Sutherland gas on a circle¶

bethe-sutherland-pbc evaluates the finite-volume spectrum of the scalar, bosonic trigonometric inverse-square gas. Here “Sutherland” means the continuum Calogero–Sutherland family, not the Uimin–Lai–Sutherland spin chain. Energies follow explicit spectral rules: there are no iterative root solves or convergence tolerances. fp64, long-double, optional fp128 and explicit MPFR precision (bits-N or digits-N) are supported.

The collision-branch tutorial compares small-ring spectra, explains the labels and illustrates the limits of a bounded scan.

bethe-sutherland-pbc 8 --length 8 --lambda 2
bethe-sutherland-pbc 4 --length 4 --lambda 0.5 --labels -1,0,0,2
bethe-sutherland-pbc 4 --length 4 --lambda 2 --levels 10 --window 2
bethe-sutherland-pbc 4 --length 4 --lambda 2 --levels all --window 2 --format csv
bethe-sutherland-pbc 4 --length 4 --lambda 2 --pseudomomenta --precision fp128

Precision and independent calculations¶

bethe-sutherland-pbc 4 --length 4 --lambda 2 --precision digits-100 --pseudomomenta
bethe-sutherland-pbc 3 --length 4 --lambda 2 --levels all --window 2 --precision bits-256

MPFR needs UNI20_ENABLE_MPREAL=ON; no MPLAPACK is required. Inputs are parsed directly at the requested precision, which also reaches pi, pseudomomenta, energies, gaps and analytic zeros. In the C++ interface, length owns the precision and lambda must have the same finite precision:

auto precision = uni20::Precision::bits(256);
uni20::mpreal length("4", precision), lambda("2", precision);
auto ground = bethe::sutherland::ground_state(3, length, lambda);
auto state = bethe::sutherland::evaluate(3, length, lambda, {-1, 0, 2});
auto scan = bethe::sutherland::spectrum(3, length, lambda, 2);

Mixed, exact or unset parameter values are rejected, including before a budget refusal. Integer coefficients in the positive gap formula remain exact under MPFR until combined with physical factors. This preserves a uniform boost's small gap even when adding it to a huge ground energy rounds it away. Spectrum ordering still compares computed gaps: arithmetic precision does not establish completeness outside the chosen label window.

Independent parameter/window scans can use the existing Async observable grid, capturing parameters and labels by value and awaiting the final result. A single cheap formula stays serial; the CLI adds no scheduler overhead. Tests compare complete states and refused scans with serial execution at independent precisions. Numerical checks apply the differential Hamiltonian to two-body Gegenbauer wavefunctions, show accuracy growth against a 512-bit reference, and resolve weak-coupling energy corrections beyond binary128 resolution. Native many-body Jastrow and Jack wavefunction checks remain in place.

Hamiltonian and collision branch¶

With \(\hbar =2m =1\), circumference \(L \gt 0\) and \(N \ge 1\), our convention is

\[ H=-\sum_i\partial_i^2 +2\lambda(\lambda-1)\left(\frac\pi L\right)^2 \sum_{i\lt j}\frac{1}{\sin^2[\pi(x_i-x_j)/L]}. \]

The input \(\lambda \ge 0\) selects the bosonic Jastrow domain, with wavefunctions behaving as \(\lvert x_{i} -x_{j} \rvert ^\lambda\) at a collision. It is not the raw potential coefficient. In particular:

\(\lambda =0\): ordinary free bosons, allowing particles to coincide.
\(\lambda =1\): hard-core bosons; the collision boundary condition remains different even though the explicit potential coefficient is also zero.
\(0\lt \lambda \lt 1\): negative inverse-square coefficient on this specified branch, not an arbitrary attractive potential with an unspecified boundary condition.

Replacing \(\lambda\) by \(1-\lambda\) leaves the potential unchanged but generally changes the domain and spectrum. We do not choose a branch by inverting that coefficient. At \(\lambda =1/2\), the Jastrow branch excludes the logarithmic collision solution. Other self-adjoint extensions are outside this interface.

The normalization and partition spectrum are given in Gurappa–Panigrahi, Eqs. (18)–(23). Their coupling \(\beta\) is our \(\lambda\); their descending partition entries become our ascending labels after reversal and an optional integer boost. The original solution is Sutherland (1971).

Labels, pseudomomenta and gaps¶

Specify \(N\) nondecreasing integers n_0 <= ... <= n_(N-1). Repeated and negative labels are allowed. The ground state has every label zero. With \(q =2\,\pi /L\),

\[ \begin{aligned} k_j&=q\left[n_j+\lambda\left(j-\frac{N-1}{2}\right)\right], \quad j=0,\ldots,N-1,\\{} E&=\sum_j k_j^2,\qquad \mathrm{momentum\_index}=\sum_jn_j,\\{} P&=q\,\mathrm{momentum\_index},\\{} E_0&=\left(\frac\pi L\right)^2\frac{\lambda^2N(N^2-1)}3. \end{aligned} \]

Continuum momentum is not reduced modulo \(2\,\pi\). The \(k_{j}\) are pseudomomenta, not separately occupied free-particle orbitals except in the appropriate free limits. At \(\lambda =1\), the free-fermion dual has integer momenta for odd \(N\) and half-integer, antiperiodic momenta for even \(N\), while the physical bosons always have periodic boundary conditions.

A uniform integer boost \(n_{j} \to n_{j} +b\) changes \(P\) by \(N \,q \,b\) and \(E\) by \(2\,q \,b \,P +N \,q ^{2}\,b ^{2}\). Reflecting and reversing the labels reverses \(P\) without changing the energy. Changing only \(L\) scales momentum as \(1/L\) and energy as \(1/L ^{2}\).

The implementation computes the gap directly from nonnegative terms,

\[ \mathrm{gap}=q^2\left[\sum_jn_j^2+ \lambda\sum_{j=1}^{N-1}j(N-j)(n_j-n_{j-1})\right]. \]

This avoids subtracting nearly equal total and ground energies. A small boost gap can remain accurate even when E0+gap rounds back to E0. Integer coefficients are accumulated before the scalar prefactors, preserving ordinary small-label degeneracies. Very large labels and nearly equal distinct gaps remain subject to the chosen floating-point precision.

What an excitation scan covers¶

--levels COUNT --window W returns the lowest COUNT states within \(-W \le n_{j} \le W\), including the ground state. --levels all returns every ordered label list in that window. One row is one bosonic eigenstate; equal energies are separate rows, and COUNT can cut through a degeneracy. Ordering uses the computed gap, then total momentum index, then lexicographic labels.

There are binomial(N+2*W,N) states in the window. Before allocating them, the tool checks this against --max-states (default 100,000). Exceeding that budget returns exit status 2 with no state rows. It does not publish a partial search as the lowest levels. Every accepted scan visits the whole window before truncation; memory grows as \(N\) times the number of states, even for a small requested COUNT. Increase budgets with that cost in mind.

The continuum spectrum is infinite. No finite-window result claims global low-energy completeness. Increase the window and compare if states outside it could matter. Ground-state and explicit-label evaluation need no scan.

The implementation bounds \(N\) at 1,000,000 and \(W\) at 1,000,000. Explicit labels and their total momentum index must fit signed 64-bit integers; a wide internal sum allows cancellation of prefixes outside that range. Invalid inputs, nonfinite computed results and positive energies/gaps underflowing to zero are errors (CLI exit 1), not valid zero energies. As with other numerical tools, extreme intermediate arithmetic can exceed the selected scalar range.

Output formats are auto, pretty, plain, csv, tsv, and json, with independent file exports through --csv FILE, --tsv FILE, and --json FILE. The primary states table has columns state_id,labels,energy,gap,momentum_index,p; integer label lists remain space-separated. --pseudomomenta adds a separate typed table state_id,index,label,k, joined to states by zero-based state_id. Use --table pseudomomenta to select it for CSV/TSV, or export both tables as JSON. See output conventions for metadata, streaming and table selection. Use --references for centralized literature, also listed in CITATIONS.md.

Library and validation¶

#include <bethe/sutherland.hpp>
namespace cs = bethe::sutherland;
auto ground = cs::ground_state(8, 8.0, 2.0);
auto excited = cs::evaluate(4, 4.0L, 0.5L, {-1, 0, 0, 2});
auto window = cs::spectrum(4, 4.0, 2.0, 2, 10); // Check window.complete.

Tests apply the coordinate-space differential Hamiltonian independently to many-particle Jastrow ground states, degree-two Jack excitations and a family of two-particle Gegenbauer excitations. Further checks cover free-boson and hard-core limits, boosts, reflection, length scaling, window counts, range errors, and native-precision CLI parsing/output. These run for every enabled scalar type, without narrowing comparisons to double.

This first slice does not expose wavefunctions, matrix elements, spinful or fermionic variants, twists, trapped/rational Calogero systems, thermodynamic response functions, or arbitrary collision domains.