Critical properties of the ground-state localization-delocalization transition in the many-particle Aubry-André model

As opposed to random disorder, which localizes single-particle wave functions in one dimension (1D) at arbitrarily small disorder strengths, there is a localization-delocalization transition for quasiperiodic disorder in the 1D Aubry-André model at a finite disorder strength. On the single-particle level, many properties of the ground state at criticality have been revealed by applying a real-space renormalization-group scheme; the critical properties are determined solely by the continued-fraction expansion of the incommensurate frequency of the disorder. Here, we investigate the many-particle localization-delocalization transition in the Aubry-André model with and without interactions. In contrast to the single-particle case, we find that the critical exponents depend on a Diophantine equation relating the incommensurate frequency of the disorder and the filling fraction which generalizes the dependence, in the single-particle spectrum, on the continued-fraction expansion of the incommensurate frequency. This equation can be viewed as a generalization of the resonance condition in the commensurate case. When interactions are included, numerical evidence suggests that interactions may be irrelevant at at least some of these critical points, meaning that the critical exponent relations obtained from the Diophantine equation may actually survive in the interacting case.

Figure 1

Scaling of the generalized fidelity susceptibility ${\chi }_{F,2+2r}$ and the superfluid fraction $\mathrm{\Gamma }$ for $\beta ={\beta }_{11}=(\sqrt{5}-1)/2$ at filling $\rho =\frac{1}{2}$ with the angle and $P_F$ specified by Table 1. (a), (b) Show how the maxima of ${\chi }_{F,2+2r}\sim L^{\mu +2zr}$ from which $\mu =2.0$ and $z=1.8285$ are extracted. In (c) and (d), the scaling of ${\chi }_F$ and $\mathrm{\Gamma }$, respectively, is consistent with $\mu =2/\nu =2.00$, and $z=1.8285$ collapses all the $\mathrm{\Gamma }$ curves onto three scaling functions.

Figure 2

In (a) and (b), the collapse of ${\chi }_F$ and $\mathrm{\Gamma }$, respectively, onto the scaling functions is plotted for $\beta ={\beta }_{22}$, and there are two separate scaling functions: one for $N$ even and one for $N$ odd. Again, $\mu =2/\nu =2.00$, and the same value of $z=2.0875$ scales all $\mathrm{\Gamma }$ curves onto each other. In (c) and (d), the same quantities are plotted for $\beta ={\beta }_{21}$ with a filling of $\rho =\frac{1}{2}$ when $N=99,239,577,1983,3363,8119$, and $\beta ={\beta }_{22}$ with a filling of $\rho =1-1/\sqrt{2}$ otherwise. Still $\nu =1$, but $z=1.575$. Up to the normalization of $\mathrm{\Gamma }(\zeta =1.0$ for ${\beta }_{21}$ and $\zeta =0.6605$ for ${\beta }_{22})$ the two scaling functions are the same, suggesting that the universality class of the ${\lambda }_c=2$ transition is the same for ${\beta }_{21}$ at half-filling and ${\beta }_{22}$ at $\rho =1-1/\sqrt{2}$. The finite-size effects are worse for ${\beta }_{22}$ because of the irrational filling. However, once $N$ is large enough for the filling to be well approximated, the collapse is very good. $\varphi$ and $P_F$ are set according to Table 1

Figure 3

The second derivative of the ground-state energy per particle at $\lambda =2,d^{2}E/d{\lambda }^2/N$, for (a) ${\beta }_{11}$, (b) ${\beta }_{22}$, and (c) ${\beta }_{21}$, is plotted against $N_F/N$, the filling fraction. A clear fractal structure emerges. Note the difference between ${\beta }_{22}$ and ${\beta }_{21}$ at half-filling, and note the similarity at the filling fraction indicated by the black vertical line at $\rho =1-1/\sqrt{2}$ for ${\beta }_{22}$ and $\rho =\frac{1}{2}$ for ${\beta }_{21}$ where the same critical exponents and scaling functions are observed (see Fig. 2).

Figure 4

At commensurate filling, the transition moves from ${\lambda }_c=2$ to 0 because the resonance condition is fulfilled for some fixed $n=\nu$. The dynamic critical exponent is always $z=1$. In (a) we have $\beta ={\beta }_{11}$ and $\rho =1-{\beta }_{11}$ with $\nu =1$. In (b) we plot several $N$ for $\beta =\{{\beta }_{11},{\beta }_{21},\frac{1}{4}\}$ at $\rho =\{2{\beta }_{11}-1,2{\beta }_{21}-1,\frac{1}{2}\}$ and we see that they have the same $\nu =2$ and $z=1$. The curves collapse onto each other when $\zeta =\{1,0.7007,0.6755\}$ and $\eta =\{1,1,1.565\}$. In (c) we have $\beta ={\beta }_{11}$ and $\rho =3{\beta }_{11}-2$ with $\nu =3$. In (d) we have $\beta ={\beta }_{11}$ and $\rho =5{\beta }_{11}-3$ with $\nu =5$. Note that the finite-size effects get worse as $\rho \to 0$.

Figure 5

(a) Even in the presence of interactions, $\mathrm{\Gamma }$ still scales onto the same curve with ${\lambda }_c\approx 2.0+0.23V,z=1.575$, and $\nu =1$. The legend applies to (a), (b), and (c): each system size $N$ corresponds to a different color, and different symbols correspond to different $V$. In (b) and (c), it is clear that the change in $\mathrm{\Gamma }$ and ${\chi }_F$ is proportional to $V$ showing that it is likely irrelevant since such proportionality would break down at larger $N$ if it were relevant. There is no collapse onto a single scaling function in (b) and (c) because there are at least two irrelevant directions that $V$ contributes to. Because the collapse is worse for $\mathrm{\Delta }\chi$ than for $\mathrm{\Delta }\mathrm{\Gamma }$, it appears that $\chi$ suffers from more finite-size effects.

Figure 6

For $\beta =1/N$ at filling $N_F=N/2$, we plot the scaling functions ${\chi }_F$ and $\mathrm{\Gamma }$ in (a) and (b), respectively. We extract exponents $\nu =1.0,z=1.245$, and ${\lambda }_c=2.0$. The fidelity susceptibility does not collapse well far from the transition, but $\mathrm{\Gamma }$ does, which is why Ref. [44] underestimated $\nu \approx 0.7$ as the scaling of the width of the curve. When we fit the maximum of ${\chi }_F\sim L^{\mu }log\left(L\right)$ we find $\mu =2.1\approx 2/\nu$ as opposed to the value of $\mu =2.25$ from Ref. [44] where such a logarithmic correction has not been included.

Figure 7

In (a) we plot the maximum of the fidelity susceptibility occurring at $\lambda ={\lambda }_{\text{max}}$. The lines indicate a fit to ${\chi }_{F,\text{max}}\sim N^{\mu }log\left(N\right)$ and $\mu \approx 2.1$ for all interactions. In (b) we plot $\mathrm{\Gamma }\left({\lambda }_{\text{max}}\right)$ and fit $\mathrm{\Gamma }\left({\lambda }_{\text{max}}\right)\sim N^{2-z}(1+a_1{N}^{-y_1})$, which all yield $z\approx 1.25$ as in the free case. (c) Depicts $[\mathrm{\Gamma }(V,{\lambda }_{\text{max}})-\mathrm{\Gamma }(0,{\lambda }_{\text{max}})]/\mathrm{\Gamma }(0,{\lambda }_{\text{max}})\equiv \mathrm{\Delta }\mathrm{\Gamma }/{\mathrm{\Gamma }}_{\text{free}}\sim N^{-y_V}$ giving us $y_V\approx 0.44–0.48$ for all $V$ when we exclude points with $N<60$.

Figure 8

We plot the scaling collapse of $\mathrm{\Gamma }$ [in (b) and (d)] and ${\chi }_F$ [in (a) and (c)] for commensurate filling ${\rho }_3=2{\beta }_{11}-1$ and two different interaction strengths, where the exponents are determined using $\mathrm{\Gamma }$ only. In the free case, $\nu =2,z=1$, and ${\lambda }_c=0$, and the scaling collapse for $\mathrm{\Gamma }$ in (b) and (d) gives $\nu \approx 2.5,z\approx 1$, and ${\lambda }_c\approx 0.05$ for $V=-0.5$ and $\nu \approx 2$ and $z\approx 1$ with ${\lambda }_c\approx 1$ for $V=-1.7$, so the exponents have not changed much. Although numerically determined, $\mathrm{\Gamma }$ close to the transition should be accurate and less prone to finite-size effects than ${\chi }_F$. However, finite-size effects are the largest source of error.