Universal Behavior beyond Multifractality in Quantum Many-Body Systems

How many states of a configuration space contribute to a wave function? Attempts to answer this ubiquitous question have a long history in physics and are keys to understanding, e.g., localization phenomena. Beyond single-particle physics, a quantitative study of the ground state complexity for interacting many-body quantum systems is notoriously difficult, mainly due to the exponential growth of the configuration (Hilbert) space with the number of particles. Here we develop quantum Monte Carlo schemes to overcome this issue, focusing on Shannon-Rényi entropies of ground states of large quantum many-body systems. Our simulations reveal a generic multifractal behavior while the very nature of quantum phases of matter and associated transitions is captured by universal subleading terms in these entropies.

Figure 1

(a) SR entropies as a function of $q$ and system size $N$ for the ground state of the Heisenberg ladder at zero field (gapped) and at half-saturated magnetization (LL) in the $S^z$ basis. As $\underset{N\to \infty }{lim}{S}_q/N=a_q$, the nontrivial behavior of $S_q/N$ signals multifractality. (b) Probability spectrum ${\zeta }_i$ at $J_{\perp }=1.0$ and half-saturated magnetization. For larger sizes, only the most probable state was recorded. The dotted line is a fit of $S_{\infty }$ to $a_{\infty }N+b_{\infty }+c_{\infty }/N$. Sizes $N=12$ and $N=20$ do not match the fit as the most probable state has periodicity 4 on each leg and only clusters of size $N=8p$ with integer $p$ obey the condition. Incommensurate clusters have a lower $p_{\max }$. Error bars are reflected by line widths. The inset shows the $J_{\perp }$ dependence of the LL parameter $K$ as obtained by fitting subleading terms in $S_1$ and $S_q$. The horizontal line displays the limit $K\to 3/4$ for $J_{\perp }\gg 1$ [30].

Figure 2

(a)–(c) Subleading coefficients $b_q$ of SR entropies in the ${\sigma }^z$ basis for the square lattice Ising model in a transverse field $h$ for different $q$, as obtained from fits to $S_q=a_{q}N+b_q+c_q/N+d_q/N^2$. We display fits over different lattice size windows such as to assess the development towards the thermodynamic limit. Subleading terms converge rapidly at criticality and display a marked jump as $h$ is tuned away from its critical value. Straight lines show the limiting cases for high ($b_q=0$) and low field ($b_q=-\ln 2$). The convergence towards the low-field thermodynamic limit $-\ln 2$ is not visible here (except for $q=\infty$), due to limited accessible system sizes, as discussed in detail in Ref. [27]. (d) The field dependence of $a_{\infty }$ reveals a clear change of slope at $h_c$, with a crossing point of $\partial a_{\infty }/\partial h$ for different finite-size fits (inset).

Figure 3

(a) $S_{\infty }$ for the $d=2$ isotropic ($\mathrm{\Delta }=1$) and anisotropic ($\mathrm{\Delta }\ne 1$) $XXZ$ model on the square lattice as a function of system size $N$. Lines are fits to the form $S_{\infty }=a_{\infty }N+l_{\infty }\ln N+b_{\infty }+c_{\infty }/N+d_{\infty }/N^2$. (b) Subleading logarithmic terms are highlighted in semilog scale by subtracting $a_{\infty }N$ from $S_{\infty }$. Reported values of $a_{\infty }$ and $l_{\infty }$ are estimates from the best fit including error bars from a bootstrap analysis.