Quenches and Relaxation in Lattice and Continuum Matter

A quench fixes an initial state with one Hamiltonian and then evolves it with another. The resulting dynamics can dephase, settle near a stationary value, approach a prethermal plateau, or agree with a thermal ensemble—but those are different statements. The distinction is controlled by the protocol, conserved sector, observable, spectrum, system size, and order of the long-time and thermodynamic limits.

Polkovnikov et al. 2011 review these protocol and ensemble distinctions for closed interacting systems.

Required background. Equilibration, thermalization, and dephasing supplies the statistical distinctions used below, and the many-body correlator dictionary fixes the real-time observables. Helpful background. The eigenstate thermalization hypothesis explains why generic nonintegrable spectra can yield thermal local data.

Post-quench amplitudes and stationary values

Let $H_i$ prepare a density operator $\rho_i$, and at $t=0$ replace it by $H_f$. For a sudden quench,

$$\rho(t)=e^{-iH_ft}\rho_i e^{iH_ft}, \qquad \langle O(t)\rangle=\operatorname{Tr}[\rho(t)O].$$

In the eigenbasis $H_f\lvert n\rangle=E_n\lvert n\rangle$,

$$\langle O(t)\rangle =\sum_{m,n}\rho_{mn}O_{nm}e^{-i(E_m-E_n)t}.$$

If energy degeneracies are treated explicitly, the infinite-time average keeps precisely the matrix elements inside degenerate energy blocks. For a nondegenerate spectrum it reduces to the diagonal ensemble,

$$\overline{\langle O\rangle} =\lim_{T\to\infty}\frac1T\int_0^T\!\mathrm dt\,\langle O(t)\rangle =\sum_n \rho_{nn}O_{nn}.$$

This identity is kinematic. It does not say that fluctuations around the average are small, that the average is Gibbsian, or that one observable’s stationary value determines another’s. Small temporal variance additionally requires sufficiently many incommensurate gaps and a state spread over many relevant eigenstates. Thermalization requires the diagonal data of local observables to agree with an appropriate thermal ensemble after all exact conserved quantities have been fixed.

The eigenstate-based mechanism for generic systems was demonstrated numerically by Rigol, Dunjko, and Olshanii 2008, whereas the charge-complete generalized-ensemble problem in integrable chains is reviewed by Essler and Fagotti 2016.

For a ramp $H[\lambda(t)]$, the same logic applies after the ramp, but the ramp rate controls the produced excitations. The sudden approximation is valid only when the protocol is short compared with the intrinsic response times of the modes under study; it can be sudden for an infrared mode and adiabatic for a high-frequency one.

Continuum quenches make this mode dependence and the ensuing correlation fronts especially explicit Calabrese and Cardy 2006.

A solvable density-wave quench

Consider spinless fermions on an infinite chain,

$$H_f=-J\sum_j(c_{j+1}^\dagger c_j+c_j^\dagger c_{j+1}),$$

prepared in the charge-density-wave product state $\lvert1010\cdots\rangle$. The single-particle propagator is

$$U_{j\ell}(t)=i^{j-\ell}J_{j-\ell}(2Jt),$$

where $J_r$ is a Bessel function. Summing the initially occupied sites gives the exact local density

$$\langle n_j(t)\rangle =\frac12\left[1+(-1)^jJ_0(4Jt)\right].$$

Thus the staggered density dephases as $t^{-1/2}$ with an oscillatory envelope, while the total density remains $1/2$. No collisions or bath are involved: phases cancel in the local sum. The stationary density agrees with an infinite-temperature value at fixed particle number, but the quadratic model retains every momentum occupation $n_k=c_k^\dagger c_k$. Other observables therefore require a generalized ensemble constrained by those occupations, not an ordinary Gibbs state.

On a ring of length $L$, discreteness eventually invalidates the continuum stationary-phase argument. The fastest lattice quasiparticles have $v_{\max}=2J$, so boundary traversal and recurrence effects enter on times of order $L/v_{\max}$, with detailed revivals set by the commensurability of the finite spectrum. A plateau observed only before that scale is finite-window evidence, not an asymptotic thermodynamic result.

Information propagation and relaxation classes

Local lattice Hamiltonians obey a Lieb–Robinson bound of the schematic form

$$\lVert[A_x(t),B_y]\rVert \le C\lVert A_x\rVert\lVert B_y\rVert e^{-\mu(d(x,y)-v_{\mathrm{LR}}t)}.$$

The velocity $v_{\mathrm{LR}}$ is an upper bound, not generally the measured front velocity. Ballistic fronts can broaden diffusively or with other model-dependent powers, and continuum theories need their own ultraviolet conditions before a finite propagation scale can be asserted.

The underlying locality bound is due to Lieb and Robinson 1972; its velocity should not be identified with a fitted quasiparticle speed without a separate calculation.

The practically useful classification is observable-by-observable:

Regime	What is established	What remains to be shown
Dephasing	Off-diagonal phases cancel in a chosen observable	Small fluctuations, ensemble form, and robustness
Equilibration	The observable stays near a stationary value for most accessible times	Whether that value is thermal
Prethermalization	A parametrically long plateau is controlled by an approximate conserved quantity or effective Hamiltonian	Its lifetime scaling and eventual heating or drift
Integrable relaxation	Local data approach a generalized ensemble	Completeness of the conserved charges and finite-size corrections
Thermalization	Local stationary data agree with the Gibbs or grand-canonical ensemble in the fixed conserved sector	Thermodynamic scaling and independence of atypical initial-state details

The order of limits matters. A thermodynamic relaxation statement normally means $L\to\infty$ before $t\to\infty$ for local observables. At any fixed finite $L$, unitary dynamics is quasiperiodic and sufficiently precise recurrence is unavoidable. Conversely, taking a time average first can erase physically important fronts and long-lived plateaus.

Validity checks

A credible model claim specifies $H_i$, $H_f$, the ramp profile if any, the prepared sector, the measured operator, and the time window. It then checks conservation laws numerically or analytically, varies $L$, compares the observation time with traversal and recurrence scales, and tests whether the same ensemble predicts an observable that was not used to fit it. Integrability breaking should move a generalized-ensemble plateau toward thermal behavior on a coupling-dependent timescale; failure to see that trend may simply mean the accessible window is too short.

Stationarity alone is weakest. A local signal can look stationary because of coarse time resolution, averaging over shots, loss of phase reference, or destructive interference, even while the full state remains far from any thermal ensemble. The strongest conclusion must therefore be phrased in terms of the measured observables and declared limits.

Exercises

1. Degenerate time average. Show that the infinite-time average of $\langle O(t)\rangle$ retains matrix elements with $E_m=E_n$. Why is the diagonal-ensemble formula basis dependent inside a degenerate eigenspace unless it is written with spectral projectors?

Solution

Time averaging gives $\lim_{T\to\infty}T^{-1}\int_0^T\mathrm dt\,e^{-i(E_m-E_n)t}=1$ when $E_m=E_n$ and zero otherwise. If $P_E$ projects onto the complete eigenspace of energy $E$, the invariant expression is $\sum_E\operatorname{Tr}(P_E\rho_iP_EO)$. Individual diagonal entries change under a basis rotation within a degenerate block, whereas the projector expression does not.

2. Dephasing exponent. Use $J_0(x)\sim\sqrt{2/(\pi x)}\cos(x-\pi/4)$ to find the late-time envelope of the staggered density in the free-chain quench. Does the result prove thermalization?

Solution

The staggered component is $(-1)^jJ_0(4Jt)/2$, so its envelope is proportional to $(Jt)^{-1/2}$. This proves dephasing of that particular local observable in the infinite chain. It does not establish a Gibbs description: the post-quench quadratic Hamiltonian conserves all $n_k$, and other local correlators retain their information through those conserved occupations.

References

• Calabrese, Pasquale, and John Cardy. “Time Dependence of Correlation Functions Following a Quantum Quench.” Physical Review Letters 96, 136801 (2006). DOI.
• Essler, Fabian H. L., and Maurizio Fagotti. “Quench Dynamics and Relaxation in Isolated Integrable Quantum Spin Chains.” Journal of Statistical Mechanics: Theory and Experiment 2016, 064002 (2016). DOI.
• Lieb, Elliott H., and Derek W. Robinson. “The Finite Group Velocity of Quantum Spin Systems.” Communications in Mathematical Physics 28, 251–257 (1972). DOI.
• Polkovnikov, Anatoli, Krishnendu Sengupta, Alessandro Silva, and Mukund Vengalattore. “Colloquium: Nonequilibrium Dynamics of Closed Interacting Quantum Systems.” Reviews of Modern Physics 83, 863–883 (2011). DOI.
• Rigol, Marcos, Vanja Dunjko, and Maxim Olshanii. “Thermalization and Its Mechanism for Generic Isolated Quantum Systems.” Nature 452, 854–858 (2008). DOI.