Off-diagonal matrix elements of local operators in many-body quantum systems

In the time evolution of isolated quantum systems out of equilibrium, local observables generally relax to a long-time asymptotic value, governed by the expectation values (diagonal matrix elements) of the corresponding operator in the eigenstates of the system. The temporal fluctuations around this value, response to further perturbations, and the relaxation toward this asymptotic value are all determined by the off-diagonal matrix elements. Motivated by this nonequilibrium role, we present generic statistical properties of off-diagonal matrix elements of local observables in two families of interacting many-body systems with local interactions. Since integrability (or lack thereof) is an important ingredient in the relaxation process, we analyze models that can be continuously tuned to integrability. We show that, for generic nonintegrable systems, the distribution of off-diagonal matrix elements is a Gaussian centered at zero. As one approaches integrability, the peak around zero becomes sharper, so the distribution is approximately a combination of two Gaussians. We characterize the proximity to integrability through the deviation of this distribution from a Gaussian shape. We also determine the scaling dependence on system size of the average magnitude of off-diagonal matrix elements.

Figure 1

Matrix structure of $|A_{\alpha \beta }|$ as function of $E_{\alpha }$ and $E_{\beta }$ for the observable $\widehat{A}=S_2^z{S}_{p+2}^z$. The diagonal matrix elements are ignored. The white bands near the edges are regions without eigenvalues. The dashed square indicates the central half of the energy range, i.e., $[\frac{3}{4}{E}_{\min }+\frac{1}{4}{E}_{\max },\frac{1}{4}{E}_{\min }+\frac{3}{4}{E}_{\max }]$ in each direction: This is the “bulk” of the spectrum on which we focus our analysis. The system size is $(L,N_{\uparrow })=(13,6)$; the Hilbert space dimension is $D=1716$. The unit of energy is set by the $xy$ coupling along the ladder legs.

Figure 2

Frequency-resolved analysis of off-diagonal elements for the $(L,N_{\uparrow })=(13,6)$ ladder. The shadings show the distribution of the values of $|A_{\alpha \beta }|$ for each frequency window; darker shading indicates more occurrences of respective $|A_{\alpha \beta }|$ values. The curves show $S_A$ as a function of the frequency $\omega$. In the bottom panel, we indicate the density of pairs $\tilde{\rho }$ for the central part of the spectrum (solid) and for the full spectrum (dashed), in arbitrary units. The marker on the horizontal axis points to the typical frequency ${\omega }_0$. The shading and solid curves are all results using the central part of the spectrum. The units of frequency are the same as the units used for energy in Fig. 1.

Figure 3

Histograms (shaded area) of the off-diagonal elements $A_{\alpha \beta }$ with [(a)–(c)] $E_{\beta }-E_{\alpha }\in (0,0.05{\omega }_0)$ and [(d)–(f)] $E_{\beta }-E_{\alpha }\in [0.25{\omega }_0,0.25{\omega }_0+\delta \omega )$, where $\delta \omega =0.05$. The solid curve is a Gaussian fit and the dashed curve is a fit of a mixture of two Gaussians, Eq. (8). The observable is $\widehat{A}=S_2^z{S}_{p+2}^z$ and the system size is $(L,N_{\uparrow })=(13,6)$. The number of state pairs in these histogram computations ranges from 4000 to $50000$.

Figure 4

Characteristics of the distribution of $A_{\alpha \beta }$. [(a) and (b)] Standard deviations ${\sigma }_1,\sigma$, and ${\sigma }_2$ of the “inner” Gaussian of Eq. (8), the full distribution, and the “outer” Gaussian, in increasing order. The thicker curve is $\sigma$. System size is $(L,N_{\uparrow })=(15,7)$. [(c) and (d)] Ratio ${\sigma }_1/{\sigma }_2$ and kurtosis $\kappa$. The dashed horizontal lines are the values for the Gaussian distribution $({\sigma }_2/{\sigma }_1=1,\kappa =3)$. We show results for $\widehat{A}=S_2^z$ in red (squares) and for $\widehat{A}=S_2^z{S}_{p+2}^z$ in blue (circles). [(e) and (f)] ${\sigma }_2/{\sigma }_1$ and $\kappa$ as a function of the Hilbert-space dimension $D$ for several values of $\lambda$. In (e), data for the smallest system size for $\lambda =5$ is absent—the procedure does not yield a solution for ${\sigma }_i$ due to the low density of states.

Figure 5

System-size scaling analysis of average $|A_{\alpha \beta }{|}^2$ through the quantities (a) ${\overline{\gamma }}_{\mathrm{low}}$, (b) ${\overline{\gamma }}_{\mathrm{all}}$, and (c) ${\overline{\gamma }}_{\mathrm{low}}/{\overline{\gamma }}_{\mathrm{all}}$ for the observable $\widehat{A}=S_2^z$ for several values of $\lambda$. The respective results for $\widehat{A}=S_2^z{S}_{p+2}^z$ are shown in panels (d)–(f). The dotted lines in (a), (b), (d), and (e) are $\overline{\gamma }=1/D$.

Figure 6

Bose-Hubbard model; observable $\widehat{A}=b_2^{†}{b}_3+b_3^{†}{b}_2$. [(a)–(c)] Frequency dependence shown through $S_A$, as in Fig. 2. Insets show fragments of the density plot of $|A_{\alpha \beta }|$ as function of $E_{\alpha }$ and $E_{\beta }$, as in Fig. 1. The system size is $(L,N_b)=(7,7)$. [(d) and (e)] Analysis of the shape of the distribution through ${\sigma }_2/{\sigma }_1$ and kurtosis $\kappa$, as in Fig. 4. [(f)–(h)] Average matrix element ${\overline{\gamma }}_{\mathrm{low}}$ and ${\overline{\gamma }}_{\mathrm{all}}$ for low and all frequencies, and their ratio, as in Fig. 5. In (f) and (g), the dotted lines are $\overline{\gamma }=4/D$.