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Let the Hamiltonian of a system on a discrete lattice be given by

$$ \mathcal{H} = \gamma \sum_\vec{x} c^\dagger_\vec{x}c^\vphantom{\dagger}_{\vec{x}+\vec{y}} + \text{h.c.}, $$

where $\gamma$ is some prefactor and $c^{(\dagger)}_\vec{x}$ is the annihilation (creation) operator for a particle at position $\vec{x}$. Now, I would like to add a delta potential to this Hamiltonian. In the ordinary Schrödinger Equation this term would just be $V(\vec{x}) = V_0 \delta(\vec{x})$ with some constant $V_0$. Is there a way to express this potential in terms of creation and annihilation operators?

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  • $\begingroup$ OK, can you represent this delta-potential in terms of creation and annihilation operators in the continuum? (You might, e.g. consider the narrow gaussian limit of the delta function.) How do singular functions present on a discrete lattice? $\endgroup$ – Cosmas Zachos Jul 19 '18 at 14:41
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I'll direct you to this question I once asked. In Quantum Mechanics a delta function potential $\delta(\mathbf{x}_{j} - \mathbf{x}_{k})$ would translate to a scalar quantum field theory with a quartic $\lambda \phi^{4}$ interaction - it's a two-body repulsive contact. I think in condensed matter language, your Hamiltonian will develop an interaction term that will have some combinations of two $c_{\mathbf{x}}^{\dagger}$ operators and two $c_{\mathbf{x}}$ operators (I want to guess the overall interaction operator will have the normal-ordered form $ \delta_{\mathbf{w}+\mathbf{x},\mathbf{y}+\mathbf{z}} c_{\mathbf{w}}^{\dagger}c_{\mathbf{x}}^{\dagger}c_{\mathbf{y}}c_{\mathbf{z}}$, although I'm not sure about this)

As mentioned in the answer by @Sean_E._Lake in the posted question, Brian Hatfield's "Quantum Field Theory of Point Particles and Strings" has a discussion about this.

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  • $\begingroup$ It is puzzling that a delta function interaction should turn into a fourth order term in the Lagrangian. Please explain. $\endgroup$ – DanielSank Jul 19 '18 at 23:37
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You can look up for the second quantized form of dirac delta in chapter 3 of many body book by Piers Coleman. In page 48, equation number 3.34 it proves that dirac delta function is equal to commutation/unticommutation relation between creation and annihilation operators in real space basis.

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