Finite-size scaling of eigenstate thermalization

According to the eigenstate thermalization hypothesis (ETH), even isolated quantum systems can thermalize because the eigenstate-to-eigenstate fluctuations of typical observables vanish in the limit of large systems. Of course, isolated systems are by nature finite and the main way of computing such quantities is through numerical evaluation for finite-size systems. Therefore, the finite-size scaling of the fluctuations of eigenstate expectation values is a central aspect of the ETH. In this work, we present numerical evidence that for generic nonintegrable systems these fluctuations scale with a universal power law $D^{-1/2}$ with the dimension $D$ of the Hilbert space. We provide heuristic arguments, in the same spirit as the ETH, to explain this universal result. Our results are based on the analysis of three families of models and several observables for each model. Each family includes integrable members and we show how the system size where the universal power law becomes visible is affected by the proximity to integrability.

Figure 1

Eigenstate expectation values (blue dots) and the microcanonical average (red curves) for the observable $\widehat{A}=S_2^z$ with (a)–(c) $\lambda =0$ and (d)–(f) $\lambda =1$ for three different system sizes $L$. The energies on the horizontal axis are scaled by system size for meaningful comparison. [The microcanonical average is nearly constant in this case, which is not representative for all observables (see e.g., Refs. [6, 14]).] The inset shows the geometry of the ladder system and site labeling. Solid and dashed lines are $H_{\mathrm{leg}}$ and $H_{\mathrm{rung}}$ couplings, respectively.

Figure 2

Fluctuation amplitudes ${\sigma }_{\Delta A}$ as a function of the control parameter $\lambda$. The ${\sigma }_{\Delta A}$ are presented for the Heisenberg $XXZ$ ladder system for several system sizes $(L,N_{\uparrow })$ (see legend) and two different observables [in (a) and (b), respectively].

Figure 3

(a)–(c) Dependence of ${\sigma }_{\Delta A}$ on the Hilbert-space dimension $D$ in the Heisenberg ladder. The different operators $\widehat{A}$ used are indicated in the legend for (d). The lines are best fits to $c_0{D}^{-e}$; the exponent estimator $e$ is indicated for the operator $C_{2,p+2}^z$. (d) The exponent estimator $e$ is plotted against $\lambda$ for the four operators. (e) Sizes up to $L_{\max }$ are used for fitting to obtain the estimator $e$. The trend is that $e\to \frac{1}{2}$ for increasing $L_{\max }$.

Figure 4

For the $XXZ$ chain in a trap, we plot (a) the exponent estimator $e$ against integrability-breaking parameter $\lambda$. The estimator $e$ is calculated with data up to $L_{\max }=18$. The estimator $e$ is calculated with data up to $L_{\max }$ sites. (b) Plot of the estimator $e$ obtained from fits with increasing $L_{\max }$. Analogous results for the Bose Hubbard chain are shown in (c) (with $L_{\max }=12$) and (d).

Figure 5

(a) Scaled participation ratios of all eigenstates with respect to the computational basis for 15-site ladder, with $\lambda =1$. The averaging region is shown by vertical bars and the average PR value is shown by the horizontal line. (b) Dependence of the scaled average PR on $D$. (c) Dependence of ${\sigma }_{\Delta A}$ on the number $D_{\mathrm{fluct}}$ of states used as input to compute this quantity. Data are for the 17-site ladder with $\widehat{A}=C_{2,p+2}^z$ for four different values of $\lambda$ [shown in the legend in (b)].

Figure 6

(a) Fluctuation amplitudes as a function of $D$ for several widths $\Delta E$ of the microcanonical window. The value $\Delta E/L=0.025$ (black curve) is used for the analysis in the rest of this work. The values $0.001$ and $0.25$ are extreme cases. (b) Average value ${\langle \Delta A_{\alpha }\rangle }_{\mathrm{c}}^2$ divided by ${\langle \Delta A_{\alpha }^2\rangle }_{\mathrm{c}}$ as a function of Hilbert-space dimension $D$ for the ladder system with $\widehat{A}=S_2^z$ for several values of $\lambda$. The lines connecting the points serve as a guide to the eye.

Figure 7

Scaled participation ratios $P_{\alpha }/D$ as a function of $E_{\alpha }$ in the Heisenberg $XXZ$ ladder system. We show the results for system sizes $L=11,13,15$ and two values of the control parameter: (a)–(c) $\lambda =0$ and (d)–(f) $\lambda =1$. The horizontal red line indicates the average scaled PR. The two vertical bars enclose the middle $20\%$ of the spectrum, the energy region for which the average is taken. Here we have chosen the computational basis, which is also an eigenbasis for most of the observables discussed in the text (e.g., $S_i^z$, $S_i^z{S}_{i+1}^z$).

Figure 8

Histograms of the fluctuations $\Delta A_{\alpha }$ in a microcanonical window. (a) Ladder system with $(L,N_{\uparrow })=(19,9)$, $\widehat{A}=C_{2,p+2}^z$, and $\lambda =1$. (b) Magnetic-trap system with $(L,N_{\uparrow })=(21,7)$, $\widehat{A}=S_2^z$, and $\lambda =2$. (c) Bose-Hubbard system with $(L,N_{\mathrm{b}})=(14,7)$, $\widehat{A}=b_2^{†}{b}_3+b_3^{†}{b}_2$, and $\lambda =0.5$. For each system we show the distributions at the three energies defined by $E/L=-0.1,0,0.1$. The dashed curves indicate normal distributions with the same variances as those of the fluctuations $\Delta A_{\alpha }$.