[Paper Review] Emergent gravity in graphene

M. A. Zubkov, G. E. Volovik|arXiv (Cornell University)|Aug 9, 2013

Graphene research and applications4 citations

TL;DR

This paper demonstrates that elastic deformations in monolayer graphene induce an emergent 2D Weitzenbök geometry and a U(1) gauge field, both arising from strain-induced variations in hopping parameters. The fermionic quasiparticles effectively propagate in a teleparallel gravity framework, with the emergent geometry and gauge field fully determined by the elastic deformation tensor, providing a condensed matter realization of emergent gravity.

ABSTRACT

We reconsider monolayer graphene in the presence of elastic deformations. It is described by the tight - binding model with varying hopping parameters. We demonstrate, that the fermionic quasiparticles propagate in the emergent 2D Weitzenbock geometry and in the presence of the emergent U(1) gauge field. Both emergent geometry and the gauge field are defined by the elastic deformation of graphene.

Motivation & Objective

• To clarify the geometric structure experienced by low-energy quasiparticles in strained monolayer graphene.
• To resolve the long-standing question of whether the emergent geometry is Riemannian or teleparallel.
• To derive the effective field theory for fermions near the floating Fermi point under elastic deformations.
• To establish the direct link between elastic strain and the emergence of both gravity-like geometry and gauge fields.

Proposed method

• Use of a tight-binding model with position-dependent hopping parameters to describe strained graphene.
• Introduction of the 'floating Fermi point' as a momentum-dependent reference point where the effective Hamiltonian vanishes.
• Expansion of the effective Hamiltonian around the floating Fermi point to extract emergent geometric and gauge structures.
• Derivation of the zweibein (vielbein) from the Fermi velocity tensor, identifying it as the geometric structure of emergent gravity.
• Expression of the emergent U(1) gauge field in terms of strain via the deformation tensor and hopping parameter variations.
• Demonstration that the emergent geometry is Weitzenböck (teleparallel) rather than Riemannian, with no spin connection in the linear approximation.

Experimental results

Research questions

• RQ1 What is the nature of the effective geometry experienced by quasiparticles in strained graphene?
• RQ2 How do elastic deformations give rise to emergent gauge and gravitational fields in the low-energy limit?
• RQ3 Is the emergent geometry in graphene Riemannian or teleparallel (Weitzenböck) in structure?
• RQ4 What is the precise relation between strain and the emergent U(1) gauge field in the system?
• RQ5 Does the spin connection contribute to the effective action in the linear response to strain?

Key findings

• Elastic deformations in graphene induce an emergent 2D Weitzenböck geometry, characterized by a zweibein derived from the Fermi velocity tensor.
• The emergent U(1) gauge field arises from strain and is expressed as $\mathbf{A}^b = -\frac{2}{3a^2}\epsilon^{ba}\sum_j \Delta_j \mathbf{l}_j^a$, matching known results from strain-induced electromagnetic fields.
• The determinant of the zweibein is unity, ensuring a flat 2D spacetime structure in the emergent geometry.
• The emergent gravity is teleparallel, with no spin connection in the linear approximation of elastic deformations.
• The volume element of the emergent 3D spacetime is $d^{(3)}V = e(\mathbf{r},t)\,d^2\mathbf{r}\,dt$, where $e = v_F(1 - \frac{1}{3}(\Delta_1 + \Delta_2 + \Delta_3))$, and $e^{-1} = \mathbf{e}^0_0$.
• The emergent geometry and gauge field are fully determined by the strain tensor via $t_a(\mathbf{r}) = t(1 - \Delta_a(\mathbf{r}))$, with $\Delta_a$ proportional to the strain $u_{ik}$ through $\beta$.