Generalised Gibbs Ensemble for the TFIM Quench

Why GGE, not Gibbs

For a generic non-integrable Hamiltonian, the eigenstate thermalisation hypothesis predicts that every local observable relaxes to the microcanonical (equivalently, canonical) Gibbs ensemble at the temperature fixed by the initial-state energy density. Free-fermion models like the TFIM are integrable — they have an extensive family of mutually commuting conserved charges $\{n_k\}$ (quasiparticle occupations). The long-time average of a local observable in a quench with these conserved charges is therefore described by the Generalised Gibbs Ensemble

\[\rho_{\rm GGE} \;=\; \frac{1}{Z}\,\exp\!\Bigl(-\sum_k \lambda_k\, n_k\Bigr),\]

with one Lagrange multiplier $\lambda_k$ per conserved mode (Rigol et al., PRL 98, 050405 (2007)). Specialising to the post-quench TFIM, $\rho_{\rm GGE}$ becomes diagonal in the post-quench quasiparticle basis with occupations frozen at their initial-state values — the diagonal ensemble.

A single canonical Gibbs ensemble cannot match every $\langle n_k\rangle`` simultaneously (only one parameter ``\beta$), so it generally fails to predict the long-time observables of an integrable quench. See Calabrese, Essler, Fagotti J. Stat. Mech. (2012) P07016 / P07022 for a detailed TFIM analysis with non-trivial counterexamples to canonical thermalisation.

Setup: TFIM $h$-quench

Pre-quench Hamiltonian:

\[H_0 \;=\; -J \sum_i \sigma^z_i\sigma^z_{i+1} \;-\; h_0 \sum_i \sigma^x_i,\]

initial state $|\psi_0\rangle = |\text{GS}(H_0)\rangle$. Post-quench Hamiltonian $H_f$ is identical with $h_0 \to h_f$ (Ising coupling $J$ is held fixed). At each momentum $k \in [0,\pi]$ the Bogoliubov diagonalisation defines the angle

\[2\theta_k(h) \;=\; \operatorname{atan2}\bigl(J\sin k,\; h - J\cos k\bigr), \qquad \Lambda_k(h) \;=\; 2\sqrt{J^2 + h^2 - 2 J h \cos k}.\]

Because $H_0$ and $H_f$ share the momentum decomposition, every mode occupation in the post-quench basis is conserved:

\[\boxed{\; n_k \;=\; \sin^2\!\bigl(\theta_k(h_0) - \theta_k(h_f)\bigr). \;}\]

Closed-form GGE expectations

The diagonal-ensemble expectation of any quadratic observable in the post-quench basis depends only on $\{n_k\}$. For the per-site energy and the bulk transverse magnetisation:

\[\boxed{\; \varepsilon_{\rm GGE} \;=\; -\frac{1}{\pi}\int_0^\pi\!\!dk\, \frac{\Lambda_k(h_f)}{2}\,\bigl(1 - 2 n_k\bigr), \;}\]

\[\boxed{\; \langle\sigma^x\rangle_{\rm GGE} \;=\; \frac{2}{\pi}\int_0^\pi\!\!dk\, \frac{h_f - J\cos k}{\Lambda_k(h_f)}\, \bigl(1 - 2 n_k\bigr). \;}\]

The factor $(1 - 2 n_k)$ replaces $\tanh(\beta\Lambda/2)$ of the equilibrium expressions. In the no-quench limit $h_0 = h_f$ one has $n_k \equiv 0$ and the right-hand sides reduce to the $T = 0$ ground-state expressions implemented in TFIM.jl / TFIM_thermal.jl.

Energy conservation

The post-quench energy is a constant of motion:

\[\langle\psi_0 \mid H_f \mid \psi_0\rangle \;=\; \langle H_f\rangle_{\rm GGE},\]

— the GGE energy is the (time-independent) initial-state expectation of $H_f$. This is a non-trivial cross-check: re-deriving $\langle\psi_0|H_f|\psi_0\rangle/N$ from the raw BdG matrix elements gives

\[\frac{\langle\psi_0|H_f|\psi_0\rangle}{N} \;=\; -\frac{1}{\pi}\int_0^\pi\!\!dk\, \Bigl[(h_f - J\cos k)\cos(2\theta_0(k)) + J\sin k\,\sin(2\theta_0(k))\Bigr],\]

which is algebraically identical to the GGE form above but uses an entirely different trigonometric branch — making it an excellent regression test for any sign / branch error in the implementation. The standalone test file test/standalone/test_tfim_gge.jl checks this explicitly.

API

m_0 = TFIM(J = 1.0, h = 2.0)   # pre-quench
m_f = TFIM(J = 1.0, h = 0.5)   # post-quench

# Per-site energy density of the relaxed state:
fetch(m_f, GGEValue(Energy()), Infinite(); initial = m_0)

# Stationary transverse magnetisation:
fetch(m_f, GGEValue(MagnetizationX()), Infinite(); initial = m_0)

The initial::TFIM keyword is required. Mismatched Ising couplings (m_0.J != m_f.J) raise a DomainError — only $h$-quenches are covered by the closed forms above.

References

M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii, Relaxation in a Completely Integrable Many-Body Quantum System: An Ab Initio Study of the Dynamics of the Highly Excited States of 1D Lattice Hard-Core Bosons, PRL 98, 050405 (2007).
P. Calabrese, F. H. L. Essler, M. Fagotti, Quantum Quench in the Transverse Field Ising Chain, J. Stat. Mech. (2012) P07016, P07022.
M. Fagotti, F. H. L. Essler, Reduced density matrix after a quantum quench, PRB 87, 245107 (2013).