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| 1 | +# Time-evolving block decimation |
| 2 | + |
| 3 | +The time-evolving block-decimation (TEBD) algorithm, introduced in |
| 4 | +[Vidal2003:slightly_entangled](@cite) is the ideal time-evolution scheme to |
| 5 | +simulate one-dimensional many-body systems that are characterised by at most |
| 6 | +nearest-neighbour interactions. |
| 7 | +With the quantum state encoded in a Vidal-form MPS, the TEBD algorithm can be |
| 8 | +straightforwardly parallelised due to the Suzuki–Trotter decomposition of the |
| 9 | +time-evolution operator. |
| 10 | + |
| 11 | +The MPSTimeEvolution package offers two variants of TEBD that differ in the |
| 12 | +order of the Suzuki-Trotter decomposition used for the time-evolution operator. |
| 13 | + |
| 14 | +Before we begin, let's introduce some objects we will need in both cases. |
| 15 | +We'll use the same initial pure state and Hamiltonian as in the [Standard |
| 16 | +TDVP1](@ref) example: |
| 17 | + |
| 18 | +```math |
| 19 | +\ket{\psi_0} = \ket{\spinup} \otimes \ket{\spindown} \otimes \ket{\spinup} |
| 20 | +\otimes \ket{\spindown} \otimes \dotsb {} |
| 21 | +``` |
| 22 | + |
| 23 | +and |
| 24 | + |
| 25 | +```math |
| 26 | +H = -\frac12 \sum_{n=1}^{N-1} \pauliz[n] \pauliz[n+1] +\sum_{n=1}^{N} \paulix[n] |
| 27 | +``` |
| 28 | + |
| 29 | +```jldoctest tdvp1 |
| 30 | +julia> using ITensorMPS, MPSTimeEvolution |
| 31 | +
|
| 32 | +julia> N = 10; s = siteinds("S=1/2", N); |
| 33 | +
|
| 34 | +julia> ψₜ = VidalMPS(s, n -> isodd(n) ? "↑" : "↓"); |
| 35 | +
|
| 36 | +julia> h = OpSum(); |
| 37 | +
|
| 38 | +julia> for n in 1:N |
| 39 | + h += "σx", n |
| 40 | + end |
| 41 | +
|
| 42 | +julia> for n in 1:N-1 |
| 43 | + h += -0.5, "σz", n, "σz", n+1 |
| 44 | + end |
| 45 | +
|
| 46 | +``` |
| 47 | + |
| 48 | +We also choose the time step and the total evolution time |
| 49 | + |
| 50 | +```jldoctest tdvp1 |
| 51 | +julia> dt = 0.01; tmax = 10; |
| 52 | +
|
| 53 | +``` |
| 54 | + |
| 55 | +We define a callback object to track the \\(z\\)-axis magnetisation on the first |
| 56 | +three sites. By setting the last argument to `10dt`, the expectation values |
| 57 | +will be computed only every tenth time step. |
| 58 | + |
| 59 | +```jldoctest tdvp1 |
| 60 | +julia> cb = ExpValueCallback("Sz(1,2,3)", s, 10dt) |
| 61 | +ExpValueCallback |
| 62 | +Operators: Sz(1), Sz(2) and Sz(3) |
| 63 | +No measurements performed |
| 64 | +
|
| 65 | +``` |
| 66 | + |
| 67 | +## First-order Suzuki-Trotter decomposition |
| 68 | + |
| 69 | +```@docs; canonical=false, collapsed=true |
| 70 | +tebd1! |
| 71 | +``` |
| 72 | + |
| 73 | +The `tebd1!` function evolves the `VidalMPS` `ψₜ` according to the specified |
| 74 | +Hamiltonian. It is enough to specfy the (full) Hamiltonian operator as an |
| 75 | +`OpSum` object: the `tebd1!` function will take care of computing the |
| 76 | +Suzuki-Trotter decomposition in odd and even terms. |
| 77 | +We also need to provide the `maxdim` and `cutoff` keyword argument in order to |
| 78 | +describe how the MPS has to be truncated after the application of the two-site |
| 79 | +time-evolution operators. |
| 80 | + |
| 81 | +```jldoctest tdvp1 |
| 82 | +julia> tebd1!(ψₜ, h, dt, tmax; cutoff=1e-12, maxdim=10, progress=false, callback=cb) |
| 83 | +
|
| 84 | +``` |
| 85 | + |
| 86 | +## Second-order Suzuki-Trotter decomposition |
| 87 | + |
| 88 | +The interface of the TEBD function with the second-order Suzuki-Trotter |
| 89 | +algorithm is exactly the same as `tebd1!`. |
| 90 | + |
| 91 | +```@docs; canonical=false, collapsed=true |
| 92 | +tebd2! |
| 93 | +``` |
| 94 | + |
| 95 | +In the second-order Suzuki-Trotter decomposition, the time-evolution operator |
| 96 | +over a time step \\(t\\) is |
| 97 | + |
| 98 | +```math |
| 99 | +U(t) = U_{odd}(t/2) U_{even}(t) U_{odd}(t/2); |
| 100 | +``` |
| 101 | + |
| 102 | +if we don't need to compute the expectation values in `cb` after each time step, |
| 103 | +we can merge the \\(U\sb{odd}(t/2)\\) in the \\(U(t)\\) across time steps, i.e. |
| 104 | + |
| 105 | +```math |
| 106 | +U(t)^k = U_{odd}(t/2) U_{even}(t) \bigl(U_{odd}(t) U_{even}(t)\bigr)^{k-1} U_{odd}(t/2) |
| 107 | +``` |
| 108 | + |
| 109 | +which avoids some tensor multiplications and unnecessary SVDs. |
| 110 | +This is automatically done by the `tebd2!` function if the measurement time step |
| 111 | +saved in the callback object is equal to \\(k>1\\) times the time step `dt`. |
| 112 | + |
| 113 | +Let's run the function, after resetting the state to its initial value, and |
| 114 | +creating a new callback object: |
| 115 | + |
| 116 | +```jldoctest tdvp1 |
| 117 | +julia> ψₜ = VidalMPS(s, n -> isodd(n) ? "↑" : "↓"); |
| 118 | +
|
| 119 | +julia> cb2 = ExpValueCallback("Sz(1,2,3)", s, 10dt); |
| 120 | +
|
| 121 | +julia> tebd2!(ψₜ, h, dt, tmax; cutoff=1e-12, maxdim=10, progress=false, callback=cb2) |
| 122 | +
|
| 123 | +``` |
| 124 | + |
| 125 | +We can see the results in the pictures below: the TEBD1 results are represented |
| 126 | +by dashed lines, and the TEDB2 results by solid lines. We find, as expected, |
| 127 | +that the two algorithm produce approximately the same results, which differ by |
| 128 | +approximately \\(10^{-5}\\) at the end of the run (which is fine, since we |
| 129 | +haven't been that careful to set up a numerical simulation with sensible |
| 130 | +parameters). |
| 131 | + |
| 132 | + |
| 133 | + |
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