Physics:DMRG of the 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 $\mathfrak{ℌ}$ with the base $\{|S,S_z⟩\}\equiv \{|1,1⟩,|1,0⟩,|1,-1⟩\}$. 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 ${\mathfrak{ℌ}}_b$, its base $\{|w_i⟩\}$ ($i:1\dots \dim ({\mathfrak{ℌ}}_b)$)and its own operators $$O_b:{\mathfrak{ℌ}}_b\to {\mathfrak{ℌ}}_b$$where

• block: ${\mathfrak{ℌ}}_B$, $\{|u_i⟩\}$, $H_B$, $S_{x_B}$, $S_{y_B}$, $S_{z_B}$
• left-site: ${\mathfrak{ℌ}}_l$, $\{|t_i⟩\}$, $S_{x_l}$, $S_{y_l}$, $S_{z_l}$
• right-site: ${\mathfrak{ℌ}}_r$, $\{|s_i⟩\}$, $S_{x_r}$, $S_{y_r}$, $S_{z_r}$
• universe: ${\mathfrak{ℌ}}_U$, $\{|r_i⟩\}$, $H_U$, $S_{x_U}$, $S_{y_U}$, $S_{z_U}$

At the starting point all four Hilbert spaces are equivalent to $\mathfrak{ℌ}$, 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 $3\times 3$ matrices, the three left-site spin operators and the three right-site spin operators, which are always $3\times 3$ 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:

${\mathbf{𝐇}}_{SB}=-J\sum _{⟨i,j⟩}{\mathbf{𝐒}}_{x_i}{\mathbf{𝐒}}_{x_j}+{\mathbf{𝐒}}_{y_i}{\mathbf{𝐒}}_{y_j}+{\mathbf{𝐒}}_{z_i}{\mathbf{𝐒}}_{z_j}$

These operators live in the superblock state space: ${\mathfrak{ℌ}}_{SB}={\mathfrak{ℌ}}_B\otimes {\mathfrak{ℌ}}_l\otimes {\mathfrak{ℌ}}_r\otimes {\mathfrak{ℌ}}_U$, the base is $\{|f⟩=|u⟩\otimes |t⟩\otimes |s⟩\otimes |r⟩\}$. For example: (convention):

$|1000\dots 0⟩\equiv |f_1⟩=|u_1,t_1,s_1,r_1⟩\equiv |100,100,100,100⟩$

$|0100\dots 0⟩\equiv |f_2⟩=|u_1,t_1,s_1,r_2⟩\equiv |100,100,100,010⟩$

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

${\mathbf{𝐇}}_{SB}={\mathbf{𝐇}}_B+{\mathbf{𝐇}}_U+\sum _{⟨i,j⟩}{\mathbf{𝐒}}_{x_i}{\mathbf{𝐒}}_{x_j}+{\mathbf{𝐒}}_{y_i}{\mathbf{𝐒}}_{y_j}+{\mathbf{𝐒}}_{z_i}{\mathbf{𝐒}}_{z_j}$

The operators are $(d*3*3*d)\times (d*3*3*d)$ matrices, $d=\dim ({\mathfrak{ℌ}}_B)\equiv \dim ({\mathfrak{ℌ}}_U)$, for example:

$⟨f|{\mathbf{𝐇}}_B|f^{\prime }⟩\equiv ⟨u,t,s,r|H_B\otimes \mathbb{𝕀}\otimes \mathbb{𝕀}\otimes \mathbb{𝕀}|u^{\prime },t^{\prime },s^{\prime },r^{\prime }⟩$

${\mathbf{𝐒}}_{x_B}{\mathbf{𝐒}}_{x_l}=S_{x_B}\mathbb{𝕀}\otimes \mathbb{𝕀}{S}_{x_l}\otimes \mathbb{𝕀}\mathbb{𝕀}\otimes \mathbb{𝕀}\mathbb{𝕀}=S_{x_B}\otimes S_{x_l}\otimes \mathbb{𝕀}\otimes \mathbb{𝕀}$

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:

• 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 $|\mathrm{\Psi }⟩=\sum {\mathrm{\Psi }}_{i,j,k,w}|u_i,t_j,s_k,r_w⟩$ is the target state, expectation value of various operators can be measured at this point using $|\mathrm{\Psi }⟩$.

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: ${\rho }_{i,j;i^{\prime },j^{\prime }}\equiv \sum _{k,w}{\mathrm{\Psi }}_{i,j,k,w}{\mathrm{\Psi }}_{i^{\prime },j^{\prime },k,w}^{*}$

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 $P_m\equiv \sum _{\alpha =1}^m{e}_{\alpha }$: $1-P_m\cong 0$.

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:

$H_{B-l}=H_B\otimes \mathbb{𝕀}+S_{x_B}\otimes S_{x_l}+S_{y_B}\otimes S_{y_l}+S_{z_B}\otimes S_{z_l}$

$S_{x_{B-l}}=\mathbb{𝕀}\otimes S_{x_l}$

$H_{r-U}=\mathbb{𝕀}\otimes H_U+S_{x_r}\otimes S_{x_U}+S_{y_r}\otimes S_{y_U}+S_{z_r}\otimes S_{z_U}$

$S_{x_{r-U}}=S_{x_r}\otimes \mathbb{𝕀}$

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:$$\begin{array}{ccc}& H_B=T{H}_{B-l}{T}^{†}& S_{x_B}=T{S}_{x_{B-l}}{T}^{†}\end{array}$$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

• White, Steven R. (1993-10-01). "Density-matrix algorithms for quantum renormalization groups". Physical Review B (American Physical Society (APS)) 48 (14): 10345–10356. doi:10.1103/physrevb.48.10345. ISSN 0163-1829. PMID 10007313. Bibcode: 1993PhRvB..4810345W. 
• White, Steven R.; Huse, David A. (1993-08-01). "Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic S=1 Heisenberg chain". Physical Review B (American Physical Society (APS)) 48 (6): 3844–3852. doi:10.1103/physrevb.48.3844. ISSN 0163-1829. PMID 10008834. Bibcode: 1993PhRvB..48.3844W. 
• Schollwöck, U. (2005-04-26). "The density-matrix renormalization group". Reviews of Modern Physics 77 (1): 259–315. doi:10.1103/revmodphys.77.259. ISSN 0034-6861. Bibcode: 2005RvMP...77..259S. 

See also

• Heisenberg model (quantum)
• Density Matrix Renormalization Group