Integrability and Generalized Gibbs Ensembles Preview

An isolated integrable many-body system can dephase and settle to stationary local behavior without forgetting enough information to reach an ordinary Gibbs ensemble. Its obstruction is structural: besides energy and familiar symmetry charges, it carries an extensive hierarchy of local or quasilocal conserved quantities.

The generalized Gibbs ensemble, or GGE, retains the expectations of these conserved quantities:

$$\boxed{ \begin{gathered} [H,Q_r]=0 \quad\Longrightarrow\quad \langle Q_r\rangle_t = \langle Q_r\rangle_0, \\ \rho_{\mathrm{GGE}} = \frac{1}{Z_{\mathrm{GGE}}} \exp\!\left( -\sum_r\lambda_r Q_r \right). \end{gathered} }$$

The multipliers $\lambda_r$ are fixed by the initial charge data. The formula is simple; identifying a physically meaningful and complete charge family is not.

A GGE is therefore neither a synonym for “nonthermal” nor a universal formula that works once enough operators are placed in an exponent. It is a model-dependent compressed description of late-time local data. Its validity must be tested against a declared observable class and a controlled thermodynamic and long-time limit.

Helpful background. Quantum quenches supplies the initial-state problem; relaxation and thermalization explains why a stationary local state need not be Gibbsian. Creation and annihilation operators are used in the free-fermion example.

Conserved charges in integrable dynamics

We use $\hbar=k_{\mathrm B}=1$ and unit lattice spacing. All finite-system entropy variations below assume a normalizable ensemble. A boundary value of the constraints can require restricting the support or taking infinite multipliers.

There is no context-free definition of quantum integrability that covers every finite system, lattice, continuum model, and quantum field theory. For translation-invariant lattice systems, a useful working signature is an extensive family

$$\left\{ Q_r^{(L)} \right\}_{r=1}^{R(L)}$$

such that

$$\begin{aligned} [H_L,Q_r^{(L)}] &=0, \\ [Q_r^{(L)},Q_s^{(L)}] &=0, \end{aligned}$$

with $R(L)$ growing with system size and with controlled locality as $L\to\infty$.

The locality requirement is essential. Every spectral projector of a finite nondegenerate Hamiltonian commutes with $H$, but declaring all projectors “integrals of motion” would make every finite matrix integrable and would say nothing useful about local physics.

Ordinary symmetry charges are not enough

A generic particle-number-conserving lattice Hamiltonian may have

$$[H,N]=0$$

without being integrable. Translation, parity, total spin, or a finite-dimensional Lie symmetry can organize exact sectors while leaving dynamics within each sector chaotic.

An integrable hierarchy is stronger. Schematically, a local charge has the form

$$Q_r^{(L)} = \sum_{j=1}^{L} q_{r,j},$$

where $q_{r,j}$ is a translate of a density supported on a finite number of nearby sites. A quasilocal charge permits tails whose norm decays sufficiently rapidly with distance. Precise norms and boundary corrections depend on the model.

Free and interacting integrability

For a quadratic model, normal-mode occupations provide transparent charges:

$$H = \sum_k \varepsilon_k n_k, \qquad [H,n_k]=0.$$

Interacting Bethe-ansatz models are not free models in disguise. Their quasiparticles acquire scattering phases, bound-state species, and state-dependent dressing. Integrability constrains many-body scattering and supplies conserved charges; it does not remove interaction effects.

Boundaries and sectors remain part of the claim

Periodic, twisted, and integrable open boundaries can support different commuting families. A generic boundary perturbation may break exact integrability even when the bulk Hamiltonian is unchanged. Ordinary symmetry sectors must also be resolved before level statistics or eigenstate comparisons are interpreted. Many-Body Quantum Chaos Preview owns the block construction, gap-ratio benchmarks, random-matrix classes, and finite-size crossover tests.

Thus “the XXZ chain is integrable” is shorthand for a specified Hamiltonian, coupling range, representation, and compatible boundary construction, not every deformation carrying the same informal model name.

Equilibration Without Ordinary Thermalization

Let

$$\rho(t) = e^{-iHt}\rho_0e^{iHt}.$$

In finite dimension, the infinite-time average is the dephased state

$$\omega = \sum_E P_E\rho_0P_E,$$

where $P_E$ projects onto the full degenerate eigenspace of energy $E$. This state retains initial populations and coherence within degenerate energy blocks.

The state $\omega$ generally contains exponentially detailed information. A statistical ensemble is useful when much of that information is invisible to a chosen family of local observables.

The stationary claim is local

Let $A$ be a fixed finite region as $L\to\infty$. A successful GGE claim may be formulated as

$$\lim_{L\to\infty} \operatorname{Tr} \left[ O_A \left( \omega_L-\rho_{\mathrm{GGE},L} \right) \right] = 0$$

for every bounded operator $O_A$ in the declared local class.

If the class contains every operator on a fixed finite-dimensional region, this is equivalent to convergence of its reduced states:

$$\left\| \omega_{A,L} - \rho_{\mathrm{GGE},A,L} \right\|_1 \longrightarrow 0.$$

Neither statement requires global trace-norm closeness. The exact diagonal ensemble can retain microscopic correlations and phase-space information that no compact GGE reproduces globally.

Generalized thermalization

It is useful to separate three claims:

Claim	Late-time comparison
equilibration	observables remain close to a stationary value for most late times
Gibbs thermalization	stationary local values agree with an energy-based Gibbs ensemble
generalized thermalization	stationary local values agree with a GGE built from a complete relevant charge family

An integrable system may satisfy the first and third while violating the second. Persistent finite-size oscillations or exact revivals can still prevent equilibration on the accessible system.

Constructing a Generalized Gibbs Ensemble

Suppose a commuting family $\{Q_r\}$ has prescribed expectations

$$q_r^{(0)} = \operatorname{Tr}(\rho_0Q_r).$$

For a finite list of constraints with a full-rank optimum, among density operators satisfying

$$\operatorname{Tr}(\rho Q_r) = q_r^{(0)},$$

the maximum-entropy state is found by varying

$$\mathcal L[\rho] = -\operatorname{Tr}(\rho\ln\rho) -\alpha \left[ \operatorname{Tr}(\rho)-1 \right] -\sum_r\lambda_r \left[ \operatorname{Tr}(\rho Q_r) -q_r^{(0)} \right].$$

Stationarity gives

$$\ln\rho = -(1+\alpha)I -\sum_r\lambda_rQ_r,$$

and therefore

$$\begin{aligned} \rho_{\mathrm{GGE}} &= \frac{e^{-\sum_r\lambda_rQ_r}} {Z_{\mathrm{GGE}}}, \\ Z_{\mathrm{GGE}} &= \operatorname{Tr} e^{-\sum_r\lambda_rQ_r}. \end{aligned}$$

The entropy maximum is unique as a density operator by strict concavity. Mutual commutation of the charges is not required for the exponential form or the first-derivative matching identity; it is required for the ordinary covariance formula below. An infinite hierarchy is understood through a convergent finite-charge sequence or an independently defined thermodynamic ensemble. This is the same constrained-inference logic developed on Maximum Entropy Principle. Dynamics enters through the choice and initial values of the conserved constraints.

Matching identities

The multipliers solve

$$-\frac{\partial\ln Z_{\mathrm{GGE}}} {\partial\lambda_r} = \langle Q_r\rangle_{\mathrm{GGE}} = q_r^{(0)}.$$

For commuting charges, the Hessian is their covariance matrix:

$$\frac{\partial^2\ln Z_{\mathrm{GGE}}} {\partial\lambda_r\partial\lambda_s} = \operatorname{Cov}_{\mathrm{GGE}}(Q_r,Q_s).$$

Null directions indicate redundant constraints or singular limits. Near a regular solution, the covariance matrix controls how sensitively the multipliers respond to charge data.

Ordinary Gibbs states are special cases

If the only retained extensive constraint is energy,

$$Q_1=H, \qquad \lambda_1=\beta,$$

the GGE reduces to the canonical state. Including a finite number of symmetry charges gives a generalized canonical or grand-canonical ensemble, for example

$$\rho \propto \exp \left[ -\beta(H-\mu N) \right].$$

That construction does not by itself imply integrability. The distinctive integrable problem is an extensive local or quasilocal hierarchy.

Choosing the Charge Family

Writing every conserved operator in an exponent is neither necessary nor useful. A physically informative family should address four questions.

Conservation

Each exact charge must satisfy

$$[H,Q_r]=0$$

for the stated finite-size sequence and boundary conditions, up to explicitly controlled boundary terms.

Locality or quasilocality

Local observables can retain memory of extensive charges with local densities. Highly nonlocal projectors may encode the full diagonal ensemble but do not provide a useful thermodynamic compression.

Independence

If

$$Q_3=aQ_1+bQ_2+cI,$$

then its multiplier only reparameterizes the same state. Independence can mean linear, functional, or thermodynamic independence depending on the representation. The criterion must be stated rather than inferred from the number of symbols.

Completeness relative to local observables

A charge family is complete when its expectation values uniquely determine the stationary macrostate relevant to the declared local observables. Completeness is stronger than having many charges and weaker than reconstructing the exact global density operator.

This definition is deliberately operational:

$$\left\{ \langle Q_r\rangle \right\} \quad\Longrightarrow\quad \left\{ \langle O_A\rangle_{\mathrm{stationary}} \right\}$$

for the intended local class.

Worked Example: Free Fermions

Consider spinless fermions on a periodic lattice with

$$H = \sum_k \varepsilon_k n_k, \qquad n_k=c_k^\dagger c_k.$$

Every $n_k$ has eigenvalues $0$ and $1$ and commutes with every other occupation. Let

$$\zeta_k = \langle n_k\rangle_0.$$

The mode-occupation GGE is

$$\rho_{\mathrm{GGE}} = \frac{1}{Z} \exp \left( -\sum_k\lambda_kn_k \right).$$

This trace is over the full fermionic Fock space: particle number is fixed in expectation, not projected to one exact finite-$N$ sector. In that ensemble the modes factorize,

$$\rho_{\mathrm{GGE}} = \bigotimes_k \rho_k,$$

with

$$\rho_k = \frac{ |0_k\rangle\langle0_k| + e^{-\lambda_k} |1_k\rangle\langle1_k| }{ 1+e^{-\lambda_k} }.$$

Matching $\langle n_k\rangle$ gives

$$\zeta_k = \frac{1}{e^{\lambda_k}+1},$$

so

$$\lambda_k = \ln \left( \frac{1-\zeta_k}{\zeta_k} \right).$$

The limiting values $\zeta_k=0$ or $1$ correspond to $\lambda_k=\pm\infty$ and pure mode factors.

Projecting the exponential to a fixed particle-number sector generally destroys the exact product form. Local agreement with a fixed-number preparation, when it holds in a thermodynamic limit, is a separate equivalence statement.

Why an ordinary Gibbs state is insufficient

A grand-canonical Gibbs state predicts

$$\zeta_k^{\mathrm{Gibbs}} = \frac{1} {e^{\beta(\varepsilon_k-\mu)}+1}.$$

It has only two adjustable parameters, $\beta$ and $\mu$. A generic quench produces a function $k\mapsto\zeta_k$ that is not Fermi–Dirac. Matching total energy and number therefore leaves other mode-weighted observables incorrect.

Local correlators

For a translation-invariant, number-conserving Gaussian stationary state, with $c_j=L^{-1/2}\sum_k e^{ikj}c_k$,

$$\langle c_i^\dagger c_j\rangle_{\mathrm{GGE}} = \frac{1}{L} \sum_k e^{ik(j-i)} \zeta_k.$$

All local Gaussian correlators follow from this two-point function by Wick reduction. Thus the conserved occupation profile determines the local reduced state.

In a quadratic quench, the momentum-space covariance evolves as

$$\Gamma_{kk'}(t) = e^{i(\varepsilon_k-\varepsilon_{k'})t} \Gamma_{kk'}(0).$$

Under suitable nondegeneracy and regularity conditions, oscillatory $k\ne k'$ contributions vanish from fixed-range real-space correlators in the long-time and thermodynamic limits. The stationary covariance retains

$$\Gamma_{kk} = \zeta_k.$$

This is dephasing, not loss of the conserved occupations. For a Gaussian initial state, the covariance determines every local correlation by Wick’s theorem throughout the evolution. For a non-Gaussian state, a separate gaussification argument is needed: suitable initial clustering and delocalizing quadratic propagation can make higher connected cumulants vanish locally. Flat bands, localized modes, and persistent long-range initial correlations can prevent this step (Gluza and collaborators, 2016).

For example, the equal mixture of the empty and completely filled lattice is stationary under any number-conserving quadratic Hamiltonian. It has $\langle n_k\rangle=1/2$ for every mode, just like the maximally mixed mode GGE, but for distinct sites its $\langle n_i n_j\rangle=1/2$ rather than $1/4$. The initial state lacks clustering. Matching all one-mode means does not make it Gaussian.

If particle-number conservation is absent, anomalous covariances such as $\langle c_i c_j\rangle$ must also be retained; occupations in a correctly chosen Bogoliubov basis can then replace the number-conserving construction.

Same energy, different short-range correlations

Take the dispersion $\varepsilon_k=-2J\cos k$ and two stationary Gaussian occupation profiles,

$$\zeta_k^{(A)}=\frac12,\qquad \zeta_k^{(B)}=\frac12+\frac14\cos(2k).$$

Both have particle density $1/2$ and energy density zero: the constant, $\cos k$, and $\cos(2k)$ are orthogonal Fourier components. Yet

$$\langle c_j^\dagger c_{j+2}\rangle_A=0,\qquad \langle c_j^\dagger c_{j+2}\rangle_B=\frac18.$$

Energy and number therefore miss a directly measurable finite-range correlation. The range-two charge below has densities $0$ and $1/4$ respectively. These are two valid stationary preparations, illustrating what the additional GGE constraints distinguish.

For $\varepsilon_k=-2J\cos k$, both stationary Gaussian states have $N/L=1/2$ and $E/L=0$. The oscillating occupation profile retains a range-two correlation invisible to energy and number. Its additional charge density is $\langle Q_2^{(c)}\rangle/L=1/4$, while the flat profile gives zero.

Mode occupations and local charges

An individual $n_k$ is delocalized in real space. For a translation-invariant chain, its Fourier moments generate finite-range charges:

$$\begin{aligned} Q_m^{(c)} &= \sum_j \left( c_j^\dagger c_{j+m} + c_{j+m}^\dagger c_j \right), \\ Q_m^{(s)} &= -i\sum_j \left( c_j^\dagger c_{j+m} - c_{j+m}^\dagger c_j \right). \end{aligned}$$

In momentum space,

$$\begin{aligned} Q_m^{(c)} &= 2\sum_k \cos(mk)n_k, \\ Q_m^{(s)} &= 2\sum_k \sin(mk)n_k. \end{aligned}$$

When the Fourier data are complete, the occupation-profile GGE and the local-charge GGE are equivalent for local observables. This equivalence is model and regularity dependent; it should be demonstrated, not assumed.

Truncated GGEs

A truncated ensemble retains charges only up to range $m_{\max}$:

$$\begin{aligned} \rho_{\mathrm{tGGE}} &\propto \exp \left[ -\sum_{m=0}^{m_{\max}} \Lambda_m \right], \\ \Lambda_m &= \lambda_m^{(c)}Q_m^{(c)} + \lambda_m^{(s)}Q_m^{(s)}. \end{aligned}$$

For a translation-invariant number-conserving Gaussian target, matching both Fourier moments through range $\ell-1$ fixes every entry of the covariance in an interval of $\ell$ sites and therefore its reduced state exactly. This statement assumes the finite-charge ensemble exists and the moments are matched exactly; it does not extend automatically to non-Gaussian targets or arbitrary interacting GGEs. Also $Q_0^{(c)}=2N$ and $Q_0^{(s)}=0$, so discard the redundant zero charge.

Degeneracy caveat

If $\varepsilon_k=\varepsilon_{k'}$, coherences within the degenerate subspace need not dephase. A complete stationary description may require conserved bilinears

$$c_k^\dagger c_{k'}$$

inside that subspace, or a basis that diagonalizes the initial covariance while respecting the Hamiltonian. Ignoring degeneracies can make an apparently reasonable occupation GGE fail.

Interacting Bethe-Ansatz Systems

In an interacting Bethe-ansatz model, a finite-density eigenstate contains a number of rapidities proportional to $L$. In the thermodynamic limit, microscopic roots are replaced by particle and hole densities

$$\rho_a(\lambda), \qquad \rho_a^{\mathrm h}(\lambda),$$

where $a$ labels quasiparticle species or bound-state strings and $\lambda$ is a rapidity.

The Bethe equations become constraints of the schematic form

$$\rho_a^{\mathrm t} = \rho_a+\rho_a^{\mathrm h} = a_a -\sum_b K_{ab}*\rho_b,$$

where

$$(K*f)(\lambda) = \int d\mu\, K(\lambda-\mu)f(\mu).$$

The kernels encode scattering. Their signs, normalizations, and species content are model dependent.

Conserved charges as functionals of root densities

An extensive charge density takes the form

$$q_r[\rho] = \sum_a \int d\lambda\, h_{r,a}(\lambda) \rho_a(\lambda),$$

where $h_{r,a}$ is the one-quasiparticle eigenvalue of charge $Q_r$. Matching the initial state means

$$q_r[\rho_*] = \lim_{L\to\infty} \frac{ \langle\psi_0|Q_r|\psi_0\rangle }{L}$$

for every charge in a complete family.

Unlike a free mode occupation, one root density influences dressed energies, velocities, and observables through coupled integral equations. The stationary macrostate is therefore an interacting thermodynamic object.

The generalized thermodynamic saddle

The number of microscopic Bethe states represented by smooth densities is controlled by the Yang–Yang entropy density

$$s_{\mathrm{YY}}[\rho] = \sum_a\int d\lambda\, \big[ \rho_a^{\mathrm t}\ln\rho_a^{\mathrm t} -\rho_a\ln\rho_a -\rho_a^{\mathrm h}\ln\rho_a^{\mathrm h} \big].$$

A GGE macrostate extremizes

$$\mathcal I[\rho] = s_{\mathrm{YY}}[\rho] -\sum_r \lambda_rq_r[\rho]$$

subject to the Bethe constraints.

Define a generalized driving term

$$w_a(\lambda) = \sum_r \lambda_rh_{r,a}(\lambda).$$

For the density constraint $\rho^{\mathrm t}=a-K\rho$ used above, variation of $s_{\mathrm{YY}}-\sum_r\lambda_rq_r$ gives

$$\epsilon_a=w_a+ \sum_b K_{ba}^{\mathrm T}* \ln(1+e^{-\epsilon_b}),$$

where $\epsilon_a=\ln(\rho_a^{\mathrm h}/\rho_a)$ and

$$(K_{ba}^{\mathrm T}*f_b)(\lambda) =\int d\mu\,K_{ba}(\mu-\lambda)f_b(\mu).$$

The plus sign follows because $\delta\rho^{\mathrm h}=-\delta\rho-K\delta\rho$; explicitly, $\delta s_{\mathrm{YY}}/\delta\rho=\epsilon-K^{\mathrm T}\ln(1+e^{-\epsilon})$. If the Bethe density equation instead uses $\rho^{\mathrm t}=a+K\rho$, the pseudoenergy equation has the familiar minus sign. Do not combine those two kernel conventions. The filling is

$$\vartheta_a = \frac{\rho_a}{\rho_a^{\mathrm t}} = \frac{1}{1+e^{\epsilon_a}}.$$

These equations show how a charge hierarchy becomes a stationary quasiparticle distribution. They are not a universal formula to transplant between models; bound-state content and kernel conventions must be rebuilt from the relevant Bethe equations.

Bethe ansatz preview

The nonequilibrium logic needs only three facts:

1. finite-volume eigenstates are labeled by interacting rapidities;
2. thermodynamic local physics is encoded by root-density functions;
3. a complete set of local and quasilocal charges must determine those functions.

The derivation of factorized scattering and the Bethe equations belongs to Exact Solutions Preview. The explicit spin-chain realization belongs to XXZ Spin Chain.

Generalized Eigenstate Thermalization

Conventional ETH proposes that local eigenstate expectation values are smooth functions of energy density and a finite set of symmetry densities. Integrable systems violate that compression: many macroscopically distinct root distributions can have the same energy density.

A generalized eigenstate-thermalization statement replaces energy by the complete integrable macrostate:

$$\langle n|O_A|n\rangle \longrightarrow \mathcal O_A[\rho]$$

for thermodynamic sequences of eigenstates whose root densities approach $\rho$.

Nearby microscopic Bethe states representing the same smooth $\rho$ then agree on local observables, while states at the same energy but with different $\rho$ need not agree.

This explains how one representative eigenstate can reproduce stationary local values in an integrable quench without satisfying conventional ETH. The “bath” for a small region is the rest of the system constrained by the full macrostate, not by energy alone.

The Quench-Action Viewpoint

For some interacting integrable quenches, initial-state overlaps with Bethe eigenstates are available in the thermodynamic limit. If

$$\left| \langle\{\lambda\}|\psi_0\rangle \right|^2 \asymp e^{-Lg_{\psi_0}[\rho]},$$

while the number of states near $\rho$ with nonzero overlap grows as

$$e^{Ls_{\mathrm{allowed}}[\rho]},$$

then the dominant stationary macrostate minimizes the quench-action functional

$$\mathcal S_{\mathrm Q}[\rho] = g_{\psi_0}[\rho] -s_{\mathrm{allowed}}[\rho]$$

subject to Bethe constraints. Count only states allowed by the overlap selection rules. For common pair-structured quenches, only parity-invariant rapidity sets contribute, and in full-rapidity conventions the allowed entropy is $\tfrac12s_{\mathrm{YY}}$, not $s_{\mathrm{YY}}$. The squared-overlap rate $g_{\psi_0}$ above already includes the factor of two relative to the real part of the log-amplitude rate. Either convention works if both factors are treated consistently.

The saddle $\rho_*$ supplies a representative state for local late-time observables. This construction uses more direct information about the initial overlaps than an initially guessed charge list.

The quench action and a complete GGE should identify the same local stationary macrostate when both apply. Disagreement is a diagnostic that the proposed GGE constraints are incomplete, incorrectly implemented, or outside their domain.

The Charge-Completeness Problem

The history of the XXZ chain provides an important warning.

Early GGEs used the familiar local charges generated by derivatives of a fundamental transfer matrix. For quenches involving bound-state strings, those charges did not uniquely determine every quasiparticle root density. Quench-action calculations and local correlators exposed the mismatch.

The selected charges were incomplete; the maximum-entropy principle was not the failing step.

Quasilocal charge families associated with additional transfer-matrix representations supplied the missing information and led to complete GGEs for the spin-$1/2$ Heisenberg chain.

What “complete” does and does not mean

In this context, complete means sufficient to fix all thermodynamic root densities relevant to local observables. It does not mean:

• every bounded operator is included as a constraint;
• the exact finite-size diagonal ensemble is reconstructed;
• the same charge basis works for every integrable model;
• boundaries, degenerate sectors, or initial-state regularity can be ignored.

Completeness is a theorem or construction to establish model by model, not an adjective to attach to a long formula.

GGE Versus the Diagonal Ensemble

For a nondegenerate spectrum and pure initial state,

$$\rho_{\mathrm{diag}} = \sum_n |c_n|^2 |n\rangle\langle n|.$$

This ensemble exactly reproduces infinite-time averages of observables. Nondegeneracy of energies is sufficient here; nondegeneracy of gaps is not needed for the first time average. It retains one population per eigenstate and is generally exponentially complex.

A GGE instead retains charge data:

$$\left\{ |c_n|^2 \right\}_{n=1}^{D} \quad\rightsquigarrow\quad \left\{ q_r^{(0)} \right\}_{r}.$$

The arrow is valid only for a chosen local observable class in a limit. It is not an equality of global states.

The comparison can be organized as follows:

Description	Memory retained	Exactness
time-evolved state	amplitudes and phases	exact unitary state
diagonal ensemble	dephased spectral blocks	exact infinite-time averages
complete GGE	relevant charge macrostate	exact or asymptotically exact for stated local observables
truncated GGE	finite subset of charges	controlled approximation only after convergence tests
Gibbs ensemble	energy and selected ordinary charges	generally insufficient in an integrable quench

Testing observables not used to fix the ensemble

First fix the charges from the initial state, then compare predictions for other observables: finite-range correlations, currents, or a small reduced density matrix. Agreement on the fitted charges is automatic. Increasing the charge range tests a truncated GGE; increasing system size and the time window tests the physical stationary limit. Those are different checks.

A useful comparison places the time-evolved values, diagonal-ensemble values, GGE predictions, and ordinary Gibbs predictions together. A GGE can correctly predict the time average while the accessible finite system still has large fluctuations. Conversely, failure on one local observable can reveal an omitted charge, surviving degeneracy coherence, or a failed Gaussian assumption.

A weak integrability-breaking perturbation offers another test: the formerly conserved charges can drift and a GGE-like intermediate state can eventually give way to ordinary thermal behavior. The lifetime requires a dynamical calculation, as discussed below.

Experiments with nearly integrable gases

Quantum Newton’s cradle

Kinoshita, Wenger, and Weiss prepared a nearly one-dimensional Bose gas whose momentum distribution remained strikingly nonthermal through thousands of collisions. The experiment established a prominent failure of rapid ordinary thermalization near an integrable regime.

It did not, by itself, reconstruct a complete GGE. Trapping, residual transverse excitations, finite observation time, and weak integrability breaking all matter. The observation supports constrained dynamics; it does not identify every persistent oscillation or recurrence with integrability.

Correlations in split one-dimensional Bose gases

Langen and collaborators measured correlation functions up to high order after splitting a one-dimensional Bose gas and found agreement with a generalized ensemble built from approximately conserved mode data. This is substantially stronger than matching a single temperature-like quantity because higher-order correlators probe the distribution of fluctuations.

The inferred charges and effective low-energy model remain part of the experimental statement. A GGE is supported when one charge-calibrated description predicts additional observables, not merely when several free fit parameters can reproduce the same dataset.

Local GGEs and Generalized Hydrodynamics

A homogeneous GGE describes a stationary macrostate. In an inhomogeneous integrable system, each slowly varying cell can instead be approximated by a local GGE with root densities

$$\rho_a(x,t,\lambda).$$

At the Euler scale, generalized hydrodynamics transports each quasiparticle species according to

$$\partial_t\rho_a + \partial_x \left( v_a^{\mathrm{eff}}[\rho] \rho_a \right) = 0.$$

The effective velocity is dressed by the local state. This differs from ordinary hydrodynamics with only particle, momentum, and energy densities because the full quasiparticle distribution is a hydrodynamic field.

Diffusive corrections, external forces, integrability breaking, and fluctuating hydrodynamics require additional terms. The equation above is an interface, not a complete derivation.

Weak Integrability Breaking

Let

$$H = H_{\mathrm{int}} + gV, \qquad |g|\ll1,$$

where

$$[H_{\mathrm{int}},Q_r]=0.$$

Under the full Hamiltonian,

$$\frac{d}{dt} \langle Q_r\rangle = ig \langle[V,Q_r]\rangle.$$

The charges are no longer exact, but they can evolve slowly. A common sequence is

$$\begin{gathered} \text{microscopic dephasing} \\ \longrightarrow \text{GGE-like plateau} \\ \longrightarrow \text{ordinary thermalization}. \end{gathered}$$

The plateau lifetime is not universally $1/|g|$ or $1/g^2$. Selection rules, resonances, dimensionality, initial state, and the observable can change the scaling. The later prethermalization page owns these lifetime mechanisms.

Comparing three stationary descriptions

Question	Ordinary Gibbs	GGE	Quench action
input data	energy and ordinary charges	complete relevant conserved charges	thermodynamic initial overlaps
natural variables	$\beta,\mu_a$	$\lambda_r$ or charge-generating functions	root-density saddle
free models	usually too coarse after a quench	often exact for local data	usually unnecessary
interacting Bethe models	generally too coarse	exact if charge family is complete	direct route when overlaps are known
global-state equality	no	no	no
main failure mode	omitted conserved memory	incomplete or ill-conditioned charges	unavailable or singular overlaps

Exercises

Exercise 1: Derive the generalized Gibbs form

Maximize

$$S(\rho) = -\operatorname{Tr}(\rho\ln\rho)$$

subject to normalization and

$$\operatorname{Tr}(\rho Q_r)=q_r.$$

Derive $\rho_{\mathrm{GGE}}$ and the matching identity for $\partial_{\lambda_r}\ln Z$.

Solution

Introduce

$$\mathcal L = S(\rho) -\alpha \left[ \operatorname{Tr}(\rho)-1 \right] -\sum_r\lambda_r \left[ \operatorname{Tr}(\rho Q_r)-q_r \right].$$

Using

$$\delta S = -\operatorname{Tr} \left[ \delta\rho(\ln\rho+I) \right],$$

stationarity for arbitrary Hermitian $\delta\rho$ gives

$$\ln\rho+I+\alpha I+\sum_r\lambda_rQ_r=0.$$

Hence

$$\rho = \frac{ e^{-\sum_r\lambda_rQ_r} }{Z}, \qquad Z = \operatorname{Tr} e^{-\sum_r\lambda_rQ_r}.$$

Differentiating under the trace and using cyclicity yields

$$-\partial_{\lambda_r}\ln Z = \operatorname{Tr}(\rho Q_r) = q_r.$$

Exercise 2: Solve one fermionic mode

For

$$\rho_k \propto e^{-\lambda_kn_k}, \qquad n_k^2=n_k,$$

derive the partition function, occupation, multiplier, and entropy in terms of $\zeta_k=\langle n_k\rangle$.

Solution

In the basis $\{|0\rangle,|1\rangle\}$,

$$Z_k = 1+e^{-\lambda_k}.$$

Therefore

$$\zeta_k = \frac{e^{-\lambda_k}} {1+e^{-\lambda_k}} = \frac{1}{e^{\lambda_k}+1},$$

so

$$\lambda_k = \ln \left( \frac{1-\zeta_k}{\zeta_k} \right).$$

The mode density matrix has eigenvalues $1-\zeta_k$ and $\zeta_k$, giving

$$S_k = -\zeta_k\ln\zeta_k -(1-\zeta_k) \ln(1-\zeta_k).$$

Exercise 3: Same energy and number, different memory

Consider four fermionic modes with energies

$$(-3,-1,1,3).$$

Compare occupation profiles

$$\boldsymbol\zeta_A = \left( \frac12,\frac12,\frac12,\frac12 \right)$$

and

$$\boldsymbol\zeta_B = (1,0,0,1).$$

Show that total number and energy agree, while

$$Q=n_1-n_2-n_3+n_4$$

distinguishes them.

Solution

Both profiles have

$$N = \sum_{a=1}^4\zeta_a = 2.$$

Their energies are

$$E_A = \frac12(-3-1+1+3) = 0$$

and

$$E_B = -3+3 = 0.$$

However,

$$\langle Q\rangle_A = 0, \qquad \langle Q\rangle_B = 2.$$

An ensemble constrained only by $E$ and $N$ cannot retain this distinction. A charge-complete GGE can.

Exercise 4: Show that Fourier charges are local

Using

$$c_j = \frac{1}{\sqrt L} \sum_k e^{ikj}c_k,$$

show that $Q_m^{(c)}$ and $Q_m^{(s)}$ are Fourier moments of $n_k$ and identify their real-space range.

Solution

Translation orthogonality gives

$$\sum_j c_j^\dagger c_{j+m} = \sum_k e^{ikm}n_k.$$

Adding the Hermitian conjugate yields

$$Q_m^{(c)} = 2\sum_k \cos(mk)n_k.$$

Taking the antisymmetric Hermitian combination gives

$$Q_m^{(s)} = 2\sum_k \sin(mk)n_k.$$

Each density connects sites separated by exactly $m$, so the charge has finite range $m$ even though each individual $n_k$ is delocalized.

Exercise 5: Detect an omitted local charge

Suppose $Q=\sum_jq_j$ is exactly conserved, the initial state and dynamics preserve translation invariance, and the initial charge density is $q_0$. A proposed stationary ensemble predicts $q_{\mathrm{trial}}\ne q_0$. Explain why the ensemble cannot describe every local observable.

Solution

For a translation-invariant state,

$$\frac{\langle Q\rangle}{L} = \langle q_j\rangle.$$

Exact conservation fixes the left side to $q_0$. If the trial ensemble gives $q_{\mathrm{trial}}\ne q_0$, it already fails on the local density $q_j$. The mismatch cannot be repaired by taking a later time because $\langle Q\rangle$ is constant.

The conclusion requires $q_j$ to belong to the declared local observable class. A highly nonlocal conserved projector would not provide the same immediate local contradiction.

Exercise 6: Weakly break integrability

For

$$H=H_{\mathrm{int}}+gV, \qquad [H_{\mathrm{int}},Q_r]=0,$$

derive the exact rate of change of $\langle Q_r\rangle$ in units $\hbar=1$. Does the result prove a relaxation time proportional to $1/|g|$?

Solution

The Heisenberg equation gives

$$\frac{d}{dt} \langle Q_r\rangle = i\langle[H,Q_r]\rangle = ig\langle[V,Q_r]\rangle.$$

This shows that the instantaneous rate begins at order $g$ when the commutator expectation is nonzero. It does not prove a universal $1/|g|$ relaxation time. Oscillatory cancellations, selection rules, kinetic phase space, resonances, and higher-order processes can produce different scalings, including rates of order $g^2$ in suitable regimes.

Exercise 7: Why many charges can still be incomplete

An interacting Bethe model has two quasiparticle species with root densities $\rho_1(\lambda)$ and $\rho_2(\lambda)$. Suppose every charge in a proposed family depends only on

$$\rho_1(\lambda)+2\rho_2(\lambda).$$

Explain why the family need not determine local observables.

Solution

If the Bethe equations and positivity permit more than one pair $(\rho_1,\rho_2)$ with the same combination $\rho_1+2\rho_2$, the proposed charges cannot distinguish those physical macrostates. A formal variation of densities is not enough unless it satisfies those constraints. The proposed charges therefore do not distinguish those macrostates.

If a local observable depends differently on the two species through dressing or form factors, its expectation values will differ even though every proposed charge agrees. Additional charges are needed whose one-particle eigenvalues resolve the species separately. This is the abstract structure behind charge-completeness problems involving Bethe strings.

Exercise 8: Design an out-of-sample GGE test

A numerical study fits $\lambda_r$ using energy, particle number, and three short-range charges, then reports agreement for those five quantities. Give a stronger validation protocol.

Solution

Agreement for fitted charges is automatic. A stronger protocol should:

1. compare against long-time and diagonal-ensemble values;
2. reserve local correlators not used in fitting, at several ranges;
3. test different operator species and, if feasible, a subsystem reduced state;
4. increase the retained charge range and verify convergence;
5. repeat over several system sizes and late-time windows;
6. compare with an ordinary Gibbs ensemble;
7. add a controlled integrability-breaking perturbation.

The study should also resolve exact symmetry sectors and report boundary conditions. This protocol tests predictive compression rather than parameter fitting.

Related topics

• Nonequilibrium Overview for the chapter-wide map.
• Relaxation and Thermalization for diagonal ensembles and equilibration criteria.
• Eigenstate Thermalization Hypothesis for the nonintegrable comparison.
• Exact Solutions Preview for transfer matrices, factorized scattering, and thermodynamic Bethe ansatz.
• XXZ Spin Chain for the canonical interacting lattice example.
• Transverse-Field Ising Model for a quadratic spin-chain example.
• Maximum Entropy Principle for constrained inference.
• Time-Dependent Correlations for stationary and dynamical correlators.
• Transport Coefficients Preview for ballistic and diffusive response language.

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