Dmrg Of Heisenberg Model

Within the study of the quantum many-body problem in physics, the DMRG analysis of the Heisenberg model is an important theoretical example applying techniques of the density matrix renormalization group (DMRG) to the Heisenberg model of a chain of spins. This article presents the "infinite" DMRG algorithm for the $S=1$ antiferromagnetic Heisenberg chain, but the recipe can be applied for every translationally invariant one-dimensional lattice.

DMRG is a renormalization-group technique because it offers an efficient truncation of the Hilbert space of one-dimensional quantum systems.

The algorithm

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 $\mathfrak$ with the base $\\equiv\$. 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 $\mathfrak_b$, its base $\$ ($i:1\dots \dim(\mathfrak_b)$)and its own operators $O_b:\mathfrak_b\rightarrow\mathfrak_b$where

*block: $\mathfrak_B$, $\$, $H_B$, $S_$, $S_$, $S_$
*left-site: $\mathfrak_l$, $\$, $S_$, $S_$, $S_$
*right-site: $\mathfrak_r$, $\$, $S_$, $S_$, $S_$
*universe: $\mathfrak_U$, $\$, $H_U$, $S_$, $S_$, $S_$
At the starting point all four Hilbert spaces are equivalent to $\mathfrak$, all spin operators are equivalent to $S_x$, $S_y$ and $S_z$ and $H_B=H_U=0$. In the following iterations, this is only true for the 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 $3\times3$ matrices, the three left-site spin operators and the three right-site spin operators, which are always $3\times3$ 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:

$\mathbf_=-J\sum_\mathbf_\mathbf_+\mathbf_\mathbf_+\mathbf_\mathbf_$

These operators live in the superblock state space: $\mathfrak_=\mathfrak_B\otimes\mathfrak_l\otimes\mathfrak_r\otimes\mathfrak_U$, the base is $\$. For example: (convention):

$, 1000\dots0\rangle\equiv, f_1\rangle=, u_1,t_1,s_1,r_1\rangle\equiv, 100,100,100,100\rangle$

$, 0100\dots0\rangle\equiv, f_2\rangle=, u_1,t_1,s_1,r_2\rangle\equiv, 100,100,100,010\rangle$

The Hamiltonian in the ''DMRG form'' is (we set $J=-1$):

$\mathbf_=\mathbf_B+\mathbf_U+\sum_\mathbf_\mathbf_+\mathbf_\mathbf_+\mathbf_\mathbf_$

The operators are $(d*3*3*d)\times(d*3*3*d)$ matrices, $d=\dim(\mathfrak_B)\equiv\dim(\mathfrak_U)$, for example:

$\langle f, \mathbf_B, f'\rangle\equiv\langle u,t,s,r, H_B\otimes\mathbb\otimes\mathbb\otimes\mathbb, u',t',s',r'\rangle$

$\mathbf_\mathbf_=S_\mathbb\otimes\mathbbS_\otimes\mathbb\mathbb\otimes\mathbb\mathbb=S_\otimes S_\otimes\mathbb\otimes\mathbb$

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:

*''The Iterative Calculation of a Few of the Lowest Eigenvalues and Corresponding Eigenvectors of Large Real-Symmetric Matrices'', Ernest R. Davidson; Journal of Computational Physics 17, 87-94 (1975)
This step is the most time-consuming part of the algorithm.

If $, \Psi\rangle=\sum\Psi_, u_i,t_j,s_k,r_w\rangle$ is the target state, expectation value of various operators can be measured at this point using $, \Psi\rangle$.

Step 3: Reduce density matrix

Form the reduced density matrix $\rho$ for the first two block system, the block and the left-site. By definition it is the $(d*3)\times(d*3)$ matrix:
$\rho_\equiv\sum_\Psi_\Psi^*_$

Diagonalize $\rho$ and form the $m\times (d*3)$ matrix $T$, which rows are the $m$ eigenvectors associated with the $m$ largest eigenvalues $e_\alpha$ of $\rho$. So $T$ is formed by the most significant eigenstates of the reduced density matrix. You choose $m$ looking to the parameter $P_m\equiv\sum_^m e_\alpha$: $1-P_m\cong 0$.

Step 4: New block and universe-block operators

Form the $(d*3)\times(d*3)$ matrix representation of operators for the system composite of the block and left-site, and for the system composite of right-site and universe-block, for example:

$H_=H_B\otimes\mathbb+S_\otimes S_+S_\otimes S_+S_\otimes S_$

$S_=\mathbb\otimes S_$

$H_=\mathbb\otimes H_U+S_\otimes S_+S_\otimes S_+S_\otimes S_$

$S_=S_\otimes\mathbb$

Now, form the $m\times m$ matrix representations of the new block and universe-block operators, form a new block by changing basis with the transformation $T$, for example:$\begin &H_B=TH_T^\dagger &S_=TS_T^\dagger \end$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.

Further reading

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See also

*Heisenberg model (quantum)
*Density Matrix Renormalization Group

References

{{Reflist

Theoretical physics
Statistical mechanics