Conformal Bootstrap for Infinite MPS

This repository is an early-stage implementation of an infinite-MPS version of the bootstrap condition discussed in bootstrap_condition_note.md.

The basic goal is:

• start from an infinite MPS $|\Psi\rangle$,
• compute its 1-site and 2-site reduced density matrices,
• build local modular Hamiltonians $$ K_i = -\log \rho_i,\qquad K_{i,i+1} = -\log \rho_{i,i+1}, $$
• construct the reconstructed infinite-chain Hamiltonian $$ H_{\mathrm{rec}}^\Psi = \sum_i (K_{i,i+1} - K_i), $$
• and search for self-consistent fixed points where the same iMPS is an eigenstate, ideally the ground state, of its own reconstructed Hamiltonian.

In practice, the fixed-point search is implemented as an iterative loop:

1. extract local RDMs from the current iMPS,
2. build the bootstrap Hamiltonian from those RDMs,
3. solve for the infinite-MPS ground state of that Hamiltonian,
4. repeat until the local data stop changing.

For the derivation, notation, and algorithmic motivation, see:

• bootstrap_condition_note.md
• imps_literature_note.md

Repository Structure

• src/IMPSBootstrap.jl

  • core Julia utilities for the iMPS bootstrap prototype
  • includes:
    • 1-site and 2-site RDM extraction from InfiniteMPS
    • regularized matrix logarithms
    • embedding a 1-site operator into 2-site Hilbert space
    • constructing a translation-invariant nearest-neighbor InfiniteMPOHamiltonian
    • wrapping MPSKit ground-state solvers
    • the self-consistency loop imps_bootstrap_fixed_point

• scripts/demo_imps.jl

  • simple sanity-check script
  • computes a transverse-field Ising iMPS ground state and prints local diagnostics

• scripts/demo_bootstrap_loop.jl

  • short Section 13 demo
  • seeds from a critical iMPS and runs a few bootstrap iterations

• scripts/run_bootstrap_experiment.jl

  • reusable CLI experiment runner
  • saves results to a TOML file for later analysis

• scripts/plot_bootstrap_result.py

  • reads a saved TOML result
  • generates summary and diagnostic plots

• results/

  • saved bootstrap runs and derived plots

• references/

  • source papers used for the project

Current Status

The current implementation already supports:

• infinite-MPS ground-state solving with MPSKit.jl
• extracting normalized 1-site and 2-site RDMs
• constructing the bootstrap local term from $K_{i,i+1}-K_i$
• running the bootstrap self-consistency loop for a one-site unit cell
• saving and plotting run diagnostics

What is still provisional:

• the exact bootstrap update rule may need refinement
• only the simplest one-site-unit-cell setup is currently implemented
• no damping/mixing or advanced convergence acceleration is included yet
• the relation between the numerically found fixed points and actual CFT data still needs systematic study

Running the Code

Julia environment:

julia --project=.

Sanity-check iMPS run:

julia --project=. scripts/demo_imps.jl

Short bootstrap demo:

julia --project=. scripts/demo_bootstrap_loop.jl

Reusable experiment run:

julia --project=. scripts/run_bootstrap_experiment.jl --max_iter=10 --seed_maxiter=30 --solver_maxiter=30 --output=results/bootstrap_run.toml

Plot a saved result:

python3 scripts/plot_bootstrap_result.py results/bootstrap_run.toml --outdir results/plots

Main Numerical Diagnostics

The most important quantities to watch are:

• delta_rho: change in local reduced density matrices between bootstrap iterations
• sigma2: variance of the reconstructed Hamiltonian in the current iMPS state
• eigensolver residual from VUMPS or IDMRG
• spectra of the 1-site and 2-site RDMs

Interpretation:

• small sigma2 means the current iMPS is close to an eigenstate of the reconstructed Hamiltonian
• small delta_rho means the bootstrap map is approaching self-consistency

Notes

• The theory note contains more detail than the code currently implements.
• The present code should be viewed as a working prototype for the Section 13 algorithm, not a finished conformal-bootstrap pipeline.