Exact diagonalization (ED) is a numerical technique used in
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
to determine the
eigenstates
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution i ...
and energy
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
of a quantum
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
. In this technique, a Hamiltonian for a discrete, finite system is expressed in matrix form and
diagonalized
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
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
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
dimension with the size of the quantum system. It is frequently employed to study lattice models, including the
Hubbard model
The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems.
It is particularly useful in solid-state physics. The model is named for John Hubbard.
The Hubbard model states that each el ...
,
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
,
Heisenberg model,
''t''-''J'' model, and
SYK model.
Expectation values from exact diagonalization
After determining the eigenstates
and energies
of a given Hamiltonian, exact diagonalization can be used to obtain expectation values of observables. For example, if
is an observable, its
thermal expectation value is
:
where
is the
partition function. If the observable can be written down in the initial basis for the problem, then this sum can be evaluated after transforming to the basis of eigenstates.
Green's functions
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differential ...
may be evaluated similarly. For example, the retarded Green's function
can be written
:
Exact diagonalization can also be used to determine the time evolution of a system after a quench. Suppose the system has been prepared in an initial state
, and then for time
evolves under a new Hamiltonian,
. The state at time
is
:
Memory requirements
The dimension of the Hilbert space describing a quantum system scales exponentially with system size. For example, consider a system of
spins localized on fixed lattice sites. 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
and
. The full system has dimension
, and the Hamiltonian represented as a matrix has size
. 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
In numerical analysis and scientific computing, a sparse matrix or sparse array is a matrix in which most of the elements are zero. There is no strict definition regarding the proportion of zero-value elements for a matrix to qualify as sparse b ...
, or using other techniques.
Comparison with other techniques
Exact diagonalization is useful for extracting exact information about finite systems. However, often small systems are studied to gain insight into infinite lattice systems. If the diagonalized system is too small, its properties will not reflect the properties of the system in the
thermodynamic limit
In statistical mechanics, the thermodynamic limit or macroscopic limit, of a system is the limit for a large number of particles (e.g., atoms or molecules) where the volume is taken to grow in proportion with the number of particles.S.J. Blundel ...
, and the simulation is said to suffer from finite size effects.
Unlike some other exact theory techniques, such as
Auxiliary-field Monte Carlo
Auxiliary-field Monte Carlo is a method that allows the calculation, by use of Monte Carlo techniques, of averages of operators in many-body quantum mechanical (Blankenbecler 1981, Ceperley 1977) or classical problems (Baeurle 2004, Baeurle 2003, ...
, exact diagonalization obtains Green's functions directly in real time, as opposed to
imaginary time
Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.
...
. Unlike in these other techniques, exact diagonalization results do not need to be numerically
analytically continued
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
. This is an advantage, because numerical analytic continuation is an ill-posed and difficult optimization problem.
Applications
* Can be used as an impurity solver for
Dynamical mean-field theory Dynamical mean-field theory (DMFT) is a method to determine the electronic structure of strongly correlated materials. In such materials, the approximation of independent electrons, which is used in density functional theory and usual band structure ...
techniques.
* When combined with finite size scaling, estimating the
ground state
The ground state of a quantum-mechanical system is its stationary state of lowest energy; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state. ...
energy and
critical exponents
Critical or Critically may refer to:
*Critical, or critical but stable, medical states
**Critical, or intensive care medicine
*Critical juncture, a discontinuous change studied in the social sciences.
*Critical Software, a company specializing in ...
of the 1D
transverse-field Ising model
The transverse field Ising model is a quantum version of the classical Ising model. It features a lattice with nearest neighbour interactions determined by the alignment or anti-alignment of spin projections along the z axis, as well as an externa ...
.
* Studying various properties of the 2D
Heisenberg model in a magnetic field, including antiferromagnetism and spin-wave velocity.
* Studying the Drude weight of the 2D Hubbard model.
* Studying out-of-time-order correlations (OTOCs) and scrambling in the SYK model.
* Simulating resonant x-ray spectra of strongly correlated materials.
Implementations
Numerous software packages implementing exact diagonalization of quantum Hamiltonians exist. These includ
QuSpinALPSDoQoEdLibedrixs and many others.
Generalizations
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.
See also
*
Lanczos algorithm
The Lanczos algorithm is an iterative method devised by Cornelius Lanczos that is an adaptation of power iteration, power methods to find the m "most useful" (tending towards extreme highest/lowest) eigenvalues and eigenvectors of an n \times n ...
References
{{reflist,
[
{{cite encyclopedia
, last1 = Weiße
, first1 = Alexander
, last2 = Fehske
, first2 = Holger
, title = Exact Diagonalization Techniques
, encyclopedia = Computational Many-Particle Physics
, year = 2008
, pages = 529–544
, publisher = Springer
, series = Lecture Notes in Physics
, volume = 739
, location =
, id =
, doi = 10.1007/978-3-540-74686-7_18
, bibcode = 2008LNP...739..529W
, isbn = 978-3-540-74685-0
]
[
{{cite encyclopedia
, last1 = Prelovšek
, first1 = Peter
, title = The Finite Temperature Lanczos Method and its Applications
, encyclopedia = The Physics of Correlated Insulators, Metals, and Superconductors
, year = 2017
, pages =
, publisher = Forschungszentrum Jülich
, series = Modeling and Simulation
, volume = 7
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, id =
, doi =
, isbn = 978-3-95806-224-5
]
External links
Quantum Simulation/Exact diagonalizationALPS full diagonalization tutorial
Correlated electrons
Materials science
Condensed matter physics
Quantum mechanics