes either weak localization or weak antilocalization depending on whether the preserved generalized time-reversal squares to $\phi$0 or $\phi$1. This provides a transport “smoking gun” for TR-invariant IVC even when orbital magnetization is absent (Wei et al., 2023).

5. Relation to superconductivity

One major reason IVC has attracted sustained attention is its repeated proximity to superconductivity. In rhombohedral trilayer graphene, one proposal is that the low-density superconducting phase SC1 arises from direct pairing of quasiparticles inside the IVC background rather than from pairing in a normal metal. In that picture, the IVC quasiparticle spectrum is gapped and multiband, the mean-field transition scale is set by the IVC-quasiparticle density of states, and the coherence length follows the empirical scaling $\phi$2 instead of the conventional $\phi$3 estimate (Chau et al., 2024).

A distinct mechanism emphasizes critical intervalley fluctuations. In rhombohedral trilayer graphene near van Hove singularities, partial nesting enhances both IVC and intervalley pairing. Renormalization-group analysis with antiferromagnetic Hund’s coupling finds that singlet intervalley superconductivity can emerge adjacent to IVC with

$\phi$4

rather than the standard $\phi$5 scaling (You et al., 2021). Closely related unrestricted Hartree–Fock and time-dependent Hartree–Fock work on ABC graphene finds a soft intervalley collective mode at the half-metal to IVC-crystal boundary, and this mode mediates a sign-changing $\phi$6-wave state in a narrow density window with $\phi$7 reaching a few hundred mK (Vituri et al., 2024).

In twisted WSe$\phi$8, the analogous logic is implemented in a multiorbital, first-principles setting. Functional renormalization group calculations find that incommensurate intervalley-coherent antiferromagnetic spin fluctuations near the displacement-field-tunable van Hove singularity drive a chiral mixed-parity $\phi$9-wave superconducting state, with a Bogoliubov Chern number $K'$00 and critical scales up to $K'$01 when longer-range interactions are included (Fischer et al., 2024). This suggests that IVC-adjacent superconductivity is not tied to a single microscopic mechanism, but several distinct theories identify intervalley coherence or soft intervalley modes as the relevant parent sector.

6. Extensions, disorder, and unresolved issues

Two recurring conceptual points remain important. First, IVC is not synonymous with observable Kekulé order in every platform. In twisted bilayer graphene, K-IVC does not exhibit Kekulé distortion in STM, whereas time-reversal-symmetric IVC does (Călugăru et al., 2021). Second, IVC is not synonymous with orbital ferromagnetism. In rhombohedral graphene, the defining experimental distinction from VI phases is precisely the absence of net orbital magnetization in IVC (Arp et al., 2023).

Disorder introduces another major subtlety. In magic-angle twisted bilayer graphene, random homostrain acts as a pair-breaking perturbation for the K-IVC state, analogous to magnetic impurities in a superconductor. The spectral gap can then be strongly suppressed, or vanish altogether, while intervalley coherence remains finite, producing a gapless K-IVC phase (Shavit et al., 2022). At the same time, an Anderson-theorem-type result shows that the K-IVC quasiparticle gap is robust against local valley-preserving perturbations that are odd under $K'$02, whereas $K'$03-even perturbations generically induce subgap states and reduce the gap (Kolář et al., 2022). The disorder phenomenology is therefore symmetry selective rather than uniformly destructive.

Beyond graphene’s two-valley setting, three-valley moiré systems reveal that intervalley coherence is not a single order but a family of orders with different phase sums, magnitude patterns, and Goldstone counting. Strong-coupling analysis and unrestricted Hartree–Fock show that $K'$04 and nematic $K'$05 states can be stabilized by both flat-metric-condition violations and superexchange, with stacking-dependent selection rules (Park et al., 15 Jun 2026). A plausible implication is that “IVC” should increasingly be treated as a symmetry class of orders rather than as one canonical phase.

Current open issues are correspondingly broad: the microscopic origin and magnitude of effective attractive interactions in IVC-mediated superconductivity, the role of collective phase dynamics beyond mean field, the quantitative effect of lattice pinning and intervalley scattering on nominally Goldstone-like modes, and the extent to which spin-orbit coupling reshuffles the competition among IVC, valley-polarized, and superconducting phases. What is already established is more limited but firm: IVC is a genuine, experimentally resolved broken-symmetry phase with identifiable order parameters, real-space and momentum-space signatures, neutral collective modes, and a recurring presence at the boundary of superconductivity across multiple material platforms (Liu et al., 2024).