DMRG

Density-matrix renormalization group (DMRG) minimizes a many-body energy by optimizing a small part of an MPS while holding the rest fixed. The reduced eigenproblem is computationally tractable because its surrounding blocks are compressed. A completed sweep is an iteration, not a certificate of the ground state; bond dimension, local residual, variance, and competing initial states all matter.

Required background. MPS canonical forms and MPO contractions define the state and Hamiltonian arrays. The variational bound is developed in Rayleigh–Ritz.

The effective local eigenproblem

Place the MPS in mixed-canonical form around adjacent sites $j,j+1$. Their combined tensor $\Theta_{\alpha st\beta}$ is a vector of length $D_{j-1}d_jd_{j+1}D_{j+1}$. Contract the MPO with left and right environments to define the action of $H_{\rm eff}$ on this vector. No dense effective matrix need be stored.

Since the surrounding bases are orthonormal,

$$E(\Theta)= \frac{\Theta^\dagger H_{\rm eff}\Theta} {\Theta^\dagger\Theta}, \qquad H_{\rm eff}\Theta=E\Theta.$$

Solve the lowest eigenpair iteratively, starting from the current tensor. Verify the local residual rather than trusting only an iteration count. Away from canonical gauge a generalized norm matrix appears, and numerical conditioning becomes worse.

Two-site updates and sweeps

After the eigensolve, reshape $\Theta$ into $(D_{j-1}d_j)\times(d_{j+1}D_{j+1})$, perform an SVD, truncate, and distribute the singular values toward the direction of the sweep. Update the newly completed environment. Reverse at the boundary.

The untruncated local minimization cannot raise the energy, but the truncation can. Thus a two-site sweep is not guaranteed to decrease the energy at every intermediate step. A one-site update avoids that truncation and keeps fixed virtual sectors, but can become trapped when the existing basis cannot introduce useful directions. Two-site warm-up or controlled subspace expansion helps; neither removes the need for independent starts.

A useful schedule increases $D$ gradually, tightens eigensolver tolerances after coarse sweeps, and finishes with repeated sweeps at the largest requested dimension.

The two-spin Heisenberg benchmark

Take $H=J\mathbf S_1\cdot\mathbf S_2$ with $\mathbf S=\boldsymbol\sigma/2$ and $J>0$. The singlet has energy $-3J/4$, the triplet has $J/4$, and a product state can reach only $-J/4$. Consequently a $D=1$ calculation that stabilizes at $-J/4$ has converged its restricted optimization but not the physical answer. A $D=2$ MPS represents the singlet exactly.

For a longer chain, compare with dense or sparse exact diagonalization at the same size, boundaries, and symmetry sector before extrapolating. Bulk thermodynamic energy formulas do not test an open finite chain without its boundary corrections.

Energy, variance, and convergence

Record the normalized energy, energy variance,

$$\sigma_H^2=\langle H^2\rangle-\langle H\rangle^2,$$

local eigensolver residuals, maximum discarded weight, and target observables after each sweep. Variance measures the state’s energy spread. Zero variance places it in one eigenspace, which may be excited; a small nonzero variance does not bound fidelity with a particular eigenvector unless the relevant spectral separation is known. Check lower energies from alternative initial states and other relevant quantum-number sectors.

Converge $D$ at fixed chain length before analyzing finite-size scaling. Linear extrapolation against discarded weight or variance can be informative only over a demonstrated asymptotic regime; it is not a general rigorous error formula. Closely spaced or symmetry-broken states also require correlators and sector diagnostics.

Exercises

For the two-spin example, compute the energy variance of $|\uparrow\downarrow\rangle$.

Solution

The state has equal singlet and zero-magnetization triplet probabilities. Its mean energy is $-J/4$, while $\langle H^2\rangle=5J^2/16$. Therefore $\sigma_H^2=J^2/4$. A converged product-state minimizer has nonzero full-Hamiltonian variance, exposing the restricted variational manifold.

References

• Schollwöck, Ulrich, “The density-matrix renormalization group in the age of matrix product states,” Annals of Physics 326, 96–192 (2011), arXiv:1008.3477 — canonical gauges, contractions, and finite-chain algorithms.
• White, Steven R., “Density matrix formulation for quantum renormalization groups,” Physical Review Letters 69, 2863–2866 (1992), doi:10.1103/PhysRevLett.69.2863 — the density-matrix truncation construction.