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Add 3-site simple update (aka 3-site cluster update) (#171)
- Implement the simple update algorithm for 3-site clusters such as in next-nearest-neighbor interactions - Define `real` and `imag` for `LocalOperator`s - Add corresponding tests - Add a commented J1-J2 simple update example --------- Co-authored-by: Lukas Devos <[email protected]> Co-authored-by: Paul Brehmer <[email protected]>
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‎.github/workflows/Tests.yml

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- examples
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- utility
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- bondenv
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- timeevol
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os:
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- ubuntu-latest
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- macOS-latest

‎docs/make.jl

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# examples pages
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examples_optimization =
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joinpath.(["heisenberg", "bose_hubbard", "xxz", "fermi_hubbard"], Ref("index.md"))
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examples_time_evolution = joinpath.(["heisenberg_su", "hubbard_su"], Ref("index.md"))
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examples_time_evolution =
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joinpath.(["heisenberg_su", "hubbard_su", "j1j2_su"], Ref("index.md"))
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examples_partition_functions =
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joinpath.(
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["2d_ising_partition_function", "3d_ising_partition_function"], Ref("index.md")

‎docs/src/assets/pepskit.bib

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publisher = {American Physical Society},
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doi = {10.1103/PhysRevB.105.195140}
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}
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@article{liu_gapless_2022,
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title = {Gapless quantum spin liquid and global phase diagram of the spin-1/2 J1-J2 square antiferromagnetic Heisenberg model},
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journal = {Science Bulletin},
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volume = {67},
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number = {10},
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pages = {1034-1041},
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year = {2022},
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issn = {2095-9273},
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doi = {10.1016/j.scib.2022.03.010},
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url = {https://www.sciencedirect.com/science/article/pii/S2095927322001001},
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author = {Wen-Yuan Liu and Shou-Shu Gong and Yu-Bin Li and Didier Poilblanc and Wei-Qiang Chen and Zheng-Cheng Gu},
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}

‎docs/src/examples/j1j2_su/index.md

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‎docs/src/examples/j1j2_su/main.ipynb

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Markdown #hide"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Three-site simple update for the $J_1$-$J_2$ model\n",
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"\n",
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"In this example, we will use `SimpleUpdate` imaginary time evolution to treat\n",
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"the two-dimensional $J_1$-$J_2$ model, which contains next-nearest-nearest neighbor interactions:\n",
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"\n",
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"$$\n",
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"H = J_1 \\sum_{\\langle i,j \\rangle} \\mathbf{S}_i \\cdot \\mathbf{S}_j\n",
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"+ J_2 \\sum_{\\langle \\langle i,j \\rangle \\rangle} \\mathbf{S}_i \\cdot \\mathbf{S}_j\n",
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"$$\n",
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"\n",
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"Here we will exploit the $U(1)$ spin rotation symmetry in the $J_1$-$J_2$ model. The goal\n",
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"will be to calculate the energy at $J_1 = 1$ and $J_2 = 1/2$, first using the simple update\n",
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"algorithm and then, to refine the energy estimate, using AD-based variational PEPS\n",
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"optimization.\n",
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"\n",
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"We first import all required modules and seed the RNG:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"using Random\n",
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"using TensorKit, PEPSKit\n",
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"Random.seed!(2025);"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Simple updating a challenging phase\n",
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"\n",
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"Let's start by initializing an `InfiniteWeightPEPS` for which we set the required parameters\n",
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"as well as physical and virtual vector spaces. We use the minimal unit cell size\n",
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"($2 \\times 2$) required by the simple update algorithm for Hamiltonians with\n",
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"next-nearest-neighbour interactions:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"Dbond, χenv, symm = 4, 32, U1Irrep\n",
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"trscheme_env = truncerr(1e-10) & truncdim(χenv)\n",
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"Nr, Nc, J1 = 2, 2, 1.0\n",
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"\n",
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"# random initialization of 2x2 iPEPS with weights and CTMRGEnv (using real numbers)\n",
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"Pspace = Vect[U1Irrep](1//2 => 1, -1//2 => 1)\n",
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"Vspace = Vect[U1Irrep](0 => 2, 1//2 => 1, -1//2 => 1)\n",
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"Espace = Vect[U1Irrep](0 => χenv ÷ 2, 1//2 => χenv ÷ 4, -1//2 => χenv ÷ 4)\n",
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"wpeps = InfiniteWeightPEPS(rand, Float64, Pspace, Vspace; unitcell=(Nr, Nc));"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"The value $J_2 / J_1 = 0.5$ corresponds to a [possible spin liquid phase](@cite liu_gapless_2022),\n",
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"which is challenging for SU to produce a relatively good state from random initialization.\n",
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"Therefore, we shall gradually increase $J_2 / J_1$ from 0.1 to 0.5, each time initializing\n",
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"on the previously evolved PEPS:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"dt, tol, maxiter = 1e-2, 1e-8, 30000\n",
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"check_interval = 4000\n",
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"trscheme_peps = truncerr(1e-10) & truncdim(Dbond)\n",
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"alg = SimpleUpdate(dt, tol, maxiter, trscheme_peps)\n",
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"for J2 in 0.1:0.1:0.5\n",
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" H = real( ## convert Hamiltonian `LocalOperator` to real floats\n",
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" j1_j2_model(ComplexF64, symm, InfiniteSquare(Nr, Nc); J1, J2, sublattice=false),\n",
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" )\n",
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" result = simpleupdate(wpeps, H, alg; check_interval)\n",
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" global wpeps = result[1]\n",
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"end"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"After we reach $J_2 / J_1 = 0.5$, we gradually decrease the evolution time step to obtain\n",
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"a more accurately evolved PEPS:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"dts = [1e-3, 1e-4]\n",
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"tols = [1e-9, 1e-9]\n",
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"J2 = 0.5\n",
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"H = real(j1_j2_model(ComplexF64, symm, InfiniteSquare(Nr, Nc); J1, J2, sublattice=false))\n",
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"for (dt, tol) in zip(dts, tols)\n",
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" alg′ = SimpleUpdate(dt, tol, maxiter, trscheme_peps)\n",
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" result = simpleupdate(wpeps, H, alg′; check_interval)\n",
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" global wpeps = result[1]\n",
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"end"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Computing the simple update energy estimate\n",
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"\n",
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"Finally, we measure the ground-state energy by converging a CTMRG environment and computing\n",
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"the expectation value, where we make sure to normalize by the unit cell size:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"peps = InfinitePEPS(wpeps)\n",
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"normalize!.(peps.A, Inf) ## normalize PEPS with absorbed weights by largest element\n",
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"env₀ = CTMRGEnv(rand, Float64, peps, Espace)\n",
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"env, = leading_boundary(env₀, peps; tol=1e-10, alg=:sequential, trscheme=trscheme_env);\n",
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"E = expectation_value(peps, H, env) / (Nr * Nc)"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Let us compare that estimate with benchmark data obtained from the\n",
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"[YASTN/peps-torch package](https://github.com/jurajHasik/j1j2_ipeps_states/blob/ea4140fbd7da0fc1b75fac2871f75bda125189a8/single-site_pg-C4v-A1_internal-U1/j20.5/state_1s_A1_U1B_j20.5_D4_chi_opt96.dat).\n",
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"which utilizes AD-based PEPS optimization to find $E_\\text{ref}=-0.49425$:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"E_ref = -0.49425\n",
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"@show (E - E_ref) / abs(E_ref);"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"## Variational PEPS optimization using AD\n",
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"\n",
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"As a last step, we will use the SU-evolved PEPS as a starting point for a `fixedpoint`\n",
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"PEPS optimization. Note that we could have also used a sublattice-rotated version of `H` to\n",
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"fit the Hamiltonian onto a single-site unit cell which would require us to optimize fewer\n",
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"parameters and hence lead to a faster optimization. But here we instead take advantage of\n",
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"the already evolved `peps`, thus giving us a physical initial guess for the optimization:"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"peps_opt, env_opt, E_opt, = fixedpoint(\n",
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" H, peps, env; optimizer_alg=(; tol=1e-4, maxiter=120)\n",
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");"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"Finally, we compare the variationally optimized energy against the reference energy. Indeed,\n",
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"we find that the additional AD-based optimization improves the SU-evolved PEPS and leads to\n",
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"a more accurate energy estimate."
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"E_opt /= (Nr * Nc)\n",
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"@show E_opt\n",
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"@show (E_opt - E_ref) / abs(E_ref);"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"---\n",
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"\n",
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"*This notebook was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).*"
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "Julia 1.11.5",
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"language": "julia",
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"name": "julia-1.11"
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},
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"language_info": {
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"file_extension": ".jl",
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"mimetype": "application/julia",
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"name": "julia",
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"version": "1.11.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 3
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}

‎examples/Cache.toml

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heisenberg_su = "299b79f36b94b301dafa0d39fc54498b44dcdf0e4988e3c5416aa6c4e9e8f584"
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xxz = "b5ab0efe6b9addb4b029250d7212d8b2e771136ca4ec4a4bfd7bccc8c688b949"
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fermi_hubbard = "13d6d827a9851bef2c47cbe8894bf6eec9e1fb3222ecc24d2a876336553439ff"
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j1j2_su = "883df8a1ccf061754538620c1086e2e13f1fd7821b8f2f489f305d0cf7098908"
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heisenberg = "f6b057dacb058656ad903778fc00832c6c8a61df930118872e04006b086030c7"

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