ETH-monotonicity in two-dimensional systems

TL;DR

The paper investigates ETH-monotonicity in two-dimensional quantum chaotic systems by analyzing the $f$-function arising in ETH matrix elements for two lattice models: hard-core bosons and the 2D transverse-field Ising model. Using exact diagonalization, it shows that $f(\bar{E})$ increases with the effective temperature $|T|$ and that its flattening rate scales with area $A \sim L^2$, and with particle number $N$ at fixed volume, while $f(\beta)$ remains non-flattening due to the intensive nature of $\beta$. The results indicate that the ETH-monotonicity phenomenon extends to higher dimensions, with energy-extensivity and local interactions underpinning the observed scaling, and reveal that the transformation of $f$ with system size is not a simple rescaling in the $(\bar{E}, \omega)$ plane. These findings provide insight into the dimensional dependence of thermalization and have implications for scaling of thermal properties in larger 2D quantum systems.

Abstract

We study a recently discovered property of many-body quantum chaotic systems called ETH-monotonicity in two-dimensional systems. Our new results further support ETH-monotonicity in these higher dimensional systems. We show that the flattening rate of the $f$-function is directly proportional to the number of degrees of freedom in the system, so as $L^2$ where $L$ is the linear size of the system, and in general, expected to be $L^d$ where $d$ is the spatial dimension of the system. We also show that the flattening rate is directly proportional to the particle (or hole) number for systems of same spatial size.

Figures (8)

• Figure 1: Plots showing the monotonically increasing behaviour of $f(T)$ as a function of $T$. Top panels: Plots for the kinetic energy operator $\mathcal{O}_{B1}$. Bottom panels: Plots for the occupation number operator at a single site $\mathcal{O}_{B2}$.
• Figure 2: Left panel: Plot of $1/f df/d\bar{E}$ for different system sizes showing that $f(\bar{E})$ flattens as the system size increases. Right panel: Plot of $1/f df/d\beta$ for different system sizes showing that $f(\beta)$ does not flatten as the system size increases. Both panels are for the kinetic energy operator $\mathcal{O}_{B1}$.
• Figure 3: Slopes of linear-fit of $f'(\bar{E})/f(\bar{E})$ as a function of $\omega$ scaled with different powers of the area $A$ of the lattice for the kinetic energy operator $\mathcal{O}_{B1}$. The flattening rate of $f(\bar{E})$ is proportional to $A$.
• Figure 4: Slopes of linear-fit of $f'(\bar{E})/f(\bar{E})$ as a function of $\omega$ scaled with different powers of the total particle number $N$ for the kinetic energy operator $\mathcal{O}_{B1}$. The flattening rate of $f(\bar{E})$ is proportional to $N$.
• Figure 5: Total magnetization $M_z=\langle \mathcal{O}_{S2} \rangle$ of energy eigenstates for different values of the transverse magnetic field $g$.