Integrable Quantum Matter and Generalized Hydrodynamics

Integrable quantum matter retains an extensive family of conserved quantities. Generalized hydrodynamics converts that microscopic structure into large-scale motion by evolving the occupations of interacting quasiparticles.

Enter this chapter

Begin with Bethe-Integrable Quantum Gases and Spin Chains to connect scattering data to Bethe quantization in gases and spin chains.

Helpful background. Bethe-Integrable Quantum Gases and Spin Chains and Generalized Hydrodynamics of Integrable Systems develop the definitions used by the chapter’s opening pages.

Topics

1. Bethe-Integrable Quantum Gases and Spin Chains — Fixes scattering, quantization, rapidity, string, energy, momentum, and boundary conventions.
2. Root Densities, Macrostates, and Model Observables — Derives thermodynamic Bethe equations, dressing, entropy, charges, and the effective velocity.
3. Integrable Charges and Generalized-Ensemble Model Tests — Tests whether local and quasilocal charges reconstruct the full macrostate.
4. Quench Action and Integrable Steady States — Combines overlap large deviations and Yang–Yang entropy to select a stationary root density.
5. Generalized-Hydrodynamic Riemann Problems and Trap Expansions — Solves Euler characteristics for bipartitions and trapped gases.
6. Diffusive GHD Corrections as Model Benchmarks — Adds positive Navier–Stokes and fluctuation corrections and separates anomalous sectors.
7. Weak Integrability Breaking and Hydrodynamic Crossover — Derives slow-charge rates and collision terms with a controlled prethermal window.
8. Finite-Size and Experimental Validation of Integrable Matter — Propagates platform, boundary, finite-size, resolution, and covariance into evidence.

How the ideas fit together

Bethe root densities specify a thermodynamic macrostate. Scattering with the occupied background dresses quasiparticle velocities, so transport depends on the entire local distribution rather than a few thermodynamic numbers Castro-Alvaredo, Doyon, and Yoshimura 2016, §§ 2–4.

Conserved charges or quench overlaps select that distribution; Euler evolution transports it. Diffusive corrections and weak integrability breaking then describe different departures from the ideal limit. Agreement with an Euler-scale density profile alone does not determine diffusive broadening, breaking rates, or the full microscopic state.

The comparison below separates the assumptions and tests needed for these different conclusions.

Integrable-matter claim test matrix

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Inputs, hydrodynamic order, and validation tests for integrable-matter descriptions.

Description or claim	Quasiparticles and charges	Dressing and entropy	Initial-state input	Euler or diffusive observable	Breaking and finite size	Validation test
Cross-chapter contract	Species, rapidity measure, physical $p$, $\varepsilon$, $K$, and charge basis fixed	One kernel sign and explicit $2\pi\hbar$ convention; Yang–Yang counting stated	GGE, overlap, LDA, or measured root density identified	Hydrodynamic order and measured quantity declared	Earliest and latest relevant breaking times, $L/v_{\max}$, boundaries, and losses bounded	Exact charge, entropy, grid, held-out, and observation-map checks
Finite-volume Bethe model	Real roots or declared string/nested species; branch quantum numbers	No thermodynamic dressing yet	Boundary condition and symmetry sector	Spectrum, energy, momentum, and form-factor input	Exponential/string and algebraic $1/L$ corrections	Exact diagonalization and state counting at several $L$
Root-density macrostate	$\rho_{\mathrm p,a}$, $\rho_{\mathrm h,a}$ for every retained species	$h^{\mathrm{dr}}=h+Kn h^{\mathrm{dr}}/(2\pi)$; $s_{\mathrm{YY}}$	Thermodynamic eigenstate or local ensemble	Charge densities, currents, susceptibilities, $v^{\mathrm{eff}}$	Species and rapidity cutoff; finite-volume spacing	Bethe equation, positivity, sum rules, cutoff convergence
Charge-complete GGE	Local and quasilocal $h_{i,a}$ span root-density space	TBA saddle and covariance from the same convention	Initial charge densities and symmetries	Stationary local observables	Finite charge set leaves inverse null directions	Rank, held-out charges, quench-action or time-evolution comparison
Quench-action steady state	Bethe species allowed by nonzero overlaps	$2\operatorname{Re}S_{\mathrm{ov}}-s_{\mathrm{YY}}$, with entropy restricted to overlap-allowed states	Normalized initial-state overlaps and selection rules	Stationary local observables and excitations	Subextensive overlap, degeneracy, recurrence	Finite-size diagonal sums, charge reconstruction, direct dynamics
Euler Riemann or trap GHD	Local fillings for all species and dressed velocities	$v^{\mathrm{eff}}=(\varepsilon')^{\mathrm{dr}}/(p')^{\mathrm{dr}}$	Left/right GGE or calibrated initial LDA profile	Ballistic density and current profile	Gradient, boundary, force, $L/v_{\max}$, breaking time	Charge conservation, grid refinement, alternate initial reconstruction
Diffusive GHD	Same species plus scattering-shift fluctuations	$L=DC$ symmetric positive; entropy production nonnegative	Smooth local macrostate and calibrated initial width	$t^{1/2}$ broadening and Navier–Stokes currents	Anomalous sector, free dispersion, finite time and resolution	Positivity, null modes, exact limits, multi-time shared fit
Weakly broken integrability	Nearly conserved charges plus exact residual charges	Slow-mode susceptibility and collision entropy	Prepared prethermal macrostate	Rate matrix or collision-corrected GHD	Resonances, secular terms, rare processes, bath, heating	$g$ scaling, size scaling, several modes and late times
Experimental integrable matter	Platform-calibrated quasiparticle model and observable	Thermodynamic and hydrodynamic solver covariance	Trap, temperature, interaction, preparation, transverse modes	Convolved density, momentum, correlation, or current data	Atom number, boundaries, loss, breaking, recurrence	Independent calibration, held-out times, alternative model, exact cutoff

Table data (JSON, version 1.0.0) · JSON schema

Where to go next

Continue to Nonequilibrium, Driven, and Open Quantum Matter to distinguish dephasing, local relaxation, and prethermal evolution after a quench.

References

• Castro-Alvaredo, O. A., Doyon, B., and Yoshimura, T. (2016). “Emergent hydrodynamics in integrable quantum systems out of equilibrium.” Physical Review X 6, 041065. doi:10.1103/PhysRevX.6.041065.