Hybrid Pseudogauge & Electrostatic Cavity in Graphene

• Hybrid pseudogauge and electrostatic cavity is defined by the coexistence of a strain-localized pseudogauge field and a gate-tunable electrostatic barrier in graphene, enabling 1D channel confinement.
• Quantization conditions from an effective Dirac Hamiltonian yield resonant transmission and valley-dependent bound states, emphasizing cavity interference effects.
• Gate-controlled tuning of the electrostatic potential modulates electron–electron interactions, paving the way for adjustable Tomonaga–Luttinger liquid behavior.

A hybrid pseudogauge and electrostatic cavity in graphene arises from the interplay between sharply localized strain (pseudogauge field) and an externally controlled electrostatic potential in a single atomic monolayer. This configuration, realized in the "graphene nanoslide" device, enables confinement and transport manipulation via distinct 1D channels with gate-tunable properties, strong sublattice and valley dependencies, and interaction-induced liquid phases. Theoretical treatment pivots on an effective Dirac Hamiltonian encapsulating both barrier types, yielding quantization and boundary conditions that dictate conductance, bound-state formation, and Tomonaga–Luttinger liquid tunability (Beule et al., 28 Dec 2025).

1. Effective Dirac Hamiltonian and Cavity Realization

In strained graphene, each valley $\tau = \pm 1$ is described by a continuum Dirac Hamiltonian incorporating both an electrostatic potential $V(x)$ (generated by a bottom gate) and a strain-induced pseudogauge field $A_s(x)$. Choosing the transverse gauge ($A_{s,x}=0$), the Hamiltonian takes the form:

$$\hat{H} = v_F\left[-i\partial_x + eA_{tot,x}\right]\tau\sigma_x + v_F\left[-i\partial_y + eA_{tot,y}\right]\sigma_y + V(x)\sigma_0$$

where $A_{tot}(x) = \tau A_s(x) e_y$, and for a sharply varying strain barrier,

$$A_s(x) \to \frac{\alpha}{e} \delta(x) e_y, \quad \alpha \simeq -\frac{\beta \Delta^2 \sin(3\theta)}{3a_0 d}$$

A narrow gate provides an electrostatic barrier $V(x) = v_F \gamma \delta(x)$.

The combination of the $\delta$-function pseudogauge and electrostatic potentials forms the hybrid cavity: a strain-dominated region spatially coincident with a tunable potential jump.

2. Boundary Conditions and Scattering Matrix Formalism

The quantum transport properties are determined by integrating the Dirac equation across the $\delta$-barriers at $x=0$, imposing boundary conditions on the spinor wavefunction:

$$\psi(0^+) = e^{\alpha\sigma_z} e^{-i\tau\gamma\sigma_x} \psi(0^-)$$

For pure pseudogauge ($\gamma=0$),

$\psi(0^+) = e^{\alpha\sigma_z} \psi(0^-)$

This leads to a valley-dependent $2\times 2$ scattering ($S$) matrix, yielding angle-resolved transmission for incoming plane waves:

$$T(\phi) = |t|^2 = \frac{\cos^2 \phi}{\cosh^2\alpha - \sin^2\phi}$$

Resonant perfect transmission occurs at values $\gamma^2 = \alpha^2 + (m\pi)^2$ (with $m\in\mathbb{Z}$), signifying quantized states defined by cavity interference and pole crossings in the scattering amplitude.

3. Quantization, Bound States, and 1D Channels

The bipolar regime, established by leads of opposite doping, supports Fabry–Pérot-type interference across the cavity with quantization dictated by:

$$\int_0^{x_0(n_1,n_2)} dx\,\sqrt{\pi |n(x)|} = \pi(m \pm \delta), \quad m\in\mathbb{Z}$$

A $\delta$-like pseudogauge barrier yields bound "snake" states, manifesting as localized valley-chiral 1D channels with dispersion:

$$E_b = \pm(v_F \, \text{sech}\,\alpha)k_y, \quad \tau k_y \alpha < 0$$

Generalizing to the hybrid case ($\gamma\neq0$), two guided modes emerge with velocities $v_\pm$:

$$E_b = v_\pm k_y,\quad v_\pm = \frac{\pm \alpha \, \text{sech}\,\zeta - \gamma}{\alpha \mp \gamma\, \text{sech}\,\zeta} v_F, \quad \zeta = \sqrt{\alpha^2 - \gamma^2}$$

The system's phase diagram, as mapped in (Beule et al., 28 Dec 2025), clarifies regimes of valley-chiral propagation ($v_+v_-<0$), counterpropagation, and flatband formation ($v_+=0$ for $\gamma = \pm \alpha\,\text{sech}\,\zeta$).

4. Gate Tuning, Valley Polarization, and Local Density of States

The electrostatic barrier strength $\gamma$ is directly gated by the applied voltage, modulating resonance width and position in transmission and enabling control over bound-state transport. Analytical results for the sublattice-resolved LDOS in the pure pseudogauge regime ($\gamma=0$) give:

$$\rho(x,E) = \rho_0\Bigl\{1 - |\sinh\alpha|\Bigl[e^{-|\alpha|}J_0(2k_F|x|) + 2\cosh\alpha \sum_{n\geq 1} e^{-2n|\alpha|} J_{2n}(2k_F|x|) - \cosh\alpha e^{-|2k_Fx\sinh\alpha|}\Bigr]\Bigr\}$$

$$\rho_A(x, E) - \rho_B(x, E) = \rho_0 \, \text{sgn}(x) |\sinh\alpha|\Bigl[e^{-|\alpha|}J_0(2k_F|x|) - 2|\sinh\alpha| \sum_{n\geq 1} e^{-2n|\alpha|} J_{2n}(2k_F|x|)\Bigr]$$

Switching on $\gamma$ breaks electron-hole symmetry and suppresses sublattice contrast at transmission resonance, providing valley and sublattice-polarized signatures near the barrier location (see (Beule et al., 28 Dec 2025), Fig.4).

5. Electron–Electron Interactions and Tomonaga–Luttinger Liquid Phenomenology

Interacting physics in the nanoslide is governed by the effective mode velocity $v$ (either $v_+$ or $v_-$), entering the Tomonaga–Luttinger parameter $K \sim [1 + g/(\pi v)]^{-1/2}$, where $g \propto e^2/\epsilon$ encodes Coulomb screening. As $v$ is gate- and strain-tunable, the hybrid cavity facilitates in situ transition between Luttinger liquid regimes ($K<1$) and chiral limits ($K\to 0$). Full bosonization analysis—incorporating intra/inter-mode couplings $g_2$, $g_4$—is deferred for future investigation (Beule et al., 28 Dec 2025).

6. Experimental Signatures and Implications

The hybrid cavity platform enables gate-tuned transport oscillations, valley and sublattice control, and confined 1D modes observable through two-terminal conductance measurements and local spectroscopies. Key phenomena are summarized in the following table:

Property	Pseudogauge Only ($\gamma=0$)	Hybrid ($\alpha,\gamma\neq0$)
Bound States	Valley-chiral snake modes	Chiral/counterpropagating/flatbands
LDOS Modulation	Strong sublattice asymmetry	Tunable asymmetry, e-h symmetry breaking
Fabry–Pérot Oscillations	Strain-induced only	Strain and gate-controlled
TLL Parameter $K$ Control	Fixed by $\alpha$	Tunable via $\gamma$

The device geometry and mechanism are detailed in Fig.1 and associated equations of (Beule et al., 28 Dec 2025), linking straintronics control to low-dimensional many-body phenomena.

7. Context and Future Directions

The graphene nanoslide represents a fundamental advance in strain-based device design, integrating pseudogauge and electrostatic barriers to fully exploit the Dirac band structure and interaction tunability. Its theoretical foundation and predicted experimental observables establish a path for valley- and sublattice-selective electronics, controlled 1D quantum liquids, and explorations of flatband localization. A plausible implication is the extension to multi-barrier architectures, complex strain textures, and strong-coupling regimes, promising continued developments in graphene straintronics and correlated electron physics (Beule et al., 28 Dec 2025).