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Phases in the 2-Dimensional Falicov-Kimball Model

<h2>Introduction</h2> <p>The dataset consists of a subset of 1,467,059 configurations obtained using Monte Carlo (MC) simulations of the Falicov-Kimball Model (FKM) on a 2-dimensional 12x12 square lattice with periodic boundary conditions.</p> <p>The FKM [1,2] is one of the simplest models of interacting fermions that displays a variety of distinct phases and phenomena. It considers two types of particles: itinerant (light) f-electrons and localized (heavy) d-electrons on a lattice. In its spineless version it can be understood as a special limit of the Hubbard model where electrons with one spin species, e.g. spin up, are localized. Its advantage over the Hubbard model is that it is accessible with a sign-problem-free Monte Carlo [3].</p> <p>The main parameters of the FKM are the hopping parameter (t), which defines the mobility of itinerant electrons across neighboring lattice sites, and the interaction parameter (U), which represents the on-site Coulomb repulsion between the itinerant and localized electrons. In our MC simulations, the parameter t has been set to 1, making U the only relevant parameter in the FKM Hamiltonian. Another important parameter in the MC simulations is the temperature (T).</p> <p>Further details of the MC simulations can be found in Ref. [3]. For each pair of U and T values, 20 independent MC simulations have been run, each producing 500 configurations. To keep the configurations in the dataset uncorrelated, just a subset as large as 5% of all the produced configurations has been randomly selected for further analysis.</p> <p>The dataset can be used to train unsupervised machine learning models for phase classification or supervised regression models for predicting simulation parameters.</p> <h2>Dataset Structure</h2> <p>The dataset consists of two CSV files:</p> <ul> <li>&nbsp;<strong>fkm_data.csv:</strong> Contains 1,467,059 configurations taken at various values of U (ranging from 0 to 12) and T (ranging from 0.005 up to 0.3). The values of U and T are chosen from a grid where U changes in steps of dU=0.25, and T changes in steps of dT=0.005. Multiple independent configurations are provided for each pair of U and T values.</li> <li>&nbsp;<strong>fkm_data_ht.csv:</strong> Contains an additional 342,320 configurations taken at higher temperatures, T, ranging from 0.35 up to 1.0 with a step of dT=0.05. The range and step of U values remain the same as in the previous file. These additional configurations can help regularize training of unsupervised machine learning models.</li> </ul> <p>Each configuration is stored as a row in the attached CSV files. A configuration vector consists of:</p> <ul> <li><strong>144 binary components</strong> (0s and 1s) indicating site occupancy by f-electrons.<br>&nbsp;In the CSV files, these components are named "f1", "f2", ..., "f144", where the number denotes the site.</li> <li><strong>144 float components</strong> (between 0 and 1) indicating the electron density of d-electrons.<br>&nbsp;In the CSV files, these components are named "d1", "d2", ..., "d144," where the number denotes the site.</li> <li><strong>2 float components</strong> specifying the corresponding values of U and T.&nbsp;<br>These values are in columns labeled "U" and "T," respectively.</li> </ul> <p>The values of U and T are given in units of the hopping parameter t.</p> <p>An accompanying Jupyter notebook, <strong>load_fkm_data.ipynb</strong>, demonstrates how to load the data and view its structure.</p> <h2>References</h2> <p>[1] J. Hubbard, Proc. Royal Soc. Lond. A 276, 238 (1963).<br>[2] L. M. Falicov and J. C. Kimball, Phys. Rev. Lett. 22, 997 (1969).<br>[3] M. M. Maśka and K. Czajka, Phys. Rev. B 74, 035109 (2006).</p>