Exact Diagonalization Methods

Exact diagonalization solves a quantum many-body model by building its Hamiltonian matrix in a chosen basis and finding its eigenvalues directly, giving numerically exact spectra for small lattices against which approximate methods are tested.

Definition

Exact diagonalization is a numerical method that computes eigenvalues and eigenvectors of a many-body Hamiltonian represented exactly in a finite basis, yielding the spectrum of a small quantum system without approximation beyond finite size.

Scope

This topic covers exact diagonalization of lattice quantum models such as Hubbard and Heisenberg systems: construction of the many-body basis, use of symmetries to block-diagonalize the Hamiltonian, and Lanczos iteration to extract low-lying states from the exponentially large but sparse matrix. It addresses the exponential wall that limits system size.

Core questions

• How is the many-body Hilbert space enumerated and the Hamiltonian built as a sparse matrix?
• How do symmetries reduce the problem into smaller blocks?
• How does the Lanczos algorithm extract the ground state from a huge sparse Hamiltonian?
• Why does the accessible system size grow only logarithmically with computer memory?

Key theories

Many-body basis construction

The Hilbert space of a lattice model is enumerated as occupation or spin configurations, and the Hamiltonian is stored as a sparse matrix because each basis state couples to only a few others.

Symmetry block-diagonalization

Conserved quantities and lattice symmetries split the Hamiltonian into independent blocks, shrinking the matrices that must be diagonalized and labeling states by their quantum numbers.

Lanczos for extreme eigenstates

The Lanczos algorithm projects the sparse Hamiltonian onto a small Krylov subspace to extract the ground state and a few excited states without forming or storing the full matrix.

Clinical relevance

Exact diagonalization gives benchmark ground states, excitation spectra and correlation functions for strongly correlated lattice models, serving as the reference for testing quantum Monte Carlo, tensor-network and other approximate many-body methods.

History

Direct diagonalization of small quantum lattices grew with computing power from the 1960s onward; the use of Lanczos iteration and symmetry reduction in the 1980s pushed accessible Hubbard and Heisenberg clusters to a few dozen sites, establishing exact diagonalization as a benchmark method.

Key figures

• Cornelius Lanczos
• Elliott Lieb
• H. Q. Lin

Seminal works

• lin1990
• lanczos1950

Frequently asked questions

Why is exact diagonalization limited to small systems?

The dimension of the many-body Hilbert space grows exponentially with the number of sites, so even with sparse storage and symmetries the matrix quickly outgrows any computer's memory, capping exact diagonalization at a few tens of sites.

What is exact diagonalization good for despite that limit?

Within reach, it gives numerically exact, unbiased results, making it the gold standard for validating approximate many-body methods and for studying small clusters where finite-size effects can be analyzed directly.