Generalized Gibbs Ensembles for Integrable Charges

An integrable generalized Gibbs ensemble is the maximum-entropy state constrained by a mutually commuting family of conserved local or quasilocal charges. It describes a stationary macrostate only when the chosen charge family is complete for the observables and preparation under study. A truncated GGE is an approximation whose omitted-charge bias must be tested; it is not the same construction as a maximum-entropy state with genuinely noncommuting charges.

The necessity of a charge-complete ensemble in an interacting integrable model is demonstrated by Ilievski et al. 2015, pp. 1–5.

Required background. Conserved Charges and Grand-Canonical States gives the ordinary exponential ensemble. Classical and Quantum Conserved Charges and Thermodynamic Bethe Ansatz and Finite-Size Ground-State Energy supply the integrable charges and macrostate language.

Maximum entropy with commuting charges

Let $\{Q_j\}$ satisfy

$$[H,Q_j]=0, \qquad [Q_i,Q_j]=0.$$

Maximizing von Neumann entropy subject to $\operatorname{Tr}(\rho Q_j)=q_j$ gives

$$\rho_{\mathrm{GGE}} =\frac1{Z_{\mathrm{GGE}}} \exp\!\left(-\sum_j\lambda_jQ_j\right), \qquad Z_{\mathrm{GGE}}= \operatorname{Tr}\exp\!\left(-\sum_j\lambda_jQ_j\right).$$

The multipliers solve

$$q_j=-\frac{\partial\log Z_{\mathrm{GGE}}}{\partial\lambda_j}.$$

Ordinary Gibbs equilibrium is the special case containing $Q_1=H$ with $\lambda_1=\beta$ and any global commuting charges with $\lambda_a=-\beta\mu_a$. In an integrable theory the additional charges constrain mode or rapidity occupations that a single temperature and chemical potential cannot encode.

From charges to a Bethe macrostate

In a thermodynamic Bethe-ansatz description, a macrostate is specified by particle and hole root densities $\rho_a(\theta)$ and $\rho_{a,h}(\theta)$ constrained by Bethe equations. Local or quasilocal charges are additive,

$$\frac{\langle Q_j\rangle}{L} =\sum_a\int\mathrm d\theta\, q_{j,a}(\theta)\rho_a(\theta).$$

Extremizing the generalized free-entropy functional gives pseudoenergies of the schematic form

$$\epsilon_a(\theta) =\sum_j\lambda_jq_{j,a}(\theta) -\sum_b K_{ab}*\log(1+e^{-\epsilon_b})(\theta),$$

with statistics- and convention-dependent logarithms. The charge data determine the macrostate only if the functions $q_{j,a}(\theta)$ span the relevant occupation information. Missing quasilocal charges can leave distinct root densities with the same retained constraints.

Completeness and representative-state equivalence

For local observables after dephasing, one often expects a representative thermodynamic state to reproduce the diagonal ensemble in the thermodynamic limit. This is not automatic. Check:

• exact rather than approximate conservation over the stated timescale;
• locality or controlled quasilocality of each charge;
• functional independence and completeness for all quasiparticle species;
• convergence of $\sum_j\lambda_jQ_j$ and normalizability;
• equivalence of local observables, not equality of full finite-volume density matrices;
• order of the large-volume and late-time limits; and
• sensitivity to charges omitted by a truncation.

The need for quasilocal charges in the XXZ chain is a canonical warning: a GGE built from an incomplete set of familiar local charges can predict the wrong stationary observables Ilievski et al. 2015.

A free-mode example

For a quadratic integrable theory with occupations $n_k$ conserved mode by mode,

$$\rho_{\mathrm{GGE}} =\frac1Z\exp\left(-\sum_k\lambda_kn_k\right).$$

For fermionic modes,

$$\langle n_k\rangle=\frac1{e^{\lambda_k}+1}, \qquad \lambda_k=\log\frac{1-\langle n_k\rangle}{\langle n_k\rangle}.$$

Matching only energy would replace the full function $\lambda_k$ by $\beta\omega_k$ and generally lose the post-quench mode distribution. Conversely, retaining every finite-volume projector is formally complete but destroys locality and predictive compression. The durable GGE question is which local or quasilocal charges determine the thermodynamic macrostate.

What a GGE does not establish

A GGE is a stationary ensemble, not a proof of relaxation into it. Dephasing, initial-state regularity, and thermodynamic limits enter the dynamical claim. Nor does the existence of integrable charges imply generalized hydrodynamics without local-equilibrium and scale-separation assumptions. Those extensions appear in the thermalization and generalized-hydrodynamics chapters.

The schematic below organizes the relationships used on this page. Inspect it with this question in mind: Which charge structures license chemical potentials or generalized equilibrium states?

A grand-canonical state requires conserved charges and a declared algebra; commuting integrable charges, noncommuting constraints, temporal holonomies, and density thresholds demand distinct constructions and checks. Connections classify the charge structure and required construction; their direction is organizational, not a causal-time ordering. Dashed marks show qualifications and failure boundaries. The diagram is schematic and not to scale.

The surrounding discussion supplies the relevant equations and checks in text form; the figure is a navigational summary.

Exercise

For two fermion modes with occupations $n_1,n_2$, compare a charge-complete GGE with a Gibbs ensemble constrained only by $H=\omega_1n_1+\omega_2n_2$.

Solution

The complete GGE has independent multipliers $\lambda_i=\log[(1-\bar n_i)/\bar n_i]$ and reproduces both target occupations. A Gibbs state requires $\lambda_i=\beta\omega_i$ for one $\beta$, which is possible only if $\log[(1-\bar n_1)/\bar n_1]/\omega_1=\log[(1-\bar n_2)/\bar n_2]/\omega_2$. Matching total energy alone does not fix the two occupations.

References

• Ilievski, Enej, Jacopo De Nardis, Bram Wouters, Jean-Sébastien Caux, Fabian H. L. Essler, and Tomaž Prosen. “Complete Generalized Gibbs Ensembles in an Interacting Theory.” Physical Review Letters 115 (2015): 157201. doi:10.1103/PhysRevLett.115.157201; Open PDF.
• Rigol, Marcos, Vanja Dunjko, Vladimir Yurovsky, and Maxim Olshanii. “Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States.” Physical Review Letters 98 (2007): 050405. doi:10.1103/PhysRevLett.98.050405.
• Vidmar, Lev, and Marcos Rigol. “Generalized Gibbs Ensemble in Integrable Lattice Models.” Journal of Statistical Mechanics 2016 (2016): 064007. doi:10.1088/1742-5468/2016/06/064007; Open PDF.