Density matrix renormalization group

Definition

Let $\mathcal{H} = (\C^d)^{\otimes N}$ and let $H$ be a Hermitian operator on $\mathcal{H}$, the Hamiltonian. Fix a bond dimension $D \ge 1$. The density matrix renormalization group (in its two-site form) is the following iterative method. It approximates the least eigenvalue of $H$ and an eigenvector for it.

1. Start from a matrix product state $\psi$ of bond dimension at most $D$.
2. Choose two adjacent sites $k, k+1$, and keep the matrices of all other sites fixed, in a gauge where the left and right parts are orthonormal.
3. Merge the matrices of sites $k$ and $k+1$ into one tensor $\Theta$. The energy $\langle \psi, H\psi \rangle / \langle \psi, \psi \rangle$ is then a Rayleigh quotient $\langle \Theta, H_{\mathrm{eff}} \Theta \rangle / \langle \Theta, \Theta \rangle$ of an effective Hermitian matrix $H_{\mathrm{eff}}$. Replace $\Theta$ by an eigenvector for the least eigenvalue of $H_{\mathrm{eff}}$.
4. Split $\Theta$ again by a singular value decomposition and keep only the $D$ largest singular values.
5. Move to the next pair of sites and repeat, sweeping from left to right and back, until the energy stops decreasing.

The squares of the singular values in step 4 are the eigenvalues of the reduced density matrix of the left part of the chain. So step 4 keeps the states of largest weight in that density matrix.