Rapidly converging methods for the location of quantum critical points from finite-size data

We analyze in detail, beyond the usual scaling hypothesis, the finite-size convergence of static quantities toward the thermodynamic limit. In this way, we are able to obtain sequences of pseudo-critical points, which display a faster convergence rate as compared to currently used methods. The approaches are valid in any spatial dimension and for any value of the dynamic exponent. We demonstrate the effectiveness of our methods both analytically, on the basis of the one dimensional $XY$ model, and numerically, considering $c=1$ transitions occurring in nonintegrable spin models. In particular, we show that these general methods are able to precisely locate the onset of the Berezinskii–Kosterlitz–Thouless transition making only use of ground-state properties on relatively small systems.

Figure 1

(Color online) The sequences of pseudocritical points obtained with the FSCM, the fast-convergent method of Eq. (7), and the HCM described by Eq. (14). Moreover, the sequences have been calculated with PRG methods, both standard and in our improved version. Continuous lines are algebraic best fits to the data.

Figure 2

(Color online) Upper panel: Searching for the zeros of $L^3{b}^{″}+\left(3L^2-16\right)b^{\prime }$ according to Eq. (14) with $J_1=1$ and $\delta L=4$. Lower panel: Extrapolations of the pseudocritical points $J_{2,L}$ (cyan line for all the points and blue line with $L=12$ excluded); the black cross marks the accepted value 0.2411 (see text for further details).