Strain-Induced Vector Potentials in Quantum Materials

• Strain-induced effective vector potential is an emergent gauge field from lattice deformations that mimics electromagnetic effects in Dirac and Weyl materials.
• It arises from modified hopping amplitudes and geometric lattice corrections, yielding pseudomagnetic fields and pseudo–Landau quantization observable in materials like graphene and borophene.
• Extended theoretical frameworks, including finite displacement and sublattice corrections, enable precise control of valley-selective transport and topological responses in strained quantum systems.

A strain-induced effective vector potential is a gauge field that arises in the low-energy electronic Hamiltonian of Dirac and Weyl materials when the crystal lattice is deformed. This emergent field mimics the action of an electromagnetic vector potential but acts in a valley-dependent (or gauge-dependent) manner, leading to so-called pseudomagnetic fields. Pseudomagnetic fields modify the momentum-space position of Dirac points, quantize energy levels into pseudo–Landau levels, and underlie a wide array of strain-engineered electronic phenomena in two- and three-dimensional quantum materials such as graphene, borophene, and topological semimetals. The effective vector potential arises from modifications to electronic hopping amplitudes and from geometric lattice corrections, and its form and magnitude are dictated by both tight-binding parameters and the symmetry of the lattice.

1. Microscopic Origin of the Strain-Induced Vector Potential

In materials like graphene and related 2D Dirac systems, strain modifies the interatomic distances, which in turn alters the nearest-neighbor hopping amplitudes in a tight-binding description. For a displacement field $\mathbf{u}(\mathbf{r})$, the strain tensor $$u_{ij} = \frac{1}{2}(\partial_i u_j + \partial_j u_i)$$ encodes the local deformation. The low-energy Hamiltonian, linearized near a Dirac point, acquires a term analogous to minimal coupling in electromagnetism: $$H = v_F \, \boldsymbol{\sigma} \cdot \left[\mathbf{p} + e\,\mathbf{A}_s(\mathbf{r})\right] + V_d(\mathbf{r}) \mathbb{I},$$ where $v_F$ is the Fermi velocity, $\boldsymbol{\sigma}$ are Pauli matrices, $\mathbf{A}_s(\mathbf{r})$ is the strain-induced vector potential (pseudovector), and $V_d(\mathbf{r})$ is the scalar deformation potential (Castro et al., 2016, Kerjouan et al., 2024, Sela et al., 2019).

The explicit form of the pseudovector potential for monolayer graphene is: $$A_x(\mathbf{r}) = \frac{\beta}{2 a} \left[u_{xx}(\mathbf{r}) - u_{yy}(\mathbf{r})\right], \quad A_y(\mathbf{r}) = -\frac{\beta}{a} u_{xy}(\mathbf{r}),$$ where $a$ is the lattice constant and $\beta = -\partial \ln t/\partial \ln a$ is the Grüneisen parameter (Castro et al., 2016, Oliva-Leyva et al., 2018).

Lattice-corrected considerations include both hopping-induced and explicit lattice deformation contributions: the latter, involving the deformation of the Brillouin zone, results in a "lattice" correction which is a pure gradient and hence curl-free, but necessary for precise quantitative evaluation in inhomogeneous strain (Kitt et al., 2011).

2. Symmetry Constraints, Generalizations, and Group-Theoretic Classification

The possible forms of strain-induced vector potentials are dictated by the microscopic symmetry of the lattice. Group-theoretic analysis identifies the allowed invariants that couple strain tensor components to the momentum operators, leading to specific strain-induced gauge structures for various crystallographic symmetries (Zabolotskiy et al., 2017, Zemouri et al., 2024). For instance, in graphene's $D_{6h}$ symmetry, the only allowed pseudovector terms are those constructed from $u_{xx}-u_{yy}$ and $u_{xy}$: $$A_x = \alpha_1 (u_{xx} - u_{yy}) + \alpha_2 u_{xy}, \quad A_y = \alpha_3 (u_{xx} - u_{yy}) + \alpha_4 u_{xy},$$ with $\alpha_{i}$ fixed by the tight-binding and Grüneisen tensors (Zemouri et al., 2024).

Symmetry-enforced Dirac semimetals exhibit richer possibilities. For cubic space groups with Dirac nodes at high-symmetry points, strain-induced vector potentials can, for example, involve only shear components, and produce valley-odd gauge fields that preserve global time-reversal symmetry (Zabolotskiy et al., 2017).

3. Extended Theoretical Frameworks: Beyond Linear Strain and Cauchy-Born Rule

Standard theories adopt the Cauchy–Born rule for mapping strain onto bond distortions. Generalizations account for:

• Finite displacements: Treating the local strain metric as a non-Euclidean manifold, incorporating finite displacement corrections via parallel transport, introduces additional terms in the transformation of atomic positions. This results in a finite-displacement tensor that can dominate the local vector potential, especially for strong, spatially nonuniform deformation (e.g., in the “point-stretch” geometry) (Crosse, 2014).
• Sublattice displacement corrections: In graphene, the presence of two atoms per cell allows for sublattice-relative displacements, which suppress the magnitude of the emergent gauge field by a factor $(1-\kappa)$, where $\kappa$ quantifies basis flexibility in the mapping $\delta_n' = (I + u)\cdot\delta_n + \Delta$ (Oliva-Leyva et al., 2018).
• 3D and multi-orbital systems: Dirac and Weyl semimetals, borophene, and multilayer systems can support multiple simultaneous vector potentials in different low-energy sectors; Hilbert-space projections reduce the problem to an effective gauge field in the relevant subspace at low energies (Zabolotskiy et al., 2016, Zemouri et al., 2024).

The table below summarizes key sources and effects for prototypical material systems:

Material/System	Key microscopic origin	Vector potential form
Graphene (monolayer)	Nearest-neighbor hopping	$(u_{xx} - u_{yy}, -2u_{xy})$
Borophene (8-Pmmn)	Tight-binding + symmetry	$(u_{xy}, u_{xx}, u_{yy})$
Bilayer graphene	Intralayer/interlayer hopping	Multiple $A^{t_0},A^{t_3},A^{t_4}$
Dirac semimetals	Shear strain	$(\epsilon_{yz},\epsilon_{zx},\epsilon_{xy})$

4. Pseudomagnetic Fields and Valley Physics

The curl of the strain-induced vector potential defines an out-of-plane pseudomagnetic field: $$B_s(\mathbf{r}) = \partial_x A_y(\mathbf{r}) - \partial_y A_x(\mathbf{r}),$$ which is odd under time reversal and opposite in sign in different valleys. This field induces pseudo–Landau quantization: $$E_n = \operatorname{sgn}(n)\, v_F \sqrt{2 e \hbar |B_s||n|},$$ analogous to the relativistic Landau problem, but with "magnetic" fields reaching hundreds of Tesla for nanoscopic strain gradients (Castro et al., 2016, Yokoyama et al., 14 Dec 2025). In systems with pronounced inhomogeneity—such as a triaxial strain field in graphene or Majorana Fermi surface systems—the resulting $B_s$ enables the formation of flat pseudo–LLs and observable quantum oscillations in the density of states and specific heat (Yokoyama et al., 14 Dec 2025).

Depending on the global and point-group symmetries, certain strain configurations can suppress $B_s$ entirely or restrict it to specific spatial profiles (Zabolotskiy et al., 2017). In carbon nanotubes, an effective pseudoflux derived from uniaxial strain leads to nanoscale analogues of the Aharonov–Bohm effect and periodic gap modulations (Rycerz et al., 2023).

5. Geometric Effects, Curved Surfaces, and Holonomy

For two-dimensional Dirac fermions subject to out-of-plane strain or curvature, the effective Hamiltonian is further modified by the geometry of the surface. The spin connection associated with the vielbein frame yields a position-dependent U(1) gauge field: $$A_\theta(r) = \frac{1}{\sqrt{1 + \left(\partial_r h(r)\right)^2}},$$ for an axially symmetric Gaussian bump $h(r)$. The corresponding pseudomagnetic field is localized and tunable by the bump's amplitude and width (Almeida et al., 25 May 2025). Geometric phases (holonomies) accrue around curved regions, leading to observable signatures in the density of states and local spectroscopy—conceptually akin to a geometric Aharonov-Bohm effect.

In curved samples, the Fermi velocity becomes position-dependent, and an effective scalar "geometric potential" appears, both of which must be taken into account alongside the vector potential in the Dirac equation (Almeida et al., 25 May 2025).

6. Experimental Realizations and Transport Signatures

Strain-engineered graphene and related materials display prominent experimental consequences of strain-induced vector potentials:

• Klein tunneling and conductance modulation: The presence of a sharp, space-dependent pseudo-vector potential at a strain barrier shifts the Dirac point locally and modifies electron transmission through generalized Snell's law conditions, resulting in quantifiable peaks and oscillations in ballistic resistance (Kerjouan et al., 2024).
• Quantized Hall response and dynamic effects: In time-dependent, spatially varying strain profiles, combined pseudomagnetic and pseudoelectric fields can yield quantized Hall currents, observable in AC transport as well as static pseudo–QH phenomena, provided pseudo–LL gaps are well resolved (Sela et al., 2019).
• Collapse of pseudo–Landau levels: Strong in-plane pseudo–electric fields generated by the deformation potential can quench pseudo–Landau quantization via an effective Lorentz-boost mechanism, leading to tunable collapse of LL structures (Castro et al., 2016).

Additional effects are predicted in borophene, Dirac semimetals, and in Kitaev spin liquid candidates, where pseudo–Landau quantization of Majorana fermions can be probed via quantum oscillations in the specific heat (Yokoyama et al., 14 Dec 2025).

7. Outlook and Future Applications

Strain-induced effective vector potentials embody key concepts in "straintronics," enabling the design of valley-selective transport, quantum Hall effects, and pseudofield-driven topological phases. For multilayer and 3D systems, the extension of symmetry-based classification strategies permits a systematic search for materials where multiple emergent vector potentials can be electrically and mechanically manipulated (Zemouri et al., 2024). The confluence of field-tunable valley physics, topological responses, and engineered nanoscale geometries positions strain-induced vector potentials as central objects in condensed matter gauge theory and device-oriented quantum material platforms.