Add 3-site simple update (aka 3-site cluster update) #171
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@@ -32,6 +32,7 @@ jobs: | |
| - examples | ||
| - utility | ||
| - bondenv | ||
| - timeevol | ||
| os: | ||
| - ubuntu-latest | ||
| - macOS-latest | ||
|
|
||
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| Original file line number | Diff line number | Diff line change |
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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "using Markdown #hide" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Three-site simple update for the $J_1$-$J_2$ model\n", | ||
| "\n", | ||
| "In this example, we will use `SimpleUpdate` imaginary time evolution to treat\n", | ||
| "the two-dimensional $J_1$-$J_2$ model, which contains next-nearest neighbor interactions:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "H = J_1 \\sum_{\\langle i,j \\rangle} \\mathbf{S}_i \\cdot \\mathbf{S}_j\n", | ||
| "+ J_2 \\sum_{\\langle \\langle i,j \\rangle \\rangle} \\mathbf{S}_i \\cdot \\mathbf{S}_j\n", | ||
| "$$\n", | ||
| "\n", | ||
| "Note that the $J_1$-$J_2$ model exhibits a $U(1)$ spin rotation symmetry, which we want to\n", | ||
| "exploit here. The goal will be to calculate the energy at $J_1 = 1$ and $J_2 = 1/2$, first\n", | ||
| "using the simple update algorithm and then, to refine the energy estimate, using AD-based\n", | ||
| "variational PEPS optimization.\n", | ||
| "\n", | ||
| "We first import all required modules and seed the RNG:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "using Random\n", | ||
| "using TensorKit, PEPSKit\n", | ||
| "Random.seed!(2025);" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Simple updating a challenging phase\n", | ||
| "\n", | ||
| "Let's start by initializing an `InfiniteWeightPEPS` for which we set the required\n", | ||
| "parameters as well as physical and virtual vector spaces. Since the $J_1$-$J_2$ model has\n", | ||
| "*next*-neighbor interactions, the simple update algorithm requires a $2 \\times 2$ unit cell:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "Dbond, χenv, symm = 4, 32, U1Irrep\n", | ||
| "trscheme_env = truncerr(1e-10) & truncdim(χenv)\n", | ||
| "Nr, Nc, J1 = 2, 2, 1.0\n", | ||
| "\n", | ||
| "# random initialization of 2x2 iPEPS with weights and CTMRGEnv (using real numbers)\n", | ||
| "Pspace = Vect[U1Irrep](1//2 => 1, -1//2 => 1)\n", | ||
| "Vspace = Vect[U1Irrep](0 => 2, 1//2 => 1, -1//2 => 1)\n", | ||
| "Espace = Vect[U1Irrep](0 => χenv ÷ 2, 1//2 => χenv ÷ 4, -1//2 => χenv ÷ 4)\n", | ||
| "wpeps = InfiniteWeightPEPS(rand, Float64, Pspace, Vspace; unitcell=(Nr, Nc));" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "The value $J_2 / J_1 = 0.5$ is close to a possible spin liquid phase, which is challenging\n", | ||
| "for SU to produce a relatively good state from random initialization. Therefore, we shall\n", | ||
| "gradually increase $J_2 / J_1$ from 0.1 to 0.5, each time initializing on the previously\n", | ||
| "evolved PEPS:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "dt, tol, maxiter = 1e-2, 1e-8, 30000\n", | ||
| "check_interval = 4000\n", | ||
| "trscheme_peps = truncerr(1e-10) & truncdim(Dbond)\n", | ||
| "alg = SimpleUpdate(dt, tol, maxiter, trscheme_peps)\n", | ||
| "for J2 in 0.1:0.1:0.5\n", | ||
| " H = real( ## convert Hamiltonian `LocalOperator` to real floats\n", | ||
| " j1_j2_model(ComplexF64, symm, InfiniteSquare(Nr, Nc); J1, J2, sublattice=false),\n", | ||
| " )\n", | ||
| " result = simpleupdate(wpeps, H, alg; check_interval)\n", | ||
| " global wpeps = result[1]\n", | ||
| "end" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "After we reach $J_2 / J_1 = 0.5$, we gradually decrease the evolution time step to obtain\n", | ||
| "a more accurately evolved PEPS:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "dts = [1e-3, 1e-4]\n", | ||
| "tols = [1e-9, 1e-9]\n", | ||
| "J2 = 0.5\n", | ||
| "H = real(j1_j2_model(ComplexF64, symm, InfiniteSquare(Nr, Nc); J1, J2, sublattice=false))\n", | ||
| "for (dt, tol) in zip(dts, tols)\n", | ||
| " alg′ = SimpleUpdate(dt, tol, maxiter, trscheme_peps)\n", | ||
| " result = simpleupdate(wpeps, H, alg′; check_interval)\n", | ||
| " global wpeps = result[1]\n", | ||
| "end" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Computing the simple update energy estimate\n", | ||
| "\n", | ||
| "Finally, we measure the ground-state energy by converging a CTMRG environment and computing\n", | ||
| "the expectation value, where we make sure to normalize by the unit cell size:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "peps = InfinitePEPS(wpeps)\n", | ||
| "normalize!.(peps.A, Inf) ## normalize PEPS with absorbed weights by largest element\n", | ||
| "env₀ = CTMRGEnv(rand, Float64, peps, Espace)\n", | ||
| "env, = leading_boundary(env₀, peps; tol=1e-10, alg=:sequential, trscheme=trscheme_env);\n", | ||
| "E = expectation_value(peps, H, env) / (Nr * Nc)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Let us compare that estimate with benchmark data obtained from the\n", | ||
| "[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", | ||
| "which utilizes AD-based PEPS optimization to find $E_\\text{ref}=-0.49425$:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "E_ref = -0.49425\n", | ||
| "@show (E - E_ref) / abs(E_ref);" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Variational PEPS optimization using AD\n", | ||
| "\n", | ||
| "As a last step, we will use the SU-evolved PEPS as a starting point for a `fixedpoint`\n", | ||
| "PEPS optimization. Note that we could have also used a sublattice-rotated version of `H` to\n", | ||
| "fit the Hamiltonian onto a single-site unit cell which would require us to optimize fewer\n", | ||
| "parameters and hence lead to a faster optimization. But here we instead take advantage of\n", | ||
| "the already evolved `peps`, thus giving us a physical initial guess for the optimization:" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "peps_opt, env_opt, E_opt, = fixedpoint(\n", | ||
| " H, peps, env; optimizer_alg=(; tol=1e-4, maxiter=120)\n", | ||
| ");" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "Finally, we compare the variationally optimized energy against the reference energy. Indeed,\n", | ||
| "we find that the additional AD-based optimization improves the SU-evolved PEPS and leads to\n", | ||
| "a more accurate energy estimate." | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "E_opt /= (Nr * Nc)\n", | ||
| "@show E_opt\n", | ||
| "@show (E_opt - E_ref) / abs(E_ref);" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "---\n", | ||
| "\n", | ||
| "*This notebook was generated using [Literate.jl](https://github.com/fredrikekre/Literate.jl).*" | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Julia 1.11.5", | ||
| "language": "julia", | ||
| "name": "julia-1.11" | ||
| }, | ||
| "language_info": { | ||
| "file_extension": ".jl", | ||
| "mimetype": "application/julia", | ||
| "name": "julia", | ||
| "version": "1.11.5" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 3 | ||
| } |
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| Original file line number | Diff line number | Diff line change | ||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| @@ -0,0 +1,121 @@ | ||||||||||
| using Markdown #hide | ||||||||||
| md""" | ||||||||||
| # Three-site simple update for the $J_1$-$J_2$ model | ||||||||||
|
|
||||||||||
| In this example, we will use [`SimpleUpdate`](@ref) imaginary time evolution to treat | ||||||||||
| the two-dimensional $J_1$-$J_2$ model, which contains next-nearest neighbor interactions: | ||||||||||
|
|
||||||||||
| ```math | ||||||||||
| H = J_1 \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j | ||||||||||
| + J_2 \sum_{\langle \langle i,j \rangle \rangle} \mathbf{S}_i \cdot \mathbf{S}_j | ||||||||||
| ``` | ||||||||||
|
|
||||||||||
| Note that the $J_1$-$J_2$ model exhibits a $U(1)$ spin rotation symmetry, which we want to | ||||||||||
| exploit here. The goal will be to calculate the energy at $J_1 = 1$ and $J_2 = 1/2$, first | ||||||||||
| using the simple update algorithm and then, to refine the energy estimate, using AD-based | ||||||||||
| variational PEPS optimization. | ||||||||||
|
|
||||||||||
| We first import all required modules and seed the RNG: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| using Random | ||||||||||
| using TensorKit, PEPSKit | ||||||||||
| Random.seed!(2025); | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| ## Simple updating a challenging phase | ||||||||||
|
|
||||||||||
| Let's start by initializing an `InfiniteWeightPEPS` for which we set the required | ||||||||||
| parameters as well as physical and virtual vector spaces. Since the $J_1$-$J_2$ model has | ||||||||||
| *next*-neighbor interactions, the simple update algorithm requires a $2 \times 2$ unit cell: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| Dbond, χenv, symm = 4, 32, U1Irrep | ||||||||||
| trscheme_env = truncerr(1e-10) & truncdim(χenv) | ||||||||||
| Nr, Nc, J1 = 2, 2, 1.0 | ||||||||||
|
|
||||||||||
| ## random initialization of 2x2 iPEPS with weights and CTMRGEnv (using real numbers) | ||||||||||
| Pspace = Vect[U1Irrep](1//2 => 1, -1//2 => 1) | ||||||||||
| Vspace = Vect[U1Irrep](0 => 2, 1//2 => 1, -1//2 => 1) | ||||||||||
| Espace = Vect[U1Irrep](0 => χenv ÷ 2, 1//2 => χenv ÷ 4, -1//2 => χenv ÷ 4) | ||||||||||
| wpeps = InfiniteWeightPEPS(rand, Float64, Pspace, Vspace; unitcell=(Nr, Nc)); | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| The value $J_2 / J_1 = 0.5$ is close to a possible spin liquid phase, which is challenging | ||||||||||
| for SU to produce a relatively good state from random initialization. Therefore, we shall | ||||||||||
| gradually increase $J_2 / J_1$ from 0.1 to 0.5, each time initializing on the previously | ||||||||||
| evolved PEPS: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| dt, tol, maxiter = 1e-2, 1e-8, 30000 | ||||||||||
| check_interval = 4000 | ||||||||||
| trscheme_peps = truncerr(1e-10) & truncdim(Dbond) | ||||||||||
| alg = SimpleUpdate(dt, tol, maxiter, trscheme_peps) | ||||||||||
| for J2 in 0.1:0.1:0.5 | ||||||||||
| H = real( ## convert Hamiltonian `LocalOperator` to real floats | ||||||||||
| j1_j2_model(ComplexF64, symm, InfiniteSquare(Nr, Nc); J1, J2, sublattice=false), | ||||||||||
| ) | ||||||||||
|
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I know this used to fail since we tried to construct |
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| result = simpleupdate(wpeps, H, alg; check_interval) | ||||||||||
| global wpeps = result[1] | ||||||||||
| end | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| After we reach $J_2 / J_1 = 0.5$, we gradually decrease the evolution time step to obtain | ||||||||||
| a more accurately evolved PEPS: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| dts = [1e-3, 1e-4] | ||||||||||
| tols = [1e-9, 1e-9] | ||||||||||
| J2 = 0.5 | ||||||||||
| H = real(j1_j2_model(ComplexF64, symm, InfiniteSquare(Nr, Nc); J1, J2, sublattice=false)) | ||||||||||
|
|
||||||||||
| for (dt, tol) in zip(dts, tols) | ||||||||||
| alg′ = SimpleUpdate(dt, tol, maxiter, trscheme_peps) | ||||||||||
| result = simpleupdate(wpeps, H, alg′; check_interval) | ||||||||||
| global wpeps = result[1] | ||||||||||
| end | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| ## Computing the simple update energy estimate | ||||||||||
|
|
||||||||||
| Finally, we measure the ground-state energy by converging a CTMRG environment and computing | ||||||||||
| the expectation value, where we make sure to normalize by the unit cell size: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| peps = InfinitePEPS(wpeps) | ||||||||||
| normalize!.(peps.A, Inf) ## normalize PEPS with absorbed weights by largest element | ||||||||||
| env₀ = CTMRGEnv(rand, Float64, peps, Espace) | ||||||||||
| env, = leading_boundary(env₀, peps; tol=1e-10, alg=:sequential, trscheme=trscheme_env); | ||||||||||
| E = expectation_value(peps, H, env) / (Nr * Nc) | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| Let us compare that estimate with benchmark data obtained from the | ||||||||||
| [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). | ||||||||||
| which utilizes AD-based PEPS optimization to find $E_\text{ref}=-0.49425$: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| E_ref = -0.49425 | ||||||||||
| @show (E - E_ref) / abs(E_ref); | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| ## Variational PEPS optimization using AD | ||||||||||
|
|
||||||||||
| As a last step, we will use the SU-evolved PEPS as a starting point for a [`fixedpoint`](@ref) | ||||||||||
| PEPS optimization. Note that we could have also used a sublattice-rotated version of `H` to | ||||||||||
| fit the Hamiltonian onto a single-site unit cell which would require us to optimize fewer | ||||||||||
| parameters and hence lead to a faster optimization. But here we instead take advantage of | ||||||||||
| the already evolved `peps`, thus giving us a physical initial guess for the optimization: | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| peps_opt, env_opt, E_opt, = fixedpoint( | ||||||||||
| H, peps, env; optimizer_alg=(; tol=1e-4, maxiter=120) | ||||||||||
| ); | ||||||||||
|
|
||||||||||
| md""" | ||||||||||
| Finally, we compare the variationally optimized energy against the reference energy. Indeed, | ||||||||||
| we find that the additional AD-based optimization improves the SU-evolved PEPS and leads to | ||||||||||
| a more accurate energy estimate. | ||||||||||
| """ | ||||||||||
|
|
||||||||||
| E_opt /= (Nr * Nc) | ||||||||||
| @show E_opt | ||||||||||
| @show (E_opt - E_ref) / abs(E_ref); | ||||||||||
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Also, I'm slightly confused by this explanation: for me this sounds like next-nearest neighbor hamiltonians have a 2x2 unitcell ground state.$2x2$ unit cell for a next-nearest-neighbour hamiltonian.
In principle the Hamiltonian still has a translation symmetry for a single site, so I don't think it follows automatically that you would need a
From what I understand, this is mostly an algorithmic restriction, and is a result of using simple update, so maybe we can slightly reword this to make that more obvious?