FR30-C · Sweep a window, lower the energy

DMRG Ground State Sweeps

How does DMRG find a ground state, and how do you know it has? DMRG sweeps a small window along the chain and replaces two sites at a time with the lowest-energy choice the rest of the chain allows, so the energy falls sweep by sweep. Convergence is checked with the energy variance, the discarded weight and, where one exists, an exact energy. A tiny variance can still belong to a symmetry-broken state.

Watch a two-site window sweep along a spin chain and lower the energy at every step. Compare the energy with an exact value, and see why a small energy variance alone does not prove you found the ground state.

Energy above exact $2.3\times10^{-10}$

Energy variance $1.9\times10^{-9}$

Largest discarded weight $1.1\times10^{-10}$

Sweeps completed $6$

Sweep the window until the energy settles

Largest bond dimension

\chi_{\max}

16

Transverse field

h

1

Chain length

L

32

Drag the gold ceiling to change the largest bond dimension.

More settings

Number of sweeps

n

6

Starting-state seed

s

3003

Model, units and sources

$H_{\rm I}=-\sum_j\sigma^x_j\sigma^x_{j+1}-h\sum_j\sigma^z_j,\quad H_{\rm XXX}=\sum_j\mathbf S_j\cdot\mathbf S_{j+1}$

Open chains, unit coupling, natural logarithms for entropy. The bond-dimension ceiling ramps up over the first three sweeps; each local update uses four Lanczos steps and keeps singular values above a relative cutoff of one part in a trillion.

Read the course chapter behind this Lab

Model assumptions

Two-site DMRG optimizes a real matrix product state against a matrix product operator, with a few Lanczos steps per local update and a bond-dimension ramp over the first sweeps. Ising energies are compared with the exact free-fermion ground energy. Heisenberg energies for eight to sixteen sites are compared with stored SciPy exact-diagonalization values. The variance is computed with the squared Hamiltonian as a matrix product operator. The dashed entropy curve is the conformal chord law with the central charge fixed and only its constant fitted.

Energy error and variance per sweep

Energy at every local update

Entanglement along the chain

Numerical diagnostics and record

\text{sweep }0

Energies, variances and parity are computed from the final matrix product state of each sweep. The record saves every local update and every sweep.

Continue the course: tensor networks for quantum many-body physics

Start with the essentials

Focus question: How does DMRG find a ground state, and how do you know it has?
One-sentence intuition: Every local update can only lower the variational energy. Whether the result is the ground state needs separate evidence: the energy variance, the discarded weight, symmetry checks and, where possible, an exact energy.

Core mathematical model

The local eigenvalue problem

H_{\rm eff}^{[j,j+1]}\,\Theta=E\,\Theta,\qquad \Theta\approx U\,\mathrm{diag}(s)\,V^{\dagger}

With the rest of the chain fixed, the energy of the two-site block is a small eigenvalue problem. A few Lanczos steps solve it, and a truncated singular value decomposition splits the block back into two tensors.

Variance as a convergence test

\sigma_E^2=\langle H^2\rangle-\langle H\rangle^2\ge0

An exact eigenstate has zero variance. A small variance means the state is close to some eigenstate, or to a mixture of nearly degenerate ones; it does not say which.

Entanglement of a critical chain

S(\ell)=\frac{c}{6}\ln\!\Big[\frac{L}{\pi}\sin\frac{\pi\ell}{L}\Big]+s_0

For an open critical chain, conformal field theory predicts this profile, with central charge one half for the Ising chain and one for the Heisenberg chain. Only the constant is fitted; small chains also show oscillations.

Common difficulties

Low variance is not proof

Typical misconceptionWhen the variance is tiny, DMRG has found the ground state.

Better mental modelNearly degenerate states, such as the two ordered states of a ferromagnet, mix with tiny variance. Check symmetries, compare different starting states and use an exact energy whenever one exists.

Run the experiment

01
Follow a sweep
Let Critical Ising chain run. Follow the gold two-site window and the entanglement bars it leaves behind.
What to observe: The energy error drops by orders of magnitude over the first sweeps, while the bars settle into an arch that is highest at the centre.
02
Shrink the bond dimension
Choose Bond dimension too small, or drag the gold ceiling down.
What to observe: The energy stalls well above the exact value, and both the variance and the discarded weight stay large. All three signals agree.
03
A converged but wrong answer
Choose Ordered phase: low variance, wrong state and compare the variance with the energy error.
What to observe: The variance is tiny, yet the energy sits above the exact ground energy. DMRG has settled into a symmetry-broken state; the parity check in the diagnostics shows it.

DMRG Ground State Sweeps