Eigenstate Thermalization Hypothesis (ETH)

• ETH is a principle in quantum statistical mechanics where eigenstate observables match smooth microcanonical ensemble averages at corresponding energies.
• It predicts that differences between adjacent eigenstate observables decay exponentially with system size, affirming thermalization in nonintegrable many-body systems.
• Periodically driven systems show enhanced thermalization, as the removal of energy conservation leads to narrower distributions of ETH indicators.

The Eigenstate Thermalization Hypothesis (ETH) is a central organizing principle in the theory of quantum statistical mechanics. It posits that, for a broad class of isolated, nonintegrable quantum many-body systems, the stationary values of few-body observables in highly-excited energy eigenstates coincide with ensemble (thermal) averages at the appropriate energy density. In this strong sense, every physically relevant eigenstate is "thermal," and thus ETH provides a fundamental mechanism by which irreversibility and equilibrium arise from microscopic unitary dynamics. Conversely, the rigorous validity of ETH in nonintegrable systems, and the behavior of possible outlier states or finite-size effects, are crucial for assessing the limits of statistical mechanics in quantum systems.

1. Formal Statement and Significance of ETH

ETH asserts that, for generic nonintegrable many-body Hamiltonians $H$ and any few-body observable $\hat O$, the expectation value in an energy eigenstate $|n\rangle$ with eigenvalue $E_n$ satisfies

$\langle n| \hat O | n \rangle = O(E_n),$

where $O(E)$ is a smooth function of energy that matches the prediction of the microcanonical (or canonical) ensemble at the same energy density. In the strong formulation—investigated directly in (Kim et al., 2014)—this thermalization property holds for every eigenstate in the bulk of the spectrum (excluding at most spectral edges/noise).

The physical implications are profound: it means that even a pure initial state, as long as it is concentrated near a given energy density, will thermalize, with long-time averages of observables equaling equilibrium predictions. Thus, ETH solves the apparent conflict between unitary quantum evolution and the emergence of thermodynamics by embedding statistical behavior into the structure of eigenstates themselves.

2. Numerical Methodology: Model Systems and Observables

To scrutinize ETH in the strong sense, (Kim et al., 2014) employs exact diagonalization of two paradigmatic 1D nonintegrable systems:

• Quantum Ising Chain with Transverse and Longitudinal Fields:

$$H = \sum_i \left[ g \sigma^x_i + h \sigma^z_i + J \sigma^z_i \sigma^z_{i+1} \right]$$

with $(g, h, J) = (0.9045, 0.8090, 1)$. The nonintegrability here is robust. Observables analyzed include single-site $\sigma^x_1, \sigma^z_1$ and two-site operators $\sigma^x_1 \sigma^x_2$, $\sigma^y_1 \sigma^y_2$.

• Hard-Core Boson Model:

$$H = \sum_i \left[ -t (b^\dagger_{i+1} b_i + h.c.) + V n_i n_{i+1} \right] + \sum_i \left[ -t' (b^\dagger_{i+2} b_i + h.c.) + V' n_i n_{i+2} \right]$$

with $t = V = t' = V' = 1$ at half filling ($N/L = 1/2$). Relevant observables included two-site density correlators $n_1 n_2$ and real/imaginary parts of the hopping operators.

For each model, the Hamiltonians were block-diagonalized by momentum and numerically diagonalized to full precision for system sizes up to those accessible with available computational resources.

3. ETH Indicators and Outlier Analysis

To go beyond microcanonical averaging, the paper introduces the ETH indicator

$$r_n \equiv \langle n+1 | \hat O | n+1 \rangle - \langle n | \hat O | n \rangle,$$

which measures the difference in observable values between adjacent eigenstates. ETH predicts that $r_n$ must vanish as system size grows, since eigenvalue spacings diminish exponentially and observables become smooth functions of energy.

"Outlier" eigenstates are defined as those maximizing $|r_n|$—representing the strongest possible deviation from ETH for a given system size and observable.

Key numerical findings are:

• Exponential reduction of deviations: Even the largest $|r_n|$ (“extreme outliers”) shrink with increasing system size $L$; the mean deviation $\langle |r| \rangle$ decays as $\langle |r| \rangle \sim \exp(-\alpha L)$, consistent with a $D^{-1/2}$ scaling in Hilbert space dimension $D$.
• Properties of outliers: Outliers tend to have low participation ratios (i.e., less delocalized eigenvectors) and often appear in special momentum sectors ($k=0$ or $k=L/2$) due to additional symmetries. However, with increasing $L$, even these outliers approach thermal values, supporting strong ETH.
• Distribution collapse: The full distribution $P(|r|)$ peaks more sharply at zero as $L$ grows, confirming that almost all eigenstates become thermal in the thermodynamic limit.

4. Effects of System Size and Distribution Properties

System size scaling is pivotal for the ETH:

System Size $L$	Mean deviation $\langle|r|\rangle$	Maximum outlier $|r|$	$P(|r|)$ shape
Small	$O(10^{-2})$	Largest	Broad, non-Gaussian
Intermediate	$O(10^{-3})$	Moderate	Peaked, but finite
Large	$O(10^{-4})$ or smaller	Very small	Sharply peaked at 0

With increasing $L$, not only the variance but the largest outlier $|r|$ becomes negligible. Graphical plots in the paper show that the spread of both typical and extreme deviations collapses exponentially.

This scaling is essential—if outliers decreased only polynomially, a nonzero fraction of nonthermal states would remain at large $L$, violating strong ETH.

5. Periodic Driving and Enhanced Thermalization

The investigation also includes the effect of periodically driving the nonintegrable Ising system, which removes strict energy conservation and generates a Floquet operator: $\hat U = \exp(-i H_x \tau/2) \exp(-i H_z \tau/2)$ with $H_x = \sum_i g \sigma^x_i$ and $$H_z = \sum_i [ h \sigma^z_i + J \sigma^z_i \sigma^z_{i+1}]$$ and period $\tau$.

In this case:

• Floquet eigenstates thermalize to infinite-temperature predictions (i.e., $\langle \sigma^x_1 \rangle \to 0$).
• Both average and outlier deviations in $| \langle \sigma^x_1 \rangle |$ are further reduced compared to the undriven (energy-conserving) case.
• The removal of conserved quantities (here, energy) "improves" thermalization, reflected in a narrower distribution of ETH indicators.

This suggests that constraints from conserved quantities can inhibit thermalization, and their removal allows faster or "stronger" approach to ETH behavior.

6. Quantitative Evidence: Formulas and Numerical Trends

Quantitatively, the paper presents:

• The ETH indicator $r_n$ and its full distribution.
• Exponential fit of $\langle|r|\rangle \sim \exp(-\alpha L)$ (with $\alpha$ empirically extracted for each operator and model).
• For Floquet systems, reduced fluctuations in both mean and maximal deviations.

For example, for the Ising chain at $L=16$ versus $L=20$, plots of $\langle n | \hat O | n \rangle$ vs. energy density show ever-decreasing scatter around the thermal curve, and tabulated lists of outlier eigenstates verify that even the worst cases become negligible with growth in $L$.

Such trends are robust across both Ising and bosonic models and for various local and few-body observables.

7. Implications for Many-Body Quantum Thermalization

These results establish numerically that, for generic nonintegrable quantum systems in one dimension:

• Strong ETH holds: all eigenstates (in the spectral bulk) are thermal, including extreme outliers, as $L \to \infty$.
• The relaxation of observables from initial states with well-defined energy density leads to equilibrium values dictated by the microcanonical ensemble, with uniqueness guaranteed by ETH.
• Symmetry constraints (e.g., total momentum, or energy in driven/undriven cases) create weak, finite-size subleading corrections but do not invalidate thermalization in the thermodynamic limit.

In periodically driven (Floquet) systems, the strengthened ETH further implies that energy conservation is not essential for quantum systems to thermalize; thus, dynamical protocols can enhance the universality of equilibrium properties.

In summary, through high-precision studies of both typical and extreme behavior, exact diagonalization, and analysis of both static and periodically driven systems, strong ETH is robustly validated for nonintegrable models, implying the universality of thermalization in such quantum systems (Kim et al., 2014).