---
title: Intervalley Coherent State (IVC)
url: https://www.emergentmind.com/topics/intervalley-coherent-state-ivc
type: topic
---

# Intervalley Coherent State (IVC)

An intervalley coherent state (IVC) is a broken-symmetry electronic phase in a multivalley system for which the many-body state acquires a valley-off-diagonal expectation value, typically of the form \(\langle c^\dagger_{K} c_{K'} \rangle \neq 0\), so that electrons occupy coherent superpositions of the \(K\) and \(K'\) valleys rather than definite valley eigenstates. In this sense, IVC is a particle–hole or excitonic condensate in valley space that breaks valley \(U(1)\) symmetry and, depending on the microscopic setting, can produce microscopic density-wave modulations at wavevector \(K-K'\), neutral Goldstone modes, anomalous transport responses, and strong competition or coexistence with valley polarization, magnetism, and superconductivity [2508.14630, 2507.18770, 2207.11281, 2411.11163].

## 1. Order parameter and symmetry structure

A general IVC order parameter is valley off-diagonal. In rhombohedral graphene this can be written in the particle–hole channel as
\[
\Phi_{\rm IVC}^{\sigma}(\mathbf Q)\sim \sum_{\mathbf k}\big\langle \hat c^\dagger_{(+,\sigma)}(\mathbf k+\mathbf Q)\hat c_{(-,\sigma)}(\mathbf k)\big\rangle,\qquad \mathbf Q=\mathbf K_+-\mathbf K_-,
\]
while in twisted WSe\(_2\) the corresponding order is written as \(\Delta_{\rm IVC}\sim \langle c^\dagger_K c_{K'}\rangle\), or after a particle–hole transformation as \(\langle c_K c_{K'}\rangle\equiv \Delta_{\rm IVC} e^{i\phi_{\rm IVC}}\) [2508.14630, 2507.18770]. In rhombohedral trilayer graphene, the same structure appears as \(\langle \psi^\dagger_{+,s,\mathbf k}\psi_{-,s',\mathbf k}\rangle\neq0\), with a corresponding valley pseudospin expectation \(\langle \tau_x+i\tau_y\rangle\neq0\) [2109.00002].

The defining broken symmetry is the relative valley phase symmetry. In twisted WSe\(_2\), valley \(U(1)\) acts as \(c_K\to e^{i\theta}c_K\), \(c_{K'}\to e^{-i\theta}c_{K'}\), and an IVC condensate selects a definite \(\phi_{\rm IVC}\), producing a Goldstone mode associated with slow variations of that phase [2507.18770]. In rhombohedral graphene, IVC can occur in either the magnetic or density particle–hole channel, so spin symmetry may be preserved or broken depending on how the spin indices are combined [2508.14630]. In TBG, one important realization is the Kramers IVC state with mean-field term
\[
h_{\rm IVC}=\Delta_0\,\sigma_y(\tau_x\cos\theta+\tau_y\sin\theta),
\]
which preserves the modified antiunitary symmetry \(\mathcal T'=i\tau_y K\) with \((\mathcal T')^2=-1\) [2207.11281].

IVC is not identical to valley polarization. Valley-polarized states have \(\langle \tau_z\rangle\neq0\) and typically carry orbital magnetization, whereas pure IVC has transverse valley pseudospin, \(\langle \tau_{x,y}\rangle\neq0\), and can have zero net valley polarization. In rhombohedral trilayer graphene, this distinction is operationally important: quarter-metal IVC states are consistent with vanishing orbital moment, while valley-imbalanced phases produce finite orbital magnetization [2310.03781].

## 2. Microscopic mechanisms and favored regimes

In rhombohedral \(N\)-layer graphene, the low-energy \(k\cdot p\) Hamiltonian has an isotropic \(k^N\) dispersion, \(\varepsilon_{\mathbf k}=\pm g_N k^N\), and a density of states \(\nu(\varepsilon)\propto |\varepsilon|^{2/N-1}\), so increasing \(N\) flattens the bands and enhances interaction-driven instabilities. In the simplified two-valley model of local intra- and intervalley repulsion, the intervalley particle–hole susceptibility is
\[
\chi^{(0),\mathrm{PH}}_{\tau,-\tau}=\frac{N-1}{N-2}\tilde\chi,
\]
which exceeds the intravalley susceptibility \(\chi^{(0),\mathrm{PH}}_{\tau,\tau}=\tilde\chi\) for \(N>2\). Within RPA, this makes IVC a natural leading instability when the interaction couples strongly to the intervalley channel; under \(SU(4)\)-symmetric interactions, the paper states that the intervalley eigenvalue exceeds the Stoner eigenvalue wherever correlated phases emerge [2508.14630].

That same work shows that in rhombohedral multilayers the IVC transition temperature follows a universal \(N\)-dependent scaling law and grows with layer number before saturating, with the model upper bound
\[
T_{c,\infty}=\frac{V}{8\pi}\frac{\gamma_1^2}{v_F^2}
\]
in RPA. It also finds that \(T_c\) is maximal at or near charge neutrality and decreases with increasing \(|\mu|\), with reentrant behavior possible when the chemical potential exceeds a threshold \(\mu_0\). In the parquet approximation, intervalley Stoner, IVC, and particle–particle instabilities compete, but thicker stacks remain increasingly susceptible to valley-related order [2508.14630].

In rhombohedral trilayer graphene near van Hove filling, inter-valley nesting plays an analogous role. One line of work finds that interactions select IVC as the preferred ordering channel over a wide parameter range, with phase boundaries that agree well with experiment on both hole- and electron-doped sides; another unrestricted Hartree–Fock study finds two closely competing incommensurate IVC phases, an IVC crystal and an IVC spiral, generated by finite-\(\mathbf q\) intervalley coherence [2109.04669, 2408.10309]. Outside graphene, intervalley coherence also appears in twisted WSe\(_2\) near the van Hove singularity, where spin–valley locking and enhanced density of states favor an intervalley excitonic condensate, and in a spinless \(p\)-orbital honeycomb lattice where intermediate interaction drives intervalley coherence with complex polar orbital ordering in a tripled Wigner–Seitz cell [2507.18770, 2505.02461].

## 3. Commensurate, incommensurate, and Kekulé manifestations

Because valley mixing transfers momentum \(\mathbf K-\mathbf K'\), IVC frequently produces real-space reconstruction. In rhombohedral trilayer graphene, spin-singlet IVC corresponds in real space to a charge-density wave at \(\mathbf Q=\mathbf K-\mathbf K'\) and triplet IVC to a spin-density wave at the same wavevector, tripling the unit cell on the effective triangular lattice [2109.00002]. In the \(p\)-orbital honeycomb model, the intermediate-coupling IVC phase likewise appears as a \(120^\circ\) orbital pseudospin pattern in a \(\sqrt3\times\sqrt3\) supercell, and quantum fluctuations then select a particular Kekulé configuration through order by disorder, producing “Kekulé orbitons” [2505.02461].

Direct STM visualizations make this structure explicit. In PtSe\(_2\)/HOPG, spectroscopic imaging reveals a Root3 by Root3 modulation pattern superimposed on a higher-order moiré superlattice, together with a small gap of \(\sim 140\) meV near the Fermi level and an anti-phase real-space conductance distribution at the two gap edges; both the modulation and the small gap disappear in PtSe\(_2\)/bilayer-graphene/SiC, where the graphene is more highly doped [2501.01622]. In rhombohedral tetralayer graphene on MoS\(_2\), STM resolves a \(\sqrt3\times\sqrt3\) supercell at approximately 60% and 70% fillings of the flat band at 77 K, while the same pattern is absent in hBN-based devices under the same conditions, pointing to a significant spin-orbit proximity effect [2411.14113].

IVC need not be commensurate. In rhombohedral trilayer graphene, STM and QPI directly resolve an incommensurate IVC state at high hole density. The additional Fourier peak occurs at \(G_{\rm IVC}\approx 18.5\pm1.0\,\mathrm{nm}^{-1}\), whereas the commensurate value would be \(G_c\approx 14.7\pm0.9\,\mathrm{nm}^{-1}\), implying an incommensurability \(q\approx 3.8\pm1.3\,\mathrm{nm}^{-1}\); the resulting order is \(C_3\)-symmetric and matches the predicted IVC-crystal phase [2411.11163].

A common misconception is that intervalley coherence always implies an observable Kekulé distortion in STM. That is not generally correct. In magic-angle TBG, the K-IVC state and its nonchiral \(\mathrm U(4)\) rotations do not exhibit Kekulé distortion in the STM signal, whereas a time-reversal-symmetric IVC state does. Valley coherence is therefore necessary for such lattice-scale Fourier weight, but not sufficient; the detailed Chern-band and symmetry structure of the occupied states matters [2110.15300].

## 4. Collective modes and response functions

Broken valley \(U(1)\) symmetry implies low-energy collective modes. In twisted WSe\(_2\), the IVC phase is modeled as an easy-plane spin–valley superfluid with Hamiltonian
\[
H=\int d^3r\,\frac12\left[A(\nabla\phi_{\rm IVC})^2+K n_z^2\right],
\]
leading to
\[
\omega(\mathbf q)=v|\mathbf q|,\qquad v=\sqrt{KA}.
\]
Experimentally, ultrafast imaging detects a fast neutral mode with velocity \(\sim 3\,\mathrm{km/s}\), consistent with the Goldstone mode, and a slower mode interpreted as a gapped amplitude mode; the fast mode disappears around \(\sim 10\) K and the slow mode around \(\sim 20\) K [2507.18770].

In graphene multilayers, time-reversal-invariant IVC also reorganizes quantum interference. Weak-field magnetoconductance can show weak localization or weak antilocalization depending on whether the surviving generalized time-reversal symmetry has \(\mathcal T^2=+1\) or \(\mathcal T^2=-1\). In that framework, the onset of intervalley coherence gaps one of the two valley Cooperons that would otherwise cancel, leaving a net low-field magnetoresistance signature that can distinguish ordinary IVC from Kramers IVC [2312.11259].

A more recent extension treats IVC as a valley-gauge-symmetry-broken phase with superconducting-like electrodynamics in the valley sector. In that setting, surface acoustic waves generate an anomalous valley current with a low-frequency power law, and the nonlinear valley conductivity acquires a contribution
\[
\sigma^{\alpha;\beta\gamma}_{\mathrm{s,NRSF}}=-\frac{1}{2\Omega^2}f^{\alpha;\beta\gamma}_{\mathrm s},
\]
where \(f^{\alpha;\beta\gamma}_{\mathrm s}\) is a nonreciprocal pseudo-superfluid density. Numerical analysis in rhombohedral graphene finds that IVC strongly enhances this response [2512.10395].

## 5. Disorder, robustness, and gap structure

For K-IVC in TBG, the relation to superconducting Bogoliubov–de Gennes structure is mathematically precise enough to yield an Anderson-type theorem. In the particle–hole basis, the mean-field Hamiltonian takes the form
\[
H(\mathbf k)=H_0(\mathbf k)\tau_z+\Delta\tau_y,
\]
and the combined chiral operation is \(\mathcal{PT}=i\tau_y\). Valley-preserving perturbations that are odd under \(\mathcal{PT}\) anticommute with the order parameter and do not reduce the quasiparticle gap, whereas \(\mathcal{PT}\)-even perturbations can generate subgap states and reduce or destroy the gap [2207.11281].

That robustness is not universal once realistic disorder channels are included. In a spinless K-IVC model for magic-angle TBG, random homostrain enters as a pseudo-gauge field that commutes with the K-IVC order parameter and therefore acts as a pair-breaking perturbation, directly analogous to magnetic disorder in a singlet superconductor. Self-consistent Born analysis shows that the spectral gap \(\omega_g\) can be strongly suppressed or even vanish while the order parameter \(\Delta_{\rm ivc}\) remains finite, producing a gapless IVC phase. In that regime the activation gap measured in transport tracks \(\omega_g\), not \(\Delta_{\rm ivc}\), offering a resolution of the large discrepancy between theoretical clean-limit K-IVC gaps and experimentally extracted activation scales [2208.03655].

This distinction between order parameter amplitude and single-particle gap is important across the field. A hard gap is therefore not a defining property of IVC. What defines the phase is the valley-off-diagonal coherence itself; the spectral gap may be robust, reduced, or absent depending on the perturbation channel [2207.11281, 2208.03655].

## 6. Competition with magnetism and superconductivity

IVC is repeatedly found at the center of the competition among particle–hole orders in rhombohedral graphene. In rhombohedral \(N\)-layer graphene, RPA supports only Stoner and IVC phases, while the parquet approximation admits a broader set of particle–hole and particle–particle instabilities. Under \(SU(4)\)-symmetric interactions, intervalley channels dominate; under generic \(SU(2)\times SU(2)\) interactions, IVC occupies a finite wedge in the \(V_0/V_1\) plane and competes with magnetic and density Stoner phases, with a crossover in the dominant particle–hole instability as layer number increases [2508.14630].

In rhombohedral trilayer graphene, Hartree–Fock analysis identifies a spin-unpolarized IVC metal as a realistic symmetry-broken normal state proximate to superconductivity. In that framework, IVC fluctuations provide a pairing glue, leading to chiral unconventional superconductivity when the fluctuations are strong; a ferromagnetic intervalley Hund’s coupling favors spin-singlet superconductivity if the normal state is spin-unpolarized, but spin-triplet pairing if the normal state is spin-polarized [2109.00002]. A related weak-coupling analysis likewise finds that interactions select IVC over a wide range, and that the same inter-valley nesting which promotes IVC also enhances inter-valley superconductivity; with antiferromagnetic Hund’s coupling, the predicted transition scale is \(T_c\sim \exp(-1/\sqrt g)\), rather than the standard \(\exp(-1/g)\), and the favored pair state is spin-singlet [2109.04669].

Another proposal places superconductivity inside the IVC phase itself rather than merely adjacent to it. In that picture, superconductivity in rhombohedral trilayer graphene arises from pairing of IVC quasiparticles in a gapped Dirac-like band structure. The mean-field transition temperature is then controlled by the density of states of IVC quasiparticles and is more suppressed than in standard BCS theory, while the coherence length obeys \(\xi\sim 1/\sqrt{T_{\rm MF}}\); the quantum metric contribution of the IVC quasiparticle bands becomes especially important near the superconductivity–IVC boundary [2404.19237]. In unrestricted Hartree–Fock plus time-dependent Hartree–Fock for ABC graphene, the half-metal to IVC-crystal transition is continuous or very weakly first order, and the associated soft inter-valley collective mode can mediate a sign-changing \(s\)-wave superconducting state with \(T_c\) reaching a few hundreds of mK in a narrow density window [2408.10309].

Taken together, these results support a general picture in which IVC is not merely one candidate among many valley orders. It is a recurrent organizing principle of correlated multivalley systems: a phase with broken valley \(U(1)\), distinctive real- and momentum-space signatures, nontrivial collective dynamics, disorder responses that can parallel or depart from superconducting analogies, and a particularly close connection to the superconducting domes of rhombohedral and moiré graphene platforms [2508.14630, 2109.00002, 2408.10309].

Source: https://www.emergentmind.com/topics/intervalley-coherent-state-ivc