Berezinskii-Kosterlitz-Thouless Renormalization Group Flow at a Quantum Phase Transition

Matthias Thamm, Harini Radhakrishnan, Hatem Barghathi, Chris Herdman, Arpan Biswas, Bernd Rosenow, and Adrian Del Maestro

arXiv:2502.18622

Abstract

We present a controlled numerical study of the Berezinskii-Kosterlitz-Thouless (BKT) transition in the one-dimensional Bose-Hubbard model at unit filling, providing evidence of the characteristic logarithmic finite-size scaling of the BKT transition. Employing density matrix renormalization group and quantum Monte Carlo simulations under periodic boundary conditions, together with a systematic finite-size scaling analysis of bipartite particle number fluctuations, we resolve boundary-induced complications that previously obscured critical scaling. We demonstrate that a suitably chosen central region under open boundaries reproduces universal RG signatures, reconciling earlier discrepancies. Finally, leveraging a non-parametric Bayesian analysis, we determine the critical interaction strength with high precision, establishing a benchmark for BKT physics in one-dimensional quantum models.

Description

This repository includes links, code, scripts, and data to generate the figures in a paper.

Requirements

The data in this project can be generated using the code in the following repositories:

1. DMRG simulations: ExtendedBH_DMRG_Fluctuations_Julia
2. QMC simulations: pigsfli

Data is included in the data directory. QMC raw data is available via a Zenodo archive:

This code requires Julia version 10.4 or higher and the IJulia package to run the Jupyter notebook. Required Julia packages can be installed by running the code in the create_figures.jl script:

using Pkg 
Pkg.activate(".")
Pkg.add(["Plots","PyFormattedStrings","NonlinearSolve","StaticArrays","Printf","Integrals","FastClosures","LaTeXStrings","DataFrames","NPZ","Measures","PyCall"])

The python code for postprocessing of the QMC data in data/pbc/QMC/postprocess requires the following packages:

• matplotlib
• numpy
• tqdm
• scipy
• zipfile-deflat64

The python code for the BO and GP analysis of the $\zeta(K)$ data in src/GP_with_BoTorch.ipynb requires the following packages:

• botorch=0.10.0
• torch
• gpytorch
• numpy
• matplotlib
• dgutils (pip install git+https://github.com/DelMaestroGroup/dgutils.git#egg=dgutils)
• scipy

Support

This work was partially supported by the National Science Foundation Materials Research Science and Engineering Center program through the UT Knoxville Center for Advanced Materials and Manufacturing (DMR-2309083). H.R. acknowledges AITennessee for financial support. Computations were performed using resources provided by the Leipzig University Computing Center and University of Tennessee Infrastructure for Scientific Applications and Advanced Computing (ISAAC).

Figures

Figure 01: BKT RG flow for Bose-Hubbard model.

Figure 02: Finite size scaling of Luttinger parameter according to BKT flow.

Figure 03: Extracting Luttinger parameter from bipartite particle number fluctuations.

Figure 04: Extracting critical point of BKT transition.

Figure S01: Periodic vs. open boundary conditions.

Figure S02: Fitting method for OBC.

Figure S03: Reproduction of literature result.

Figure S04: GP mean combined with the confidence interval of the posterior distribution..

These figures are relesed under CC BY-SA 4.0 and can be freely copied, redistributed and remixed.