Generalized-Hydrodynamic Riemann Problems and Trap Expansions

Generalized hydrodynamics (GHD) evolves a slowly varying local Bethe macrostate rather than a few thermal densities. At Euler order, every quasiparticle rapidity and species is advected with its state-dependent dressed velocity. Partitioning protocols become self-consistent Riemann problems; trap releases add an initial LDA map and, while the trap remains on, a model-specific force in rapidity space.

Required background. Root densities, macrostates, and observables fixes the dressing convention and effective velocity. Generalized hydrodynamics for integrable systems supplies the conservation-law framework.

Helpful background. Quench action and integrable steady states supplies microscopic initial macrostates for homogeneous regions.

Euler GHD and characteristic velocities

For quasiparticle species $a$, the Euler equation without external force is

$$\partial_t\rho_{\mathrm p,a}(x,\lambda,t) +\partial_x\!\left[ v_a^{\mathrm{eff}}(x,\lambda,t) \rho_{\mathrm p,a}(x,\lambda,t) \right]=0,$$

where at every $(x,t)$

$$v_a^{\mathrm{eff}}(\lambda) =\frac{(\varepsilon_a')^{\mathrm{dr}}(\lambda)} {(p_a')^{\mathrm{dr}}(\lambda)}.$$

The local filling obeys the convective form

$$\partial_t n_a+v_a^{\mathrm{eff}}\partial_x n_a=0$$

under the same regularity and convention assumptions. Dressing is recomputed from the local vector of fillings; using a velocity dressed in the initial state throughout the evolution is generally inconsistent.

Charge conservation follows by multiplying the root-density equation by a bare one-particle charge $h_{i,a}$ and integrating:

$$\partial_t\mathsf q_i+\partial_x\mathsf j_i=0,$$

with

$$\mathsf q_i=\sum_a\int\mathrm d\lambda\, \rho_{\mathrm p,a}h_{i,a}, \qquad \mathsf j_i=\sum_a\int\mathrm d\lambda\, \rho_{\mathrm p,a}v_a^{\mathrm{eff}}h_{i,a}.$$

These equalities are essential numerical checks. Violating a charge at grid refinement can signal an inconsistent dressing solve, species truncation, flux discretization, or boundary treatment.

The partitioning Riemann problem

Join two homogeneous GGEs at $x=0$:

$$n_a(x,\lambda,0)= \begin{cases} n_{a,L}(\lambda),&x<0,\\ n_{a,R}(\lambda),&x>0. \end{cases}$$

At Euler scale the solution depends on the ray $\zeta=x/t$. Substitution gives

$$\left[v_a^{\mathrm{eff}}(\lambda;\zeta)-\zeta\right] \partial_\zeta n_a(\lambda;\zeta)=0.$$

The entropy solution is written implicitly as

$$n_a(\lambda;\zeta)= \begin{cases} n_{a,L}(\lambda), &v_a^{\mathrm{eff}}(\lambda;\zeta)>\zeta,\\ n_{a,R}(\lambda), &v_a^{\mathrm{eff}}(\lambda;\zeta)<\zeta, \end{cases}$$

with the equality set fixed by continuity or a numerical entropy prescription. Because $v^{\mathrm{eff}}$ depends on the unknown filling at the same ray, this is a nonlinear fixed-point problem. Iterating velocities without checking branch changes can converge to a spurious solution near characteristic crossings.

Bertini et al. 2016, main text and Castro-Alvaredo, Doyon, and Yoshimura 2016, §§2–4 independently established this Euler-scale hydrodynamics from local Bethe states and currents.

The validity figure places the Euler solution before diffusion, finite-size, resolution, and weak-breaking tests. Inspect the first dashed exit: agreement at microscopic time or across a discontinuity only a few correlation lengths wide is not an Euler-scale validation.

Validity map for Euler GHD. The exact velocity formula does not remove hydrodynamic scale separation, initial-state, boundary, force, numerical, or platform requirements. Original schematic, not to scale.

Traps and force terms

For a Lieb–Liniger gas in a smooth one-body potential $V(x,t)$, the semiclassical acceleration of wave-number rapidity is

$$\dot k=-\frac{1}{\hbar}\partial_xV.$$

The forced kinetic equation is

$$\partial_t\rho_{\mathrm p} +\partial_x(v^{\mathrm{eff}}\rho_{\mathrm p}) +\partial_k(\dot k\,\rho_{\mathrm p})=0.$$

For a generic integrable model or a field coupling to another charge, the rapidity force must be dressed and matched to the quasiparticle momentum; it cannot be copied from the Galilean formula. Doyon and Yoshimura 2017, §§3–4 formulate GHD in inhomogeneous external fields. Turning off a trap at $t=0$ sets the subsequent force to zero but leaves an inhomogeneous initial macrostate

$$n(x,k,0)= n_{\mathrm{eq}}\!\left[k;\mu_0-V(x),T,\ldots\right]$$

only where LDA and preparation equilibrium were valid. A box edge, narrow cloud, or critical region can require gradient or microscopic treatment.

Euler GHD requires a separation

$$\ell_{\mathrm{micro}}\ll L_{\mathrm{grad}}, \qquad t_{\mathrm{micro}}\ll t, \qquad t\ll\min(t_{\mathrm{rec}},t_{\mathrm{break}},t_{\mathrm{loss}}).$$

It gives ballistic-scale profiles. Front broadening, diffusion, fluctuations, finite-size recurrences, transverse excitations, and residual longitudinal trapping belong to later corrections.

The canonical integrable-matter claim test matrix records initial input, Euler observable, force, breaking scale, finite size, and validation. A reproducible verification workflow solves Riemann and trap protocols with charge and grid-convergence checks.

Exercise

Free-fermion partitioning solution. Let $K=0$, so $v(k)=\hbar k/m$. Show that the Riemann solution is

$$n(k;\zeta)= n_L(k)\,\Theta\!\left(\frac{\hbar k}{m}-\zeta\right) +n_R(k)\,\Theta\!\left(\zeta-\frac{\hbar k}{m}\right).$$

Write the density on the ray.

Solution

With $K=0$, the velocity is state independent, so each mode solves

$$\partial_t n(k)+\frac{\hbar k}{m}\partial_xn(k)=0.$$

Tracing the characteristic backward from $(x,t)$ gives initial coordinate $x_0=x-\hbar kt/m=t(\zeta-\hbar k/m)$. It originated on the left when $x_0<0$ and on the right when $x_0>0$, yielding the displayed result. In the spinless convention, the particle density is

$$\mathsf n(\zeta)= \int_{\mathbb R}\frac{\mathrm dk}{2\pi}\, \left[ n_L(k)\Theta\!\left(\frac{\hbar k}{m}-\zeta\right) +n_R(k)\Theta\!\left(\zeta-\frac{\hbar k}{m}\right) \right].$$

Interactions replace the fixed threshold $\hbar k/m=\zeta$ by a self-consistent dressed-velocity condition.

References

• Bertini, B., Collura, M., De Nardis, J., and Fagotti, M. (2016). “Transport in out-of-equilibrium XXZ chains: Exact profiles of charges and currents.” Physical Review Letters 117, 207201. doi:10.1103/PhysRevLett.117.207201.
• 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.
• Doyon, B., and Yoshimura, T. (2017). “A note on generalized hydrodynamics: Inhomogeneous fields and other concepts.” SciPost Physics 2, 014. doi:10.21468/SciPostPhys.2.2.014.