Dmrg of Heisenberg model

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 H with the base { | S , S z ⟩ } ≡ { | 1 , 1 ⟩ , | 1 , 0 ⟩ , | 1 , − 1 ⟩ } . With this base the spin operators are S x , S y and S z for the single site. For every "block", the two blocks and the two sites, there is its own Hilbert space H b , its base { | w i ⟩ } ( i : 1 … dim ⁡ ( H b ) )and its own operators O b : H b → H b :

At the starting point all four Hilbert spaces are equivalent to H , all spin operators are equivalent to S x , S y and S z and H B = H U = 0 . 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 3 × 3 matrices, the three left-site spin operators and the three right-site spin operators, which are always 3 × 3 matrices. The Hamiltonian 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:

H S B = − J ∑ ⟨ i , j ⟩ S x i S x j + S y i S y j + S z i S z j

These operators live in the superblock state space: H S B = H B ⊗ H l ⊗ H r ⊗ H U , the base is { | f ⟩ = | u ⟩ ⊗ | t ⟩ ⊗ | s ⟩ ⊗ | r ⟩ } . For example: (convention):

| 1000 … 0 ⟩ ≡ | f 1 ⟩ = | u 1 , t 1 , s 1 , r 1 ⟩ ≡ | 100 , 100 , 100 , 100 ⟩

| 0100 … 0 ⟩ ≡ | f 2 ⟩ = | u 1 , t 1 , s 1 , r 2 ⟩ ≡ | 100 , 100 , 100 , 010 ⟩

The Hamiltonian in the DMRG form is (we set J = − 1 ):

H S B = H B + H U + ∑ ⟨ i , j ⟩ S x i S x j + S y i S y j + S z i S z j

The operators are ( d ∗ 3 ∗ 3 ∗ d ) × ( d ∗ 3 ∗ 3 ∗ d ) matrices, d = dim ⁡ ( H B ) ≡ dim ⁡ ( H U ) , for example:

⟨ f | H B | f ′ ⟩ ≡ ⟨ u , t , s , r | H B ⊗ I ⊗ I ⊗ I | u ′ , t ′ , s ′ , r ′ ⟩

S x B S x l = S x B I ⊗ I S x l ⊗ I I ⊗ I I = S x B ⊗ S x l ⊗ I ⊗ I

Step 2: Diagonalize the superblock Hamiltonian

At this point you must choose the eigenstate of the Hamiltonian for which some observables is calculated, this is the target state . At the beginning you can choose the ground state and use some advanced algorithm to find it, one of these is described in:

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

If | Ψ ⟩ = ∑ Ψ i , j , k , w | u i , t j , s k , r w ⟩ is the target state, expectation value of various operators can be measured at this point using | Ψ ⟩ .

Step 3: Reduce Density Matrix

Form the reduce density matrix ρ for the first two block system, the Block and the left-site. By definition it is the ( d ∗ 3 ) × ( d ∗ 3 ) matrix: ρ i , j ; i ′ , j ′ ≡ ∑ k , w Ψ i , j , k , w Ψ i ′ , j ′ , k , w Diagonalize ρ and form the m × ( d ∗ 3 ) matrix T , which rows are the m eigenvectors associated with the m largest eigenvalue e α of ρ . So T is formed by the most significant eigenstates of the reduce density matrix. You choose m looking to the parameter P m ≡ ∑ α = 1 m e α : 1 − P m ≅ 0 .

Step 4: New, Block and Universe Block, operators

Form the ( d ∗ 3 ) × ( d ∗ 3 ) 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: H B − l = H B ⊗ I + S x B ⊗ S x l + S y B ⊗ S y l + S z B ⊗ S z l

S x B − l = I ⊗ S x l

H r − U = I ⊗ H U + S x r ⊗ S x U + S y r ⊗ S y U + S z r ⊗ S z U

S x r − U = S x r ⊗ I

Now, form the m × m matrix representations of the new Block and Universe-Block operators, form a new block by changing basis with the transformation T , for example:

At this point the iteration is ended and the algorithm goes back to step 1. The algorithm stops successfully when the observable converges to some value.