Majorana dHvA in Strained Kitaev Liquids

• Majorana de Haas–van Alphen effect is a strain-induced quantum oscillation in Kitaev spin liquids, where a pseudomagnetic field quantizes Majorana excitations into discrete Landau levels.
• The phenomenon arises from a triaxial strain pattern that modulates Kitaev couplings, generating valley-odd gauge fields and forming electron-like and hole-like Majorana pockets.
• Oscillatory density of states and specific heat, described by a Lifshitz–Kosevich framework, serve as key thermodynamic signatures of Landau quantization in this charge-neutral system.

Searching arXiv for the cited paper and directly related references. arXiv Search Query: (Yokoyama et al., 14 Dec 2025) The Majorana de Haas–van Alphen effect denotes a strain-driven quantum-oscillation phenomenon for charge-neutral Majorana quasiparticles in a Kitaev spin liquid with a Majorana Fermi surface. In the realization studied in "Strain-induced quantum oscillation in Kitaev spin liquid with Majorana-Fermi surface" (Yokoyama et al., 14 Dec 2025), triaxial lattice strain generates an effective vector potential and a uniform pseudo-magnetic field that Landau-quantize itinerant Majorana fermions. The resulting pseudo-Landau levels produce oscillations of the density of states and the specific heat at very low temperatures, in close analogy to the conventional de Haas–van Alphen effect in metals, but with a neutral, valley-odd, strain-induced gauge coupling rather than minimal coupling to an electromagnetic field.

1. Microscopic setting in the Kitaev model

The starting point is the isotropic spin-$1/2$ Kitaev model on the honeycomb lattice,

$$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$

where $\gamma\in\{x,y,z\}$ labels bond types and $\langle ij\rangle_\gamma$ are nearest neighbors on $\gamma$-bonds. In the Majorana representation used in the paper, the spin operator is written as

$S_i^\alpha = i\, b_i^\alpha c_i,$

with one itinerant Majorana $c_i$ and three gauge Majoranas $b_i^\alpha$ per site, together with a static $\mathbb{Z}_2$ link variable $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$. In a fixed flux sector, and in particular in the flux-free ground state, the model reduces to free Majorana hopping,

$$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$0

which yields Dirac cones at the honeycomb $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$1 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$2 points for $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$3 (Yokoyama et al., 14 Dec 2025).

The paper considers perturbations that generate Majorana Fermi surfaces near these Dirac points. Two microscopic routes are identified. A staggered Zeeman field $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$4, associated with proximity to weak antiferromagnetism, induces an effective three-spin term that becomes next-nearest-neighbor Majorana hopping at low energies. A uniform electromagnetic field with a cross term $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$5 likewise generates a three-spin interaction that maps to next-nearest-neighbor Majorana hopping. In both cases, time-reversal and inversion symmetries are broken.

In momentum space, these perturbations shift the Dirac cones at $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$6 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$7 in opposite directions in energy: the $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$8 cone moves up and the $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$9 cone moves down by a valley-dependent offset $\gamma\in\{x,y,z\}$0. This produces small electron-like and hole-like Majorana pockets around $\gamma\in\{x,y,z\}$1 and $\gamma\in\{x,y,z\}$2, respectively. The corresponding low-energy linearized Hamiltonian near valley $\gamma\in\{x,y,z\}$3 is

$\gamma\in\{x,y,z\}$4

where $\gamma\in\{x,y,z\}$5 is the Dirac velocity, $\gamma\in\{x,y,z\}$6 act in sublattice space, and $\gamma\in\{x,y,z\}$7. The simultaneous presence of $\gamma\in\{x,y,z\}$8 and $\gamma\in\{x,y,z\}$9 at zero energy is the immediate origin of the Majorana Fermi pockets.

2. Strain as an axial gauge field

The quantizing field in the Majorana de Haas–van Alphen effect is not an external magnetic field but a pseudogauge field generated by lattice strain. In the model, strain modulates the Kitaev couplings on each nearest-neighbor bond according to

$\langle ij\rangle_\gamma$0

with magnetoelastic constant $\langle ij\rangle_\gamma$1, lattice constant $\langle ij\rangle_\gamma$2, and bond vector

$\langle ij\rangle_\gamma$3

In the Dirac continuum limit, the nonuniform $\langle ij\rangle_\gamma$4 appears as an effective valley-odd gauge field $\langle ij\rangle_\gamma$5 that minimally couples to the Majorana Dirac Hamiltonian, in direct analogy to strained graphene (Yokoyama et al., 14 Dec 2025).

The effective vector potential has the form

$\langle ij\rangle_\gamma$6

where

$\langle ij\rangle_\gamma$7

is the strain tensor and $\langle ij\rangle_\gamma$8 is the displacement field. The specific strain pattern used is the triaxial profile

$\langle ij\rangle_\gamma$9

with strain amplitude $\gamma$0. This yields

$\gamma$1

and therefore a uniform pseudo-magnetic field perpendicular to the plane,

$\gamma$2

so that $\gamma$3 in lattice units.

A defining feature is the axial character of the coupling: quasiparticles near $\gamma$4 and $\gamma$5 experience opposite signs of $\gamma$6. The strained system therefore preserves overall time-reversal invariance while still exhibiting Landau quantization within each valley. This valley-odd structure is central to the Majorana version of de Haas–van Alphen physics, because it produces mirror-related electron-like and hole-like sectors rather than a single charged carrier species.

3. Landau quantization of Majorana quasiparticles

Under a uniform axial field, the low-energy Majorana Dirac fermions form discrete pseudo-Landau levels. For each valley $\gamma$7, the spectrum is

$\gamma$8

The zeroth Landau level is pinned at

$\gamma$9

When $S_i^\alpha = i\, b_i^\alpha c_i,$0, the $S_i^\alpha = i\, b_i^\alpha c_i,$1 level sits at zero energy; with finite $S_i^\alpha = i\, b_i^\alpha c_i,$2, it shifts valley-dependently but persists under triaxial strain. The paper’s local density of states shows that the zero-mode peak survives and moves slightly away from zero when $S_i^\alpha = i\, b_i^\alpha c_i,$3 (Yokoyama et al., 14 Dec 2025).

For sufficiently large $S_i^\alpha = i\, b_i^\alpha c_i,$4, the dispersion near the small pockets becomes approximately parabolic, and a nonrelativistic description is appropriate: $S_i^\alpha = i\, b_i^\alpha c_i,$5 with cyclotron frequency $S_i^\alpha = i\, b_i^\alpha c_i,$6 and valley-dependent chemical potential $S_i^\alpha = i\, b_i^\alpha c_i,$7. The paper also gives the areal degeneracy per Landau level as

$S_i^\alpha = i\, b_i^\alpha c_i,$8

for each valley, where $S_i^\alpha = i\, b_i^\alpha c_i,$9 is the sample area.

Because Majorana fermions are real rather than complex, the spectrum is symmetric in $c_i$0, and thermodynamic counting involves both signs. In the flux-free sector the Landau levels are sharp, whereas in the two-flux sector bound states appear between Landau levels. Semiclassically, the oscillations are governed by Onsager quantization,

$c_i$1

where $c_i$2 is the extremal orbit area in momentum space. The phase offset is $c_i$3 for Dirac pockets because of the Berry phase, and $c_i$4 for nonrelativistic pockets.

4. Oscillatory thermodynamics and the de Haas–van Alphen analogy

The defining observable consequence of the pseudo-Landau quantization is an oscillatory contribution to the density of states and to the specific heat as functions of $c_i$5, or equivalently $c_i$6. The paper describes this using a Lifshitz–Kosevich-type form adapted to the pseudo-field,

$c_i$7

with harmonic index $c_i$8, phase $c_i$9, temperature damping

$b_i^\alpha$0

and Dingle factor

$b_i^\alpha$1

The oscillatory part of the specific heat follows from the grand potential through

$b_i^\alpha$2

and inherits the same periodicity and damping factors (Yokoyama et al., 14 Dec 2025).

The fundamental frequency is set by the extremal area of the Majorana Fermi surface at zero energy,

$b_i^\alpha$3

This differs from the conventional metallic expression $b_i^\alpha$4: for neutral Majoranas the effective coupling is to the strain-generated pseudofield, so $b_i^\alpha$5 does not appear, and quantization is controlled by $b_i^\alpha$6. In the numerical results, the density of states at $b_i^\alpha$7 shows clear oscillations versus strain $b_i^\alpha$8 when $b_i^\alpha$9, and the very-low-$\mathbb{Z}_2$0 specific heat $\mathbb{Z}_2$1 likewise oscillates with $\mathbb{Z}_2$2, with increasing amplitude as $\mathbb{Z}_2$3 grows.

A common misconception is that de Haas–van Alphen-type oscillations require charged carriers. In this setting, the oscillatory response is instead driven by a pseudogauge field derived from strain; no Lorentz force on electric charge is involved. The analogy to conventional dHvA is therefore thermodynamic and semiclassical rather than electromagnetic.

5. Distinctive features relative to conventional dHvA

Several features distinguish the Majorana effect from ordinary electron dHvA while preserving the central logic of Landau quantization of closed Fermi-surface orbits (Yokoyama et al., 14 Dec 2025).

Aspect	Majorana de Haas–van Alphen effect	Conventional de Haas–van Alphen effect
Source of quantizing field	Pseudo-magnetic field $\mathbb{Z}_2$4 from triaxial strain; axial coupling with opposite signs in valleys	Real magnetic field $\mathbb{Z}_2$5; minimal coupling with electric charge $\mathbb{Z}_2$6
Carriers	Charge-neutral Majorana quasiparticles; spectrum symmetric in $\mathbb{Z}_2$7; Dirac zero-mode persists	Charged electrons; no zero-mode in parabolic bands
Frequency	$\mathbb{Z}_2$8 with $\mathbb{Z}_2$9; periodic in $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$0	$u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$1; periodic in $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$2
Phase	Berry phase $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$3 for Dirac pockets; valley-dependent contributions and possible bound-state features in flux sectors	$u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$4 for parabolic bands; $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$5 for Dirac electrons
Tuning parameter	Strain amplitude $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$6	Magnetic field $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$7

The Dirac zero-mode is especially distinctive. In the strained Majorana system a robust zeroth Landau level remains present, shifted by $u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$8 when the perturbation-induced pocket offset is finite. The axial coupling is likewise nonstandard: the two valleys contribute with opposite pseudo-field signs, so the oscillatory response combines electron-like and hole-like pockets that are mirror images in spectrum but identical in level spacing.

The paper also notes that valley mixing and flux-sector bound states can modulate the oscillation phase and amplitude. This places the Majorana effect closer to a topological Dirac-Landau problem than to the textbook case of a parabolic electron band.

6. Quantitative scale, measurement protocols, and limitations

For the triaxial profile

$u_{ij}=ib_i^\alpha b_j^\alpha=\pm 1$9

the pseudo-field magnitude is $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$00, and the magnetic length is

$$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$01

A representative numerical choice is flake radius $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$02 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$03, which gives $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$04 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$05. The Dirac Landau-level spacing near $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$06 scales as

$$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$07

Taking $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$08 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$09–10 meV in candidate materials as an order-of-magnitude estimate yields $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$10–6 meV for $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$11, corresponding to 20–70 K. In the calculations, the oscillations in $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$12 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$13 are resolved at very low temperature, with $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$14 (Yokoyama et al., 14 Dec 2025).

The oscillation period in inverse pseudo-field is

$$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$15

Because $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$16 is controlled by the pocket size near $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$17, and the pocket size is controlled by $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$18, increasing $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$19 enlarges the pockets and increases the oscillation amplitude. Tiny pockets imply a small $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$20 and therefore slow oscillations in $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$21; larger pockets imply faster oscillations.

The proposed measurements are specific heat versus strain amplitude $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$22, seeking periodic oscillations in $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$23 at very low temperature, and STM measurements of the local density of states in strained monolayer flakes, seeking pseudo-Landau levels near the flake center. Nanobubble-like triaxial strain profiles are proposed as a way to produce a nearly uniform $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$24 in a central region. Smooth phonon contributions to specific heat provide a background that varies smoothly in $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$25 and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$26, without $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$27 periodicity; magnetic backgrounds such as magnons are not expected to show axial-field-induced quantization at zero applied magnetic field. Valley-dependent electron-like and hole-like pockets and the shifted Dirac zero-mode are therefore identified as distinctive signatures.

The analysis is subject to explicit assumptions and limitations. The flux-free sector is assumed to dominate at low $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$28, with flux excitations neglected except when explicitly studied. The model is isotropic, and Heisenberg and $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$29 interactions and other perturbations are ignored. The calculations use open-boundary hexagonal flakes and impose $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$30 to avoid sign changes of $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$31 at the edges. The mapping to $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$32 is a linearized Dirac description valid for small strains and smooth displacement fields, and the precise mapping from magnetoelastic coefficients to $$H_K=-J\sum_{\langle ij\rangle_\gamma} S_i^\gamma S_j^\gamma,$$33 is material dependent. Within those assumptions, the Majorana de Haas–van Alphen effect provides a route to probing a charge-neutral Majorana Fermi surface through strain-controlled Landau quantization and the resulting oscillatory density of states and thermodynamics.