Dmrg of Heisenberg model - AbsoluteAstronomy.com

This example presents the infinite DMRG
Density matrix renormalization group
The density matrix renormalization group is a numerical variational technique devised to obtain the low energy physics of quantum many-body systems with high accuracy. It was invented in 1992 by Steven R...

algorithm. It is about

antiferromagnetic Heisenberg chain

Heisenberg model (quantum)

The Heisenberg model is a statistical mechanical model used in the study of critical points and phase transitions of magnetic systems, in which the spin of the magnetic systems are treated quantum mechanically...

, but the recipe can be applied for every translationally invariant one-dimensional lattice

Lattice (group)

In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...

. DMRG is a renormalization-group

Renormalization group

In theoretical physics, the renormalization group refers to a mathematical apparatus that allows systematic investigation of the changes of a physical system as viewed at different distance scales...

technique because it offers an efficient truncation of the Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

of one-dimensional quantum systems

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

The Starting Point

To simulate an infinite chain, starting with four sites. The first is the Block site , the last the Universe-Block site and the remaining are the added sites , the right one is "added" to the Universe-Block site and the other to the Block site.

The Hilbert space for the single site is

with the base

. With this base the spin

Spin (physics)

In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

operators are

and

for the single site. For every "block", the two blocks and the two sites, there is its own Hilbert space

, its base

(

)and its own operators

Block: , , , , ,
left-site: , , , ,
right-site: , , , ,
Universe: , , , , ,

At the starting point all four Hilbert spaces are equivalent to

, all spin operators are equivalent to

and

. This is always (at every iterations) true only for left and right sites.

Step 1: Form the Hamiltonian matrix for the Superblock

The ingredients are the four Block operators and the four Universe-Block operators, which at the first iteration are

matrices

Matrix (mathematics)

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...

, the three left-site spin operators and the three right-site spin operators, which are always

matrices. The Hamiltonian

Hamiltonian system

In physics and classical mechanics, a Hamiltonian system is a physical system in which forces are momentum invariant. Hamiltonian systems are studied in Hamiltonian mechanics....

matrix of the superblock (the chain), which at the first iteration has only four sites, is formed by these operators. In the Heisenberg antiferromagnetic S=1 model the Hamiltonian is:

These operators live in the superblock state space:

, the base is

. For example: (convention):

The Hamiltonian in the dmrg form is (we set

The operators are

matrices,

, for example:

Step 2: Diagonalize the superblock Hamiltonian

At this point you must choose the eigenstate

Eigenvalue, eigenvector and eigenspace

The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

of the Hamiltonian for which some observable

Observable

In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value off...

s is calculated, this is the target state . At the beginning you can choose the ground state

Stationary state

In quantum mechanics, a stationary state is an eigenvector of the Hamiltonian, implying the probability density associated with the wavefunction is independent of time . This corresponds to a quantum state with a single definite energy...

and use some advanced algorithm to find it, one of these is described in:

The Iterative Calculation of a Few of the Lowest Eigenvalues and Corresponding Eigenvectors
Eigenvalue, eigenvector and eigenspace
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix...

of Large Real-Symmetric Matrices , Ernest R. Davidson
Ernest R. Davidson
Ernest R. Davidson, born October 12, 1936 in Terre Haute, Indiana, is Professor of Chemistry, University of Washington, Seattle, WA, USA. He graduated from Wiley High School, Terre Haute and Rose Polytechnic Institute....

; Journal of Computational Physics 17, 87-94 (1975)

This step is the most time-consuming part of the algorithm.

If

is the target state, expectation value

Expected value

In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

of various operators can be measured at this point using

Step 3: Reduce Density Matrix

Form the reduce density matrix

for the first two block system, the Block and the left-site. By definition it is the

matrix:

Diagonalize

Diagonalization

In mathematics, diagonalization may refer to:* Diagonal matrix, which is in a form with nonzero entries only on the main diagonal* Diagonalizable matrix, which can be put into a form with nonzero entries only on the main diagonal...

and form the

matrix

, which rows are the

eigenvectors associated with the

largest eigenvalue

. So

is formed by the most significant eigenstates of the reduce density matrix. You choose

looking to the parameter

Step 4: New, Block and Universe Block, operators

Form the

matrix representation of operators for the system composite of Block and left-site, and for the system composite of right-site and Universe-Block, for example:

Now, form the

matrix representations of the new Block and Universe-Block operators, form a new block by changing basis with the transformation