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Equal-coupling Bose–Fermi mixture

The worked tutorial compares repulsion and selected compositions, with plots, downloadable roots and Lieb–Liniger checks.

The library bethe/bose_fermi.hpp implements periodic ground states of an equal-mass gas containing scalar bosons and spinless fermions. Bose–Bose and Bose–Fermi contact strengths must be equal and repulsive. This is a distinct graded model, not another color of the SU(kappa) Fermi gas. The frontend is bethe-bose-fermi-pbc.

Command-line use

bethe-bose-fermi-pbc --bosons 2 --fermions 3 --length 5 --c 1 --roots
bethe-bose-fermi-pbc --bosons 4 --fermions 0 --length 4 --c 2 --json bosons.json
bethe-bose-fermi-pbc --bosons 1 --fermions 1 --length 1 --c 1e-40 --precision fp128
bethe-bose-fermi-pbc --bosons 2 --fermions 3 --length 5 --c 1 --precision digits-100 --roots
bethe-bose-fermi-pbc --references

Both populations, circumference, and coupling are required. The defaults are fp64 and ground-state calculation; there is no excitation mode yet. --max-iterations and --tolerance expose the library controls described below. The common output options support screen presentation, CPU timing, metadata, and JSON/CSV/TSV files, including streaming and --no-retain. Literature is printed only with --references, not ordinary help.

Alongside fp64, platform-native long double and optional fp128, the frontend accepts explicit MPFR precision (bits-N or digits-N) when MPFR scalars are configured. This covers the existing mixed, pure-species, free and vacuum paths. More arithmetic bits do not extend the supported shells or guarantee convergence within the iteration budget.

states contains total energy, momentum/index, residual, iterations, and convergence status. With --roots, interacting calculations add charge_roots and auxiliary_roots; the pure-boson reduction has an empty auxiliary table. Free calculations instead add free_modes, with species, integer mode, and physical k. Bosons each occupy mode zero. Half-integer labels are decimal in exports. Use --json result.json for all tables, or named destinations such as --csv-table charge_roots=charge.csv.

Failed solves leave energy, physical momentum, and numerical roots missing (JSON null, CSV/TSV empty); requested Bethe labels remain available. Exit codes are 0 for convergence, 2 for numerical failure, and 1 for invalid input or output errors. Parameters and supported shells are checked before opening output files, including with --force.

Hamiltonian and supported sectors

In units \(\hbar ^{2}/(2m)=1\), circumference \(\ell\), the first-quantized Hamiltonian is

\[ H=-\sum_j\partial_j^2+2c\sum_{i\lt j}\delta(x_i-x_j),\qquad c\ge0. \]

Antisymmetry removes the contact interaction between identical fermions. For \(c \gt 0\) and both species present, the current branch requires odd N_f; \(N_{b}\) may be odd or even. Pure bosons, pure fermions, vacuum, and \(c =0\) accept arbitrary counts. Unequal masses/couplings, attraction, other interacting periodic shells, excitations, boundaries, and thermodynamics are not supported.

The equations and normalization follow Imambekov–Demler, Annals of Physics (2006), equations (3), (28)–(34); the original short report is Phys. Rev. A (2006). For N=N_b+N_f charge roots k and M=N_b auxiliary roots lambda:

\[ \begin{aligned} \ell k_j+\sum_a2\arctan\!\left(\frac{2(k_j-\lambda_a)}c\right)&=2\pi I_j,\\{} \sum_j2\arctan\!\left(\frac{2(\lambda_a-k_j)}c\right)&=2\pi J_a,\\{} E&=\sum_jk_j^2. \end{aligned} \]

Our \(J\) is the negative of the paper's auxiliary logarithmic label. There is no auxiliary–auxiliary scattering term.

Our parity audit gives 2I = M (mod 2) and 2J = N (mod 2) from the rational equations. The centered choices \(I_{j} =j -(N -1)/2\), \(J_{a} =a -(M -1)/2\) satisfy both precisely when N_f is odd. We reject even interacting mixed shells rather than treating a centered solution with incompatible parity as periodic. This branch has zero total momentum. At c=0, even fermion populations have two lowest filled shells; we choose the positive-momentum representative, momentum_index=N_f/2, with P=2*pi*momentum_index/ell.

C++ API

#include <bethe/bose_fermi.hpp>

auto state = bethe::bose_fermi::ground_state(2, 3, 5.0, 1.0);
// N_b, N_f, ell, c; double, long double, or enabled Uni20 fp128.
if (state.converged) {
    auto energy = *state.energy;
    auto const& k = state.momenta;
    auto const& lambda = state.auxiliary;
}

SolverOptions<Real> is the shared solver control: default residual tolerance 32 epsilon and 10000 accepted Newton updates. Zero updates only checks the seed. Failures return iteration_limit, stalled, or precision_limit, with no energy. Root arrays on failure, when available, are diagnostics only. Invalid parameters and unsupported shells throw before solving.

Runtime precision requires options and physical inputs at the same finite precision; exact-only, unset and mixed-precision inputs are rejected:

auto p = uni20::Precision::bits(256);
bethe::bose_fermi::SolverOptions<uni20::mpreal> options(p);
auto state = bethe::bose_fermi::ground_state(
    2, 3, uni20::mpreal("5", p), uni20::mpreal("1", p), options);

The shared continuum Newton driver uses normal Uni20 dispatch for MPFR updates, with CPU LU available when MPLAPACK is disabled. No global precision or exponent-range setting is changed.

The pure-boson branch delegates to Lieb–Liniger. Free states store boson zero momenta followed by the occupied fermion modes in momenta; exact integer fermion labels are in free_fermion_modes. These are not interacting charge labels. The mixed branch returns ordered real charge and auxiliary roots and their separate half-integer labels. Pure-boson labels are Lieb–Liniger labels; no auxiliary roots are introduced for this reduction.

For independent compositions or couplings, capture each request and its options by value in the existing bounded Async grid. Each solve owns its Newton scratch and runs its dependent iterations sequentially. Wait on the final grid result; failed solves retain their diagnostics and missing energy. The same grid can contain different explicit precisions. No additional scheduler or model-specific execution wrapper is needed.

Numerical method and verification

Our mixed solver uses dimensionless roots \(q =\ell \,k\), \(g =c \,\ell\), and auxiliary coordinates scaled by max(1,g). Reflection reduction enforces exact paired roots and explicit central zeros. The analytic Jacobian couples only the two root families. It uses the shared rational scattering kernel, compensated sums, normal Uni20 LU dispatch, and physical-domain damped Newton driver. Small-g rank phases are canceled analytically before summing complementary angles; charge residuals are divided by max(sqrt(g),abs(q)) there, and otherwise equations are scaled by N. Returned physical roots are checked again after dimensional conversion. Unresolved subnormal or overflowing energies are not published.

Tests include original complex rational equations, numerical differentiation of the Jacobian, scaling, free-shell energies, weak-coupling slopes, strong fermionization, and native-precision two-particle energies. One fermion agrees with the Lieb–Liniger ground energy; one boson agrees with the corresponding Gaudin–Yang impurity. A separate momentum-space contact Hamiltonian for two bosons and three fermions supplies decreasing variational upper bounds as the cutoff grows. Those finite-cutoff energies are not claimed to be exact.

MPFR tests exercise 128, 256 and 333 bits, including original complex equations, pure-boson and one-fermion Lieb–Liniger reductions, free energies, failed solves, finite-precision central zeros and serial/concurrent agreement. Independent 512-bit two-body bisection checks accuracy growth beyond binary128. At \(c=10^{-60}\) and length one, 333-bit arithmetic resolves \(E/(2c)=1-c/12+O(c^2)\); binary128 cannot resolve that correction. The mixed \(N_b=2,N_f=3\) calculation also resolves the first-order energy shift \(14c\) above its nonzero free Fermi sea at that coupling. Input, root and state exports retain the selected precision, including free boson zeros.