Interacting bosons in a lattice

\[H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)\]

Phases of the BHM

Mott Insulator

\[\small H = \frac{U}{2}\sum_i n_i(n_i - 1) \hspace{0.5cm}\longrightarrow\hspace{0.5cm} |\Psi_{MI}\rangle = \prod_{i=1}^M |n\rangle\]

Superfluid

\[\small H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j\hspace{0.5cm}\longrightarrow\hspace{0.5cm}|\Psi_{SF}\rangle= \frac{1}{N!} (\sum_{i=1}^M a_i^{\dagger})^N |{0}\rangle \hspace{0.5cm}\]

Exact Diagonalization

Consider N bosons in M lattice sites.

• \[\text{Enumerate basis set: }\hspace{0.5cm} |n_1, n_2, ..., n_M\rangle \hspace{0.5cm} s.t. \hspace{0.5cm} \sum_{i=1}^M n_i = N\]
  
• \[\text{Construct hamiltonian matrix: }\hspace{0.5cm} \langle n_1, n_2, ..., n_M| \hspace{0.1cm}\hat{H} \hspace{0.1cm}| n_1', n_2', ..., n_M'\rangle\]
  
• \[\text{Diagonalize the hamiltonian: } \hspace{0.5cm} H|\psi\rangle = E|\psi\rangle\]

Results (ED)

Sparsity of the Hamiltonian

10 bosons; 10 lattice sites.

Dimensionality scaling

\[\small\text{Dim }(\mathcal{H}) = \frac{(N+M-1)!}{N!(M-1)!}\]

Order of Magnitude

Mean Field Theory

\[\small \hat{a}_i = \Psi + \delta\hat{a}_i \hspace{1cm} \Bigg |\hspace{1cm}\mathcal{O}(\delta \hat{a}_i ^2) \approx 0\]

\[\small \underbrace{H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)}_{\text{coupled lattice sites}} \hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi \} = \sum_i-zt \cdot (\Psi^*a_i + \Psi a_i^{\dagger} - |\Psi|^2) + \frac{U}{2}n_i(n_i -1)}_{\text{de-coupled lattice sites}}\]

Solution (MFT)

To find the equilibrium state of the system, we must minimize \(G = H - \mu N + TS\).

\[\frac{\partial \langle G\rangle}{\partial \Psi} = \frac{\partial \langle H - \mu N\rangle}{\partial \Psi}\Bigg|_{T=0K} = 0\]

\[ \Psi = \langle \psi_{gs} |\hat{a} | \psi_{gs} \rangle\]
\[\text{Minimize G} \equiv \text{Self-consistently diagonalize } H\{\Psi\}\]

Algorithm (MFT)

Results (MFT)

Results (MFT, contd.)

Cluster Mean Field Theory

\[\small \underbrace{H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1)}_{\text{coupled lattice sites}} \hspace{0.5cm}\longrightarrow\hspace{0.5cm} \underbrace{H \{\Psi_i \} = \sum_C H_{exact} + \sum_{C, C'}H_{MFT}\{ \Psi_i \}}_{\text{de-coupled clusters of sites}}\]

Results (CMFT)

Dipolar bosons in a lattice

\[H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1) + V \sum_{\langle i, j \rangle} n_i n_j\]

Solution (MFT)

\[\small H = -t\sum_{\langle i, j \rangle} a_i^{\dagger}a_j + \frac{U}{2}\sum_i n_i(n_i - 1) + V\sum_{\langle i, j \rangle} n_i n_j\] \[\hat{n}_i = \rho_i + \delta \hat{n}_i\hspace{0.5cm}\Bigg\downarrow \hspace{0.5cm}\hat{a}_i = \Psi_i + \delta\hat{a}_i\] \[\small H_A \{\Psi_A, \Psi_B,\rho_A, \rho_B \} = -zt \cdot (\Psi_B^*a_A + \Psi_B a_A^{\dagger} - \Psi_A^*\Psi_B) + zV\cdot(\rho_Bn_A - \rho_A\rho_B) + \frac{U}{2}n_A(n_A -1) \\ \small H_B \{\Psi_A, \Psi_B,\rho_A, \rho_B \} = -zt \cdot (\Psi_A^*a_B + \Psi_A a_B^{\dagger} - \Psi_B^*\Psi_A) + zV\cdot(\rho_An_B - \rho_B\rho_A) + \frac{U}{2}n_B(n_B -1)\]
\[\small H \{\Psi_A, \Psi_B,\rho_A, \rho_B\}= \sum_{i \in A} H_i + \sum_{j \in B} H_j\]

Solve self-consistently: \(\hspace{0.5cm}\Psi_i = \langle \psi_{gs, i}|\hat{a}_i |\psi_{gs, i}\rangle \hspace{0.5cm};\hspace{0.5cm} \rho_i = \langle \psi_{gs, i}|\hat{n}_i |\psi_{gs, i}\rangle \hspace{0.5cm}; \hspace{0.5cm} i \in \{A, B\}\)

Results (MFT)

Results (MFT, contd.)

What next?

Path Integral QMC

• Evaluate the partition function \(Z = Tr(e^{\beta\hat{H}})\).

• \[\begin{align} Z &= \sum_{|n_1\rangle} \langle n_1 |\left (e^{\frac{\beta}{M}\hat{H}} \right )^M |n_1 \rangle \\ &= \lim_{M\to \infty} \ \ \sum_{|n_1\rangle} \langle n_1 |\left (1 + \frac{\beta}{M}\hat{H} \right)^M |n_1 \rangle \\ &\approx \sum_{\{|n_i\rangle\}} \langle n_1 |\left (1 + i\Delta t\hat{H} \right) |n_2 \rangle \cdot \langle n_2 |\left (1 + i\Delta t\hat{H} \right) |n_3 \rangle \dots \langle n_M |\left (1 + i\Delta t\hat{H} \right) |n_1 \rangle \\ &\approx \sum_{C_i \in C} w(C_i) \end{align}\]

• Map d-dimensional quantum system to (d+1) classical system (\(\beta \rightarrow iM\Delta t\)).

• Formulated as a classical system determined by a configuration - string of M basis elements; \(C \equiv |n_1 \rangle \rightarrow |n_2 \rangle \rightarrow \dots \rightarrow |n_M \rangle\rightarrow |n_1 \rangle\).

Worldline representation

Variational Ansatz - ANN

• Ansatz for the wave-function: \(\Psi = \sum_{n} \Psi(n) |n\rangle\) such that \(\Psi(n)\) is captured by a neural network.

• 

• Train the network weights to minimize \(\langle \hat{H} \rangle\).

Mapping experiment to BHM parameters