The open-source iFluid framework made the generalized-hydrodynamic (GHD) equations of one-dimensional integrable models numerically accessible, but the accuracy of its solvers was left unquantified in three regimes. First, for the XXZ spin chain the implementation covers only the sector B<0B<0, Δ≥1\Delta\ge 1, where the thermodynamic Bethe ansatz requires infinitely many Bethe-string quasiparticle families, and the tower is truncated in practice on the unquantified assertion that each additional family has a 'diminishing effect'. Second, at low temperature the filling functions develop hard Fermi-sea edges whose fine structure is smeared by fixed-grid quadratures and interpolation. Third, evolution on large or semi-infinite domains relies on extrapolating fillings at grid boundaries, a practice the authors themselves flag as 'usually ill-advised'. The problem is to supply GHD solvers with quantified, controllable errors: truncation-error analysis for the infinite string hierarchy, discretization and interpolation schemes with controlled errors in cold regimes with sharp edges, and robust boundary treatments avoiding extrapolation. High-order schemes and exact quadratures have been developed for the Lieb-Liniger model, but the truncation error of the infinite Bethe-string hierarchy and the discretization error in cold Fermi-edge regimes have not been quantified.

Background

A one-dimensional quantum many-body system is called integrable if it can be diagonalized by the Bethe ansatz: its exact eigenstates are labelled by collections of rapidities, parameters that play the role of wave-vectors for elementary excitations (quasiparticles). In the thermodynamic limit, an integrable state is characterized by root densities ρ(λ)\rho(\lambda), the density of occupied rapidities, or equivalently by filling functions ϑ(λ)=ρ(λ)/ρs(λ)∈[0,1]\vartheta(\lambda)=\rho(\lambda)/\rho^{\mathrm{s}}(\lambda)\in[0,1], the fraction of available quasiparticle states that are occupied (ρs\rho^{\mathrm{s}} is the density of states). The thermodynamic Bethe ansatz (TBA) encodes the thermodynamics: single-particle energies and the two-body scattering kernel Θ(λ−μ)\Theta(\lambda-\mu) determine, through dressing relations, the effective velocity veff(λ)=(∂λϵ)dr(∂λp)dr,v^{\mathrm{eff}}(\lambda)=\frac{(\partial_\lambda\epsilon)^{\mathrm{dr}}}{(\partial_\lambda p)^{\mathrm{dr}}}, where a dressed quantity fdrf^{\mathrm{dr}} solves the linear integral equation fdr=f+∫T(λ−μ) ϑ(μ) fdr(μ) dμf^{\mathrm{dr}}=f+\int T(\lambda-\mu)\,\vartheta(\mu)\,f^{\mathrm{dr}}(\mu)\,d\mu with kernel T=Θ′/(2π)T=\Theta'/(2\pi).

Generalized hydrodynamics (GHD), introduced in 2016 by Castro-Alvaredo, Doyon and Yoshimura and by Bertini, Collura, De Nardis and Fagotti, describes the large-scale dynamics of integrable systems in local equilibrium. Because integrable systems possess infinitely many conservation laws, they do not thermalize in the conventional sense; instead, the filling function itself is transported, satisfying the Euler-scale kinetic equation ∂tϑ(t,x,λ)+veff[ϑ](t,x,λ) ∂xϑ(t,x,λ)=0\partial_t\vartheta(t,x,\lambda)+v^{\mathrm{eff}}[\vartheta](t,x,\lambda)\,\partial_x\vartheta(t,x,\lambda)=0 (with an effective-acceleration term when external couplings vary in space and time). GHD is now a standard quantitative tool, benchmarked directly against cold-atom experiments such as the one-dimensional Bose gas (Lieb-Liniger model) on an atom chip.

'Introducing iFluid' by Møller and Schmiedmayer (SciPost Phys. 8, 041 (2020)), released the first open-source framework for solving these equations numerically. Its authors were explicit that the numerical methodology was immature: (i) 'only little effort has been put into comparing different algorithms for solving the GHD equations', and at very cold temperatures the filling 'resembles that of a Fermi sea, whose hard walls and edges complicates numerical solutions'; (ii) the XXZ spin-chain implementation (a chain of spin-1/21/2 degrees of freedom with anisotropic exchange Δ\Delta and magnetic field BB) is 'only valid in the sector of B<0B<0 and Δ≥1\Delta\ge 1', where the Bethe-ansatz solution requires infinitely many families of quasiparticles: bound states of kk elementary magnons (single spin-flip excitations), called Bethe strings, with coupled root densities {ρk(λ)}k=1∞\{\rho_k(\lambda)\}_{k=1}^{\infty} and nested scattering kernels. In practice the tower is truncated to a few families 'as each additional root density has diminishing effect' — an assertion with no quantitative error analysis; (iii) for partitioning protocols (two semi-infinite reservoirs joined at time zero), the finite computation grid requires extrapolating fillings at the domain boundary, which is 'usually ill advised' and only tolerable when each reservoir is homogeneous; (iv) measured inaccuracies 'in the interpolation and the finite number of grid points' make it 'very hard to resolve the fine structure of the characteristics'. The task is to make GHD numerics for the XXZ chain carry quantified, controllable errors in exactly these regimes. The Bethe-string truncation issue is generic: nested models — models whose Bethe-ansatz solution requires several coupled families of rapidities, such as the Yang-Gaudin model of spin-1/2 fermions with contact interactions — and the sine-Gordon field theory likewise have infinite towers of bound-state or nested-string species.

Problem Statement

Develop GHD solvers for integrable models whose errors are quantified and controllable, closing the accuracy gaps explicitly left open by the iFluid framework (Møller and Schmiedmayer, 2020). The objective is certified-accuracy numerical solution of the GHD (TBA) equations, addressing the following source-traceable accuracy gaps: (1) for the XXZ spin chain in the massive regime Δ≥1\Delta\ge 1, whose Bethe-ansatz description contains the infinite tower of Bethe-string quasiparticle families k=1,2,…k=1,2,\dots (the same infinite-tower structure arises in nested and related models, such as the Yang-Gaudin model and the sine-Gordon field theory, cited here as illustrations of the genericity of the tower structure rather than as additional required targets; the required result is for the XXZ chain), establish and quantify the error committed by retaining only a finite number NN of families, as an explicit quantitative function or bound in NN, temperature, magnetic field and observable, in regimes including those where many families contribute (for instance, near-zero magnetic field, where the ground state itself populates the whole tower, and moderate-to-high temperature, where higher strings are thermally excited); (2) design and analyze spatial/rapidity discretization and interpolation schemes whose errors are controlled in cold regimes where filling functions develop hard Fermi-sea edges and fine multi-edge structure, e.g. by adaptive meshes, edge tracking, or matched asymptotics, replacing the current unquantified 'inaccuracies in the interpolation and the finite number of grid points'; (3) provide boundary treatments for large or semi-infinite domains (partitioning protocols, trap releases) that avoid the extrapolation of fillings at grid boundaries or else quantify its error. Because the objective is certified error control, each of (1)-(3) must quantify its error explicitly (through a bound or a function of the governing parameters), and answers must demonstrate the claimed error control against the independent benchmarks available in the regime studied (exact solutions, published experimental or numerical reference profiles) and state the parameter and temperature ranges over which the claimed error control holds. All three parts (1)-(3) are mandatory: a complete resolution must address all of them, and addressing only a subset does not by itself resolve the task.

References

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