FR30-B · Entanglement grows, the window closes

TEBD Quench and the Trusted Time Window

How long can a matrix product state follow a quantum quench? Only until the entanglement it must carry outgrows its bond dimension. After a quench, entanglement grows roughly linearly in time, while a bond dimension can hold only an amount equal to its logarithm. Doubling the bond dimension therefore extends the reliable time by a fixed step, not by a factor of two.

Start a spin chain in a simple product state, switch on the transverse field and follow it with TEBD. An exact free-fermion solution marks the moment the truncated simulation stops being trustworthy.

Trusted until $t^{\ast}=2.7$

Largest local error now $0$

Centre entanglement $0$

Discarded weight so far $0$

Follow the quench until the window closes

Bond dimension

\chi

16

Transverse field

h

1

Time step

\Delta t

0.05

Drag the gold ceiling to change the bond dimension; click the carpet to replay a moment.

More settings

Chain length

L

16

Model, units and sources

$H=-\sum_{j=1}^{L-1}\sigma^x_j\sigma^x_{j+1}-h\sum_{j=1}^{L}\sigma^z_j$

Open chain, unit coupling, time in units of the inverse coupling, natural logarithms for entropy. The carpet colour is the matrix-product-state magnetization; dashed diagonals show the fastest quasiparticle speed, twice the smaller of field and coupling.

Read the course chapter behind this Lab

Model assumptions

The open transverse-field Ising chain with unit coupling starts with every spin up along the field axis. Second-order Trotter steps evolve a complex matrix product state under a fixed bond-dimension ceiling. The exact reference maps the chain to free Majorana fermions and evolves their covariance matrix, so it carries neither Trotter nor truncation error. The trusted window ends at the first time any local magnetization differs from the exact value by more than one thousandth. Time is measured in units of the inverse coupling.

Centre entanglement against time

Local error and discarded weight

Trusted time against bond dimension

Numerical diagnostics and record

t=0

The exact reference is the free-fermion covariance evolution, independent of the Trotter gates. The record saves every frame of both calculations and the trusted-time scan.

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

Physics tutorial

TEBD Quench and the Trusted Time Window

BackgroundA product state has no entanglement, so a matrix product state with a single strand per bond stores it exactly. After a sudden change of the Hamiltonian, pairs of quasiparticles spread through the chain and entangle distant regions.

Why it mattersTEBD applies short two-site gates and truncates after each one. It is fast and accurate as long as the discarded weight stays small. The transverse-field Ising chain can also be solved exactly, so this Lab can measure when the truncated run fails.

Start with the essentials

Focus question: How long can a matrix product state follow a quantum quench?
One-sentence intuition: After a global quench the entanglement across a cut grows roughly linearly in time, while a bond dimension can carry at most its logarithm. The trusted time grows only with the logarithm of the bond dimension.

Core mathematical model

The quench Hamiltonian

H=-J\sum_{j=1}^{L-1}\sigma^x_j\sigma^x_{j+1}-h\sum_{j=1}^{L}\sigma^z_j,\qquad |\psi(0)\rangle=|{\uparrow\uparrow\cdots\uparrow}\rangle

The coupling is set to one. Every spin starts up along the field axis, which is not an eigenstate once the coupling acts.

Second-order Trotter step

e^{-iH\Delta t}=\prod_{b=1}^{L-1}e^{-ih_b\Delta t/2}\prod_{b=L-1}^{1}e^{-ih_b\Delta t/2}+O(\Delta t^3)

Each bond term carries its share of the field. A forward and a backward sweep of half steps make the error per step third order, so the error at a fixed time scales as the square of the time step.

How much entanglement a bond can carry

S_c(t)=-\sum_\alpha\lambda^{(c)}_\alpha(t)\ln\lambda^{(c)}_\alpha(t)\le\ln\chi

With at most a given number of Schmidt weights, the entropy reaches its largest value when they are all equal. Once the exact entropy approaches this ceiling, truncation must discard real weight.

Common difficulties

A small discarded weight is not a certificate

Typical misconceptionIf every truncation discards almost nothing, the simulation must still be accurate.

Better mental modelDiscarded weights accumulate over thousands of gates, and the Trotter error is separate from truncation. Compare with an exact result or with a larger bond dimension and a smaller step.

Run the experiment

01
Watch the window close
Let Critical quench run and watch the carpet fill from the top. The dashed line marks where the local error first exceeds one thousandth.
What to observe: The centre entropy follows the exact curve, then flattens against the gold ceiling. The error grows quickly from that point on.
02
Starve the bonds
Choose Too few strands, or drag the gold ceiling on the bond bars down.
What to observe: The window closes much earlier. The plot of trusted time against bond dimension gains about the same step for each doubling.
03
Separate truncation from time step
Choose Coarse time step and compare the discarded weight with the local error.
What to observe: The error now appears even while the discarded weight stays small. That error comes from the Trotter step, not from truncation.

TEBD Quench and the Trusted Time Window

TEBD Quench and the Trusted Time Window

Core mathematical model

The quench Hamiltonian

Second-order Trotter step

How much entanglement a bond can carry

Common difficulties

A small discarded weight is not a certificate

Run the experiment

Watch the window close

Starve the bonds

Separate truncation from time step

Related experiments and articles

Tensor networks and DMRG, from entanglement to quantum quenches