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Hamiltonian_Zoo.lyx
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Hamiltonian_Zoo.lyx
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#LyX 2.3 created this file. For more info see http://www.lyx.org/
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\begin_body
\begin_layout Title
Hamiltonian Zoo
\end_layout
\begin_layout Author
\bar under
Nike Dattani
\end_layout
\begin_layout Author Email
n.dattani@cfa.harvard.edu
\end_layout
\begin_layout Affiliation
Harvard-Smithsonian Center for Astrophysics
\end_layout
\begin_layout Author
\bar under
Tim Li
\end_layout
\begin_layout Affiliation
HPQC Labs
\end_layout
\begin_layout Author
\bar under
Joshua Tyler Cantin
\end_layout
\begin_layout Affiliation
University of British Columbia
\end_layout
\begin_layout Author
and
\bar under
Whoever Else Contributes
\end_layout
\begin_layout Affiliation
No Matter Where They're From
\end_layout
\begin_layout Date
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
today
\end_layout
\end_inset
\end_layout
\begin_layout Section
Model Hamiltonians
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
Madelung model? Madelung constant? Zoo of constants?
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Coulomb
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Coulomb}}=\frac{q_{1}q_{2}}{4\pi\epsilon_{0}r}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsection
Electronic Structure
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
https://doi.org/10.1146/annurev.physchem.53.082201.124330 See Eq 36 (coriolis,
centrifugal, anharmonic Hamiltonians)
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Molecular Hamiltonian
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm BO}}=-\sum_{A}\frac{1}{2M_{A}}\nabla_{A}^{2}-\sum_{i}\frac{1}{2}\nabla_{i}^{2}-\sum_{iA}\frac{Z_{A}}{r_{iA}}+\sum_{i>j}\frac{1}{r_{ij}}+\sum_{B>A}\frac{Z_{A}Z_{B}}{R_{AB}}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Born-Oppenheimer Model
\end_layout
\begin_layout Subsubsection
Watson Model
\end_layout
\begin_layout Subsection
Fine and Hyperfine Structure
\end_layout
\begin_layout Subsubsection
Stark
\end_layout
\begin_layout Subsubsection
Zeeman
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Zeeman}}=-\frac{\mu_{B}\left(g_{l}\vec{L}+g_{s}\vec{S}\right)}{\hbar}\cdot\vec{B}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Complete magnetic dipole interaction
\end_layout
\begin_layout Standard
\size scriptsize
\begin_inset Formula
\begin{align}
H_{{\rm Magnetic}\,\text{Dipole}} & =\frac{g_{I}\mu_{N}\mu_{B}\mu_{0}}{2\pi}\left(\frac{1}{L_{z}}\sum_{i}\frac{\hat{l}_{zi}}{r_{i}^{3}}\vec{I}\cdot\vec{L}+\frac{2g_{s}}{S_{z}}\sum_{i}\frac{\hat{s}_{zi}}{r_{i}^{3}}\left(3\left(\vec{I}\cdot\hat{r}\right)\left(\vec{S}\cdot\hat{r}\right)-\vec{I}\cdot\vec{S}\right)+\frac{g_{s}}{3S_{z}}\sum_{i}\hat{s}_{zi}\delta^{3}\left(\vec{r}_{i}\right)\vec{I}\cdot\vec{S}\right)
\end{align}
\end_inset
\end_layout
\begin_layout Subsubsection
Frosch-Foley
\end_layout
\begin_layout Standard
\color black
\begin_inset Formula
\begin{equation}
H_{{\rm Frosch-Foley,1}}=\frac{e\hbar}{2mc}\frac{2\mathbf{\mu}\cdot\mathbf{L}}{r_{1}^{3}}+\frac{{\rm i}2\mu_{0}}{\hbar}\mathbf{S}\cdot[\mathbf{p}\times\mathbf{A}]+\frac{2e^{2}\hbar}{4m^{2}c^{3}}\mathbf{S}\cdot[\mathbf{\epsilon^{\prime}\times\mathbf{A}]}.\label{eq:eq2.6of1952FroschFoley}
\end{equation}
\end_inset
\begin_inset Newline newline
\end_inset
\begin_inset Formula
\begin{equation}
H_{{\rm Frosch-Foley,2}}=aI_{z}L_{z}+b\vec{I}\cdot\vec{S}+cI_{z}S_{z},\label{eq:Eq.4.7of1952FroschFoley}
\end{equation}
\end_inset
\color inherit
where
\end_layout
\begin_layout Standard
\color black
\begin_inset Formula
\begin{eqnarray}
a & \equiv & 2g_{I}\mu_{0}\mu_{N}\sum_{i}\Big\langle\frac{1}{r_{i}^{3}}\Bigr\rangle,\\
b & \equiv & 2g_{I}\mu_{0}\mu_{N}\sum_{i}\Big\langle\frac{2e^{2}mc^{2}\mathcal{E}_{r}}{\left(E+eV+mc^{2}\right)^{2}}\frac{\left(\nicefrac{1}{2}\sin^{2}\chi+\cos^{2}\chi\right)}{r_{i}^{2}}-\frac{1}{2}\left(\frac{3\cos^{2}\chi-1}{r_{i}^{3}}\right)\Bigr\rangle,\,{\rm and}\\
c & \equiv & 2g_{I}\mu_{0}\mu_{N}\sum_{i}\Big\langle\frac{2e^{2}mc^{2}\mathcal{E}_{r}}{\left(E+eV+mc^{2}\right)^{2}}\frac{\left(\nicefrac{1}{2}\sin^{2}\chi-\cos^{2}\chi\right)}{r_{i}^{2}}+\frac{3}{2}\left(\frac{3\cos^{2}\chi-1}{r_{i}^{3}}\right)\Bigr\rangle.
\end{eqnarray}
\end_inset
\end_layout
\begin_layout Subsubsection
Spin-orbit
\end_layout
\begin_layout Subsubsection
Orbit-orbit
\end_layout
\begin_layout Subsubsection
Spin-other-orbit (Gaunt)
\end_layout
\begin_layout Subsubsection
Fermi Contact
\end_layout
\begin_layout Subsection
Special Relativity
\end_layout
\begin_layout Subsubsection
Dirac
\end_layout
\begin_layout Standard
\color black
\begin_inset Formula
\begin{equation}
H=\beta mc^{2}-eV+c\mathbf{\alpha}\cdot(\mathbf{p}-\frac{e}{c}\mathbf{A}).\label{eq:diracHamiltonian}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Klein-Gordon
\end_layout
\begin_layout Subsubsection
Darwin (1
\begin_inset Formula $e^{-}$
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Darwin-1}}=\alpha_{{\rm FS}}^{2}\frac{\pi Z}{2}\sum_{i}\delta\left(\vec{r}_{i}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Darwin (2
\begin_inset Formula $e^{-}$
\end_inset
)
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Darwin-2}}=\alpha_{{\rm FS}}^{2}\pi\sum_{i<j}\delta\left(\vec{r}_{ij}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Mass Velocity
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Mass-Velocity}}=-\alpha_{{\rm FS}}^{2}\frac{1}{8}\sum_{i}\nabla_{i}^{4}.
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Breit-Pauli
\end_layout
\begin_layout Subsubsection
Dirac-Coulomb-Breit
\end_layout
\begin_layout Subsection
Quantum Electrodynamics
\end_layout
\begin_layout Subsubsection
Aracki-Sucher
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Aracki-Sucher}}=-\alpha_{{\rm FS}}^{3}\frac{7}{6\pi}\sum_{i>j}\lim_{a\rightarrow0}\left(\frac{\theta\left(r_{ij}-a\right)}{r_{ij}^{3}}+4\pi\left(\gamma+\ln a\right)\delta(\vec{r}_{ij})\right)
\end{equation}
\end_inset
\begin_inset Note Note
status open
\begin_layout Plain Layout
where
\begin_inset Formula $\gamma$
\end_inset
is the Euler-Mascheroni constant, and
\begin_inset Formula $\theta(\cdot)$
\end_inset
is the Heaviside step function.
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
One-Loop
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm 1-loop}}=\alpha_{{\rm FS}}^{4}\pi Z^{2}\left(\frac{427}{96}-{\rm ln}2\right)\sum_{i}\delta\left(\vec{r}_{i}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Schwinger-Dyson
\end_layout
\begin_layout Subsection
Nuclear Motion
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H_{{\rm Nuc}}=-\frac{\hbar^{2}}{2}\sum_{i=1}^{N}\sum_{\alpha=1}^{3}\frac{1}{M_{i}}\frac{\partial^{2}}{\partial R_{i\alpha}^{2}}+V(\text{\ensuremath{\boldsymbol{R}_{1},\ldots,\boldsymbol{R}_{N})}}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Rigid Rotor Harmonic Oscillator
\end_layout
\begin_layout Subsubsection
Centrifugal distortion
\end_layout
\begin_layout Subsubsection
Centrifugal distortion
\end_layout
\begin_layout Subsection
Long-range Interactions
\end_layout
\begin_layout Subsubsection
van der Waals
\end_layout
\begin_layout Subsubsection
Lenard-Jones
\end_layout
\begin_layout Subsubsection
Casimir-Polder
\end_layout
\begin_layout Subsubsection
Resonance dipole-dipole
\end_layout
\begin_layout Subsubsection
Meath
\end_layout
\begin_layout Subsubsection
Axilrod-Teller
\end_layout
\begin_layout Subsection
Hubbard Models
\end_layout
\begin_layout Subsubsection
Non-interacting Hubbard model
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H=\sum_{\boldsymbol{{\rm k}}}\sum_{\sigma}\left(\varepsilon_{\boldsymbol{{\rm k}}}-\mu\right)n_{\boldsymbol{{\rm k}},\sigma}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
1D Hydrogen chain Hubbard model
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H=-t\sum_{\langle i,j\rangle}\sum_{\sigma}\left(\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{j,\sigma}+\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{i,\sigma}\right)+U\sum_{i=1}^{N}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow},\,\,\,t>0,U>0
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
1D Hydrogen chain Hubbard model with
\begin_inset Formula $\mu$
\end_inset
term
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H=-t\sum_{\langle i,j\rangle}\sum_{\sigma}\left(\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{j,\sigma}+\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{i,\sigma}\right)+U\sum_{i=1}^{N}\hat{n}_{i\uparrow}\hat{n}_{i\downarrow}-\mu\sum_{i}\left(n_{i\uparrow}+n_{i\downarrow}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
With particle-hole symmetry
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H=-t\sum_{\langle i,j\rangle}\sum_{\sigma}\left(\hat{c}_{i,\sigma}^{\dagger}\hat{c}_{j,\sigma}+\hat{c}_{j,\sigma}^{\dagger}\hat{c}_{i,\sigma}\right)+U\sum_{i=1}^{N}\left(\hat{n}_{i\uparrow}-\frac{1}{2}\right)\left(\hat{n}_{i\downarrow}-\frac{1}{2}\right)-\mu\sum_{i}\left(n_{i\uparrow}+n_{i\downarrow}\right)
\end{equation}
\end_inset
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
is completely equivalent to the original HH (https://www.cond-mat.de/events/correl
16/manuscripts/scalettar.pdf)
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Bose-Hubbard (Boson Hubbard)
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H=-t\sum_{\langle i,j\rangle}\hat{b}_{i}^{\dagger}\hat{b}_{j}+\frac{1}{2}U\sum_{i}\hat{n}_{i}\left(\hat{n}_{i}-1\right)-\mu\sum_{i}\hat{n}_{i}
\end{equation}
\end_inset
\end_layout
\begin_layout Subsubsection
Fermi-Hubbard (Fermi Hubbard)
\end_layout
\begin_layout Subsubsection
Bose–Fermi–Hubbard
\end_layout
\begin_layout Subsubsection
Jaynes-Cummings-Hubbard
\end_layout
\begin_layout Subsubsection
Tavis-Cummings-Hubbard
\end_layout
\begin_layout Subsubsection
Periodic Anderson Model
\end_layout
\begin_layout Standard
\begin_inset ERT
status open
\begin_layout Plain Layout
\backslash
begin{align}
\end_layout
\begin_layout Plain Layout
H=&-t
\backslash
sum_{
\backslash
langle i,j
\backslash
rangle}
\backslash
sum_{
\backslash
sigma}
\backslash
left(
\backslash
hat{c}_{i,
\backslash
sigma}^{
\backslash
dagger}
\backslash
hat{c}_{j,
\backslash
sigma}+
\backslash
hat{c}_{j,
\backslash
sigma}^{
\backslash
dagger}
\backslash
hat{c}_{i,
\backslash
sigma}
\backslash
right)+V
\backslash
sum_{
\backslash
langle i,j
\backslash
rangle}
\backslash
sum_{
\backslash
sigma}
\backslash
left(c_{j
\backslash
sigma}^{
\backslash
dagger}d_{i
\backslash
sigma}+d_{i
\backslash
sigma}^{
\backslash
dagger}c_{j
\backslash
sigma}
\backslash
right)
\backslash
nonumber
\backslash
\backslash
\end_layout
\begin_layout Plain Layout
&+U
\backslash
sum_{i=1}^{N}
\backslash
left(
\backslash
hat{n}_{di
\backslash
uparrow}-
\backslash
frac{1}{2}
\backslash
right)
\backslash
left(
\backslash
hat{n}_{di
\backslash
downarrow}-
\backslash
frac{1}{2}
\backslash
right)-
\backslash
mu
\backslash
sum_{i}
\backslash
left(n_{di
\backslash
uparrow}+n_{di
\backslash
downarrow}+n_{ci
\backslash
uparrow}+n_{ci
\backslash
downarrow}
\backslash
right),
\end_layout
\begin_layout Plain Layout
\backslash
end{align}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
where
\begin_inset Formula $d_{i\sigma}$
\end_inset
is the annihilation operator for an electron in the localized band at site
\begin_inset Formula $i$
\end_inset
,
\begin_inset Formula $c_{i\sigma}$
\end_inset
is the annihilation operator for an electron in the conduction band at
site i,
\begin_inset Formula $V$
\end_inset
is termed the hybridization and is the transition amplitude of an electron
moving from the localized band at site
\begin_inset Formula $i$
\end_inset
to the conduction band at site
\begin_inset Formula $i$
\end_inset
or vice versa.
\end_layout
\begin_layout Standard
\begin_inset Note Note
status open
\begin_layout Plain Layout
Should include diagram of geometry (Fig 5 of Ref [1]).
This Hami is Eqn.
24 of Ref [1]
\end_layout
\begin_layout Plain Layout
Refs:
\end_layout
\begin_layout Plain Layout
[1] https://www.cond-mat.de/events/correl16/manuscripts/scalettar.pdf
\end_layout
\begin_layout Plain Layout
Contributors:
\end_layout
\begin_layout Plain Layout
JTC
\end_layout
\end_inset
\end_layout
\begin_layout Subsection
Disorder Models
\end_layout
\begin_layout Subsubsection
Tight-binding Model with Onsite Disorder
\end_layout
\begin_layout Standard
\begin_inset Formula
\begin{equation}
H=\sum_{i}\omega_{i}\hat{c}_{i}^{\dagger}\hat{c}_{i}+t\sum_{\langle i,j\rangle}\hat{c}_{i}^{\dagger}\hat{c}_{j},
\end{equation}
\end_inset
where
\begin_inset Formula $\omega_{i}$
\end_inset
is drawn from a unifrom distribution of width
\begin_inset Formula $W$
\end_inset
, the disorder strength, and the angular brackets indicate a sum over nearest-ne
ighbours only.
\end_layout
\begin_layout Standard
\begin_inset Note Comment
status open
\begin_layout Plain Layout
Need to generalize to arbitrary distributions.
\end_layout
\begin_layout Plain Layout
Refs:
\end_layout
\begin_layout Plain Layout
L.
J.
Root and J.
L.
Skinner J.
Chem.
Phys.
89, 3279-3284 (1988).
DOI:10.1063/1.454933
\end_layout
\begin_layout Plain Layout
Contributors:
\end_layout
\begin_layout Plain Layout
JTC
\end_layout
\end_inset
\end_layout
\begin_layout Subsubsection
Tight-binding Model with Continuous Off-diagonal Disorder
\end_layout
\begin_layout Subsubsection
Tight-binding Model with Binary Off-diagonal Disorder
\end_layout
\begin_layout Subsubsection
Long-range Disorder Models
\end_layout
\begin_layout Subsection
Mean-field Hamiltonians
\end_layout