Minimal energy ensemble Monte Carlo algorithm for the partition function of fermions coupled to classical fields

Models of noninteracting fermions coupled to auxiliary classical fields are relevant to the understanding of a wide variety of problems in many-body physics, e.g., the description of manganites, diluted magnetic semiconductors, or strongly interacting electrons on lattices. We present a flat-histogram Monte Carlo algorithm that simulates a statistical ensemble that allows one to directly acquire the partition function at all temperatures for such systems. The defining feature of the algorithm is that it utilizes the complete thermodynamic information from the full energy spectrum of noninteracting fermions available during sampling of the configuration space of the classical fields. We benchmark the method for the classical Ising and Potts models in two dimensions, as well as the Falicov-Kimball model describing itinerant electrons interacting with heavy ions.

Figure 1

Specific heat $C_V$ per site for the 2D Ising model using our algorithm (black) and the WL (red dashed) algorithm. The lattice size is $60\times 60$ with PBCs. The upper inset shows the difference between these lines. The lower inset shows the averaged error in our sampling (black line) for an $8\times 8$ system with PBCs. The red dashed line depicts the error generated by temperature shifting exact data (see the text).

Figure 2

Specific heat $C_V$ per site for the $30\times 30$ site 2D ten-state Potts model using our algorithm (black) and the WL (red dashed) algorithm. The lower inset shows the difference between these lines. The upper inset is a close-up of the transition region.

Figure 3

Specific heat $C_V$ per site for the 2D Falicov-Kimball model at half filling obtained with our algorithm (black line) and the Metropolis (red circles) algorithm. The electron-ion on-site repulsion $U/t$ is equal to (from the left panel to the right) 0.25 (weak coupling), 1 (moderate coupling), and 8 (strong coupling). The lattice size is $16\times 16$ with PBCs. The insets show a comparison of internal energies.