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Effective Hamiltonians modeling many-layered materials are often tuned using some sort of bias voltage. For instance, in a $4\times 4$ Hamiltonian matrix to describe biased bilayer graphene using some basis, one often sees $diag(-U,-U,+U,+U)$ added. By doing this, the (1,1) matrix element has $-U$ added to it, the (3,3) matrix element has $+U$, and so on. Tuning $U$ can cause interesting phenomena, such as bandgap closing, etc. The physical justification for using $U$ requires the presence of more than one layer: $U$ can act as a bias voltage, for example, that modifies the coupling between the two layers and the four energy bands.

Now, my question is: What is the physical justification/consequence of using such a parameter in a monolayer model? I was reading Ref. [1], which gives a "Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides". Can I use a similar parameter $U$ - such as $diag(-U,+U,+U)$ or some other variant - in this case as well? Numerically, doing this will modify the band structure, but what does this correspond to physically? We don't have multiple layers in this case, and so the idea of the bias voltages tuning interlayer properties is not applicable here. So, I am trying to understand what exactly altering the diagonal means physically (what changes in the model besides the diagonal? is this a known experimental procedure?).

I am relatively new to this field and I still do not understand the orbital bases used in Ref. [1] too well. However, based on Eqs. 5,8 and 9, it seems as if there is a constant material-dependent parameter added to the diagonal already $diag(\epsilon_1,\epsilon_2,\epsilon_2)$. But I am not sure what these $\epsilon_i$ correspond to physically, or mean. It feels as if these parameters can move the high-symmetry points around because I think they might be some sort of hopping, but this is just a guess.

Any intuition will be appreciated!

EDIT: After months, the comment+answer makes sense to me now.

[1] https://arxiv.org/abs/1305.6089

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    $\begingroup$ In essence, you are considering tailoring your low-energy effective Hamiltonian. Possibly the adding diagonal matrix can be understood as strain effects. For example, seeing chapter 14.5 in this book: amazon.com/2D-Materials-Properties-Phaedon-Avouris/dp/… $\endgroup$
    – Jack
    Jan 19 at 13:56
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I will assume that we are working with a Hamiltonian in some tight-binding-like position basis (where each row/column in $H$ is a localized orbital).

Essentially the constant you are adding acts as a site-dependent potential. If it's uniform, it has no real effect. You can think of adding a bunch of different "shapes" to the potential to have different effects. For example, if you add a $-U$ on one site, that's effectively attractive. If you make a pattern like $+U,-U,+U,-U...$, you can predispose the system towards some sort of pattern.

Physically, these local $U$'s could correspond to any number of things. Defects or substitutions in the lattice could create isolated local potentials. Interactions with a substrate could produce potentials in some pattern. An external electric field could produce a pattern of $U$'s that looks like a ramp.

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