Many-Body Bases and Classical Simulation

Finite many-body computation begins by specifying the Hilbert space and operator action. Solver sophistication cannot repair a wrong fermionic sign, doubled bond, missing symmetry sector, or uncontrolled boson cutoff. This chapter develops those foundations and connects them to exact diagonalization and circuit simulation.

Enter this chapter

Begin with tensor products in code for subsystem ordering and local contractions. Then choose the appropriate basis construction: spin chains, fermions, or bosonic occupations. These are distinct operator algebras even when stored configurations look similar.

Sparse many-body Hamiltonians turns local rules into matrix entries or matrix-vector actions. Symmetry-sector decomposition reduces the problem only when the symmetry is exact. Exact diagonalization then adds residual checks, degeneracy handling, and finite-size interpretation.

Tight-binding models specializes assembly to finite-orbital hopping systems. Disorder and localization adds ensemble and size analysis. Nonrelativistic field operators explains how mode integrals become occupation-space operators.

Choose a representation by structure

A full spin-$1/2$ state has dimension $2^L$, while fixed magnetization reduces it to $\binom{L}{N_\uparrow}$. Fixed-number fermions have the same combinatorial count but different signs. Bosons with $N$ particles in $M$ unrestricted modes have dimension $\binom{N+M-1}{N}$.

These reductions do not change the model when the conserved sector is chosen correctly. Local occupation cutoffs, by contrast, can change the model and require convergence.

For example, four identical fermions in eight modes need $\binom84=70$ amplitudes, while four bosons in eight modes need $\binom{11}4=330$. Restricting each bosonic mode to at most one particle also gives 70 configurations, but changes its operator algebra and forbids double occupation. Matching dimensions therefore does not justify reusing a fermionic or spin implementation without changing its transition rules. Even for an exact conserved sector, a particle-removing observable maps to a neighboring sector; its matrix is generally rectangular.

For circuits, statevector simulation stores generic pure states. Stabilizer simulation uses polynomial binary data for Clifford circuits and Pauli measurements. Error-correction simulation combines fault sampling, syndrome extraction, decoding, and logical failure classification. Resource estimation accounts for memory, accuracy, and architecture assumptions.

When full-state storage fails, continue to tensor-network algorithms or Monte Carlo. Their approximations and accessible observables differ; neither is a universal substitute for diagonalization.

Small-system benchmarks

A useful verification ladder starts with one operator, then one bond, then one complete small Hamiltonian. For a Heisenberg bond,

$$E_{\rm singlet}=-3J/4,\qquad E_{\rm triplet}=J/4.$$

For a noninteracting two-site hopping problem, single-particle energies are $\pm t$; filling both fermion modes gives energy zero. For two bosons hopping between two sites, the noninteracting energies are $-2t,0,2t$. Together these tests distinguish spin normalization, Pauli exclusion, and bosonic enhancement.

Check matrix action and observables as well as spectra. A transposed complex Hermitian matrix has the same eigenvalues but can reverse phase-sensitive predictions. Compare the full and reduced bases on a small instance before extending system size.

The tests should challenge different assumptions. Unequal tensor-factor dimensions expose reversed axes; a complex local operator exposes a missing conjugation; short translation orbits expose incorrect momentum projection; and a finite boson cutoff exposes products that leave and re-enter the retained space. The corresponding pages supply small calculations that can be reproduced independently. Two codes sharing the same transition routine can share its mistakes; a dense tensor construction or analytic limit checks that routine independently.

Reading a numerical claim

A controlled result identifies the basis, sector, boundary conditions, truncations, solver residuals, and physical observable. Finite-size trends need several sizes and a justified extrapolation; disorder results need independent realizations and uncertainty; circuit results need a specified gate and noise model.

The word “exact” never removes floating-point error or the distinction between a finite model and an infinite system. Keeping those layers explicit makes small-system computation useful as a reference for more approximate methods.

References

• Aaronson, S., and Gottesman, D., “Improved simulation of stabilizer circuits,” Physical Review A 70, 052328 (2004), arXiv:quant-ph/0406196.
• Weinberg, P., and Bukov, M., “QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems. Part I: spin chains,” SciPost Physics 2, 003 (2017), arXiv:1610.03042.
• Weinberg, P., and Bukov, M., “QuSpin. Part II: bosons, fermions and higher spins,” SciPost Physics 7, 020 (2019), arXiv:1804.06782.