Implement a new model

BAFQMC can be developed beyond the two triangular-lattice models distributed here. This guide maps a new Hamiltonian onto the actual solver, ED, and measurement code. It covers changes to the lattice, hopping, pairing, flavors, and interactions. For a parameter scan of an already implemented model, use the calculation recipes. The observable reference defines the current measured operators and output normalizations that extensions can build on.

Start from the Hamiltonian the researcher requested and carry the extension through a working example and independent numerical checks. The worked anisotropic-lattice example below is an implementation blueprint; the current executable has no generic lattice or hopping-matrix input flag.

Specify sites, operators, and the Hamiltonian

Distinguish unit cells, physical sites, flavor modes, and Nambu components. For a crystal with \(n_{\mathrm{sub}}\) sites per unit cell,

\[ N_{\mathrm{cell}}=L_xL_y,\qquad N_s=n_{\mathrm{sub}}N_{\mathrm{cell}},\qquad \mathbf r_{\mathbf R a}=R_1\mathbf a_1+R_2\mathbf a_2+\boldsymbol\delta_a. \]

Here \(a\) is the sublattice index and \(\boldsymbol\delta_a\) is its position within the cell. With \(F\) boson flavors there are \(M=F N_s\) annihilation operators. A full Nambu representation has \(2M\) components. The existing number-conserving solver propagates one flavor and reconstructs its conjugate partner; its matrix dimension is consequently \(N_s\) for the current model.

Write a site/flavor index convention and an explicit bond list, including periodic image displacements. State whether each bond appears once with its Hermitian conjugate or as two directed entries. Small periodic clusters can have repeated image bonds and self-image bonds: summing their amplitudes and deduplicating their endpoints are different Hamiltonians. The current code sums the image contributions.

A useful general quadratic specification, using the paper's negative pairing convention, is

\[ H_0 =\sum_{\alpha\beta} a_\alpha^+ h^{(0)}_{\alpha\beta}a_\beta -\frac12\sum_{\alpha\beta} \left(\Delta_{\alpha\beta}a_\alpha^+a_\beta^+ +\Delta_{\alpha\beta}^{*}a_\beta a_\alpha\right)+C, \qquad h^{(0)}=\bigl(h^{(0)}\bigr)^\dagger,\quad \Delta=\Delta^T. \]

The physical Hamiltonian and its grand-canonical counterpart are

\[ H=H_0+H_{\mathrm{int}},\qquad H_\mu=H-\mu N,\qquad N=\sum_\alpha a_\alpha^+a_\alpha, \qquad h=h^{(0)}-\mu I. \]

The combined indices \(\alpha,\beta\) include physical site and flavor, and \(h\) is the normal matrix used for the thermal trace. Hermitian hopping requires the reverse amplitude to be the complex conjugate. Bosonic pairing is symmetric under exchange of the two combined indices; the factor \(1/2\) avoids double counting. State the physical pairing phase explicitly. The present solver uses a positive real interflavor pair coefficient, related to the paper's negative coefficient by \(c_{\mathrm{code}}=-c_{\mathrm{paper}}\).

Specify the interaction in operators, including its linear and constant terms. For example, \(n^2=n(n-1)+n\), so replacing one by the other also changes the chemical-potential contribution. Keep physical energy and grand-canonical energy separately defined throughout the implementation. For extensions of the paper's main model, use \(b,c\) for the two flavors and \(U=U_2\) with \(U_1=0\) for the relative-density interaction.

Derive the decoupling and its sign protection

For an interaction written as a sum of Hermitian quadratic channels \(H_{\mathrm{int}}=\sum_\ell g_\ell Q_\ell^2\), derive the HS representation used in each Trotter factor with time step \(\Delta\tau\). A continuous Gaussian channel has the identity

\[ e^{-\Delta\tau g Q^2} =\int_{-\infty}^{\infty}\frac{d\phi}{\sqrt{2\pi}} e^{-\phi^2/2} e^{\sqrt{-2\Delta\tau g}\,\phi Q}. \]

For \(g<0\) the field coefficient is real; for \(g>0\) it is imaginary. If the channels or kinetic factors do not commute, document their order and the Trotter approximation. Carry every normal-ordering shift and scalar factor into the weight, rather than placing all information in the matrix propagator.

For each independent field configuration, establish the applicable time-reversal symmetry (TRS), reflection positivity (RP), or conjugate-sector construction after decoupling. Write the symmetry operator or reflection, its action on sites/flavors/Nambu indices, and the conditions on each generated quadratic factor. For a TRS route, record the antiunitary square and all additional hypotheses of the sign-positivity criterion being used. For an RP route, specify the reflected partition and the required form and sign of cross-partition couplings. Symmetry of the original interaction alone does not check these configuration-level conditions.

The simplest existing construction gives an explicit example. For equal real hopping of the two flavors, \(U_1\leq0\) and \(U_2\geq0\) generate conjugate single-flavor factors at the same field values. If \(B_c(\phi)=B_b(\phi)^*\), then, within the convergent trace domain,

\[ w(\phi)=p_{\mathrm{HS}}(\phi) \left|\det\left[I-B_b(\phi)\right]\right|^{-2}\geq0. \]

The number-conserving code uses this relation directly. A model with different flavor hoppings, flavor mixing, or a new HS channel needs its own matrix representation and weight derivation whenever that relation changes. Taking an absolute value of a new determinant is not a proof of its sign protection. For complex hopping, this shortcut requires conjugate normal matrices, \(h_c=h_b^*\), together with conjugate HS factors. Equal complex hopping with the same flux in both flavors generally does not satisfy that condition. Replacing one flavor's flux by its opposite changes the requested Hamiltonian; assess the actual same-flux model using its applicable symmetry construction. If a proposed Hamiltonian lies outside an established class, identify which condition is affected and develop the requested model accordingly; do not silently change its physical couplings to recover the old class.

If a requested model instead requires complex-weight sampling, implement that estimator explicitly. The paired solver's local ratio-phase diagnostics are not phase reweighting: its current measurements are ordinary averages. A complex-weight calculation needs the full configuration phase \(w(\phi)=|w(\phi)|e^{i\theta(\phi)}\) and the corresponding estimator

\[ \langle O\rangle =\frac{\langle O(\phi)e^{i\theta(\phi)}\rangle_{|w|}} {\langle e^{i\theta(\phi)}\rangle_{|w|}}. \]

This entails phase-aware measurement and analysis, including the scalar and branch contributions to \(\theta\), or a separately established positive construction for the same physical Hamiltonian.

Establish the physical trace and matrix representation

Sign protection and trace convergence are separate parts of the construction. Specify the parameter region in which the grand-canonical physical model has a finite thermal trace, and establish the domain required by the decoupled quadratic traces. A finite ED occupation cutoff defines a finite-dimensional reference; convergence toward an untruncated model is a further calculation.

For a quadratic model, positivity of the Hermitian bosonic energy matrix is a useful sufficient condition for a finite thermal trace. One sufficient bound for the quadratic form above is

\[ \lambda_{\min}(h)>\|\Delta\|_2. \]

Real Bogoliubov frequencies alone do not establish positivity of that energy matrix. For the current \(U_1=0\) construction with uniform onsite pairing and real hopping, the corresponding sufficient bound is

\[ \mu<\varepsilon_{\min}-|\Delta|, \]

where \(\varepsilon_{\min}\) is the hopping-band minimum. The relative-density HS factors are unitary. The triangular benchmark has \(\varepsilon_{\min}=-3t\); a different graph needs its own band bound. New nonunitary HS channels require the corresponding trace-domain analysis.

Keep the quadratic energy matrix distinct from the bosonic commutator matrix that is exponentiated. In the full Nambu formulation the latter is generally non-Hermitian even for a Hermitian physical Hamiltonian. The paired solver's exp_general_matrix uses general-complex diagonalization. A new pairing matrix must retain the correct particle/hole signs and conjugations; using a Hermitian eigensolver on that commutator matrix changes the propagator.

The paired representation also has a scalar normal-ordering contribution. For the current total-density HS field \(x=\sqrt{-2U_1\Delta\tau}\,\phi_1\), the scalar factor is \(e^{-x}\), and a proposal contributes \(e^{-(x'-x)}\) to ratio_constant. Re-derive this factor for a different quadratic generator or Nambu convention.

The local determinant factor currently uses det_Pblock**(-0.5d0). The code records the phase and samples the magnitude; it does not maintain an explicit branch-continuation state. Establish the square-root branch associated with the physical trace for a new paired model, anchored in a known convergent limit, and test the full ratio including its scalar. If the model needs branch tracking, implement and verify that state as part of the extension. Changing the determinant exponent alone does not convert between a full and a reduced Nambu representation.

Change the actual implementation

The following map points to active source files. Both components use similar names, with different matrix dimensions and update formulas.

Responsibility	Files and symbols	Required coordination
Input and dimensions	`src/<solver>/src/calc_basic.f90`: `read_input`, `Params_set`	Read and broadcast new parameters; separate cells, sites, flavors, and sectors; record them in run metadata. `RT=1` is currently assigned here.
Crystal and graph	`src/<solver>/src/lattice.f90`: `Lattice_make`	Update `L_bonds`, `LT_bonds`, site/sector index maps, real/reciprocal vectors, and Fourier phases.
Quadratic propagation	`src/<solver>/src/non_interact.f90`: `def_hamT`, `opT_set`; paired `exp_general_matrix`	Build the new hopping/pairing matrix and its inverse-time factors, keeping the physical and commutator conventions distinct.
Fields and channel schedule	`fields.f90`: `AuxConf_make`; `model.f90`: `Model_init`; `local_sweep.f90`	Update field support, channel allocation, initial/restart I/O, and both sweep directions. The active schedule explicitly uses two channels.
HS factors	`operator_Hubbard.f90`: `opU_set`, `opU_get_delta`, `opU_mmult_L/R`	Implement the derived operator, field measure, scalar, and local matrix change. In the paired code, coupling sign currently selects the channel type.
Ratios and Green updates	`localU.f90`: `LocalU_metro`; `multiply.f90`; `stabilization.f90`	Match update support and rank, derive the determinant ratio, and verify stabilized propagation against direct matrix products.
Measurements	`obser_equal.f90`: `Obs_equal_calc`; `fourier_trans.f90`	Change kinetic, interaction, pairing, and momentum estimators with the Hamiltonian; carry definitions into file output and analysis.
Campaigns and references	`src/<solver>/run_paper.py`, `src/<solver>/benchmarks/campaign*.py`, ED files below	Generate consistent solver/ED inputs, validate the implemented model, and report every new parameter and reference cutoff.

Consult the source directly through the number-conserving component and paired component. In particular:

A sublattice count is not a complete lattice definition. The current Norb=1 bond construction explicitly targets destination orbital 1. A kagome extension needs three physical sublattices, intracell positions, intercell bonds, and consistent orbital indices. Audit allocations and normalizations using Lq, Ndim, Nsite, Norb, Nsub, and Nbond. Sublattice count, coordination number, and forward-bond storage count are distinct. Nbond=3 and the spatial/time columns of LT_bonds are currently fixed to the triangular graph. Check tensor extents in Fourier helpers such as m_write_k_3 and m_write_reciprocal_3 against their callers; orbital-correlation dimensions must follow the new orbital indexing. The paired Nsec=4 labels \((b,c,b^+,c^+)\); it is not the sublattice count.
Quadratic and interaction changes affect different update support. New deterministic hopping or bond pairing may retain the existing onsite density-HS support. A bond HS field, flavor-off-diagonal channel, or new interaction changes it. The number-conserving LocalU_metro applies a rank-one diagonal-site update. The paired version selects four sectors of one site and uses only diagonal entries of its \(4\times4\) Delta. Adding off-diagonal entries to that array alone does not implement the required Woodbury update.
Channel signs encode operators. Paired opU_set selects total density for negative coupling and relative density for positive coupling. Opposite signs or additional operators require explicit channel definitions; renaming Op_U1 or changing a JSON number does not change that dispatch.
Measurements contain the old Hamiltonian explicitly. Kinetic energy sums the scalar RT over L_bonds; paired energy uses onsite RDelta*pair_equal. Update these when propagation changes. The current paired implementation extracts the full four-sector contractions; it does not reconstruct its interacting paired \(c\) sector by conjugating \(b\).
Momentum and normalization follow the new crystal. Use \(e^{i\mathbf q\cdot(\mathbf r_{\mathbf R a}-\mathbf r_{\mathbf R'b})}\), including sublattice positions, and state whether the result is a matrix in sublattice indices or a summed physical structure factor. Audit every \(N_s^{-1}\) and \(N_s^{-2}\) factor. The triangular K index is not a generic ordering momentum; current noncommensurate Fortran runs leave its output zero while paired ED omits it. That zero represents an unavailable K estimator. Define and implement the new model's actual momenta.

The existing paired time-dependent observable routine Obs_tau_calc is a placeholder. A research task involving imaginary-time correlations needs an implemented estimator and its propagation checks in addition to the equal-time workflow.

Extend ED and analysis with the solver

The number-conserving ED driver contains _hamiltonian_static, _hopping_two_species, _translation_permutations, the two-species basis construction, and K-point operators. Its public validator currently restricts interacting reference calculations to \(3\times3\). Rework those restrictions and symmetry blocks when introducing other sizes or graphs; do not retain translation blocks that the new Hamiltonian no longer conserves. The free reference in src/number_conserving/benchmarks/campaign_analysis.py also contains the triangular dispersion and needs the new spectrum.

For pairing, update geometry_triangle.py, the general driver's build_basis, build_hamiltonian, and observable operators, and the tensor ED implementation. The present basis has two flavors per physical site. Complex hopping/pairing also requires complex coefficients and a suitable Hamiltonian dtype. Preserve the distinction between physical site count and QuSpin's basis.Ns, which is the many-body Hilbert-space dimension.

Add a model identifier and explicit parameter fields in the new input schema when several implementations coexist. Extend the readers and validators that consume those fields. A new manifest flag has an effect only after those readers and numerical kernels implement it. Keep generated tables self-describing: model, geometry, parameters, units, normalization, sampling, SEM, ED cutoffs, and energy convention. Use separate examples and campaign outputs so the original triangular cases and new model are both runnable. Update reference data intentionally when definitions or calculations change, and explain the change alongside the result.

Worked blueprint: anisotropic triangular hopping

Consider a requested extension with the same two flavors and onsite channels, but three independent real hopping amplitudes:

\[ H_{t}=\sum_{\mathbf R,\sigma}\sum_{\nu=1}^{3}t_\nu \left(a_{\mathbf R\sigma}^+a_{\mathbf R+\mathbf d_\nu,\sigma} +a_{\mathbf R+\mathbf d_\nu,\sigma}^+a_{\mathbf R\sigma}\right), \qquad \mathbf d_1=(1,0),\quad\mathbf d_2=(0,1),\quad\mathbf d_3=(-1,1). \]

The displacement pairs are integer cell coordinates. Keep the physical triangular primitive vectors from the algorithm guide. The dispersion becomes

\[ \varepsilon(\mathbf k)=2\left[ t_1\cos(\mathbf k\cdot\mathbf a_1) +t_2\cos(\mathbf k\cdot\mathbf a_2) +t_3\cos\bigl(\mathbf k\cdot(\mathbf a_2-\mathbf a_1)\bigr) \right]. \]

This change preserves the conjugate-flavor relation for equal real hoppings and the original onsite HS channels. For \(U_1=0\) and uniform real onsite pairing, a simple sufficient thermal bound is

\[ \mu<-2\left(|t_1|+|t_2|+|t_3|\right)-|\Delta|. \]

It follows from a lower bound on the hopping spectrum and need not be the tight band minimum. As a concrete implementation test, use \((t_1,t_2,t_3)=(1,0.8,0.35)\), \(\mu=-5\), \(\beta=1\) and first set \(U_1=U_2=0\). Test \(\Delta=0\) and, for the paired extension, \(\Delta=0.1\). Then exercise the existing relative-density interaction at \(U_2=0.7\). These are proposed extension tests, not bundled runnable cases.

An agent implementing this request should:

Add and broadcast the three amplitudes in the input representation; retain the isotropic values \((1,1,1)\) as the original-model default. Connect each L_bonds(:,nu) entry to its amplitude in def_hamT.
Make the kinetic estimator use the identical amplitudes and Hermitian conjugates. Keep onsite HS algebra and its local support unchanged after checking the conjugate-sector relation for the new quadratic factors.
Update both ED hopping constructions and the analytic free dispersion. Choose a small finite-occupation reference or extend the current NC ED size interface as needed; record its basis definition.
Compute free thermal density, energy, and momentum occupation independently from the dispersion. For pairing, use \(\omega_{\mathbf k}=\sqrt{(\varepsilon_{\mathbf k}-\mu)^2-\Delta^2}\) in the stable Gaussian case. Check kinetic energy against derivatives with respect to each \(t_\nu\).
Verify a nonuniform fixed HS field by direct dense multiplication, then run an interacting BAFQMC/ED comparison. Recover the original triangular answers at \((1,1,1)\) with the existing paper cases.
Add a small curated example, exact commands, resource measurements, and a model definition to the documentation. Keep larger generated scans in their own output directory.

Setting \(t_3=0\) removes the diagonal graph bonds. To describe a physical square lattice as well, set orthogonal primitive vectors and update the reciprocal coordinates and momentum observables. This illustrates why graph connectivity and its real-space embedding are separate pieces of a model.

Independent verification and deliverables

Build a small dense reference directly from the new model definition. It should assemble site operators, hopping, pairing, and interaction terms independently of the production helpers. Compare Hamiltonian spectra and thermal observables in a common finite Fock basis; also check Hermiticity, pairing symmetry, energy decomposition, and relevant thermodynamic or gauge identities. The approach in test_ed_physics.py is a starting point.

For explicit nonuniform fields, form the ordered short-time matrices and their full products independently. Compare Green functions, the complete weight including scalars, and proposed-update ratios. Test several field configurations and stabilization intervals. Extend test_fixed_field_solver.py to the new geometry/channel; its current coverage checks interacting HS propagation but does not test a nonzero Metropolis proposal. A change to the update formula additionally needs old/new dense ratios and updated Green matrices for deterministic nonzero proposals.

Use analytic free or Bogoliubov limits as in test_gaussian_solvers.py, then perform an interacting run that actually exercises the new terms. Report means, SEM, and reference differences without a universal statistical sigma gate. Test the new model explicitly: passing the old triangular benchmarks only establishes that those cases remain consistent.

Run the existing checks alongside the added model tests:

make check
make physics

Use PYTHON_ED=/path/to/quspin/python when ED has a separate environment. These checks have independent physical references; a new model should add equally direct coverage. Full 22-point production remains available through python3 reproduce.py when required by the requested validation scope.

Deliver a documented Hamiltonian and lattice, the HS/symmetry and trace derivation appropriate to its parameters, working solver and reference implementations, a runnable small example, independent tests, and measured computational cost. Record intentional changes to existing inputs or results as part of normal development. The researcher should be able to run both the original examples and the new physical model from this checkout.