Why a quench outruns every matrix product state

A sudden change of the Hamiltonian fills the chain with pairs of quasiparticles that fly apart at finite speed. Each pair that straddles a cut adds entanglement across it, so the entanglement grows linearly in time. A bond of dimension chi can carry at most the logarithm of chi. The trusted time therefore grows only with the logarithm of the bond dimension: every doubling adds about the same fixed amount.

Sandbox Physics · 2026-10-29

A global quench fills the chain with pairs

In the quench Lab every spin starts up along the field axis. That is the ground state for an infinitely strong field. At time zero the field is set to the coupling, and the starting state now has extra energy spread evenly over the whole chain. This is a global quench.

Calabrese and Cardy explained what happens next with a simple picture. Every point of the chain emits pairs of quasiparticles that are entangled with each other and fly off in opposite directions. The entanglement of an interval with the rest counts the pairs that have one member inside and one outside. It therefore grows linearly in time until the fastest pairs have crossed the interval, at a time of half its length divided by the maximum speed, and then saturates at a value proportional to the length.

S_\ell(t)\approx\begin{cases}a\,t, & t<\ell/2v_{\max}\\ b\,\ell, & t>\ell/2v_{\max}\end{cases}

A speed limit for correlations

The picture rests on a general fact. In a chain with local interactions, information spreads no faster than a finite speed, up to tails that fall off exponentially outside the light cone. Bravyi, Hastings and Verstraete used this Lieb–Robinson bound to show that correlations and entanglement are distributed at a finite speed, and that the rate at which a block gains entropy scales with its boundary.

A cut in a chain has a boundary of one point, so its entanglement can grow at most linearly in time. Schollwöck draws the consequence in Section 7.4: many quenches reach this linear growth, and since the bond dimension needed grows exponentially with the entanglement, it must grow exponentially in time. At a fixed bond dimension, the precision instead deteriorates.

S(t)\le S(0)+c\,t\quad\Longrightarrow\quad \chi(t)\gtrsim e^{S(t)}\ \text{grows exponentially in }t

Worked example

Check the exact growth rate

For the Ising chain the growth rate is known exactly. Fagotti and Calabrese derived it in their Eq. 2 from the free-fermion solution: each momentum mode moves at its group velocity and contributes the binary entropy of how much the quench mixed it. For a single cut in a long chain, the rate is the integral of speed times entropy over all modes. With the Lab Hamiltonian and this starting state, the integral at a field equal to the coupling evaluates to two divided by pi, about 0.6366.

The exact free-fermion solution of a 120-spin chain tests this. Its middle-cut entropy grows with a fitted slope of 0.6360 between times two and eight. An independent rebuild with 240 spins gives 0.6365. At a field of three, the integral gives 0.1165 and the chain gives 0.1157. Growth this regular is exactly what limits a matrix product state.

\frac{dS}{dt}=\int_{-\pi}^{\pi}\frac{dk}{2\pi}\,|v_k|\,H(\cos\Delta_k),\qquad H(x)=-\tfrac{1+x}{2}\ln\tfrac{1+x}{2}-\tfrac{1-x}{2}\ln\tfrac{1-x}{2},\qquad \frac{dS}{dt}\Big|_{h=J}=\frac{2}{\pi}

Why each doubling buys the same time

A bond that keeps a fixed number of Schmidt terms can carry at most the logarithm of that number. If the entropy grows at a steady rate, doubling the bond dimension raises the ceiling by the logarithm of two and so buys a fixed extra time. The Lab measures the trusted time, the first moment the local magnetization differs from the exact value by more than one thousandth, at bond dimensions four, eight, sixteen and 32. It finds 1.25, 1.95, 2.7 and 3.45: about three quarters of a time unit per doubling.

The crude ceiling argument predicts 1.09 per doubling, the logarithm of two divided by the growth rate. The measured step is shorter, so the argument gets the logarithmic form right but not the constant. The chain length leaves the window where it is: at bond dimension sixteen it closes at 2.7 for twelve, sixteen and twenty spins. What sets it is how fast entanglement builds up near each cut.

t^\ast(\chi)\approx t_0+\kappa\ln\chi:\quad \kappa_{\rm measured}\ln2\approx1.06\times0.693\approx0.73,\qquad \frac{\ln2}{2/\pi}\approx1.09

An experiment to think through

Predict, change, explain

Open the quench Lab on the critical quench. Predict the trusted time at bond dimension eight before you move the slider.
Drag the gold ceiling on the bond bars, or move the slider, through four, eight, sixteen and 32.
Explain why the trusted-time plot gains about the same step for each doubling, using the linear growth of the centre entanglement.

Follow a quench until the simulation stops being trusted

The Labs simulate open spin-1/2 chains with energies in units of the coupling and time in units of the reduced Planck constant divided by the coupling. Each Lab states its model and limits in its notes and can export the data behind its readouts.

Try explaining it yourself

1. If the measured step of about 0.75 per doubling held further, how long would bond dimension 64 stay trusted?

Read an explained answer

About 3.45 plus 0.75, roughly 4.2. Treat this as a prediction to test, not a measurement; the Lab slider stops at 32.

2. Would a longer chain at the same bond dimension stay trusted for longer?

Read an explained answer

No. The Lab finds the same trusted time, 2.7, for twelve, sixteen and twenty spins at bond dimension sixteen. Entanglement builds up near each cut at the same rate whatever the total length.

3. Does the linear growth limit only TEBD, or every method that stores the state as a matrix product state?

Read an explained answer

Every such method. The exact state itself carries the growing entanglement, and chapter four showed that no state with too few Schmidt terms at a cut can come closer than the dropped tail. The best any algorithm can do is reach that floor.

Sources to read alongside

U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)
The standard review. Section 3 uses the area law to explain why DMRG works in one dimension; Section 4.1 builds matrix product states from successive singular value decompositions and bounds the truncation error in Eq. 58; Section 6 sets out the ground-state search as a variational sweep; Section 7 covers time evolution and its limits.
P. Calabrese and J. Cardy, Evolution of entanglement entropy in one-dimensional systems, Journal of Statistical Mechanics P04010 (2005)
After a global quench the entanglement of a block grows linearly in time and then saturates; the authors explain this with pairs of entangled quasiparticles that fly apart at finite speed.
M. Fagotti and P. Calabrese, Evolution of entanglement entropy following a quantum quench: analytic results for the XY chain in a transverse magnetic field, Physical Review A 78, 010306(R) (2008)
Solves the quench of the transverse-field chain exactly with free fermions, gives the growth rate as an integral over quasiparticles and bounds the entropy by the logarithm of two per site.
S. Bravyi, M. B. Hastings and F. Verstraete, Lieb–Robinson bounds and the generation of correlations and topological quantum order, Physical Review Letters 97, 050401 (2006)
Uses the Lieb–Robinson bound to show that local Hamiltonians spread correlations no faster than a finite speed, so entanglement across a cut can grow at most linearly in time.