Emergent gravity in graphene
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 Weitzenbock geometry and in the presence of the emergent gauge field. Both emergent geometry and the gauge field are defined by the elastic deformation of graphene.
It can be shown [1, 2, 3, 4, 5, 6], that in D systems with fermions near the Fermi - points the two - component spinors effectively appear. The remaining components of the original fermion field are in general case massive, and, therefore, decouple in the low energy limit. These Weil spinors are described by action
In this action the collective mode
may contain the shift of the Fermi point resulted in the
effective gauge field and the effective spin connection
. The modes may be considered as the vielbein describing the gravitational degrees of freedom
2 The tight - binding model with varying hopping parameters. Floating Fermi - point as the emergent gauge field.
The carbon atoms of graphene form a honeycomb lattice with two sublattices A and B (of the triangular form). We denote the lattice spacing by . Let us introduce vectors that connect a vertex of the sublattice A to its neighbors (that belong to the sublattice B): , , .
We suppose that the hopping parameter varies, so that its value depends on the particular link connecting two adjacent points of the honeycomb lattice. This is caused by elastic deformations. We have three values of at each point. The Hamiltonian has the form
In addition, we define the following variables: . In momentum space the effective Hamiltonian has has the form , where
while is the area of momentum space, and . We imply, that the variations of are given by
We introduce the notation for the ”floating Fermi point”. It is defined in such a way, that for , the function vanishes: . For the case of homogenious variations of hopping parameters when this definition gives the true Fermi point. For the interpretation is not so obvious. In general case the eigenvectors of the one - particle Hamiltonian do not correspond to the definite value of momentum. We have the complicated wave - packets instead. However, in the next section it will be shown that in the linear approximation of (as a function of , ) the term proportional to may be neglected in the considered theory. This brings us back to the interpretation of as the Fermi point. Now we take into account that hopping parameters themselves depend on . That’s why is the momentum - dependent (”floating”) Fermi point. One can easily find , where in addition to the fixed Fermi - point the emergent gauge field appears with the components :
3 Expansion near the floating fermi - point
Next, we expand around , . The result has the form:
where (as well as ) depends on and is given by
Here we introduced  the new tensor : . One can see, that the emergent field is related to the field as follows:
We define the new spinors: . In order to return to the coordinate space we should use the following rule . As a result the hamiltonian has the form , where
Here the field is defined in coordinate space and is related to the variables according to Eq. (3.2). The product in these equations should be understood as . The additional gauge field appears: . This additional field is to be compared with the emergent gauge field . One can see that . Therefore, this is not reasonable to keep this additional field together with in the field - theoretical description, where all dimensional quantities are to be much larger than the lattice spacing . This shows that even in case of the variations of depending on the position in coordinate space we may omit the derivatives of in the effective low energy field - theoretical Hamiltonian (that is we may omit the terms in momentum space representation). The field is related to the dreibein as follows: ; , , where . Here the determinant of the zweibein , where . At the same time the three - dimensional determinant of is equal to . The three - volume element is . The value of is given by .
4 Elastic deformations as a source of emergent gravity
The graphene sheet is parametrized by variable . The classical elasticity theory has the displacements as degrees of freedom (). The three - dimensional coordinates of the graphene sheet are given by
At the graphene is flat. The emergent metric of elasticity theory is given by . According to  the elastic deformations of graphene result in the change of the hopping elements, which determine the effective geometry experienced by fermions. The simplest connection between the deformations and the hopping elements (), which is allowed by symmetry, is
The dimensionless phenomenological parameter is determined by the microscopic physics. As it was mentioned above, the given consideration works for the displacements that are not necessarily small. However, we imply that . This is the requirement that the derivatives of are small being multiplied by .
The emergent geometry and emergent gauge field are given by and Eq. (3.3) with . This results in the usual expression for the strain - induced electromagnetic field (see [10, 12, 13, 14, 15] and references therein). As for our values of , they differ from the expression for the anisotropic Fermi velocity calculated in . The zweibein is given by . It is constructed in such a way that the determinant of the zweibein . The D volume element corresponds to the function . This function is related to the - component of the dreibein as . The determinant of the dreibein is equal to .
Here we considered the long - standing problem about the type of the geometry experienced by fermionic quasiparticles in graphene in the presence of elastic deformations. In some of the previous works on this subject it was supposed that such an emergent geometry is Riemannian . Later, the derivation of the space - dependent Fermi velocity in the presence of strain was undertaken . Here we present the direct derivation of the emergent geometry. In our approach the expansion of the effective hamiltonian is performed near to the ”floating” Fermi point (where the hamiltonian vanishes in the limit of homogenious elastic deformations). We demonstrate, that the emergent geometry is described by the zweibein. Also, the emergent gauge field appears (its expression through strain coincides with the one derived previously ) while the spin connection does not appear in the approximation linear in elastic deformations
The authors kindly acknowledge discussions of emergent gravity with D.I.Diakonov, and useful correspondence with M.Vozmediano. This work was partly supported by RFBR grant 11-02-01227, by the Federal Special-Purpose Programme ’Human Capital’ of the Russian Ministry of Science and Education. GEV acknowledges a financial support of the Academy of Finland and its COE program, and the EU FP7 program (228464 Microkelvin).
- It is worth mentioning that there is the conceptually different approach to emergent gravity, in which the bilinear combinations or are identified with the vielbein for some tensorial operators . Corresponding constructions were considered in the context of relativistic field theory in [7, 8] and in the context of condensed matter physics in .
- Since the spin connection is not forbidden by symmetry, it should appear in the next approximations.
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