Eigenstate Thermalization Hypothesis

Why can one highly excited pure energy eigenstate reproduce thermal measurements? The eigenstate thermalization hypothesis, or ETH, answers by proposing a special structure for experimentally simple observables in the energy eigenbasis of a generic interacting many-body Hamiltonian.

Diagonal ETH says that nearby eigenstates give nearly the same local measurements; off-diagonal ETH describes the small matrix elements governing temporal fluctuations and equilibrium correlations. Neither identifies a globally pure eigenstate with a mixed thermal density operator.

ETH is one of the central organizing ideas for thermalization in isolated quantum matter. It is also a hypothesis with a precise domain, not a theorem that every nonintegrable Hamiltonian must satisfy.

Helpful background. Relaxation and thermalization separates small temporal fluctuations from agreement with an ensemble; the microcanonical ensemble provides the energy-shell comparison.

Eigenstate thermalization and its origins

Deutsch showed in 1991 that weak generic mixing of an integrable many-body basis can make eigenstate expectation values approach microcanonical predictions with exponentially small deviations. Srednicki connected this mechanism to chaotic eigenfunctions in 1994, introduced the phrase “eigenstate thermalization,” and later formulated the matrix-element ansatz now commonly called ETH.

The modern many-body program accelerated after numerical work by Rigol, Dunjko, and Olshanii showed that individual eigenstates of a nonintegrable lattice model reproduce thermal values while the corresponding integrable model does not. Subsequent work separated several logically different claims:

• almost-all-eigenstate statements;
• every-eigenstate statements;
• off-diagonal matrix-element statistics;
• reduced-density-matrix statements;
• dynamical consequences for specified initial states.

Keeping these versions distinct prevents a great deal of confusion.

Energy sectors and local observables

Consider a sequence of finite systems labeled by volume $V$,

$$H_V|n\rangle = E_n|n\rangle.$$

The thermodynamic question concerns a limit such as

$$\begin{gathered} V\to\infty \\ \text{at fixed energy density}, \qquad e=\frac{E}{V}. \end{gathered}$$

ETH is not a property of an unlabeled finite matrix. A meaningful claim must specify:

1. a family of Hamiltonians $H_V$;
2. an exact symmetry sector;
3. an energy-density interval;
4. a class of observables;
5. a norm or statistical measure of deviations;
6. a thermodynamic limiting procedure.

Resolve exact sectors first

If

$$[H_V,Q_a]=0,$$

then the energy eigenbasis decomposes into sectors labeled by the conserved quantum numbers $q_a$. ETH must be tested within one irreducible sector, or within a physically declared mixture of sectors.

Combining sectors can produce:

• exact crossings mistaken for anomalous level statistics;
• branches with different conserved densities;
• duplicated eigenstate expectation values;
• apparent outliers that are ordinary sector dependence.

Particle number, magnetization, momentum, parity, reflection, and gauge constraints supply common restrictions. Noncommuting symmetry generators cannot all be simultaneously diagonalized: use an appropriate commuting set and irreducible representations. The relevant decomposition is developed in Number Operators and Conserved Quantities.

For an operator that changes a symmetry charge, all diagonal elements in a fixed-charge basis can vanish identically. This alone is a trivial test of ETH. Its off-diagonal correlations involve different sectors and their respective densities of states. The single-sector ansatz below applies to the sector-preserving block; sector-changing blocks need that extension.

Choose a bulk energy window

Let

$$\mathcal W_V(E,\Delta E) = \left\{ n: |E_n-E|\le\frac{\Delta E}{2} \right\}$$

inside one sector, and let

$$D_{\mathcal W} = \left| \mathcal W_V(E,\Delta E) \right|$$

be the number of states in the window.

The window should contain many levels while remaining narrow on thermodynamic scales:

$$D_{\mathcal W}\gg1, \qquad \frac{\Delta E}{V}\to0.$$

One should generally avoid spectral edges unless edge behavior is the subject of the study. Ground states, low-lying quasiparticles, mobility edges, and phase-coexistence windows need separate analysis.

Declare the observable class

Standard ETH concerns local, few-body, or suitably normalized sums of local observables. Examples include

$$\begin{gathered} Z_j, \qquad Z_jZ_{j+1}, \\ \frac{1}{V}\sum_j Z_j, \qquad \frac{1}{V}\sum_j h_j. \end{gathered}$$

The normalization matters. An extensive operator and its density have different finite-size scaling.

ETH cannot hold uniformly for every operator. The eigenprojector

$$P_n=|n\rangle\langle n|$$

distinguishes one eigenstate perfectly and is manifestly nonlocal in the physical degrees of freedom. A statement that omits the observable class is therefore incomplete.

The Microcanonical Benchmark

For an observable $A$, define its eigenstate expectation values

$$A_{nn} \equiv \langle n|A|n\rangle$$

and the shell average

$$A_{\mathrm{mc}}(E,\Delta E) = \frac{1}{D_{\mathcal W}} \sum_{n\in\mathcal W} A_{nn}.$$

The microcanonical variance is

$$\sigma_{A,\mathrm{mc}}^2 = \frac{1}{D_{\mathcal W}} \sum_{n\in\mathcal W} \left| A_{nn}-A_{\mathrm{mc}} \right|^2.$$

At finite size, neither the variance nor the largest deviation is generally zero. ETH is a scaling statement about what happens as $V$ increases.

Diagonal ETH

The diagonal statement is that eigenstate expectation values of a simple observable become a smooth thermodynamic function of energy density:

$$A_{nn} = \mathcal A(E_n) + \delta A_{nn},$$

where $\mathcal A(E)$ varies smoothly on the scale occupied by the initial state and $\delta A_{nn}$ becomes small in the relevant thermodynamic sense.

Equivalently, a single typical eigenstate in a narrow shell satisfies

$$\langle n|A|n\rangle \simeq A_{\mathrm{mc}}(E_n,\Delta E).$$

This is the sense in which an eigenstate is “thermal.” It concerns the declared observables, not equality of global density matrices.

Weak ETH

Weak ETH requires the fraction of nonthermal eigenstates in a shell to vanish. A common variance formulation is

$$\lim_{V\to\infty} \sigma_{A,\mathrm{mc}}^2 = 0.$$

For any fixed tolerance $\epsilon>0$, define

$$\begin{aligned} \mathcal B_\epsilon^{(V)} &= \left\{ n\in\mathcal W: \left|A_{nn}-A_{\mathrm{mc}}\right|>\epsilon \right\}, \\ f_{\mathrm{bad}}^{(V)}(\epsilon) &= \frac{\left|\mathcal B_\epsilon^{(V)}\right|} {D_{\mathcal W}}. \end{aligned}$$

Chebyshev’s inequality gives

$$f_{\mathrm{bad}}^{(V)}(\epsilon) \le \frac{ \sigma_{A,\mathrm{mc}}^2 }{ \epsilon^2 }.$$

Thus vanishing shell variance implies that almost all eigenstates are thermal for $A$. It does not control the most extreme outlier.

Strong ETH

Strong ETH asks that every eigenstate in the specified bulk window become thermal:

$$r_{A,\max}^{(V)} \equiv \max_{n\in\mathcal W} \left| A_{nn}-A_{\mathrm{mc}} \right| \longrightarrow 0.$$

Strong ETH implies weak ETH. The converse is false: a vanishing fraction of order-one outliers contributes negligibly to the variance but keeps the maximum deviation finite.

Different authors use “strong ETH” for closely related but not identical statements. Always report the norm, window, observable set, and order of limits instead of relying on the label alone.

The Full Matrix-Element Ansatz

Set $\hbar=k_{\mathrm B}=1$ from here through the remaining formulas. Thus $\omega=E_m-E_n$ is an energy difference and also the angular frequency in these units. Take $A$ to be Hermitian; assume unresolved exact degeneracies have been excluded, or replace individual eigenvectors by an explicitly specified invariant-block formulation. For

$$A_{mn} \equiv \langle m|A|n\rangle,$$

define

$$\bar E = \frac{E_m+E_n}{2}, \qquad \omega = E_m-E_n.$$

The standard ETH ansatz is

$$\boxed{ A_{mn} = \mathcal A(\bar E)\delta_{mn} + e^{-S(\bar E)/2} f_A(\bar E,\omega) R_{mn}. }$$

Here:

• $\mathcal A(\bar E)$ is the smooth thermal expectation value;
• $S(\bar E)$ is the dimensionless thermodynamic entropy of the relevant sector;
• $f_A(\bar E,\omega)$ is a smooth operator-dependent spectral envelope;
• $R_{mn}$ has zero local mean and order-one local variance after appropriate binning.

The ansatz is statistical. It does not say that an individual deterministic Hamiltonian contains literal independent random numbers.

Entropy and the density of states

Let $\rho(E)$ be a smoothly binned density of states in one sector, measured per unit energy. Choose a fixed reference energy $E_{\mathrm{ref}}>0$ and define the entropy convention by

$$e^{S(E)}=\rho(E)E_{\mathrm{ref}},$$

so the logarithm has a dimensionless argument. In a many-body bulk window,

$$S(E) \sim V\,s(e).$$

Therefore

$$e^{-S(E)/2} \sim e^{-V s(e)/2}$$

is exponentially small in volume.

Changing $E_{\mathrm{ref}}$ rescales $f_A$ so that $e^{-S/2}f_A$ is unchanged. The bin width used to estimate $\rho(E)$ should contain many levels while resolving the smooth density; it is not an extra physical energy scale. We keep $E_{\mathrm{ref}}$ explicit in the spectral formula below.

Why the amplitude is entropy suppressed

The entropy factor has a simple counting motivation. Project $A$ into a shell of dimension

$$D_{\mathcal W} \sim e^S.$$

Let $P_{\mathcal W}$ project onto that shell and define $A_{\mathcal W}=P_{\mathcal W}AP_{\mathcal W}$. For a bounded local observable, its projected shell-averaged Hilbert–Schmidt weight is at most order one:

$$\frac{1}{D_{\mathcal W}} \operatorname{Tr}_{\mathcal W} \left( A_{\mathcal W}^\dagger A_{\mathcal W} \right) \leq\|A\|_\infty^2=O(1).$$

Expanding the trace,

$$\frac{1}{D_{\mathcal W}} \sum_{m,n\in\mathcal W} |A_{mn}|^2 = O(1).$$

The projection matters: $\operatorname{Tr}(P_{\mathcal W}A^\dagger A P_{\mathcal W})$ would also include transitions to states outside the shell. Suppose the off-diagonal weight within the shell is itself of order $D_{\mathcal W}$ and is broadly distributed over order $D_{\mathcal W}^2$ entries. Its typical squared element then scales as

$$\overline{|A_{mn}|^2} \sim \frac{1}{D_{\mathcal W}} \sim e^{-S}.$$

Hence a typical amplitude scales as $e^{-S/2}$. This counting argument motivates the entropy factor; it does not prove smoothness, Gaussianity, or independence.

The diagonal correction

For $m=n$, the full ansatz gives

$$A_{nn} = \mathcal A(E_n) + e^{-S(E_n)/2} f_A(E_n,0)R_{nn}.$$

The smooth term is order one for a local observable, while eigenstate-to-eigenstate fluctuations are exponentially small in the idealized full-ETH scaling regime. Finite-size systems can show sizeable corrections, algebraic prefactors, long tails, or crossover behavior.

Hermiticity constraints

For Hermitian $A$,

$$A_{nm} = A_{mn}^*.$$

The random-looking factors and smooth envelope must respect this relation. One convenient convention imposes

$$R_{nm} = R_{mn}^*,$$

with $f_A(\bar E,-\omega)=f_A(\bar E,\omega)^*$. A common choice makes $f_A$ real and even and absorbs the remaining phases into $R_{mn}$. Treating all $R_{mn}$ as unrelated samples would violate the operator’s exact algebra.

Smooth does not mean constant

The envelope $f_A(\bar E,\omega)$ may contain physically important frequency structure:

• microscopic resonances;
• transport peaks;
• hydrodynamic singularities or long-time tails;
• gaps and thresholds;
• selection-rule zeros.

ETH does not erase dynamics. It organizes dynamics into smooth thermodynamic functions and entropy-scaled fluctuations.

From Diagonal ETH to Thermal Values

Let the initial pure state be

$$|\psi_0\rangle = \sum_n c_n|n\rangle, \qquad p_n=|c_n|^2.$$

After dephasing, the long-time expectation value is

$$\overline{\langle A(t)\rangle} = \sum_n p_n A_{nn}.$$

Insert diagonal ETH:

$$\overline{\langle A(t)\rangle} = \sum_n p_n\mathcal A(E_n) + \sum_n p_n\delta A_{nn}.$$

Define

$$\begin{aligned} \bar E &= \sum_n p_nE_n, \\ (\Delta E)^2 &= \sum_n p_n(E_n-\bar E)^2. \end{aligned}$$

If the occupied energy density is narrow,

$$\frac{\Delta E}{V} \longrightarrow 0,$$

then a Taylor expansion is useful when the derivatives are controlled on the populated window and higher moments make its remainder small:

$$\sum_n p_n\mathcal A(E_n) = \mathcal A(\bar E) + \frac{1}{2} \mathcal A''(\bar E) (\Delta E)^2 +\cdots.$$

The linear term vanishes by definition of $\bar E$. For a local observable whose thermodynamic value is a smooth function of $e=E/V$,

$$\mathcal A''(E) = O(V^{-2}),$$

away from singular points. Thus a subextensive energy width can make the curvature correction vanish.

The ETH fluctuation term obeys the elementary bound

$$\left| \sum_n p_n\delta A_{nn} \right| \le \max_{n:p_n\ne0} |\delta A_{nn}|.$$

Strong ETH makes this small for every narrow-energy preparation in the window. Weak ETH requires an additional condition: the initial state must not place appreciable weight on the rare nonthermal set.

Three sources of error in a thermal mean

A useful decomposition is

$$\overline{\langle A\rangle} - A_{\mathrm{mc}}(\bar E) = \epsilon_{\mathrm{width}} + \epsilon_{\mathrm{ETH}} + \epsilon_{\mathrm{shell}}.$$

Here:

• $\epsilon_{\mathrm{width}}$ measures variation of the smooth function over the occupied energies;
• $\epsilon_{\mathrm{ETH}}$ measures eigenstate fluctuations or rare-state weight;
• $\epsilon_{\mathrm{shell}}$ measures the finite-window convention and finite-size microcanonical error.

The three terms distinguish broad initial-energy support, nonthermal eigenstates, and a poorly resolved comparison shell.

From microcanonical to canonical

ETH directly identifies eigenstates with an energy-matched microcanonical prediction. Replacing that prediction by

$$\rho_\beta = \frac{e^{-\beta H}}{Z}$$

requires ensemble equivalence for the chosen observable and thermodynamic regime. ETH alone does not prove canonical–microcanonical equivalence, especially at phase coexistence, for long-range interactions, or for macroscopic observables.

Off-Diagonal ETH and Temporal Fluctuations

The exact time-dependent expectation value is

$$\langle A(t)\rangle = \sum_{m,n} c_m^*c_n e^{i(E_m-E_n)t} A_{mn}.$$

Subtract its dephased value:

$$\delta A(t) = \sum_{m\ne n} c_m^*c_n e^{i(E_m-E_n)t} A_{mn}.$$

If the relevant energy gaps are nondegenerate, the infinite-time variance is

$$\overline{|\delta A(t)|^2} = \sum_{m\ne n} p_mp_n |A_{mn}|^2.$$

Off-diagonal ETH gives the scale

$$|A_{mn}|^2 \sim e^{-S(\bar E)} |f_A(\bar E,\omega)|^2.$$

If an exponentially small uniform upper bound on $|A_{mn}|^2$ holds throughout the occupied window, then the variance is exponentially small for every set of probabilities, because $\sum_{m\ne n}p_mp_n\leq1$. A broad initial state is not necessary for that argument. A typical-size statement from ETH is weaker: it must also exclude significant weight on anomalously large entries.

Important qualifications remain:

• degeneracies and repeated gaps alter the time-average formula;
• typical matrix-element scaling does not exclude specially weighted anomalous entries; a single eigenstate is stationary and already has zero temporal variance;
• special phases among coefficients can create long transients;
• low-frequency hydrodynamic structure can produce slow relaxation;
• recurrence is not removed in a finite spectrum.

ETH is therefore part of a thermalization argument, not a substitute for every dynamical hypothesis.

One Eigenstate Can Be Thermal Without Relaxing

An energy eigenstate evolves only by a phase:

$$e^{-iHt}|n\rangle = e^{-iE_nt}|n\rangle.$$

Its observable expectation value is stationary:

$$\langle n|A(t)|n\rangle = A_{nn}.$$

If diagonal ETH holds, that stationary value is thermal for $A$. There is no relaxation because the state was already locally thermal.

Conversely, a superposition can equilibrate by dephasing even when diagonal ETH fails. It may settle to a nonthermal diagonal ensemble. This is why equilibration and eigenstate thermalization are distinct claims.

Correlations and the Spectral Envelope

The off-diagonal envelope is connected to equilibrium correlation functions for Hermitian, sector-preserving $A$. For a single eigenstate $|n\rangle$, define the connected correlator

$$C_n(t) = \langle n| A(t)A(0) |n\rangle - A_{nn}^2.$$

Its spectral representation is

$$C_n(t) = \sum_{m\ne n} e^{-i(E_m-E_n)t} |A_{mn}|^2.$$

With

$$C_n(\omega) = \int_{-\infty}^{\infty} dt\, e^{i\omega t} C_n(t),$$

one obtains

$$C_n(\omega) = 2\pi \sum_{m\ne n} |A_{mn}|^2 \delta \left( \omega-E_m+E_n \right).$$

At finite size the spectrum is a sum of delta functions, so a ratio at an arbitrary point is not a smooth function. Average over frequency bins containing many transitions but narrow on the envelope’s variation scale, then replace the sum by $\rho(E_n+\omega)d\omega$. With the entropy convention above, ETH gives

$$C_n(\omega) \simeq \frac{2\pi}{E_{\mathrm{ref}}} \times e^{ S(E_n+\omega) - S(E_n+\omega/2) } \times \left| f_A(E_n+\omega/2,\omega) \right|^2.$$

For $\omega$ small on thermodynamic scales,

$$S(E_n+\omega) - S(E_n+\omega/2) \simeq \frac{\beta\omega}{2},$$

so

$$C_n(\omega) \simeq \frac{2\pi}{E_{\mathrm{ref}}} e^{\beta\omega/2} |f_A(E_n,\omega)|^2.$$

For Hermitian $A$ and the corresponding envelope symmetry,

$$\frac{C_n(-\omega)}{C_n(\omega)} \simeq e^{-\beta\omega}.$$

This is the smoothed thermodynamic ETH prediction for detailed balance wherever the denominator is nonzero. It is not an exact KMS identity for one finite-system eigenstate. Exact conventions depend on Fourier signs, entropy coarse graining, and whether a connected or symmetrized correlator is used. Time-Dependent Correlations, Spectral Functions, and the Fluctuation–Dissipation Theorem own those conventions and consequences.

Pairwise ETH is not the whole dynamics

The standard ansatz controls one- and two-eigenstate data: diagonal values and pairwise matrix elements. Operator spreading, entanglement growth, out-of-time-order correlators, and other higher-point quantities can depend on correlations among four or more eigenstates.

Thus:

$$\begin{gathered} \text{standard ETH} \nRightarrow \\ \text{complete statistics of chaotic dynamics}. \end{gathered}$$

Modern extensions study these multi-eigenstate correlations. Many-Body Quantum Chaos Preview and Scrambling and OTOCs Preview discuss the associated higher-point correlations.

Subsystem ETH

Let the system factor as

$$\mathcal H = \mathcal H_A\otimes\mathcal H_B,$$

and define an eigenstate reduced density operator

$$\rho_A^{(n)} = \operatorname{Tr}_B |n\rangle\langle n|.$$

The reduced microcanonical state is

$$\rho_{A,\mathrm{mc}} = \operatorname{Tr}_B \rho_{\mathrm{mc}}.$$

Subsystem ETH asks, under specified scaling of $A$ relative to the full system, that

$$D\left( \rho_A^{(n)}, \rho_{A,\mathrm{mc}} \right) \longrightarrow 0,$$

where

$$D(\rho,\sigma) = \frac12 \|\rho-\sigma\|_1.$$

This is stronger than agreement for one observable. It implies agreement for every bounded measurement confined to subsystem $A$:

$$\left| \operatorname{Tr} \left[ O_A \left( \rho_A^{(n)} - \rho_{A,\mathrm{mc}} \right) \right] \right| \le 2 \|O_A\|_\infty D\left( \rho_A^{(n)}, \rho_{A,\mathrm{mc}} \right).$$

The subsystem limit matters

The cleanest formulation holds for fixed subsystem size while

$$V\to\infty.$$

If

$$f = \frac{V_A}{V}$$

remains finite, global purity, energy conservation, and complement constraints matter at leading or subleading order. A half-system reduced state is not governed by the same approximation as a fixed local region.

Subsystem ETH can imply a volume-law eigenstate entanglement entropy with thermal leading coefficient. That derivation belongs to Thermal Entropy versus Entanglement Entropy and Volume Laws.

ETH, Typicality, and Ensemble Equivalence

Three statements are often blended together:

Canonical typicality

Most randomly sampled pure states in a high-dimensional constrained shell are locally close to the shell state.

Eigenstate thermalization

The specially selected vectors that diagonalize a physical Hamiltonian are locally thermal and have structured matrix elements.

Ensemble equivalence

Different mixed equilibrium ensembles give the same predictions for a declared observable class in a thermodynamic limit.

None follows from the definition of another. Their conclusions can coincide in generic systems, but their measures, assumptions, and failure modes differ.

A Concrete Diagnostic Model

A standard nonintegrable testbed is the mixed-field Ising chain

$$H = J \sum_{j=1}^{L-1} Z_jZ_{j+1} + h_x \sum_{j=1}^{L} X_j + h_z \sum_{j=1}^{L} Z_j.$$

For generic nonzero $J$, $h_x$, and $h_z$, this is a nonintegrable testbed. The longitudinal field both breaks spin-flip symmetry and spoils the quadratic Jordan–Wigner solution of the transverse-field model. The symmetry itself is not the cause of integrability: many nonintegrable models retain a spin-flip symmetry. Open uniform boundaries retain reflection symmetry, so reflection sectors must still be separated.

Possible probes include

$$A_1 = Z_{\lfloor L/2\rfloor}, \qquad A_2 = Z_{\lfloor L/2\rfloor} Z_{\lfloor L/2\rfloor+1},$$

and the intensive energy-density components

$$A_3 = \frac{1}{L} \sum_j X_j.$$

The Transverse-Field Ising Model owns the integrable limit and its exact structure. Here the mixed-field deformation is only a diagnostic laboratory.

How to Test Diagonal ETH

1. Resolve all exact symmetries

Block diagonalize by particle number, momentum, parity, reflection, or other exact labels. Document whether the observable preserves or connects those sectors.

2. Choose energy-density windows

Use a window centered at fixed

$$e = \frac{E}{L}$$

as $L$ varies. Report both $\Delta E$ and the number of states $D_{\mathcal W}$.

3. Plot eigenstate expectation values

Plot

$$A_{nn} \quad\text{against}\quad e_n=\frac{E_n}{L}.$$

In an ETH regime, the cloud narrows around a smooth curve in the spectral bulk. Visual smoothness is orientation, not a scaling result.

4. Subtract a local smooth benchmark

Define a local shell estimate

$$\widehat{\mathcal A}_L(E_n) = \frac{1}{N_n} \sum_{ m: |E_m-E_n|<\Delta/2 } A_{mm}.$$

Then analyze residuals

$$\delta A_{nn}^{(L)} = A_{nn} - \widehat{\mathcal A}_L(E_n).$$

The smoothing width must contain many levels but resolve thermodynamic variation. Exclude the point being tested from its local fit, or show that its weight in the fit is negligible; otherwise a small bin artificially shrinks residuals. Vary both the bin and the fitting order.

5. Track both typical and extreme statistics

Useful diagnostics include

$$\sigma_{\mathrm{diag}}^2(L) = \frac{1}{D_{\mathcal W}} \sum_{n\in\mathcal W} \left| \delta A_{nn}^{(L)} \right|^2,$$

the median absolute deviation,

$$\operatorname{MAD}(L) = \operatorname{median}_{n\in\mathcal W} \left| \delta A_{nn}^{(L)} \right|,$$

and the maximum

$$r_{\max}(L) = \max_{n\in\mathcal W} \left| \delta A_{nn}^{(L)} \right|.$$

The variance probes weak ETH. The maximum probes strong ETH and is much more sensitive to rare states and finite-size drift.

6. Inspect the full distribution

A shrinking variance can coexist with long tails. Report quantiles or the empirical distribution

$$P_L(\delta A)$$

instead of compressing every result to one number.

How to Test Off-Diagonal ETH

For each pair $m\ne n$, bin matrix elements by

$$\bar E = \frac{E_m+E_n}{2}$$

and

$$\omega = E_m-E_n.$$

Within a narrow bin, estimate

$$\overline{|A_{mn}|^2}_{\bar E,\omega}.$$

The entropy-rescaled variance is

$$\mathcal F_A(\bar E,\omega) = e^{S(\bar E)} \overline{|A_{mn}|^2}_{\bar E,\omega}.$$

ETH predicts that $\mathcal F_A$ approaches a smooth function proportional to $|f_A|^2$.

Distribution tests

Define rescaled elements

$$\widetilde R_{mn} = \frac{ e^{S(\bar E)/2}A_{mn} }{ \sqrt{\mathcal F_A(\bar E,\omega)} }.$$

One can test:

• zero mean;
• unit variance;
• real or complex distribution appropriate to symmetries;
• higher moments;
• correlations between neighboring matrix elements.

Approximate Gaussianity is common in random-matrix-like regimes, but it is not the entire ETH content and need not hold at every frequency.

Low-frequency structure

Near $\omega=0$, transport and conservation laws can dominate. A diffusive density produces long timescales and a narrow spectral structure. The entries $R_{mn}$ cannot then be treated as independent down to arbitrary energy scales.

This is not a failure of thermodynamics. The spectral envelope and matrix-element correlations retain information about locality and hydrodynamics.

The exact toy example below with $a=0$ and $\delta=1$: $D-1$ diagonal values are zero and one is unity. Their root-mean-square deviation from the shell mean shrinks as $D^{-1/2}$, while the largest deviation approaches one. A shrinking typical width does not establish strong ETH, and an initial state need not weight eigenstates uniformly.

Exact Toy Example: Weak ETH Without Strong ETH

Consider synthetic diagonal data for a shell containing $D\geq2$ eigenstates. This is a logical example distinguishing the definitions, not a construction of a local many-body Hamiltonian. Let

$$A_{nn} = a$$

for $D-1$ states and

$$A_{**} = a+\delta$$

for one exceptional state $|*\rangle$, with $\delta$ independent of $D$.

The shell mean is

$$A_{\mathrm{mc}} = a+\frac{\delta}{D}.$$

The variance is

$$\sigma_{A,\mathrm{mc}}^2 = \frac{D-1}{D} \left( \frac{\delta}{D} \right)^2 + \frac{1}{D} \left[ \delta \left( 1-\frac{1}{D} \right) \right]^2,$$

so

$$\sigma_{A,\mathrm{mc}}^2 = \delta^2 \frac{D-1}{D^2} \longrightarrow 0.$$

Weak ETH holds. But

$$r_{\max} = |\delta| \left( 1-\frac{1}{D} \right) \longrightarrow |\delta|,$$

so strong ETH fails.

Now choose an initial state with probability $p_*$ on the exceptional eigenstate. Its diagonal expectation is

$$\overline{\langle A\rangle} = a+p_*\delta.$$

If $p_*$ remains order one, the state stays nonthermal even though almost every shell eigenstate is thermal. A rare set can have negligible counting fraction while carrying substantial weight in a physically chosen preparation.

When ETH Fails or Needs Qualification

Integrability

Integrable systems possess extensively many conserved quantities. Eigenstate expectation values can depend on the corresponding charge densities, so energy alone does not label the local thermodynamic macrostate. Ordinary Gibbs ETH fails, although generalized eigenstate statements may hold after all relevant charges are fixed.

Integrability and Generalized Gibbs Ensembles Preview owns generalized stationary ensembles and their relation to the conserved-charge hierarchy.

Many-body localization

An idealized many-body localized phase has quasilocal integrals of motion, persistent local memory, and nonthermal highly excited eigenstates. Local observables can vary strongly between nearby energies, and eigenstate entanglement can obey an area law rather than a thermal volume law.

Many-Body Localization Preview explains the effective description, stability questions, and finite-size limitations.

Hilbert-space fragmentation

Kinetic or gauge constraints can split a formal symmetry sector into dynamically disconnected Krylov sectors:

$$\mathcal H_q = \bigoplus_\alpha \mathcal K_{q,\alpha}.$$

An ETH-like statement may hold inside each sufficiently large component while failing across their union. The accessible component is part of the conserved information.

Rare embedded eigenstates and scars

Local translationally invariant Hamiltonians can be constructed with a small set of exact nonthermal eigenstates embedded in an otherwise thermal spectrum. Quantum many-body scars provide physically important families of atypical low-entanglement eigenstates and can support revivals from specially chosen initial states.

Such systems can satisfy weak ETH while violating strong ETH. They also show that strong ETH is sufficient but not logically necessary for thermalization of most experimentally generic initial states.

Symmetry breaking and phase coexistence

Finite systems with an exact symmetry can have cat-like eigenstates even where thermodynamic pure phases break that symmetry. Order parameters, boundary conditions, sector projections, and the order of limits must be handled together.

Near first-order thermal transitions, multiple macroscopic branches can coexist at similar energy density. A single smooth function $\mathcal A(E)$ may be an inadequate finite-size description until the phase label and limiting prescription are specified.

Spectral edges and low-energy states

ETH is primarily a finite-energy-density statement. Ground states, topological sectors, quasiparticles above a vacuum, and low-entropy spectral edges need not resemble thermal bulk eigenstates.

Nonlocal and state-selective observables

Projectors onto eigenstates, Wilson-loop-scale probes, global order parameters, and operators whose support grows with system size can retain information invisible to local observables. The observable support and normalization must accompany the claim.

Long-range interactions

For slowly decaying interactions, conventional locality bounds, additivity, ensemble equivalence, and finite-size scaling can change. One should not import short-range ETH scaling without checking the interaction exponent and thermodynamic normalization.

Relation to quantum chaos

Level repulsion and random-matrix spectral correlations concern the eigenvalues; ETH concerns matrix elements of specified physical observables. They often occur together in generic nonintegrable models, but one test does not establish the other. A local conserved density can have slow hydrodynamic structure even in a spectrum with random-matrix statistics. See many-body quantum chaos for symmetry-resolved spectral diagnostics and Thouless scales.

What Experiments Can Test

Directly preparing and comparing many individual many-body eigenstates is difficult. Experiments more often test consequences compatible with ETH:

• local observables approach energy-matched ensemble values;
• different initial states with the same conserved densities approach the same local predictions;
• small subsystems acquire thermal reduced states while the global state remains pure;
• entanglement entropy approaches a thermal volume-law value;
• integrability breaking changes stationary behavior;
• rare preparations exhibit anomalous revivals or memory.

Quantum-gas microscopy experiments have reconstructed local reduced states and entanglement in small isolated systems, finding local thermal behavior compatible with ETH while preserving global purity. Superconducting-qubit experiments have similarly explored entanglement and ergodic dynamics in controlled few-qubit systems.

The three-qubit superconducting experiment used periodic kicks, so it probes Floquet eigenstate dynamics rather than directly testing an energy-conserving Hamiltonian ETH ansatz. Neither those data nor finite gas experiments establish a universal statement for all macroscopic Hamiltonians.

Exercises

1. Sector mixing can mimic ETH failure

Suppose a Hamiltonian has two exact sectors, $q=\pm1$. In a narrow energy window, an observable commuting with the symmetry has

$$A_{nn} \simeq \begin{cases} a+\delta,&q=+1,\\ a-\delta,&q=-1. \end{cases}$$

Assume each sector separately has vanishing within-sector variance and contains the same number of states. Compute the variance obtained by pooling the sectors. Does it vanish?

Solution

The pooled mean is

$$A_{\mathrm{pool}} = a.$$

Neglecting the vanishing within-sector variances, every state differs from the pooled mean by magnitude $|\delta|$. Therefore

$$\sigma_{\mathrm{pool}}^2 = \delta^2.$$

It does not vanish. The pooled data falsely suggest ETH failure even though each irreducible sector satisfies diagonal ETH. Sector resolution is part of the definition, not optional preprocessing.

2. Motivate the entropy factor

A microcanonical shell has dimension $D=e^S$. A bounded local observable satisfies

$$\frac{1}{D} \operatorname{Tr}_{\mathcal W} (A_{\mathcal W}^\dagger A_{\mathcal W}) = c$$

where $A_{\mathcal W}=P_{\mathcal W}AP_{\mathcal W}$ and $c=O(1)$. Assume an order-one fraction of this weight is off diagonal and distributed over order $D^2$ entries. Estimate the typical magnitude of $A_{mn}$ for $m\ne n$.

Solution

The trace identity gives

$$\sum_{m,n\in\mathcal W} |A_{mn}|^2 = cD.$$

If order $D^2$ entries carry comparable off-diagonal weight, then

$$\overline{|A_{mn}|^2} \sim \frac{cD}{D^2} = \frac{c}{D}.$$

Thus

$$|A_{mn}|_{\mathrm{typ}} \sim \sqrt c\,D^{-1/2} = \sqrt c\,e^{-S/2}.$$

This estimates the scale but does not establish the smooth envelope or statistical independence.

3. Energy-width correction

Let a local thermal value depend smoothly on energy density:

$$\mathcal A(E) = g\left(\frac{E}{V}\right).$$

Show that the leading width correction for a state centered at $\bar E$ is

$$\frac{g''(\bar e)}{2} \frac{(\Delta E)^2}{V^2}, \qquad \bar e=\frac{\bar E}{V}.$$

What condition makes it vanish?

Solution

Differentiating with respect to total energy,

$$\mathcal A'(E) = \frac{1}{V} g'\left(\frac{E}{V}\right),$$

and

$$\mathcal A''(E) = \frac{1}{V^2} g''\left(\frac{E}{V}\right).$$

The linear term averages to zero because

$$\sum_n p_n(E_n-\bar E)=0.$$

Hence the leading correction is

$$\frac12 \mathcal A''(\bar E) (\Delta E)^2 = \frac{g''(\bar e)}{2} \frac{(\Delta E)^2}{V^2}.$$

It vanishes when $\Delta E/V\to0$ and $g''$ remains finite.

4. One outlier

For the toy shell with $D-1$ values equal to $a$ and one value equal to $a+\delta$, calculate the bad-state fraction for any tolerance satisfying

$$0<\epsilon<|\delta|.$$

Compare it with the maximum deviation as $D\to\infty$.

Solution

For sufficiently large $D$, only the exceptional state differs from the shell mean by more than $\epsilon$. Thus

$$f_{\mathrm{bad}}(\epsilon) = \frac{1}{D} \longrightarrow 0.$$

Meanwhile,

$$r_{\max} \longrightarrow |\delta|.$$

The example satisfies weak ETH but violates strong ETH.

5. Temporal variance from off-diagonal ETH

Assume nondegenerate energy gaps and a state with probabilities $p_n$. Starting from

$$\delta A(t) = \sum_{m\ne n} c_m^*c_n e^{i(E_m-E_n)t} A_{mn},$$

derive the infinite-time variance. Then use

$$|A_{mn}|^2 \le C e^{-S_*}$$

throughout the occupied window to bound it.

Solution

Multiplying by the complex conjugate produces terms labeled by two energy gaps. Nondegenerate gaps make all unequal-gap terms vanish under infinite-time averaging, leaving

$$\overline{|\delta A(t)|^2} = \sum_{m\ne n} p_mp_n |A_{mn}|^2.$$

Using the bound,

$$\overline{|\delta A(t)|^2} \le C e^{-S_*} \sum_{m\ne n} p_mp_n.$$

Since

$$\sum_{m\ne n}p_mp_n = 1-\sum_n p_n^2 \le 1,$$

one obtains

$$\overline{|\delta A(t)|^2} \le C e^{-S_*}.$$

The assumptions exclude gap degeneracies and anomalous low-frequency matrix elements.

6. Subsystem ETH and measurements

Suppose

$$D\left( \rho_A^{(n)}, \rho_{A,\mathrm{mc}} \right) \le \varepsilon.$$

Bound the difference in expectation values of a Hermitian observable $O_A$ with

$$\|O_A\|_\infty\le1.$$

Solution

Trace-norm duality gives

$$\left| \operatorname{Tr} \left[ O_A \left( \rho_A^{(n)} - \rho_{A,\mathrm{mc}} \right) \right] \right| \le \|O_A\|_\infty \left\| \rho_A^{(n)} - \rho_{A,\mathrm{mc}} \right\|_1.$$

Because trace distance is half the trace norm,

$$\left\| \rho_A^{(n)} - \rho_{A,\mathrm{mc}} \right\|_1 \le 2\varepsilon.$$

Therefore the expectation-value difference is at most $2\varepsilon$.

7. Detailed balance from the ETH envelope

Assume

$$C_n(\omega) \simeq \frac{2\pi}{E_{\mathrm{ref}}} e^{\beta\omega/2} |f_A(E_n,\omega)|^2$$

and

$$|f_A(E,-\omega)|^2 = |f_A(E,\omega)|^2.$$

Derive the detailed-balance ratio.

Solution

Replacing $\omega$ by $-\omega$ gives

$$C_n(-\omega) \simeq \frac{2\pi}{E_{\mathrm{ref}}} e^{-\beta\omega/2} |f_A(E_n,\omega)|^2.$$

Dividing,

$$\frac{C_n(-\omega)}{C_n(\omega)} \simeq e^{-\beta\omega}.$$

The result uses the chosen Fourier convention. Reversing the convention reverses which spectrum appears in the numerator.

8. Design an ETH test

Design a finite-size ETH study for the mixed-field Ising chain. State at least six checks required before claiming strong ETH for a local spin observable.

Solution

A credible study would:

1. specify $J$, $h_x$, $h_z$, boundaries, and the sequence of lengths $L$;
2. resolve reflection and every other exact symmetry;
3. choose a bulk window at fixed energy density and report its level count;
4. define the local observable and its normalization;
5. subtract a documented smooth microcanonical benchmark;
6. report both typical width and maximum deviation versus $L$;
7. vary the smoothing and shell widths;
8. inspect residual distributions and outliers rather than only their variance;
9. compare more than one local observable;
10. separate diagonal from off-diagonal tests.

A strong-ETH test must establish that the maximum deviation tends to zero, not merely that most points form a narrow cloud.

Further Connections

• Nonequilibrium Overview places ETH among quench, integrability, localization, and driving mechanisms.
• Quantum Quenches defines the initial-state energy distribution to which diagonal ETH is applied.
• Relaxation and Thermalization separates dephasing error from ensemble-identification error.
• Microcanonical Ensemble defines the shell average used in every diagonal test.
• Ensemble Equivalence controls the later replacement of microcanonical by canonical predictions.
• Time-Dependent Correlations develops the spectral data encoded by the off-diagonal envelope.
• Thermal Entropy versus Entanglement Entropy owns the reduced-state and entropy consequences.

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