Eigenstate Thermalization for Local versus Translationally Invariant Observables

TL;DR

The work analyzes the eigenstate thermalization hypothesis (ETH) for local and translationally invariant observables in a nonintegrable spin-1 tilted-field Ising chain under open and periodic boundary conditions. It shows that diagonal ETH functions can coincide for a center-site local observable and its TI sum, yet their spectral functions differ due to inter-site correlations among off-diagonal matrix elements, with the difference characterized by a decomposition into contributions labeled by $\Delta_\ell=\tfrac{2\pi}{L}\ell$. For periodic systems, translational symmetry confines TI matrix elements to blocks with $\kappa_{mn}=|k_m-k_n|=0$, while local observables sample nonzero $\kappa_{mn}$, and the authors formulate an extended ETH that explicitly includes $\kappa_{mn}$, $\bar{E}_{mn}$, and $\omega_{mn}$. This extension demonstrates that TI and local spectral functions generally differ, clarifies when one can replace local by TI observables, and suggests computational simplifications by focusing on quasimomentum blocks. Overall, the results reveal new off-diagonal ETH structure in translationally invariant systems and quantify when boundary conditions and correlations affect ETH predictions.

Abstract

Local observables and their translationally invariant counterparts are generally thought as providing the same predictions for experimental measurements. This is used in the context of their expectation values, which are indeed the same in clean systems (up to finite-size effects), but also in the context of their correlation functions, which need not be the same. We examine this intuition from the perspective of the eigenstate thermalization hypothesis. Specifically, we explore the diagonal matrix elements and the spectral functions of local and translationally invariant observables in the spin-1 tilted field Ising chain with periodic and open boundary conditions. We discuss in which ways those observables are different and in which contexts they can be thought as being the same. Furthermore, we unveil a novel form of off-diagonal eigenstate thermalization in translationally invariant systems that applies to pairs of energy eigenstates with different quasimomenta.

Figures (7)

• Figure 1: Diagonal matrix elements $O_{mm}$ vs the energy density $E_m/L$ for the local observable $\hat{O}^j=\hat{S}_x^j$ at site $j=\tfrac{L}{2}$ for OBCs, and its translationally invariant counterpart $\hat{\mathscr{O}}=\tfrac{1}{L}\sum_j\hat{S}_x^j$ for OBCs and PBCs, in chains with $L=10$. (Inset) Relative difference $\Delta O_{L}(\beta)$ between $O_{L}(\beta)$ in a chain with $L$ sites and its thermodynamic limit value $O_\text{TL}(\beta)$ for $\beta=0.5$, see text. $O_\text{TL}(\beta)$ is obtained fitting the PBC results for the largest four chains to $O_{L}(\beta)=O_\text{TL}(\beta)+ae^{-bL}$ with fitting parameters $O_\text{TL}(\beta),\ a$, and $b$. That value is used for OBCs to fit $\Delta O_{L}(\beta)$ for $\hat{O}^j$ to $ae^{-bL}$, and for $\hat{\mathscr{O}}$ to $aL^{-b}$, with fitting parameters $a$ and $b$. The fits are shown as lines.
• Figure 2: Spectral functions for the local observable $\hat{O}^j=\hat{S}_x^j$ at site $j=\lceil L/2\rceil$ and its translationally invariant counterpart $\hat{\mathscr{O}}=\tfrac{1}{L}\sum_j \hat{S}_x^j$ in chains with OBCs \ref{['eq:H_OBC']}. (Inset) Contributions to $|f_{\mathscr{O}}(E_\infty,\omega)|^2$, see Eq. \ref{['eq:termsinTI']}, from terms with $|j-l|=0$ through 5 ($d_0$ through $d_5$ in the legend) in chains with $L=10$.
• Figure 3: Spectral functions for: (a) $\hat{O}^j=\hat{S}_x^j$ at site $j=L/2$, and (b) its translationally invariant counterpart $\hat{\mathscr{O}}=\tfrac{1}{L}\sum_j \hat{S}_x^j$ in chains with PBCs and OBCs, and $L=10$.
• Figure 4: (a) Spectral functions for $\hat{O}^j=\hat{S}_x^j$ at site $j=\lceil L/2\rceil$ and its translationally invariant counterpart $\hat{\mathscr{O}}=\tfrac{1}{L}\sum_j \hat{S}_x^j$ in chains with PBCs \ref{['eq:H_PBC']}. (inset) $|f_{\mathscr{O}}(E_\infty,\omega)|^2$ in chains with OBCs and PBCs for $L=9,\,10$. (b) Contributions to $|f_{O^j}(E_\infty,\omega)|^2$ from the six terms with different values of $\Delta_\ell$ in Eq. \ref{['eq:PBC-local-spectral-function']} for $L=11$. Open squares represent the rescaled ETH functions $\tfrac{1}{L}|\mathrm{f}_{O^j}|^2$ ($\tfrac{2}{L}|\mathrm{f}_{O^j}|^2$) in Eq. \ref{['eq:ETH_TI']} calculated using 100 pairs of states with $\bar{E}_{mn}\simeq E_\infty$ and $\omega_{mn}\simeq\omega$ in blocks with quasimomenta $k_m=\tfrac{2\pi}{L}$ and $k_n=\tfrac{2\pi}{L}$ ($k_n=\tfrac{4\pi}{L},\,\tfrac{6\pi}{L}$).
• Figure S1: Spectral functions for the local observable $\hat{O}^j=\hat{S}_x^j\hat{S}_x^{j+1}$ with $j=\lceil\tfrac{L}{2}\rceil$ and its translationally invariant counterpart $\hat{\mathscr{O}}=\tfrac{1}{N}\sum_{j=1}^N \hat{S}_x^j\hat{S}_x^{j+1}$ in chains with: (a) OBCs [see Eq. \ref{['eq:H_OBC']}] for which $N=L-1$, and (b) PBCs [see Eq. \ref{['eq:H_PBC']}] for which $N=L$. Inset in (a): Contributions to $|f_{\mathscr{O}}(E_\infty,\omega)|^2$, see Eq. \ref{['eq:termsinTI']} where one needs to change $L\rightarrow L-1$, from terms with $|j-l|=0$ through 5 ($d_0$ through $d_5$ in the legend) in chains with $L=10$. Inset in (b): Contributions to $|f_{O^j}(E_\infty,\omega)|^2$ from different values of $\Delta_\ell$ in Eq. \ref{['eq:PBC-local-spectral-function']} for $L=11$.