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When studying quantum systems, we often use "hopping" Hamiltonians to represent the system energy. For example, we might represent a 1D ring of N sites with a single particle as:

$H = -t\left[\sum_{i}^{N-1}(\sigma^+_i\sigma^-_{i+1}+\sigma^-_i\sigma^+_{i+1}+)+\sigma^+_L\sigma^-_1+\sigma^-_L\sigma^+_1\right]$

Or a graphene lattice as:

$H=-t\sum_{i, j}\left(a^\dagger_ib_j+b_j^\dagger a_i\right)$

I'm struggling to understand how a series of hopping operators can produce the energy of a system. Any insights will be greatly appreciated!

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The other answers relating hopping to kinetic energy are good but I want to emphasize why hopping Hamiltonians are physically meaningful to begin with and can actually be exact in principle (though not in practice).

Tight binding models with electron hopping have their physical origin in Wannier functions. Wannier functions $w_{n\ell}(r)$ are localized around atomic sites in a crystal and form an orthonormal basis of the Hilbert space. So they are exactly the basis which allows us to consider hopping between localized orthogonal states (i.e. we can write terms in a Hamiltonian that look like $|w_{n\ell}\rangle\langle w_{n'\ell'}|=c_{n\ell}^\dagger c_{n'\ell'}$)


Wannier Functions: A Localized Basis

To give more detail, recall that the one-electron wavefunction in a lattice is a Bloch function $$\psi_{nk}(r)=e^{ik\cdot r}u_{nk}(r)$$ where $u_{nk}(r)$ has the periodicity of the lattice (i.e. we have $u_{nk}(r-R_\ell)=u_{nk}(r)$ for any lattice vector $R_\ell$). We typically impose the periodic gauge condition so that for any reciprocal lattice vector $G$, we have $u_{n,k+G}(r)=e^{-iG\cdot r}u_{nk}(r)$. This ensures that our Bloch functions are periodic in $k$-space: $ \psi_{n,k+G}(r)=\psi_{nk}(r) $. As a consequence of this periodicity, we can expand $\psi_{nk}(r)$ in a Fourier series: $$ \psi_{nk}(r) = \sum_{\ell} e^{ik\cdot R_\ell}w_{n\ell}(r) $$ The functions $w_{n\ell}(r)$ are the Wannier functions. Inverting the above Fourier series, we can define them explicitly by $$ w_{n\ell}(r) = \frac{\Omega}{(2\pi)^d}\int_{BZ}d^dk\:e^{-ik\cdot R_\ell}\psi_{nk}(r) $$ where $\Omega$ is the unit cell volume. Note that by definition of Bloch functions we can write $w_{n\ell}(r)=w_n(r-R_\ell)$. Their utility comes from two important properties:

  • They are orthonormal: $\displaystyle\langle w_{n\ell}| w_{n'\ell'}\rangle = \frac{(2\pi)^d}{\Omega}\int d^dr\:w_{n\ell}^*(r) w_{n'\ell'}(r) = \delta_{nn'}\delta_{\ell\ell'}$

  • If $\psi_{nk}$ is smooth in $k$-space, we can take* $w_{n\ell}$ to be exponentially localized around an atomic site in the unit cell at $R_\ell$

The point is: Wannier functions provide an orthonormal basis of localized functions!


*More generally we have to consider generalized Wannier functions which involve unitarily mixed Bloch functions for an isolated set of bands. But the point remains the same

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I think the hopping term can be understood as the kinetic energy of the system. For example in the Hubbard model, there is a term $-t\sum_{<ij>\sigma}(c^{\dagger}_{i\sigma}c_{j\sigma}+c^{\dagger}_{j\sigma}c_{i\sigma})$ in which each term means destructing a fermion with spin $\sigma$ on site $i(j)$ and creating one on site $j(i)$. This means the fermion moves from one site to the other site. I think it's similar for other examples in your question. Indeed, this term can be understood as the second-quantized form of the kinetic energy operator.

For example, we can consider a single particle basis $\{|i>\}$ and the kinetic energy operator can be written as $t=p^2/{2m}$, then in this basis $t=\sum_{i,j}t_{ij}|i><j|$ where $t_{ij}$ is the matrix element $<i|t|j>$. The total KE $T=\sum_{ij}t_{ij}\sum_{\alpha=1}^{N}|i>_{\alpha}<j|_{\alpha}$(N is the total number of particles). In second quantization, the total KE has the form $T=\sum_{ij}t_{ij}c^{\dagger}_ic_j$

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The inspiration for the hopping Hamiltonians are the tight-binding Hamiltonians in condensed matter physics (hint - this is where one should look for derivations). One typically solves such a Hamiltonian in terms of plane waves, expands the dispersion near the band bottom (effective mass approximation) and thus obtains the usual kinetic term $\hbar^2k^2/(2m)$ (although one may prefer expanding the dispersion in other points - typically near the Fermi level - e.g., in graphene or one treating Luttinger liquids.)

The Hamiltonian is usually called hopping rather than tight-binding in the contexts where the underlying lattice is fictitious rather than real, i.e., where the form of the Hamiltonian is a matter of computational convenience rather than an approximation to real lattice.

Where the real difficulty lies is that such Hamiltonians cannot be rigorously obtained from the expression for the true energy/potential. One typically considers sites as isolated atomic orbitals and postulates ad-hoc coupling between them. The coupling parameters are then estimated experimentally or using more sophisticated band structure calculation techniques.

Another term used for this type of Hamiltonians is transfer Hamiltonian - typically in a context where discrete states are coupled via tunneling to continuous energy bands.

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