eBHM - Mean Field

Algorithm

Self-consistency loop: \[H \equiv H\{\psi_i, \rho_i\} \hspace{1cm}\forall i \in \text{unit cell}\] \[H \cdot|\Psi_{gs}\rangle = E_{gs} \cdot |\Psi_{gs}\rangle\] \[\psi_i = \langle \Psi_{gs} | \hat{a}_i | \Psi_{gs} \rangle \hspace{1cm} \rho_i = \langle \Psi_{gs} | \hat{n}_i | \Psi_{gs} \rangle\]

Recast in terms of a multi-variate function:

\[f(\{\psi_i, \rho_i\}) \rightarrow \text{Diagonalize } H\{\psi_i, \rho_i\} \rightarrow \text{Evaluate }\Psi_{gs} \text{ expectation values} \rightarrow \{\psi_i', \rho_i'\}\]

Self consistency \(\equiv\) Find fixed point of the function.

\[f(\{\psi_i^*, \rho_i^*\}) = \{\psi_i^*, \rho_i^*\}\]

Naive attempt at a solution

Given a function \(f(x)\), find a fixed point \(x^*\) such that \(f(x^*) = x^*\).

\[f(x^{(0)}) = x^{(1)}\] \[f(x^{(1)}) = x^{(2)}\] \[\vdots\] \[f(x^{(n)}) = x^{*}\]

Repeatedly apply \(f\) on an initial guess \(x^{(0)}\) till convergence; (a.k.a fixed point iteration)

\[f(f(f(...f(x)))) \rightarrow x^*\]

Seems easy enough?

Problem 1: No convergence

Manifests as 2-cycles, like so:

\[f(\{\psi_A, \psi_B, \rho_A, \rho_B\}) = \{\psi_A', \psi_B', \rho_A', \rho_B'\}\] \[f(\{\psi_A', \psi_B', \rho_A', \rho_B'\}) = \{\psi_A, \psi_B, \rho_A, \rho_B\}\]

Exploit patterns in the 2-cycle to guess the fixed point.

Try a different initial guess.

Problem 2: Sub-linear convergence

Deceptively similar to a 2-cycle in short iterations:

\[f(\{\psi_A, \psi_B, \rho_A, \rho_B\}) \approx \{\psi_A', \psi_B', \rho_A', \rho_B'\}\] \[f(\{\psi_A', \psi_B', \rho_A', \rho_B'\}) \approx \{\psi_A, \psi_B, \rho_A, \rho_B\}\]

Relative error with the actual value scales sub-linearly with the number of iterations. \[\underbrace{f(f(f(...f(\{\psi_A, \psi_B, \rho_A, \rho_B\}))))}_{\text{a looooot of times}} \rightarrow \{\psi_A^*, \psi_B^*, \rho_A^*, \rho_B^*\}\]

Solution

Use more sophisticated solvers (Anderson/Nesterov acceleration).

Not so fast!

Strange artefacts for larger \(zV/U\).

Problem 3: Multiple fixed points (a.k.a. local minima)

• Plot heatmap of number of iterations required to converge.

• Indicates that the artefacts are not convergence issues, but rather local minima.

Solution

• Start with a grid of \(N\) initial guesses \(\{\psi_i, \rho_i\}_k\)
  
• Converge to fixed points \(\{\psi_i', \rho_i'\}_k\)
  
• \(\{\psi_i^*, \rho_i^*\} = \underset{1 \leq k \leq N}{\text{argmin }} E_{gs}\{\psi_i', \rho_i'\}_k\)
  

• 

Triangular lattice

Possible line of inquiry: Is it easier to observe SS in triangular lattice?

Extracting phase boundaries

Naive method: Compute for a grid of parameter values and find the points where the order parameter jumps.

• 

• Precise method: Use a bisection algorithm. Precision scales as \(~2^{-n}\) for \(n\) iterations.

Result: Square lattice

Cleaner phase boundaries with arbitrary precision.

Neural Network variational ansatz

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 \(\langle \hat{H} \rangle\).

Problem 1: …local minima

Requires better gradient descent algorithm. Not working on this at the moment.

Quantum Monte Carlo: Stochastic Series Expansion (SSE)

Start with a spin-1/2 heisenberg model;

\[H = -J \sum_{\langle i, j\rangle} \vec{S_i} \cdot \vec{S_j}\]
\[H = -J \sum_{\langle b \rangle} \frac{1}{2} (S_{i(b)}^+S_{j(b)}^- + S_{j(b)}^+S_{i(b)}^-) + S_{i(b)}^z S_{j(b)}^z\]
\[H = -J\sum_b \underbrace{H_{b, 1}}_{\text{off-diagonal}} + \underbrace{H_{b, 2}}_{\text{diagonal}}\]

Compute the partition function

\[Z = Tr(\exp(-\beta H))\]
\[Z = Tr\left[ \sum_{n=0}^{\infty} \frac{(-\beta)^n}{n!} \cdot \left(\sum_b H_{b, 1} + H_{b, 2}\right)^n \right ]\]
\[Z = \sum_{n=0}^{\infty} \frac{(-\beta)^n}{n!} \cdot \sum_{|\alpha\rangle}\sum_{S_n} \langle \alpha | \left (\prod_{\{b, i\} \in S_n} H_{b, i} \right) | \alpha \rangle = \sum_{C_i \ \in\ \mathcal{C}} w(C_i)\]
Define configuration of the system, \(C_i \equiv [|\alpha\rangle, S_n]\).

Sample these \(C_i \in \mathcal{C}\) ergodically to compute diagonal observables.

Expansion truncation

\[\langle E \rangle = -\frac{\langle n \rangle}{\beta} \hspace{0.5cm} \text{and} \hspace{0.5cm} C_v = \langle n^2 \rangle - \langle n \rangle^2 - \langle n \rangle\]

We can maintain a cut-off \(n_{max}\) dynamically as the simulation progresses and introduce negligible error.

Mapping to eBHM

XXZ spin-\(1/2\) model:

\[H = \frac{J_x}{2} \sum_{\langle i, j \rangle} (S_{i}^+S_{j}^- + S_{j}^+S_{i}^-) + J_z\sum_{\langle i, j \rangle} S_{i}^z S_{j}^z + h_z \sum_i S_i^z\] eBHM w/ hard-core bosons:

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

Map the operators like so: \[S_i^+ \equiv a_i^{\dagger} \hspace{1cm} S_i^z \equiv (n_i - 1/2)\] Analogous quantities: \[t \equiv \frac{J_x}{2} \hspace{1cm} V \equiv J_z \hspace{1cm} \mu = J_z - h_z\]

Finite-size scaling effects

\[Q(t, L) = L^{\zeta/\nu} \cdot g(tL^{1/\nu}) \hspace{2cm} t = (T - T_c)/T_c\]

Results for the 2D Ising model

Obtaining the phase boundary

Binder’s cumulant: \[U_L = 1 - \frac{\langle m^4 \rangle_L}{3\langle m^2 \rangle^2_L} \hspace{1cm} \text{and} \hspace{1cm} U_L(t=0) = U_{L'}(t=0) \hspace{0.2cm} \ \forall \ L, L'.\]

Plot binder’s cumulant of the order parameter for different lattice sizes and find the intersection point.

Results for heisenberg model

• Transition point does not seem to match?

• Finite-size scaling not discernible visually?

Supplementary: Convergence of MFT in BHM

Convergence is monotonic!

A single iteration with an initial guess of \(~10^{-6}\) is sufficient to determine the phase.