\chapter{Introduction}
\section{Birth of a quantum era}
The entire human civilization was brought to believe that nature were certain and deterministic prior the 20$^\mathrm{th}$ century.
This belief is in fact the principle behind Newtonian classical mechanics \cite{classical_mechanics}, which lays the foundation of the tallest architecture and the fastest bullet train on Earth.
Further evidence for the belief came in 1861-2 when James Clerk Maxwell unified classical electromagnetism under his four Maxwell equations \cite{classical_electromagnetism}.
At that time, science and technology could all be explained by classical mechanics.
Its extraordinary beauty had in fact blinded many scientists up to the mid-twentieth century, including the mighty Albert Einstein who once made the following comment.
%
\begin{quote}
``God does not play dice with the universe.''
\quotation{--- Albert Einstein, {\it{The Born-Einstein Letters 1916-55}}}
\\
\end{quote}
%
\noindent The beginning of the 20$^\mathrm{th}$ century marked a revolutionary period among the scientific community, where a series of microscopic experiments\footnote{Classical mechanics became inadequate in explaining a series of microscopic experiments. Some examples include the black-body radiation, the photoelectric effect, the Compton effect, the discretization of atomic energy spectra, and the quantization of angular momentum. \cite{quantum_mechanics1}}, one after another, failed to comply by the law of classical mechanics. \cite{quantum_mechanics1}
Evidence provided by these experiments slowly convinced some scientists at that time that all microscopic particles are waves.
In 1926, Erwin Schr\"dingier formulated the wave interpretation of quantum mechanics by the proposal of his Schr\"odingier equation \cite{schrodinger1}
%
\begin{equation}
i\hbar \frac{\partial}{\partial t} \psi ({\bf{x}},t) = \left( - \frac{\hbar^2}{2m} \nabla^2 + V({\bf{x}}, t) \right) \psi ({\bf{x}},t)
\end{equation}
%
that dictates the particle wavefunction $\psi ({\bf{x}},t)$ in the presence of an external potential $V ({\bf{x}},t)$.
In the same year, Max Born proposed a probabilistic interpretation for $| \psi ({\bf{x}},t)|^2$ \cite{born1}, such that the underlying microscopic particle could never be located with certainty at any spatial position in general.
Surprisingly, nature reveals itself to be stochastic and unpredictable at the microscopic level, and all subsequent experiments prove quantum mechanics to be fundamentally correct.
Other remarkable scientists, who had greatly contributed to the development of quantum mechanics, include Werner Heisenberg, Paul Dirac, and Richard Feynmann.
%
\begin{quote}
`` ... nobody understands quantum mechanics.''
\quotation{--- Richard Feynmann, {\it{The Character of Physical Law 1965}}}
\\
\end{quote}
%
\noindent Quantum mechanics exhibits many peculiar phenomenons that are not easily understood.
Microscopically, the wave nature of particles makes them identical or indistinguishable, leading the many-body wavefunction to be either symmetric or antisymmetric under particle permutation \cite{quantum_mechanics}.
As a consequence, identical particles could either behave like bosons exhibiting symmetric statistics \cite{bose}, or fermions exhibiting antisymmetric statistics \cite{fermi,dirac}.
In fact, bosons and fermions are the only two types of particles that exist in nature.
Their statistical effects give rise to many new phases of matter, like superfluidity and antiferromagnetism \cite{statistical_mechanics}.
After a century, quantum mechanics has become the fundamental pillar of all modern technology today.
\section{Quantum engineering of materials}
The modern description of solids came in 1928 when Felix Bloch invented the electronic band theory for crystals
based on lattice translational symmetries in quantum mechanics \cite{bloch0}.
A remarkable consequence is the classification of semiconductor materials characterized by a finite band gap of the order of a few eV, distinct from either a conductor or an insultor \cite{physics_of_semiconductor_devices}.
The historical moment came in 1947 when Bardeen, Shockley, and Brattain invented the first germanium-based semiconductor point contact npn-transistor that marked the beginning of the robust semiconductor industry \cite{transistor}.
Today, we are in a nanotechnology era, whereby modern commercial transistors size about 45 nm in size, and modern graphic cards have over 3 billion transistors beautifully engineered as integrated circuits \cite{how_transistors_work}.
Excitingly, researchers have been successful in creating new exotic transistors which are based on carbon nanotubes in 1998 \cite{carbon_nanotube}, and biomolecules like DNA, RNA, proteins in 2013 \cite{biotransistor}.
The theoretical description of these conventional materials is completely known from Kohn-Sham density functional theory \cite{KS}, including higher-order diagrammatic corrections \cite{GW}.
Its success enables scientists to make accurate predictions of new semiconductor and biological materials.
\\ \\
\noindent An entirely different class of strongly correlated materials exists in nature which yields fascinatingly captivating properties beyond those merely from conventional materials.
These materials are derived from transition metals or/ and rare earth metals, having incompletely filled d- or f- electron orbitals respectively with narrow energy bands \cite{fulde1}.
Typical examples include 1) ferromagnetic materials such as nickel Ni, cobalt Co, and iron Fe; 2) ferrimagnetic materials such as hematite Fe$_2$O$_3$, and yttrium iron garnet Y$_3$Fe$_2$(FeO$_4$)$_3$; and 3) antiferromagnetic materials such as chromium Cr, and nickel oxide NiO. \cite{fm1_wiki}
Of particular engineering interest lies probably in the high temperature cuprate superconductors that exhibit zero electrical resistivity when cooled below some particular transition temperature T$_c$ \cite{fulde1}.
Since its first discovery by Bednorz and M\"uller in 1986 \cite{bednorz}, subsequent discoveries have revealed unconventional superconductivity up to a temperature of 133K in mercury-barium-calcium-based cuprates HgBa$_2$Ca$_2$Cu$_3$O$_8$ \cite{schilling}.
Unconventional superconductivity has also been recently found in iron pnictides \cite{kamihara} which is another class of strongly correlated materials.
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\noindent The physics of strongly correlated materials remains theoretically uncertain after many decades since their discoveries.
The inadequency of the single-electron picture makes density functional theory alone unsuitable for studying strongly correlated materials qualitatively.
Instead, physicists turn to toy models and hope that they could capture most of the essential physics of strongly correlated materials.
Unfortuantely, the Hubbard model \cite{hubbard}
%
\begin{equation}
\label{hubbard_model}
\hat{H} = -t \sum_{\langle i, j \rangle , \sigma} \hat{c}^\dag_{i\sigma} \hat{c}_{j\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow} \,\, ,
\end{equation}
%
having been believed to capture unconventional superconductivity in doped cuprates for instance, turns out to be too challenging for the most powerful machines to be solved numerical exactly via Quantum Monte Carlo methods due to the notorious sign problem till today.
As a consequence, there exists no quantitative phase diagram for the Hubbard model to be directly compared with experiments, therefore seriously handicapping our understanding towards these materials, needless to mention further about prediction and engineering.
Recently, optical lattices have been proposed to be alternative quantum emulators \cite{ole}, in hoping that they could solve such quantum models and bring greater insights to strongly correlated materials.
\section{Optical lattice as a quantum emulator}
\subsection{Optical lattice}
\begin{figure}[htp]
\centering
\includegraphics[width=7cm]{./figures/optical-lattice-m-yamashita.jpg}
\caption{Atomic gases in a 2D optical lattice. Counter-propagating laser light form a standing wave, or an optical lattice, due to wave interference effect. The lattice intensity is $V_0$, the wavelength $\lambda$ and thus spacing $d=\lambda/2$. {\it{This figure is modified from its original version, courtesy of M. Yamashita.}}\cite{yamashita1}.}
\label{bosonsol_optical_lattice_experiment}
\end{figure}
\noindent The physics of wave interference, generated by counter-propagating laser light, dictates the presence of a standing wave. \cite{quantum_optics}
The environment of standing waves is called an optical lattice with intensity $V_0$ and spacing $d=\frac{\lambda}{2}$, which is conceptually illustrated in figure \ref{bosonsol_optical_lattice_experiment}.
For instance, the potential of an isotropic 3D optical lattice reads
%
\begin{equation}
V({\bf{x}}) = \sum_{x^i=x,y,z} V_0 \, \sin^2 (k x^i)
\end{equation}
%
with wavevector $k=\frac{2\pi}{\lambda}=\frac{\pi}{d}$ \cite{greiner1}.
In addition, quantum particles in an optical lattice have to be confined by a trapping potential $V_T({\bf{x}})$, otherwise they would fly apart.
In the Greiner experiment, the confinement has been realised with a tight Gaussian laser focus, as schematically illustrated by figure \ref{gaussian_waist}.
Here, the 3D optical lattice is radially confined by the trapping envelope
%
\begin{equation}
\label{eq_trapping_envelope}
V_0 \exp \left( -\frac{2r^2}{w_0^2} \right)
\end{equation}
%
of waist (or $1/e^2$-radius) $w_0$ \cite{greiner1}.
%
\begin{figure}[h]
\centering
\includegraphics[width=6cm]{./figures/gaussian-waist-optical-potential.jpg}
\caption{Schematic illustration of the Gaussian confinement of bosons.\cite{greiner1} Top: A pair of counter-propagating laser beams interfere in the $x$-direction to form an enveloped standing wave of intensity $V_0$, with trapping envelope of waist $w_0$. Bottom: Near the centre, the optical potential is almost uniformly periodic, i.e. $V_0 \sin^2(kx)$, with an additional parabolic trapping term $\frac{V_0}{w_0^2}x^2$.}
\label{gaussian_waist}
\end{figure}
%
Particles in current experiments are trapped in the vicinity close to the center of the gaussian trap, therefore effectively confined by the first order parabolic trapping term.
Taking into account also all other parabolic trapping, the trapping potential can be effectively written as
%
\begin{equation}
V_T({\bf{x}}) = V_T {\bf{x}}^2
\end{equation}
%
where $V_T$ is the strength of the parabolic trapping.
\subsection{Single particle in an optical lattice}
To better conceptualise the physics, let us hypothetically consider a single particle of mass {\it{m}} in an optical lattice.
Within non-relativisitic regime, its quantum mechanical nature is captured by the Schr\"dingier equation in reciprocal space $\hat{H}_{\bf{k}} u_{\bf{k}} = E_{\bf{k}} u_{\bf{k}}$. \cite{solid_state}
Due to the periodicity of the optical lattice, the Bloch Hamiltonian is given by
%
\begin{equation}
\hat{H} = \frac{\hbar^2}{2m} \left( -i \nabla + {\bf{k}} \right)^2 + V({\bf{x}}) \,\, ,
\end{equation}
%
where the {\bf{k}}-points in reciprocal space take values depending individually on different lattice geometries.
Figure \ref{bosonsol_single_particle_energy_band_structure} illustrates the energy band structure $E_{\bf{k}}$ of an optical lattice, where one could clearly observe the opening of a energy band gap $\Delta$ as the shallow lattice (small $V_0$) becomes deep (large $V_0$).
%
\begin{figure}[htp]
\centering
\includegraphics[width=14cm]{./figures/single-particle-energy-band-structure.jpg}
\caption{Single particle energy band structure $E_{\bf{k}}$ of a simple cubic lattice. Energy band gap $\Delta$ opens up with increasing lattice intensity $V_0$, i.e. from $1.0 E_R$ (left), to $2.2 E_R$ (centre), and finally $3.4 E_R$ (right).
Here, the high-symmetry {\bf{k}}-points of a simple cubic lattice are namely $\Gamma(0,0,0)$, $X(\frac{\pi}{d},0,0)$, $M(\frac{\pi}{d},\frac{\pi}{d},0)$ and $R(\frac{\pi}{d},\frac{\pi}{d},\frac{\pi}{d})$. In addition, energy is expressed in units of recoil energy $E_R=\frac{h^2}{2m\lambda^2}$. Typically, for a Rb-87 particle in an optical lattice with $\lambda=800nm$, the recoil energy is $E_R = 172 nK$.}
\label{bosonsol_single_particle_energy_band_structure}
\end{figure}
%
\noindent \\ \\
\noindent In the limit of larger $V_0$ such that $\Delta \gg k_BT$ where T is the thermodynamical temperature, only the ground state band becomes relevant.
By taking the Fourier transform of ground state Bloch functions $u^{(0)}_{\bf{k}}({\bf{x}})$, i.e.
%
\begin{equation}
w({\bf{x}} - {\bf{x}}_i) = \frac{1}{\sqrt{\Omega}} \sum_{\bf{k}} u^{(0)}_{\bf{k}} ({\bf{x}}) e^{i{\bf{k}} \cdot ({\bf{x}} - {\bf{x}}_i)} \,\, ,
\end{equation}
%
where $\Omega$ is the volume of the primitive unit cell, we obtain an orthonormal basis of Wannier functions $\{w({\bf{x}} - {\bf{x}}_i)\}$ that are maximally localized at every individual lattice site. \cite{jaksch1}
Next, we define the hopping strength of a particle from site {\it{i}} to {\it{j}} as
%
\begin{equation}
t_{ij} = - \int d{\bf{x}} \, w({\bf{x}} - {\bf{x}}_i) \left( -\frac{\hbar^2}{2m} \nabla^2 + V({\bf{x}}) \right) w({\bf{x}} - {\bf{x}}_j)
\end{equation}
%
and the onsite energy $\epsilon_i = t_{ii}$.
Often, only the hopping strength between nearest neighbouring sites is relevant to the physics, and is simply denoted by $t$ for the case of homogeneous lattice.
\noindent Two identical particles in an optical lattice interact through short-range potential, hereby modelled by contact interaction in real space, i.e.
%
\begin{equation}
U({\bf{x}}, {\bf{x}}') = \frac{4\pi a_s \hbar^2}{m} \delta({\bf{x}}, {\bf{x}}') \,\, ,
\end{equation}
%
where the scattering mechanism is effectively described by the s-wave scattering length $a_s$, which can be easily tuneable in experiments via Feshbach resonance technique. \cite{greiner1,chin1}
Next, we define interaction strength among site {\it{i}}, {\it{j}}, {\it{k}}, {\it{l}} as
%
\begin{equation}
U_{ijkl} = \frac{4\pi a_s \hbar^2}{m} \int d{\bf{x}} \, w({\bf{x}} - {\bf{x}}_i) \, w({\bf{x}} - {\bf{x}}_j) \, w({\bf{x}} - {\bf{x}}_l) \, w({\bf{x}} - {\bf{x}}_k) \,\, .
\end{equation}
%
Often, only the onsite interaction strength is relevant to the physics, and is simply denoted by U for the case of homogeneous lattice.
\subsection{Bosons in an optical lattice}
%
\begin{figure}[h]
\centering
\includegraphics[width=7cm]{./figures/boson-hubbard-model-phase-diagram.jpg}
\caption{Phases of boson Hubbard model for 3D lattices. \cite{trotsky} At lower temperatures $T/t$, an homogeneous bosonic optical lattice exhibits superfluid (Mott insulator) phase in the limit of small (large) interaction $U/t$. At higher temperatures $T/t$, the system becomes normal fluid.}
\label{boson_hubbard_model_phase_diagram}
\end{figure}
%
\noindent The hamiltonian for single-component bosons in an optical lattice is
%
\begin{equation}
\label{many_body_hamiltonian_1}
\hat{H}
= \int d{\bf{x}} \, \hat{\psi}^\dag ({\bf{x}}) \left( -\frac{\hbar^2}{2m} \nabla^2 + V({\bf{x}}) \right) \hat{\psi} ({\bf{x}}) \,\,
+\,\, \frac{1}{2} \int d{\bf{x}} \, d{\bf{x}}' \hat{\psi}^\dag ({\bf{x}}) \hat{\psi}^\dag ({\bf{x}}') \, U({\bf{x}}, {\bf{x}}') \, \hat{\psi} ({\bf{x}}') \hat{\psi} ({\bf{x}})
\end{equation}
%
where the field operators
%
\begin{equation}
\hat{\psi}({\bf{x}}) = \sum_i w({\bf{x}} - {\bf{x}}_i) \, \hat{b}_i
\end{equation}
%
are expanded in the wannier basis $\{ w({\bf{x}} - {\bf{x}}_i) \}$ for every site $i$ in the optical lattice \cite{quantum_mechanics}.
Here, $\hat{b}_i$ and $\hat{b}^\dag_i$ are annihilation and creation operators respectively at site $i$ that satisfy the
following commutation relations
%
\begin{equation}
[\hat{b}_i,\hat{b}_j]=0 \,\,\,\,\, , \,\,\,\,\, [\hat{b}^\dag_i,\hat{b}^\dag_j]=0 \,\,\,\,\, , \,\,\,\,\, [\hat{b}_i,\hat{b}^\dag_j]=\delta_{ij}\,\,.
\end{equation}
\\
\noindent In the limit of deep lattice (large $V_0$), the bosonic homogeneous optical lattice \ref{many_body_hamiltonian_1} can be effectively mapped to the boson Hubbard model \cite{fisher}
%
\begin{equation}
\label{boson_hubbard_model}
\hat{H} - \mu \hat{N} = -t \sum_{\langle i,j \rangle} \hat{b}^\dag_i \hat{b}_j + \frac{U}{2} \sum_i n_i(n_i-1) - \mu \sum_i n_i
\end{equation}
%
where the summation $\langle i,j \rangle$ extends over all nearest neighbouring lattice sites, and the chemical potential $\mu$ determines the total number of bosons in the system.
The boson Hubbard model \ref{boson_hubbard_model} has been completely solved at least numerically \cite{trotsky}, with its phase diagram illustrated in figure \ref{boson_hubbard_model_phase_diagram} for different interactions $U/t$ and temperatures $T/t$.
The first indirect evidence of a superfluid-insulator phase transition was derived from the time-of-flight images obtained from the experiment by M. Greiner {\it{et al}} in 2002 \cite{greiner2}, which had been numerically confirmed by Quantum Monte Carlo simulations 7 years later \cite{trotsky}.
%
\begin{figure}
\centering
\includegraphics[width=7cm]{./figures/experiment-comparison-density-profile.jpg}
\caption{Comparison between theory and experiment. Average density profile $\langle n \rangle$ with respect to radial distance $r$ in units of lattice spacing $d$. Blue dots - optical lattice experiment\cite{chin1}: 9400 Cs-133 bosons are confined in a 2D square optical lattice with intensity $V_0=5E_R$ and trapping strength $V_T=0.01nK$. Band structure calculations give the hopping strength $t=4.18nK$ and onsite repulsion strength $U=10.79nK$. Red line - numerical-exact Quantum Monte Carlo simulation\cite{shiang1}: Single-band boson Hubbard model with interaction $U/t = 2.58$, temperature $T/t=5.98$ and trapping strength $V_T/t=0.00239$, thereby exhibiting normal-fluid phase. An excellent agreement has been observed.}
\label{experiment-comparison-density-profile}
\end{figure}
%
Recent experimental advancement has enabled density probing within single-site resolution, \cite{chin2, bloch1} and quantitative agreement has been found in the direct comparison between theory and experiment as illustrated in figure \ref{experiment-comparison-density-profile}.
\\ \\
In the presence of parabolic trapping, the trapped boson Hubbard hamiltonian \cite{jaksch1} reads
%
\begin{equation}
\label{trapped_boson_hubbard_hamiltonian}
\hat{H} = -t \sum_{\langle i,j \rangle} \hat{b}^\dag_i \hat{b}_j + \frac{U}{2} \sum_i n_i(n_i-1) - \sum_i (\mu - V_T {\bf{x}}_i^2)n_i \,\, .
\end{equation}
%
\subsection{Fermions in an optical lattice}
The hamiltonian for 2-component fermions in an optical lattice is
%
\begin{eqnarray}
\label{many_body_hamiltonian_2}
\hat{H}
&=& \sum_{\sigma = \uparrow, \downarrow} \int d{\bf{x}} \, \hat{\psi}^\dag_\sigma ({\bf{x}}) \left( -\frac{\hbar^2}{2m} \nabla^2 + V({\bf{x}}) \right) \hat{\psi}_\sigma ({\bf{x}}) \,\, \nonumber \\
& & +\,\, \frac{1}{2} \sum_{\sigma, \sigma' = \uparrow, \downarrow} \int d{\bf{x}} \, d{\bf{x}}' \hat{\psi}^\dag_\sigma ({\bf{x}}) \hat{\psi}^\dag_{\sigma'} ({\bf{x}}') \, U({\bf{x}}, {\bf{x}}') \, \hat{\psi}_{\sigma'} ({\bf{x}}') \hat{\psi}_\sigma ({\bf{x}})
\end{eqnarray}
%
where the field operators
%
\begin{equation}
\hat{\psi}({\bf{x}}) = \sum_i w({\bf{x}} - {\bf{x}}_i) \, \hat{c}_i
\end{equation}
%
are expanded in the wannier basis $\{ w({\bf{x}} - {\bf{x}}_i) \}$ for every site $i$ in the optical lattice \cite{quantum_mechanics}.
Here, $\hat{c}_i$ and $\hat{c}^\dag_i$ are annihilation and creation operators respectively at site $i$ that satisfy the
following anti-commutation relations
%
\begin{equation}
\{\hat{c}_i,\hat{c}_j\}=0 \,\,\,\,\, , \,\,\,\,\, \{\hat{c}^\dag_i,\hat{c}^\dag_j\}=0 \,\,\,\,\, , \,\,\,\,\, \{\hat{c}_i,\hat{c}^\dag_j\}=\delta_{ij}\,\,.
\end{equation}
\noindent In the limit of deep lattice (large $V_0$), the fermionic homogeneous optical lattice \ref{many_body_hamiltonian_2} can be effectively mapped to the Hubbard model \cite{hubbard}
%
\begin{equation}
\hat{H} = -t \sum_{\langle i, j \rangle , \sigma} \hat{c}^\dag_{i\sigma} \hat{c}_{j\sigma} + U \sum_i n_{i\uparrow} n_{i\downarrow} \tag{\ref{hubbard_model}}
\end{equation}
%
where the summation $\langle i,j \rangle$ extends over all nearest neighbouring lattice sites, and $\sigma=\uparrow,\downarrow$.
Unlike the bosonic case, the Hubbard model \ref{hubbard_model}, being the simplest correlation model for fermions, remains generally unsolved till today.
Only for specific cases like half-filling ($\langle n \rangle = 1$) could the Hubbard model \ref{hubbard_model} be solved exact numerically via Quantum Monte Carlo methods \cite{muramatsu1}.
At half-filling towards larger interaction $U/t$, the Hubbard model is driven from the Mott insulating phase to the antiferromagnetic phase with decreasing temperature $T/t$ \cite{muramatsu1}.
Unfortunately, this cannot yet be confirmed by current experiments, due to tough challenges in cooling the fermionic optical lattice system beyond the Neel temperature \cite{fermion_ol_experiment}.
\\ \\
\noindent Away from half-filling is perhaps where the Hubbard model \ref{hubbard_model} is being the most interesting with the conjecture of a superconductivity phase at low temperature.
With a belief of relevance to unconventional superconductivity, the Hubbard model is probably one of the many scientific puzzles that remains urgently to be solved.
Numerically, the notorious negative-sign problem forbids the use of Quantum Monte Carlo methods, which then brings up the interest of designing fermionic optical lattices to be quantum emulators of the Hubbard model \cite{ole}.