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.
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 →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 →Eight chapters, with examples and experiments
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 →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 →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 →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 →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 →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 →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 →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 →