Probing Symmetries of Quantum Many-Body Systems through Gap Ratio Statistics

The statistics of gap ratios between consecutive energy levels is a widely used tool—in particular, in the context of many-body physics—to distinguish between chaotic and integrable systems, described, respectively, by Gaussian ensembles of random matrices and Poisson statistics. In this work, we extend the study of the gap ratio distribution $P(r)$ to the case where discrete symmetries are present. This is important since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place. Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks. We check our formulas against simulations from large random matrices, showing excellent agreement. We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically driven spin systems. In all these models, the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces. We finally discuss the relevance of our analysis for existing results in the literature, as well as its practical usefulness, and point out possible future applications and extensions.

Popular Summary

The idea of describing properties of complicated physical systems, such as complex atomic nuclei, using random numbers dates back to the 1950s. One application is in quantum mechanics, where random numbers are a tool for making accurate predictions about the statistics of discrete energy levels a system can assume. More precisely, the statistical distribution of the spacings between successive energy levels can be compared with distributions from random matrices, which provides a signature of whether the system behaves in a regular or a chaotic way. Here, we take this toolkit a step further and use it to probe for symmetries, sometimes hidden, in certain physical systems.

Because of additional subtleties in the case of interacting particles, the focus in recent years has shifted away from the statistics of the spacings between energy levels and toward the statistics of ratios between successive spacings, which are by now a widely used tool to compare experimental or numerical observations with theoretical predictions. Extra symmetries can modify these statistics to be somewhere between the “regular” and “chaotic” predictions. We show that it is possible to extend the theory of spacing ratio statistics to account for the presence of additional symmetries.

Our result provides a tool to not only get a signature of chaos or regularity in systems with symmetries but also uncover these symmetries if they were previously unnoticed.

Figure 1

Four configurations of consecutive spacings considered in Sec. 3b. Different colors correspond to distinct spectra. The three central levels are the ones from which the ratios are calculated; the outer levels are at the same height to indicate that their relative position is irrelevant.

Figure 2

Distribution of the ratio of consecutive level spacings $P(r)$ for (from top to bottom) GOE, GUE, and GSE ensembles and $m=1$ to $m=4$ blocks (from bottom to top at $r=0$). Solid lines are the surmises obtained from Sec. 3, except the $m=1$ case, for which the corresponding surmise Eq. (7) is taken from Ref. [18]. Points are numerical results from random matrices of size at least $N_{\mathrm{tot}}=2000$, averaged over $3.6\times {10}^5$ histograms.

Figure 3

Top panel: $P_2^{\mathrm{GOE}}(r)$, for various density ratios $\alpha$ (see text). Note that ${\alpha }_{\varphi }=1/(1+{\varphi }^2)$ (in golden color in both plots) is the value that corresponds to the anyonic chain application discussed in Sec. 4d. Bottom panel: $⟨r{⟩}_2^{\mathrm{GOE}}$ as a function of $\alpha$. In both plots, solid lines are predictions from the surmises of Sec. 3, and points are numerical results obtained on random matrices of size at least $N_{\mathrm{tot}}=2000$, averaged over $3.6\times {10}^5$ realizations.

Figure 4

Probability distribution of $I_m^{1/4}$ for $N/m$ GOE blocks with $m=2$ (blue) and $m=3$ (orange). Histograms are obtained from 20 000 values, each of which is calculated from 40 realizations of matrices of size $N=180$ (solid lines) and from 120 realizations for size $N=48$ (dashed lines). Vertical blue and orange lines indicate the theoretical predictions for $m=2$ and 3, respectively; the blue line is the mid-value $\frac{1}{2}(I_2^{1/4}+I_3^{1/4})$.

Figure 5

The $Q$-state clock model. Top panel: average gap ratio $⟨r⟩$ for the model of Eq. (62) for different values of $Q$, as a function of Hilbert space size $|\mathcal{H}|$. Open symbols denote simulations where the ${\mathbb{Z}}_Q$ symmetry is resolved, in which case the Hilbert space size is the block size $Q^{L-1}$ (we averaged data over all $Q$ equivalent blocks). Filled symbols denote results for the full Hilbert space of size $Q^L$. The dashed lines represent the values of $⟨r{⟩}_m^{\mathrm{GOE}}$ obtained in Sec. 3 (taken from Table 2), while $⟨r{⟩}^{\mathrm{GOE}}$ is the numerical estimate for the GOE distribution taken from Ref. [18]. The precision of our numerics allows us to clearly distinguish the case $m=4$ blocks with $⟨r{⟩}_{m=4}^{\mathrm{GOE}}$ from the Poisson value $⟨r{⟩}^{\text{Poisson}}$, also represented in the plot. Simulation parameters are $\epsilon =0.5$, $J=1$, $\mathrm{\Gamma }=0.8$, and $g=0.5$. For $Q=2$, instead of the time-reversal and charge conjugation breaking term in Eq. (62), which vanishes when $Q=2$, we add a next-nearest-neighbor interaction $g_2{\sum }_j{\sigma }_j{\sigma }_{j+2}$ in order to break the mapping to a free-fermion model (we take $g_2=0.5$). Statistics are obtained by focusing on $200Q$ eigenstates in the middle of the full spectrum, except for the smaller sizes where about $20Q$ eigenstates were considered. Results are averaged over more than 4000 realizations of disorder, except for the largest size where 1000 realizations were used. For $Q=2$, 3, 4, we obtained results on chains of sizes up to $L=17$, 11, 9, respectively. Bottom panel: probability distribution of the gap ratio $P(r)$, as obtained from simulations of chains of sizes $L=16$, 10, 8 for $Q=2$, 3, 4, respectively. Simulation parameters are the same as in the top panel. The solid lines represent the surmises $P_m^{\mathrm{GOE}}(r)$ obtained from the analytical computations in Sec. 3.

Figure 6

Probability distribution of the gap ratio from the Heisenberg model of Eq. (63) with $L=18$ spins, with periodic boundary conditions. Data are averaged over all realizations of the binary transverse field $h_j=\pm 1/2$ ($2^{18}$ in total, 3914 of which are nonequivalent up to symmetries) and over 150 eigenstates around infinite temperature energy $E=(E_{\min }+E_{\max })/2$. The red solid line represents the analytical prediction, Eq. (64): a linear combination of surmises $P_m^{\mathrm{GOE}}(r)$ obtained from the analytical computations in Sec. 3. For comparison, the green solid line shows the predicted distribution when two block contributions are not taken into account. Inset: difference $\delta P(r)$ between the numerical data and these surmises.

Figure 7

Top panel: average gap ratio as a function of $J$ in the time-independent and driven spin-$1/2$ chain model, Eq. (65). Bottom panel: gap ratio distributions at $J=1$ for the time-independent and driven case (points), and comparison with the surmise distribution for $m=1$ and $m=2$ GOE blocks (solid lines). In the time-independent case, the average is performed over 2000 disorder realizations (except at the $J=1$ point where 5000 realizations are used), and 100 eigenstates around infinite-temperature energy $E=(E_{\min }+E_{\max })/2$, for the system of size $L=14$, while in the driven case, the average is performed over 2000 disorder realizations and all eigenstates of the system of size $L=12$.

Figure 8

RSOS model. Top panel: average gap ratio $⟨r⟩$ for the RSOS model in Eq. (71), as a function of Hilbert space size. Blue circles are the results for the full spectrum of the Hamiltonian (69); squares and triangles are those in the $Y=(1\pm \sqrt{5}/2)$ sectors. The dashed lines represent the results for the (single-block) GOE distribution of Ref. [18] and from the surmise computations in Sec. 3 for $m=2$ GOE blocks with size ratio ${\varphi }^{-2}$. Data in the $Y=(1+\sqrt{5})/2$ sector were obtained by comparing the energy spectrum in the loop representation with noncontractible loop weight $2\cos (\pi /5)$ and the RSOS representation. Data in the $Y=(1-\sqrt{5})/2$ were obtained by considering the rest of the states in the RSOS representation. We focus on mid-spectrum eigenstates of the RSOS spectrum, obtaining about 300 eigenstates for every disorder realization. Results are averaged over between 300 and 1000 realizations of disorder of strength $\epsilon =0.2$. Bottom panel: probability distribution of the gap ratio $P(r)$, as obtained from simulations of a RSOS chain of size $L=22$. The solid line is the surmise $P_{m=2,\alpha =1/(1+{\varphi }^2)}^{\mathrm{GOE}}(r)$ obtained from the analytical computations in Sec. 3.

Figure 9

$R(\mathcal{O})$ for $\mathcal{O}=S_{L/2}^z$ (bottom data) and $\mathcal{O}=S_{L/2}^x$ (top data) for the spin-1 model Eq. (a1), as a function of energy difference $\omega =|E_{\alpha }-E_{\beta }|$ between eigenstates $|\alpha ⟩$ and $|\beta ⟩$. Inset: average gap ratio $⟨r⟩$ as a function of system size (here, $\epsilon =1$).