A picture of me.

Tom Hodson

Maker, Baker Programmer Reformed Physicist RSE@ECMWF

3 The Long Range Falicov-Kimball Model


This chapter expands on work presented in

  [1] One-dimensional long-range Falicov-Kimball model: Thermal phase transition and disorder-free localization, Hodson, T. and Willsher, J. and Knolle, J., Phys. Rev. B, 104, 4, 2021,

The code is available online  [2].

Johannes had the initial idea to use a long-range Ising term to stabilise order in a 1D Falicov-Kimball model. Josef developed a proof of concept during a summer project at Imperial along with Alexander Belcik. I wrote the simulation code and performed all the analysis presented here.

Chapter Summary

This chapter is organised as follows. First, I will introduce the Long-Range Falicov-Kimball (LRFK) model and motivate its definition. Second, I will present the methods used to solve it numerically, including Markov chain Monte Carlo and finite size scaling. I will then present and interpret the results obtained.

The Model

Dimensionality is crucial for the physics of both localisation and phase transitions. We have already seen that the 1D standard Falicov-Kimball (FK) model cannot support an ordered phase at finite temperatures and therefore has no Finite-Temperature Phase Transition (FTPT).

On bipartite lattices in dimensions two and above, the FK model exhibits a finite temperature phase transition to an ordered Charge Density Wave (CDW) phase  [3]. In this phase, the spins order anti-ferromagnetically, breaking the \(\mathbb{Z}_2\) symmetry. In 1D, however, Peierls’s argument  [4,5] states that domain walls only introduce a constant energy penalty into the free energy while bringing an entropic contribution logarithmic in system size. Hence, the 1D model does not have a finite temperature phase transition. However, 1D systems are much easier to study numerically and admit simpler realisations experimentally. We therefore introduce a long-range coupling between the ions in order to stabilise a CDW phase in 1D.

Figure 1: The Long-Range Falicov-Kimball (LRFK) Model is a model of classical spins S_i coupled to spinless fermions \hat{c}_i where the fermions are mobile with hopping t and the fermions are coupled to the spins by an Ising type interaction with strength U. The difference from the standard FK model is the presence of a long-range interaction between the spins J_{ij}S_i S_j.

We interpret the FK model as a model of spinless fermions, \(c^\dagger_{i}\), hopping on a 1D lattice against a classical Ising spin background, \(S_i \in {\pm \frac{1}{2}}\). The fermions couple to the spins via an onsite interaction with strength \(U\) which we supplement by a long-range interaction, \[ J_{ij} = 4\kappa J\; (-1)^{|i-j|} |i-j|^{-\alpha}, \]

between the spins, see fig. 1. The additional coupling is very similar to that of the long-range Ising (LRI) model. It stabilises the antiferromagnetic (AFM) order of the Ising spins which promotes the finite temperature CDW phase of the fermionic sector.

The hopping strength of the electrons, \(t = 1\), sets the overall energy scale and we concentrate throughout on the particle-hole symmetric point at zero chemical potential and half-filling  [6].

\[\begin{aligned} H_{\mathrm{FK}} = & \;U \sum_{i} S_i\;(c^\dagger_{i}c^{\phantom{\dagger}}_{i} - \tfrac{1}{2}) -\;t \sum_{i} (c^\dagger_{i}c^{\phantom{\dagger}}_{i+1} + \textit{h.c.)}\\ & + \sum_{i, j}^{N} J_{ij} S_i S_j. \label{eq:HFK}\end{aligned}\]

Without proper normalisation, the long-range coupling would render the critical temperature strongly system size dependent for small system sizes. Within a mean field approximation, the critical temperature scales with the effective coupling to all the neighbours of each site, which for a system with \(N\) sites is \(\sum_{i=1}^{N} i^{-\alpha}\). Hence, the normalisation \(\kappa^{-1} = \sum_{i=1}^{N} i^{-\alpha}\), renders the critical temperature independent of system size in the mean field approximation. This greatly improves the finite size behaviour of the model.

Taking the limit \(U = 0\) decouples the spins from the fermions, which gives a spin sector governed by a classical long-range Ising model. Note, the transformation of the spins \(S_i \to (-1)^{i} S_i\) maps the AFM model to the FM one. As discussed in the background section, Peierls’ classic argument can be extended to long-range couplings to show that, for the 1D LRI model, a power law decay of \(\alpha < 2\) is required for a FTPT. This is because the energy of defect domain scales with the system size when the interactions are long-range and can overcome the entropic contribution. A renormalisation group analysis supports this finding and shows that the critical exponents are only universal for \(\alpha \leq 3/2\)  [79]. In the following, we choose \(\alpha = 5/4\) to avoid the additional complexity of non-universal critical points.

Next Section: Methods


T. Hodson, J. Willsher, and J. Knolle, One-Dimensional Long-Range Falikov-Kimball Model: Thermal Phase Transition and Disorder-Free Localization, Phys. Rev. B 104, 045116 (2021).
M. M. Maśka and K. Czajka, Thermodynamics of the Two-Dimensional Falicov-Kimball Model: A Classical Monte Carlo Study, Phys. Rev. B 74, 035109 (2006).
R. Peierls, On Ising’s Model of Ferromagnetism, Mathematical Proceedings of the Cambridge Philosophical Society 32, 477 (1936).
T. Kennedy and E. H. Lieb, An Itinerant Electron Model with Crystalline or Magnetic Long Range Order, Physica A: Statistical Mechanics and Its Applications 138, 320 (1986).
C. Gruber and N. Macris, The Falicov-Kimball Model: A Review of Exact Results and Extensions, Helvetica Physica Acta 69, (1996).
D. Ruelle, Statistical Mechanics of a One-Dimensional Lattice Gas, Commun.Math. Phys. 9, 267 (1968).
D. J. Thouless, Long-Range Order in One-Dimensional Ising Systems, Phys. Rev. 187, 732 (1969).
M. C. Angelini, G. Parisi, and F. Ricci-Tersenghi, Relations Between Short-Range and Long-Range Ising Models, Phys. Rev. E 89, 062120 (2014).