Phase diagram detection via Gaussian fitting of number probability distribution

We investigate the number probability density function that characterizes subportions of a quantum many-body system with globally conserved number of particles. We put forward a linear fitting protocol capable of mapping out the ground-state phase diagram of the rich one-dimensional extended Bose-Hubbard model: The results are quantitatively comparable with more sophisticated traditional and machine learning techniques. We argue that the studied quantity should be considered among the most informative bipartite properties, being moreover readily accessible in atomic gases experiments.

Figure 1

Phase diagram of the EBH model traced with the residuals of the PDF's quadratic fit $p\left(m\right)\propto e^{-\beta m^2}$ for a $L=64$ system. Details about the phase transitions along the dashed and dotted cuts are shown in Fig. 2. The PDFs correspondent to the configurations marked with the colored shapes are displayed below in Fig. 3.

Figure 2

Residuals of a Gaussian fit of the PDF $p\left(m\right)\propto e^{-\beta m^2}$ for different system's sizes $L=16,32,64,128,256$ from light to dark color in the proximity of (a) the gapped-to-gapped phase transitions MI-HI-CDW along the cut $U=5$ and (d) the gapless-to-gapped phase transitions SF-MI (BKT) for $V=0$. In the insets, the location of the residuals' minimum is fitted against the inverse of the system's size in order to extrapolate the critical $V_C$ for the MI-HI phase transition (b) and the HI-CDW transition (c).

Figure 3

The logarithm of the PDF (black points) is plotted for a representative configuration of the (a) SF phase [red star; $(U,V)=(0.5,0.5)$], (b) MI phase [green pentagon; $(U,V)=(5.5,0.5)$], (c) CDW phase (gray square; $(U,V)=(2,4.5)$], and (d) HI phase [brown diamond; $(U,V)=(5,3.3)$]. The gray lines show the fit of the PDF in terms of Gaussian envelopes and their shifts, as discussed in the main text. For both MI and CDW, the behavior is linear for small $\left|m\right|$, in compliance with horizontally shifted parabolas: on the right we show the shift of parabolas' minima from $m=0$ as a function of the inverse distance from the critical point for MI (e) and CDW (f). It is apparent that they tend, respectively, to 1 and 2 deep in the phases, as predicted by perturbation theory in the text. For the HI, the dominant feature is a vertical shift between parabolas for the different parities of $m$: (g) shows a magnification thereof.