One-dimensional long-range Falikov-Kimball model: Thermal phase transition and disorder-free localization

Disorder or interactions can turn metals into insulators. One of the simplest settings in which to study this physics is given by the Falikov-Kimball (FK) model, which describes itinerant fermions interacting with a classical Ising background field. Despite the translational invariance of the model, inhomogeneous configurations of the background field give rise to effective disorder physics which lead to a rich phase diagram in two (or more) dimensions with finite-temperature charge-density wave (CDW) transitions and interaction-tuned Anderson versus Mott localized phases. Here, we propose a generalized FK model in one dimension with long-range interactions which shows a similarly rich phase diagram. We use an exact Markov chain Monte Carlo method to map the phase diagram and compute the energy-resolved localization properties of the fermions. We compare the behavior of this transitionally invariant model to an Anderson model of uncorrelated binary disorder about a background CDW field which confirms that the fermionic sector only fully localizes for very large system sizes.

Figure 1

Phase diagrams of the long-range 1D FK model. (a) The $\mathit{TJ}$ plane at $U=5$: the CDW ordered phase is separated from a disordered Mott insulating (MI) phase by a critical temperature $T_c$, linear in $J$. (b) The $\mathit{TU}$ plane at $J=5$: the disordered phase is split in two: at large (small) $U$ there is an MI (Anderson) phase characterized by the presence (absence) of a gap at $E=0$ in the single-particle energy spectrum. $U_c$ is independent of the temperature. At $U=0$ the fermions are decoupled from the spins, forming a simple Fermi gas. (c) The order parameters, $\langle m^2\rangle$ (solid line) and $1-f$ (dashed line), describing the onset of the CDW phase of the long-range 1D FK model at low temperature with staggered magnetization $m=N^{-1}{\sum }_i{(-1)}^i{S}_i$ and fermionic order parameter $f=2N^{-1}|{\sum }_i{(-1)}^i\langle c_i^{†}{c}_i|\rangle$. (d) Crossing of the Binder cumulant, $B=\langle m^4\rangle /{\langle m^2\rangle }^2$, with system size provides a diagnostic that the phase transition is not a finite-size effect; it is used to estimate the critical lines shown in (a) and (b). All plots use system sizes $N=[10,20,30,50,70,110,160,250]$ and lines are colored from $N=10$ in dark blue to $N=250$ in yellow. Parameter values are $U=5,J=5,$ and $\alpha =1.25$ except where explicitly noted.

Figure 2

Energy-resolved DOS($\omega$) and $\tau$ [scaling exponent of IPR($\omega$) vs system size $N$]. The left column shows the Anderson phase $U=2$ at a high temperature, $T=2.5$, and the CDW phase at a low temperature, $T=1.5$. IPRs are evaluated for one of the in-gap states ${\omega }_0/U=0.057$ and the center of the band ${\omega }_1/U=0.81$. The right column shows instead the Mott and CDW phases at $U=5$ with ${\omega }_0/U=0.24$ and ${\omega }_1/U=0.571$. For all the plots $J=5,\alpha =1.25$, and the fits for $\tau$ use system sizes greater than 60. The measured ${\tau }_0,{\tau }_1$ for each figure are (a) $0.06\pm 0.01,0.02\pm 0.01$; (b) $0.04\pm 0.02,0.00\pm 0.01$; (c) $0.05\pm 0.03,0.30\pm 0.03$; and (d) $0.06\pm 0.04,0.15\pm 0.05$. We show later that the apparent scaling of the IPR with system size can be explained by the changing defect density with system size rather than due to delocalization of the states.

Figure 3

The density of states (a, c) and scaling exponent $\tau$ (b, d) as a function of the energy and temperature. (a) and (b) show the system transitioning from the CDW phase to the gapless Anderson insulating one at $U=2$, while (c) and (d) show the CDW–to–gapped Mott phase transition at $U=5$. Regions where the DOS is close to 0 are shown in white. The scaling exponent $\tau$ is obtained from fits to $\mathrm{IPR}|\left(N\right)=A{N}^{-\lambda }$ for a range of system sizes. $U=5,J=5,\alpha =1.25$.

Figure 4

A comparison of the full FK model to a simple binary disorder model (DM) with a CDW wave background perturbed by uncorrelated defects at density $0<\rho <1$ matched to the largest corresponding FK model. As in Fig. 2, the energy-resolved DOS($\omega$) and $\tau$ are shown. The DOSs match well and these data make clear that the apparent scaling of the IPR with the system size is a finite-size effect due to weak localization [14], hence all the states are indeed localized as one would expect in one dimension. The disorder model ${\tau }_0,{\tau }_1$ are (a) $0.01\pm 0.05,-0.02\pm 0.06$; (b) $0.01\pm 0.04,-0.01\pm 0.04$; (c) $0.05\pm 0.06,0.04\pm 0.06$; and (d) $-0.03\pm 0.06,0.01\pm 0.06$. The lines are fit on system sizes $N>400$.