BAFQMC

From local positivity to global symmetry

Bosonic auxiliary-field quantum Monte Carlo

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Documentation · Get started · The algorithm · Reproduce results · Guide for agents

Worldline and stochastic-series-expansion methods organize sign-free sampling around local matrix-element positivity. BAFQMC establishes nonnegative bosonic weights through global symmetry after Hubbard–Stratonovich decoupling. Reflection positivity (RP) and time-reversal symmetry (TRS) open a new computable region: frustrated bosonic models protected by these symmetries.

This repository puts the construction to work: finite-temperature solvers for two-flavor bosons on a triangular lattice, with and without onsite pairing; exact diagonalization (ED); and a complete workflow to reproduce the paper's benchmarks. Run new calculations. Extend the method. Build on the code. Everything here is available under MIT.

What can I simulate?

The supplied solvers calculate finite-temperature properties of two boson flavors on a periodic triangular lattice. In the paper's notation, the main benchmark Hamiltonian is

$$\begin{aligned} \hat H ={}& t\sum_{\langle ij\rangle} \left(\hat b_i^+\hat b_j+\hat b_j^+\hat b_i +\hat c_i^+\hat c_j+\hat c_j^+\hat c_i\right)\\\ &-\sum_i\left(\Delta\,\hat b_i^+\hat c_i^+ +\Delta^*\,\hat b_i\hat c_i\right) +U\sum_i\left(\hat n_{b,i}-\hat n_{c,i}\right)^2. \end{aligned}$$

Here $\hat n_{b,i}=\hat b_i^+\hat b_i$ and $\hat n_{c,i}=\hat c_i^+\hat c_i$. Simulations use the grand-canonical ensemble $Z=\mathrm{Tr}e^{-\beta(\hat H-\mu\hat N)}$, with $\hat N=\sum_i(\hat n_{b,i}+\hat n_{c,i})$. Positive $t=1$ gives frustrated hopping. The number-conserving solver sets $\Delta=0$; the Nambu solver supports real onsite pairing.

The lattice has $N_s=L_xL_y$ sites and primitive vectors

$$\mathbf a_1=(1,0),\qquad \mathbf a_2=\left(\frac12,\frac{\sqrt3}{2}\right).$$

Nearest-neighbor bonds run along $\mathbf a_1$, $\mathbf a_2$, and $\mathbf a_2-\mathbf a_1$, including their reverse directions and periodic images. The benchmarks use $3\times3$ clusters and measure $\rho$, $-E$, $S_{\mathrm{SF}}(K)$, and $S_{\mathrm{DW}}(K)$.

The code also implements the two density-interaction channels used in the Supplemental Material:

$$\hat H_U=\sum_i\left[ U_1(\hat n_{b,i}+\hat n_{c,i})^2 +U_2(\hat n_{b,i}-\hat n_{c,i})^2\right], \qquad U_1\leq0,\quad U_2\geq0.$$

The main model is $U_1=0$, $U_2=U$. The input pairing strength is the same real $\Delta$ as in the paper; the code uses the equivalent operator phase $c_{\mathrm{code}}=-c_{\mathrm{paper}}$. The model, lattice, and parameter guide explains this mapping, the ensemble, and the reference calculations.

Want to study another model? The model-development guide and dedicated agent skill explain how to extend the lattice, hopping, interactions, and pairing, with matching ED and physics tests.

Work with your agent

Clone this repository and open it in your preferred coding agent. The agent instructions and task recipes provide the implementation map, physical conventions, commands, and checks. Start with a request such as:

Read AGENTS.md, set up BAFQMC in Linux or WSL, and run the small BAFQMC + ED
installation check. Tell me where the results are.

Reproduce all paper benchmarks. Use the published parameters and seeds,
save the new results in a dedicated directory, and show me the final figures.

I want to study the triangular-lattice model at U = 1, beta = 4 and mu = -5
while varying the pairing strength. Prepare a separate campaign, check one
small case with ED, and explain the computing budget before production.

Extend the paper's main model to a nearest-neighbor kagome lattice at
t = 1, U = 1, Delta = 0, beta = 4, mu = -5. Read the new-model skill, derive
the HS symmetry, implement the geometry and matching ED and observables,
and validate a small case. Report the computing budget before production.

Add an observable for my research. Derive its estimator using the existing
Green-function convention, implement it in BAFQMC and ED, and verify it on a
small system.

Describe the physics you want to study and the computing resources you can use. Your agent can work through the details in the linked guides.

Reproduce the paper

After environment setup, run:

python3 reproduce.py

This builds the solvers, runs fresh BAFQMC and ED calculations for all 22 benchmark points, processes the measurements, and produces both figures. Allow approximately 12–24 hours, 16 GiB RAM, and 8 GiB free disk on a modern desktop CPU. The default uses one MPI rank and one numerical-library thread, with cases run sequentially. The resource guide gives measured timings and instructions for resuming a campaign.

python3 reproduce.py --plan        # Show the scope and resource estimate
python3 reproduce.py --mode smoke  # Small BAFQMC + ED installation check

Small processed benchmark data, including means and standard errors, are included. Large simulation outputs are generated locally and excluded from Git.

Explore

• Get started: install, run, and find your results.
• Algorithm and conventions: models, symmetry, and observables.
• Output observables: operator definitions and file conventions.
• Extend to other models: from Hamiltonian to tested implementation.
• Research tasks for agents: new calculations and extensions.
• Benchmark details: parameters, data, and reproduction modes.
• Contributing: develop and validate changes.

Paper

arXiv: 2609.XXXXX (forthcoming).