Exact diagonalization (ED) is a numerical technique used in physics to determine the eigenstates and energy eigenvalues of a quantum Hamiltonian.
In this technique, a Hamiltonian for a discrete, finite system is expressed in matrix form and diagonalized using a computer.
Exact diagonalization is only feasible for systems with a few tens of particles, due to the exponential growth of the Hilbert space dimension with the size of the quantum system.
of a given Hamiltonian, exact diagonalization can be used to obtain expectation values of observables.
can be written Exact diagonalization can also be used to determine the time evolution of a system after a quench.
The dimension of the on-site basis is 2, because the state of each spin can be described as a superposition of spin-up and spin-down, denoted
This implies that computation time and memory requirements scale very unfavorably in exact diagonalization.
In practice, the memory requirements can be reduced by taking advantage of symmetry of the problem, imposing conservation laws, working with sparse matrices, or using other techniques.
If the diagonalized system is too small, its properties will not reflect the properties of the system in the thermodynamic limit, and the simulation is said to suffer from finite size effects.
Unlike some other exact theory techniques, such as Auxiliary-field Monte Carlo, exact diagonalization obtains Green's functions directly in real time, as opposed to imaginary time.
Unlike in these other techniques, exact diagonalization results do not need to be numerically analytically continued.
This is an advantage, because numerical analytic continuation is an ill-posed and difficult optimization problem.
[3] Numerous software packages implementing exact diagonalization of quantum Hamiltonians exist.
These include ALPS[permanent dead link], DoQo, EdLib, edrixs, Quanty and many others.
Exact diagonalization results from many small clusters can be combined to obtain more accurate information about systems in the thermodynamic limit using the numerical linked cluster expansion.