Monte Carlo Instantons in the Double Well Potential

Numerical simulations of instantons in the quantum mechanical double well potential using Euclidean lattice Monte Carlo. The implementation follows:

T. Schäfer - Instantons and Monte Carlo Methods in Quantum Mechanics

arXiv:hep-lat/0411010

Overview

This project studies tunneling and instanton physics using several complementary numerical approaches:

• Euclidean Monte Carlo path integral sampling
• Cooling to extract instanton content
• Instanton liquid models (RILM / IILM)
• Adiabatic switching for non-Gaussian corrections
• Correlation functions and energy gap extraction

The code aims to reproduce the figures and numerical experiments presented in Schäfer’s lecture notes.

Physical Model

We consider the double well Hamiltonian

$$ H = \frac{p^2}{2m} + (x^2-\eta^2)^2 $$

After rescaling (2m = 1), the discretized Euclidean action becomes

$$ S = \sum_i \left[ \frac{(x_i-x_{i-1})^2}{4a} + a(x_i^2-\eta^2)^2 \right] $$

with periodic boundary conditions. The classical instanton solution is

$$ x_I(\tau) = \eta \tanh \left( \frac{\omega}{2}(\tau-\tau_0) \right), \qquad \omega = 4\eta $$

and the classical instanton action

$$ S_0 = \frac{4\eta^3}{3} $$

Computational Layers

The project contains four conceptual layers.

1. Spectral quantum mechanics

Direct diagonalization of the Hamiltonian.

2. Full Monte Carlo path integral

Metropolis sampling of Euclidean paths.

3. Semiclassical instanton extraction

Cooling and instanton density measurements.

4. Instanton liquid effective models

Analytical multi-instanton configurations (RILM, IILM, streamline).

Repository Structure

.
├── CMakeLists.txt
├── LICENSE
├── README.md
├── bin/              # compiled executables
├── build/            # CMake build directory
├── data/             # generated CSV data
├── figures/          # generated plots
├── plots/            # python plotting scripts
├── docs/             # documentation (Doxygen)
├── tests/            # unit tests
├── src
│   ├── analysis
│   │   ├── analysis_driver.*
│   │   ├── cooling_evolution.*
│   │   ├── ensemble.*
│   │   ├── fig10_rilm.*
│   │   ├── fig11_gauss.*
│   │   ├── fig12_13_heating.*
│   │   ├── fig14_16.*
│   │   ├── fig15.*
│   │   ├── fig17_iilm.*
│   │   ├── fig7.*
│   │   ├── fig8.*
│   │   ├── fig9.*
│   │   ├── qmidens.*
│   │   └── qmidens.*
│   ├── core
│   │   ├── instanton.*
│   │   ├── lattice.*
│   │   ├── metropolis.*
│   │   ├── observables.*
│   │   ├── potential.*
│   │   └── potential.*
│   ├── main.*
│   ├── models
│   │   ├── fig14_ia_interaction.*
│   │   ├── heating.*
│   │   ├── iilm.*
│   │   ├── rilm.*
│   │   └── rilm.*
│   └── utils
│       ├── io.*
│       ├── parameters.*
│       ├── periodic.*
│       ├── statistics.*

Build

Configure

cmake -S . -B build -DCMAKE_BUILD_TYPE=Debug

Compile

cmake --build build -j

Generated executables:

bin/montecarlo
bin/test_*

Command Line Interface

The main executable runs different numerical experiments.

./bin/montecarlo <command> [options]

Optional arguments

--seed <int>
--etas a,b,c

Example

./bin/montecarlo eta-scan --etas 1.0,1.2,1.4,1.6

Main Commands

Command	Description
basic	typical Euclidean path + cooled path
ensemble	correlators and probability distributions
cooling-evolution	instanton density vs cooling
fig7	averaged cooling analysis
fig8	instanton density vs η
fig9	switching paths
fig11	Gaussian effective potential
rilm	random instanton liquid model
heated-rilm	heated RILM
iilm	interacting instanton liquid
fig14	IA interaction and streamline
fig15	streamline paths
fig16	IA separation histogram
fig17	instanton positions
qmidens	non-Gaussian corrections
eta-scan	scan in η

---

Generating Figures

Typical workflow

./bin/montecarlo <command>
python3 plots/plot_figX.py

Example

./bin/montecarlo fig7
python3 plots/plot_fig7.py

Figure Reproduction Map

Schäfer figure	Method
Fig.1	Schrödinger spectral calculation
Fig.2	Monte Carlo path
Fig.3	probability distribution
Fig.4	correlators
Fig.5	adiabatic switching
Fig.6	cooled correlators
Fig.7	cooling evolution
Fig.8	instanton density vs η
Fig.9	switching paths
Fig.10	RILM correlators
Fig.11	Gaussian effective potential
Fig.12	RILM vs heated RILM
Fig.13	heated RILM correlators
Fig.14–16	IA interaction and streamline
Fig.17	instanton liquid positions

Testing

Run all unit tests

ctest --test-dir build --output-on-failure

Tests verify:

• action consistency
• Metropolis acceptance rule
• cooling monotonicity
• boundary conditions
• potential symmetry

Documentation

Generate API documentation with

cmake --build build --target docs

Then open

docs/api/html/index.html

Rebuild

Incremental rebuild

cmake --build build -j

Clean rebuild

rm -rf build
cmake -S . -B build
cmake --build build -j

Author

Marco Benazzi, Theoretical Physics MSc Project

Started: 2025