---
title: "Tensor networks and DMRG, from entanglement to quantum quenches"
description: "Learn how a matrix product state compresses a spin chain, how DMRG finds ground states and how long TEBD can follow a quench. Eight chapters with three Labs."
canonical_url: "https://sandboxphysics.com/stories/tensor-networks/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/tensor-networks/"
last_modified: "2026-10-30"
---

[Tensor networks · A course with experiments](https://sandboxphysics.com/stories/tensor-networks/)

FROM ONE CUT TO A WHOLE QUENCH

# Tensor networks and DMRG, from entanglement to quantum quenches

A chain of 32 spins has more than four billion amplitudes, yet the DMRG Lab stores its ground state in about 90 kilobytes, and its energy is off by only a few ten-billionths of the coupling. The reason is entanglement: a ground state carries little of it across any cut. Learn to measure that entanglement, store a state as a matrix product state, find ground states with DMRG and follow a quench with TEBD until growing entanglement ends the simulation.

Sandbox Physics · 2026-10-29

Storing a state · Chapters 1–4

## Why a spin chain can be compressed

Count the amplitudes of a chain, cut it in two to read its entanglement, build a matrix product state from repeated cuts and bound the error when small weights are thrown away.

[Start with the counting →](https://sandboxphysics.com/stories/tensor-networks/many-spins-many-amplitudes/)

Using the state · Chapters 5–8

## Ground states, quenches and their limits

Sweep DMRG to a ground state and check it against exact answers, evolve a quench with two-site gates, see why growing entanglement ends every simulation, then read a published error figure yourself.

[Start with DMRG →](https://sandboxphysics.com/stories/tensor-networks/dmrg-variational-sweeps/)

## Eight chapters, with examples and experiments

1. [01 L1 Why a chain of spins outgrows any computer, and why its ground state does not If 32 spins need billions of amplitudes, how can a laptop find the ground state of such a chain at all? Read and try →](https://sandboxphysics.com/stories/tensor-networks/many-spins-many-amplitudes/)
2. [02 L1 Measure entanglement with one matrix decomposition How do you put a number on how strongly the left and right halves of a chain depend on each other? Read and try →](https://sandboxphysics.com/stories/tensor-networks/schmidt-decomposition/)
3. [03 L2 Turn a wave function into a chain of small matrices How does a chain of singular value decompositions turn a list of two to the power L amplitudes into a product of small matrices, and how many numbers does that need? Read and try →](https://sandboxphysics.com/stories/tensor-networks/matrix-product-states/)
4. [04 L2 How much does cutting the bonds cost? When you keep only a fixed number of Schmidt weights at every cut, how large is the error, and which measurements does it spoil? Read and try →](https://sandboxphysics.com/stories/tensor-networks/truncation-and-local-errors/)
5. [05 L3 Find a ground state by sweeping, and decide whether to believe it How does DMRG find the lowest-energy matrix product state, and what evidence tells you the result is the true ground state? Read and try →](https://sandboxphysics.com/stories/tensor-networks/dmrg-variational-sweeps/)
6. [06 L3 Evolve a matrix product state in time with two-site gates How does TEBD move a matrix product state forward in time, and how do you tell its two kinds of error apart? Read and try →](https://sandboxphysics.com/stories/tensor-networks/tebd-real-time/)
7. [07 L3 Why a quench outruns every matrix product state Why can a matrix product state follow a quench only for a limited time, and why does more bond dimension buy so little extra? Read and try →](https://sandboxphysics.com/stories/tensor-networks/entanglement-growth-and-light-cones/)
8. [08 L4 Read Figure 3(a) of García-Ripoll's time-evolution paper What does Figure 3(a) of García-Ripoll's 2006 paper establish about Trotter time steps, and how far can you reproduce it yourself? Read and try →](https://sandboxphysics.com/stories/tensor-networks/reading-garcia-ripoll-figure-3/)

## Keep the experiments beside the chapters

Each Lab compares a tensor-network calculation with an exact answer computed a different way, so every error you see is measured, not estimated.

[Cut the bonds of a spin chain and watch the error ↗](https://sandboxphysics.com/labs/entanglement-compression/) [Sweep DMRG down to the ground-state energy ↗](https://sandboxphysics.com/labs/dmrg-ground-state-sweeps/) [Follow a quench until the simulation stops being trusted ↗](https://sandboxphysics.com/labs/tebd-quench-time-window/)