Information lattice approach to the metal-insulator transition

Abstract

Correlation functions and correlation lengths are frequently used to describe phase transitions in quantum systems, but they require an explicit choice of observables. The recently introduced information lattice instead provides an observable-independent way to identify where and at which scale information is contained in quantum lattice models. Here, we use it to study the difference between the metallic and insulating regime of one-dimensional tight-binding chains. We find that the information per scale follows a power law in metals at low temperature and that Friedel-like oscillations are visible in the information lattice. At high temperature or in insulators at low temperature, the information per scale decays exponentially. Thus, the information lattice is a useful tool for analyzing the metal-insulator transition.

Figures (14)

• Figure 1: Sketch of Rice-Mele Hamiltonian of length $N=8$ (bottom left), with weak and strong bonds $t_1$, $t_2$ and alternating on-site potential $V$. The corresponding information lattice can be visualized as a pyramid (top left), here for $t_1=1$, $t_2=0.5$, $V=0$, at $k_B T=1/3$ and $\mu=1$. It shows that most information is contained at the level of the dimers ($\ell=1$ and strong bonds). The row sum $i^\ell$ of the information is shown on the right axis. The electronic spectrum (panel right) for this short chain is discrete; the formation of continuous bands happens in the limit of large $N$. The curve next to the spectrum shows $n_\text{FD}(1-n_\text{FD})$ to illustrate which energy levels are partially occupied at this temperature and chemical potential $\mu$.
• Figure 2: Illustration of three subsets $C_n^\ell$ at the left boundary of a one-dimensional lattice. $C^1_{1/2}$ is a strict subset of $C^4_2$, i.e., $C^1_{1/2} \subset C^4_3$, so the information contained in $C^{1}_{1/2}$ is removed when calculating $i^4_3$ according to \ref{['def:informationlattice']}.
• Figure 3: Information per scale $i^\ell$ as a function of the scale $\ell$, shown as a log-plot on the left and as a log-log plot in the middle to identify exponential versus power-law scaling. Here, $N=500$, $t_1=t_2=1$, $V=0.5$. The red data corresponds to $\mu=1$, where the chemical potential lies inside a band and the system is metallic, while the black data is for $\mu=0$, an insulating system with the chemical potential inside the gap. For both chemical potentials, we show high temperature (triangles) and low temperature (circles). The corresponding spectra and the function $n_F (1-n_F)$ are shown for the same high (dotted line) and low (solid line) temperatures. At high temperature (triangles), $i^\ell$ is similar in the the metallic and insulating phase and decays exponentially with $\ell$. At low temperature, the insulator still decays exponentially but with a longer decay length $\xi$, whereas the metal starts to show power-law behaviour. The dashed black line shows the scaling Ansatz. Note that the bipartite nature of the system leads to distinct values for even and odd $\ell$.
• Figure 4: Inverse decay length $\xi^{-1}$ as a function of $k_B T$, for a chain of length $N=201$ with $t_1=t_2=1$. (a) $V=0.5$ and two different chemical potentials, corresponding to a metal and an insulator as illustrated in the density of states on the right. $\xi$ is obtained by a fitting $i^\ell\sim \exp(-\ell/\xi)$ using even values of $\ell$ only. (b) $V$ controls the size of the gap in the insulator at fixed $\mu=0$, $t_1=t_2=1$ and $N=201$, and therefore has a major impact on the temperature-dependence of $\xi$.
• Figure 5: Dependence of $i^\ell_n$ on temperature in the SSH model with $t_1=1$, $t_2=1.2$, $\mu=0.75$ and $N=80$. The solid lines show power laws and are drawn as a guide to the eye.