Quench Action and Integrable Steady States

The quench action determines an integrable steady macrostate directly from overlaps between the initial state and post-quench Bethe eigenstates. It combines the exponentially small weight of an individual eigenstate with the exponentially large number of states sharing a root density. The saddle describes stationary local observables under dephasing assumptions; it does not turn the diagonal ensemble into an ordinary thermal ensemble.

Required background. Integrable charges and generalized-ensemble tests supplies macrostate completeness and the comparison with a GGE.

Helpful background. Equilibration, thermalization, and dephasing supplies the time-averaging and local-observable limits.

Overlap large deviations and Yang–Yang entropy

Expand the initial state in energy eigenstates,

$$\lvert\Psi_0\rangle=\sum_\alpha c_\alpha\lvert\alpha\rangle, \qquad c_\alpha=\langle\alpha\vert\Psi_0\rangle.$$

The diagonal expectation of an observable is

$$\langle O\rangle_{\mathrm{DE}} =\sum_\alpha |c_\alpha|^2 O_{\alpha\alpha}.$$

For Bethe states with macrostate $\boldsymbol\rho=\{\rho_{\mathrm p,a}\}$, suppose the overlap has the thermodynamic large-deviation form

$$-\frac1L\ln \left|\langle\boldsymbol\rho\vert\Psi_0\rangle\right| \longrightarrow \mathcal S_{\mathrm{ov}}[\boldsymbol\rho].$$

The number of microscopic states near $\boldsymbol\rho$ grows as $\exp[Ls_{\mathrm{YY}}(\boldsymbol\rho)]$, possibly reduced by initial-state selection rules. Therefore the leading functional in the diagonal sum is

$$\mathcal S_{\mathrm Q}[\boldsymbol\rho] =2\operatorname{Re}\mathcal S_{\mathrm{ov}}[\boldsymbol\rho] -s_{\mathrm{YY}}[\boldsymbol\rho],$$

subject to Bethe root-density constraints and any exactly fixed charges. The stationary macrostate satisfies

$$\frac{\delta\mathcal S_{\mathrm Q}} {\delta\rho_{\mathrm p,a}(\lambda)} \bigg|_{\boldsymbol\rho_*}=0.$$

The factor two comes from $|c_\alpha|^2$. Omitting it changes the saddle. If only parity-invariant Bethe states have nonzero overlap, the entropy term must count that restricted set rather than all microscopic states.

Caux and Essler 2013, main text introduced this saddle-point construction for interacting integrable quenches. Exact overlap formulas, their normalization, zero modes, and string selection rules are model and initial-state data.

The structure figure shows how the overlap route and charge route meet at the same macrostate when both are complete. Inspect the entropy label: the quench action includes state counting explicitly, whereas matching a few charges does not reconstruct it by itself.

Quench-action dictionary. Overlap normalization, allowed species and symmetries, entropy counting, thermodynamic limit, and dephasing are all required to turn microscopic coefficients into a stationary local macrostate. Original schematic, not to scale.

Explicit free-fermion saddle

For independent post-quench modes $k$, let the initial Gaussian state have occupation probability $n_0(k)$, with no unresolved anomalous coherence in the stationary basis. A macrostate with occupation $n(k)$ has relative-entropy quench action per unit length

$$\mathcal S_{\mathrm Q}[n] =\int\frac{\mathrm dk}{2\pi} \left[ n\ln\frac{n}{n_0} +(1-n)\ln\frac{1-n}{1-n_0} \right].$$

Varying gives

$$\frac{\delta\mathcal S_{\mathrm Q}}{\delta n(k)} =\frac1{2\pi} \ln\frac{n(k)[1-n_0(k)]} {n_0(k)[1-n(k)]},$$

so the unique interior saddle is $n_*(k)=n_0(k)$. This simple result already contains both overlap probabilities and binomial entropy. It is also a GGE with mode-dependent multiplier

$$\beta(k)=\ln\frac{1-n_0(k)}{n_0(k)}.$$

For an interacting Bethe gas, the occupation modes are coupled through root-density constraints and the overlap functional, so the saddle becomes a nonlinear integral equation rather than a pointwise identity.

Limits, excitations, and steady-state claims

The leading extensive saddle fixes stationary local one-point functions. Subextensive overlap terms, determinants, and nearby particle–hole excitations control normalization, finite-size corrections, and time-dependent correlation functions. Degenerate eigenstates can retain off-diagonal coherences; selection rules or bound-state strings omitted from the overlap formula can change the saddle itself.

Order of limits matters:

$$\lim_{t\to\infty}\lim_{L\to\infty} \langle O(t)\rangle$$

need not equal the reverse order because finite systems recur. A steady local macrostate can emerge for microscopic times $t_{\mathrm{micro}}\ll t\ll t_{\mathrm{rec}}(L)$ without the global pure state converging in norm.

A quench-action saddle agrees with a GGE only if the latter’s charges are complete on the relevant state class. Agreement on energy and particle number is not a completeness test. Direct time evolution, exact finite-size diagonal sums, charge reconstruction, and sum rules provide independent checks.

Brockmann et al. 2014, §§3–5 derive normalized overlaps and the quench action for a Néel-to-XXZ quench, illustrating the importance of parity restrictions and strings. Pozsgay et al. 2014, main text compare steady-state constructions and expose the failure of an incomplete GGE.

The canonical integrable-matter claim test matrix retains the initial-state input, entropy, charge completeness, and finite-size ceiling. A reproducible verification workflow checks overlap saddle equations and charge reconstruction.

Exercise

Convexity of the free-mode quench action. Verify the saddle above and compute the second functional derivative for $0<n_0(k)<1$.

Solution

The first derivative vanishes only when

$$\frac{n(1-n_0)}{n_0(1-n)}=1,$$

which gives $n=n_0$. The local second derivative is

$$\frac{\delta^2\mathcal S_{\mathrm Q}} {\delta n(k)\delta n(q)} =\frac{\delta(k-q)}{2\pi} \left[\frac1{n(k)}+\frac1{1-n(k)}\right] =\frac{\delta(k-q)}{2\pi n(k)[1-n(k)]}.$$

It is positive for $0<n<1$, so the saddle is a minimum. Modes with $n_0=0$ or $1$ lie on the boundary and are fixed rather than varied as interior degrees of freedom.

References

• Brockmann, M., Wouters, B., Fioretto, D., De Nardis, J., Vlijm, R., and Caux, J.-S. (2014). “Quench action approach for releasing the Néel state into the spin-$1/2$ XXZ chain.” Journal of Statistical Mechanics: Theory and Experiment 2014, P12009. doi:10.1088/1742-5468/2014/12/P12009.
• Caux, J.-S., and Essler, F. H. L. (2013). “Time evolution of local observables after quenching to an integrable model.” Physical Review Letters 110, 257203. doi:10.1103/PhysRevLett.110.257203.
• Pozsgay, B., Mestyán, M., Werner, M. A., Kormos, M., Zaránd, G., and Takács, G. (2014). “Correlations after quantum quenches in the XXZ spin chain: Failure of the generalized Gibbs ensemble.” Physical Review Letters 113, 117203. doi:10.1103/PhysRevLett.113.117203.