{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Fermi-Hubbard Model II" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "using Pkg; Pkg.activate()\n", "\n", "using KadanoffBaym, FFTW, Interpolations\n", "using LinearAlgebra, BlockArrays\n", "\n", "using JLD" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "using PyPlot\n", "# PyPlot.plt.style.use(\"./paper.mplstyle\")\n", "using LaTeXStrings" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Model" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Hamiltonian\n", "\n", "$$\n", "\\begin{align}\\begin{split}\n", " \\hat{H} &= - J \\sum_{\\langle{i,\\,j}\\rangle}\\sum_\\sigma \\hat{c}^{\\dagger}_{i,\\sigma} \\hat{c}^{\\phantom{\\dagger}}_{i+1,\\sigma} + U\\sum_{i=1}^L \\hat{c}^{\\dagger}_{i,\\uparrow} \\hat{c}^{\\phantom{\\dagger}}_{i,\\uparrow} \\hat{c}^{\\dagger}_{i,\\downarrow} \\hat{c}^{\\phantom{\\dagger}}_{i,\\downarrow}, \n", "\\end{split}\\end{align}\n", "$$\n", "\n", "### Green functions\n", "\n", "$$\n", " G^>_{\\uparrow,ij}(t, t') = -i \\left\\langle \\hat{c}^{\\phantom{\\dagger}}_{i,\\uparrow}(t) \\hat{c}^{{\\dagger}}_{i,\\uparrow}(t') \\right\\rangle\\\\\n", " G^>_{\\downarrow,ij}(t, t') = -i \\left\\langle \\hat{c}^{\\phantom{\\dagger}}_{i,\\downarrow}(t) \\hat{c}^{{\\dagger}}_{i,\\downarrow}(t') \\right\\rangle\\\\\n", "$$\n", "\n", "### Self-energies\n", "\n", "Hartree-Fock:\n", "$$\n", " \\Sigma^{\\mathrm{HF}}_{\\uparrow,\\,ij}(t, t') = {\\mathrm{i}}\\delta_{ij}\\delta(t - t') G^<_{\\downarrow,ii}(t, t)\\\\\n", " \\Sigma^{\\mathrm{HF}}_{\\downarrow,\\,ij}(t, t') = {\\mathrm{i}}\\delta_{ij}\\delta(t - t') G^<_{\\uparrow,ii}(t, t)\n", "$$\n", "\n", "\n", "Second-order Born approximation:\n", "$$\n", " \\Sigma_{ij, \\uparrow} (t, t') = U^2 G_{ij, \\uparrow}(t, t') G_{ij, \\downarrow}(t, t') G_{ji, \\downarrow}(t', t),\\\\\n", " \\Sigma_{ij, \\downarrow}(t, t') = U^2 G_{ij, \\downarrow}(t, t') G_{ij, \\uparrow}(t, t') G_{ji, \\uparrow}(t', t)\n", "$$\n", "\n", "\n", "$T$-matrix approximation:\n", "$$\n", " \\Sigma_{ij, \\uparrow} (t, t') = i U^2 T_{ij}(t, t') G_{ji, \\downarrow}(t', t),\\\\\n", " \\Sigma_{ij, \\downarrow}(t, t') = i U^2 T_{ij}(t, t') G_{ji, \\uparrow}(t', t)\n", "$$\n", "\n", "$$\n", " \\boldsymbol{T}(t, t') = \\boldsymbol{\\Phi}(t, t') - U \\int_{\\mathcal{C}}\\mathrm{d}s\\; \\boldsymbol{\\Phi}(t, s) \\boldsymbol{T}(s, t')\n", "$$\n", "\n", "$$\n", " \\Phi_{ij}(t, t') = -i G_{ij, \\uparrow}(t, t') G_{ij, \\downarrow}(t, t')\n", "$$\n", "\n", "$$\n", " \\boldsymbol{T}^\\lessgtr(t, t') = \\boldsymbol{\\Phi}^\\lessgtr(t, t') - U \\int_{0}^{t}\\mathrm{d}s\\; [\\boldsymbol{\\Phi}^>(t, s) - \\boldsymbol{\\Phi}^<(t, s)] \\boldsymbol{T}^\\lessgtr(s, t') - U \\int_{0}^{t'}\\mathrm{d}s\\; \\boldsymbol{\\Phi}^\\lessgtr(t, s)[\\boldsymbol{T}^<(s, t') - \\boldsymbol{T}^>(s, t')]\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Solving" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "function integrate1(hs::Vector, t1, t2, A::GreenFunction, B::GreenFunction, C::GreenFunction; tmax=t1)\n", " retval = zero(A[t1,t1])\n", "\n", " @inbounds for k in 1:tmax\n", " @views LinearAlgebra.mul!(retval, A[t1, k] - B[t1, k], C[k, t2], hs[k], 1.0)\n", " end\n", " return retval\n", "end\n", "\n", "function integrate2(hs::Vector, t1, t2, A::GreenFunction, B::GreenFunction, C::GreenFunction; tmax=t2)\n", " retval = zero(A[t1,t1])\n", "\n", " @inbounds for k in 1:tmax\n", " @views LinearAlgebra.mul!(retval, A[t1, k], B[k, t2] - C[k, t2], hs[k], 1.0)\n", " end\n", " return retval\n", "end" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "function fixed_point(F::Function, x0::AbstractArray; \n", " mixing::Float64=0.5, \n", " abstol::Float64=1e-12, \n", " maxiter::Int=1000, \n", " verbose::Bool=true, \n", " norm=x -> LinearAlgebra.norm(x, Inf)\n", " )\n", " \n", " x_old = copy(x0)\n", "\n", " step = 0\n", " while step < maxiter\n", " x = F(x_old)\n", " res = norm(x - x_old)\n", " if verbose\n", " @info \"step: $step // res: $res\"\n", " end\n", " if res < abstol\n", " break\n", " end\n", " @. x_old = mixing * x + (1.0 - mixing) * x_old\n", " step += 1\n", " end\n", "\n", " if step == maxiter\n", " @warn \"No convergence reached.\"\n", " end\n", "\n", " return x_old\n", "end" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Lattice size\n", "L = 8\n", "\n", "# Allocate the initial Green functions (time arguments at the end)\n", "GL_u = GreenFunction(zeros(ComplexF64, L, L, 1, 1), SkewHermitian)\n", "GG_u = GreenFunction(zeros(ComplexF64, L, L, 1, 1), SkewHermitian)\n", "GL_d = GreenFunction(zeros(ComplexF64, L, L, 1, 1), SkewHermitian)\n", "GG_d = GreenFunction(zeros(ComplexF64, L, L, 1, 1), SkewHermitian);" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Initial conditions\n", "N_u = zeros(L)\n", "N_d = zeros(L)\n", "\n", "N_u[1:4] = 0.1 .* [1, 1, 1, 1]\n", "N_d[1:4] = 0.1 .* [1, 1, 1, 1]\n", "\n", "N_u[5:8] = 0.0 .* [1, 1, 1, 1]\n", "N_d[5:8] = 0.0 .* [1, 1, 1, 1]\n", "\n", "GL_u[1, 1] = 1.0im * diagm(N_u)\n", "GG_u[1, 1] = -1.0im * (I - diagm(N_u))\n", "GL_d[1, 1] = 1.0im * diagm(N_d)\n", "GG_d[1, 1] = -1.0im * (I - diagm(N_d));" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "Base.@kwdef struct FermiHubbardDataTM{T}\n", " GL_u::T\n", " GG_u::T\n", " GL_d::T\n", " GG_d::T\n", "\n", " ΣL_u::T = zero(GL_u)\n", " ΣG_u::T = zero(GG_u)\n", " ΣL_d::T = zero(GL_d)\n", " ΣG_d::T = zero(GG_d)\n", "\n", " TL::T = zero(GL_u)\n", " TG::T = zero(GG_u)\n", "\n", " ΦL::T = zero(GL_u)\n", " ΦG::T = zero(GG_u)\n", "end\n", "\n", "data = FermiHubbardDataTM(GL_u=GL_u, GG_u=GG_u, GL_d=GL_d, GG_d=GG_d)\n", "\n", "# Initialize T-matrix\n", "data.TL[1, 1] = -1.0im .* GL_u[1, 1] .* GL_d[1, 1]\n", "data.TG[1, 1] = -1.0im .* GG_u[1, 1] .* GG_d[1, 1]\n", "data.ΦL[1, 1] = data.TL[1, 1]\n", "data.ΦG[1, 1] = data.TG[1, 1];" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "Base.@kwdef struct FermiHubbardModel{T}\n", " # interaction strength\n", " U::T\n", "\n", " # 8-site 3D cubic lattice\n", " h = begin\n", " h = BlockArray{ComplexF64}(undef_blocks, [4, 4], [4, 4])\n", " diag_block = [0 -1 0 -1; -1 0 -1 0; 0 -1 0 -1; -1 0 -1 0]\n", " setblock!(h, diag_block, 1, 1)\n", " setblock!(h, diag_block, 2, 2)\n", " setblock!(h, Diagonal(-1 .* ones(4)), 1, 2)\n", " setblock!(h, Diagonal(-1 .* ones(4)), 2, 1)\n", "\n", " h |> Array\n", " end\n", "\n", " H_u = h\n", " H_d = h\n", "end\n", "\n", "# Interaction parameter\n", "const U₀ = 2.0\n", "model = FermiHubbardModel(U = t -> U₀);" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Right-hand side for the \"vertical\" evolution\n", "function fv!(model, data, out, times, h1, h2, t, t′)\n", " # Unpack data and model\n", " (; GL_u, GG_u, GL_u, GG_d, ΣL_u, ΣG_u, ΣL_d, ΣG_d) = data\n", " (; H_u, H_d, U) = model\n", "\n", " # Real-time collision integrals\n", " ∫dt1(A, B, C) = integrate1(h1, t, t′, A, B, C)\n", " ∫dt2(A, B, C) = integrate2(h2, t, t′, A, B, C)\n", " \n", " # The interaction varies as a function of the forward time (t+t')/2\n", " U_t = U((times[t] + times[t′])/2)\n", " \n", " # Hartree-Fock self-energies\n", " ΣHF_u(t, t′) = im * U_t * Diagonal(GL_d[t, t])\n", " ΣHF_d(t, t′) = im * U_t * Diagonal(GL_u[t, t])\n", " \n", " # Equations of motion\n", " out[1] = -1.0im * ((H_u + ΣHF_u(t, t′)) * GL_u[t, t′] + \n", " ∫dt1(ΣG_u, ΣL_u, GL_u) + ∫dt2(ΣL_u, GL_u, GG_u)\n", " )\n", "\n", " out[2] = -1.0im * ((H_u + ΣHF_u(t, t′)) * GG_u[t, t′] + \n", " ∫dt1(ΣG_u, ΣL_u, GG_u) + ∫dt2(ΣG_u, GL_u, GG_u)\n", " )\n", "\n", " out[3] = -1.0im * ((H_d + ΣHF_d(t, t′)) * GL_d[t, t′] + \n", " ∫dt1(ΣG_d, ΣL_d, GL_d) + ∫dt2(ΣL_d, GL_d, GG_d)\n", " )\n", "\n", " out[4] = -1.0im * ((H_d + ΣHF_d(t, t′)) * GG_d[t, t′] +\n", " ∫dt1(ΣG_d, ΣL_d, GG_d) + ∫dt2(ΣG_d, GL_d, GG_d)\n", " ) \n", " \n", " return out\n", "end\n", "\n", "function fd!(model, data, out, times, h1, h2, t, t′)\n", " fv!(model, data, out, times, h1, h2, t, t)\n", " out .-= adjoint.(out)\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## T-matrix" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Callback function for the self-energies\n", "function T_matrix!(model, data, times, h1, h2, t, t′)\n", " # Unpack data and model\n", " (; GL_u, GG_u, GL_d, GG_d, TL, TG, ΣL_u, ΣG_u, ΣL_d, ΣG_d, ΦL, ΦG) = data\n", " (; U) = model\n", " \n", " # Real-time collision integral\n", " ∫dt1(A, B, C) = integrate1(h1, t, t′, A, B, C)\n", " ∫dt2(A, B, C) = integrate2(h2, t, t′, A, B, C)\n", " \n", " # Resize self-energies etc. when Green functions are resized\n", " if (n = size(GL_u, 3)) > size(ΣL_u, 3)\n", " resize!(ΣL_u, n)\n", " resize!(ΣG_u, n)\n", " resize!(ΣL_d, n)\n", " resize!(ΣG_d, n)\n", " \n", " resize!(TL, n)\n", " resize!(TG, n)\n", " resize!(ΦL, n)\n", " resize!(ΦG, n)\n", " end\n", " \n", " # The interaction varies as a function of the forward time (t+t')/2\n", " U_t = U((times[t] + times[t′])/2)\n", " \n", " # Set all Φs at the very first t′ since they are all known by then\n", " if t′ == 1\n", " for t′ in 1:t\n", " ΦL[t, t′] = -1.0im .* GL_u[t, t′] .* GL_d[t, t′]\n", " ΦG[t, t′] = -1.0im .* GG_u[t, t′] .* GG_d[t, t′]\n", " end\n", " end \n", " \n", " # Solve VIEs implicitly (uncached version)\n", " TL[t, t′], TG[t, t′] = fixed_point([ΦL[t, t′], ΦG[t, t′]]; mixing=0.5, verbose=false) do x\n", " TL[t, t′], TG[t, t′] = x[1], x[2]\n", " \n", " [\n", " ΦL[t, t′] - U_t * (∫dt1(ΦG, ΦL, TL) + ∫dt2(ΦL, TL, TG)),\n", " ΦG[t, t′] - U_t * (∫dt1(ΦG, ΦL, TG) + ∫dt2(ΦG, TL, TG))\n", " ]\n", " end\n", " \n", "# # Solve VIEs implicitly (cached version)\n", "# TL[t, t′], TG[t, t′] = let\n", "# ∫dt_(x...) = (x[1] < 1 || x[2] < 1) ? zero(x[3][1,1]) : ∫dt(x...)\n", " \n", "# ΦL_ = ΦL[t, t′]\n", "# ΦG_ = ΦG[t, t′]\n", " \n", "# I1 = integrate1(h1, t, t′, ΦG, ΦL, TL; tmax=t-1)\n", "# I2 = integrate2(h2, t, t′, ΦL, TL, TG; tmax=t′-1)\n", "# I3 = integrate1(h1, t, t′, ΦG, ΦL, TG; tmax=t-1)\n", "# I4 = integrate2(h2, t, t′, ΦG, TL, TG; tmax=t′-1)\n", " \n", "# L_ = ΦL[t, t′] - U_t * (I1 + I2)\n", "# G_ = ΦG[t, t′] - U_t * (I3 + I4)\n", " \n", "# fixed_point([L_, G_]; mixing=0.5, verbose=false) do x\n", "# TL[t, t′], TG[t, t′] = x[1], x[2]\n", "\n", "# [\n", "# L_ - U_t * (h1[t] * (ΦG[t,t] - ΦL[t,t]) * TL[t,t′] + h2[t′] * ΦL[t,t′] * (TL[t′,t′] - TG[t′,t′])),\n", "# G_ - U_t * (h1[t] * (ΦG[t,t] - ΦL[t,t]) * TG[t,t′] + h2[t′] * ΦG[t,t′] * (TL[t′,t′] - TG[t′,t′]))\n", "# ]\n", "# end\n", "# end \n", " \n", " # Define the self-energies\n", " ΣL_u[t, t′] = 1.0im .* U_t^2 .* TL[t, t′] .* transpose(GG_d[t′, t])\n", " ΣL_d[t, t′] = 1.0im .* U_t^2 .* TL[t, t′] .* transpose(GG_u[t′, t])\n", " \n", " ΣG_u[t, t′] = 1.0im .* U_t^2 .* TG[t, t′] .* transpose(GL_d[t′, t])\n", " ΣG_d[t, t′] = 1.0im .* U_t^2 .* TG[t, t′] .* transpose(GL_u[t′, t])\n", "end" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "tmax = 8 # 32;\n", "tols = [(1e-8, 1e-10), (1e-6, 1e-8), (1e-4, 1e-6)];" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": true }, "outputs": [], "source": [ "for (rtol, atol) in tols\n", " @time sol = kbsolve!(\n", " (x...) -> fv!(model, data, x...),\n", " (x...) -> fd!(model, data, x...),\n", " [data.GL_u, data.GG_u, data.GL_d, data.GG_d],\n", " (0.0, tmax);\n", " callback = (x...) -> T_matrix!(model, data, x...),\n", " atol = atol,\n", " rtol = rtol,\n", " dtini=1e-10,\n", " stop = x -> (println(\"t: $(x[end])\"); flush(stdout); false)\n", " );\n", "\n", " save(\"FH_3D_T_matrix_sol_U_$(U₀)_tmax_$(tmax)_atol_$(atol)_rtol_$(rtol).jld\",\n", " \"solution\", sol,\n", " \"GFs\", data.GL_u.data .+ data.GL_d.data)\n", "end" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Second Born approx." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "# Callback function for the self-energies\n", "function second_Born!(model, data, times, _, _, t, t′)\n", " # Unpack data and model\n", " (; GL_u, GG_u, GL_d, GG_d, ΣL_u, ΣG_u, ΣL_d, ΣG_d) = data\n", " (; U) = model\n", " \n", " # Resize self-energies when Green functions are resized \n", " if (n = size(GL_u, 3)) > size(ΣL_u, 3)\n", " resize!(ΣL_u, n)\n", " resize!(ΣG_u, n)\n", " resize!(ΣL_d, n)\n", " resize!(ΣG_d, n) \n", " end\n", " \n", " # The interaction varies as a function of the forward time (t+t')/2\n", " U_t = U((times[t] + times[t′])/2)\n", " \n", " # Define the self-energies\n", " ΣL_u[t, t′] = U_t^2 .* GL_u[t, t′] .* GL_d[t, t′] .* transpose(GG_d[t′, t])\n", " ΣL_d[t, t′] = U_t^2 .* GL_u[t, t′] .* GL_d[t, t′] .* transpose(GG_u[t′, t])\n", " \n", " ΣG_u[t, t′] = U_t^2 .* GG_u[t, t′] .* GG_d[t, t′] .* transpose(GL_d[t′, t])\n", " ΣG_d[t, t′] = U_t^2 .* GG_u[t, t′] .* GG_d[t, t′] .* transpose(GL_u[t′, t])\n", "end" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": true }, "outputs": [], "source": [ "# Callback function for the self-energies\n", "function second_Born!(model, data, times, _, _, t, t′)\n", " # Unpack data and model\n", " (; GL_u, GG_u, GL_d, GG_d, ΣL_u, ΣG_u, ΣL_d, ΣG_d) = data\n", " (; U) = model\n", " \n", " # Resize self-energies when Green functions are resized \n", " if (n = size(GL_u, 3)) > size(ΣL_u, 3)\n", " resize!(ΣL_u, n)\n", " resize!(ΣG_u, n)\n", " resize!(ΣL_d, n)\n", " resize!(ΣG_d, n) \n", " end\n", " \n", " # The interaction varies as a function of the forward time (t+t')/2\n", " U_t = U((times[t] + times[t′])/2)\n", " \n", " # Define the self-energies\n", " ΣL_u[t, t′] = U_t^2 .* GL_u[t, t′] .* GL_d[t, t′] .* transpose(GG_d[t′, t])\n", " ΣL_d[t, t′] = U_t^2 .* GL_u[t, t′] .* GL_d[t, t′] .* transpose(GG_u[t′, t])\n", " \n", " ΣG_u[t, t′] = U_t^2 .* GG_u[t, t′] .* GG_d[t, t′] .* transpose(GL_d[t′, t])\n", " ΣG_d[t, t′] = U_t^2 .* GG_u[t, t′] .* GG_d[t, t′] .* transpose(GL_u[t′, t])\n", "end\n", "\n", "# Solve second Born for comparison\n", "@time sol_2B = kbsolve!(\n", " (x...) -> fv!(model, data, x...),\n", " (x...) -> fd!(model, data, x...),\n", " [data.GL_u, data.GG_u, data.GL_d, data.GG_d],\n", " (0.0, tmax);\n", " callback = (x...) -> second_Born!(model, data, x...),\n", " atol = 1e-8,\n", " rtol = 1e-6,\n", " dtini=1e-10,\n", " stop = x -> (println(\"t: $(x[end])\"); flush(stdout); false)\n", ");" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Example plots" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [ "GFs = []\n", "ts = []\n", "\n", "for (i, (rtol, atol)) in enumerate(tols)\n", " loaded_data = load(\"FH_3D_T_matrix_sol_U_$(U₀)_tmax_$(tmax)_atol_$(atol)_rtol_$(rtol).jld\")\n", " push!(GFs, loaded_data[\"GFs\"])\n", " push!(ts, loaded_data[\"solution\"].t)\n", "end;" ] }, { "cell_type": "code", "execution_count": null, "metadata": { "scrolled": true }, "outputs": [], "source": [ "xpad = 8\n", "ypad = 5\n", "\n", "figure(figsize = (8, 3))\n", "\n", "ax = subplot(121)\n", "idx_1 = 1\n", "ms = 4\n", "\n", "colors = [\"C2\", \"C0\", \"C3\"]\n", "styles = [\"s\", \"D\", \"o\"]\n", "ls = [\"-\", \"-.\", \"--\"]\n", "\n", "plot([], [], label=\"\\$\\\\texttt{rtol}\\$\", c=\"w\")\n", "for (i, (rtol, atol)) in enumerate(tols)\n", " plot(ts[i], [imag(GFs[i][idx_1, idx_1, k, k]) for k in eachindex(ts[i])], \n", " label = L\"\\texttt{%$(rtol)}\", lw=1.5, styles[i], ms=ms, c = colors[i])\n", "end\n", "\n", "plot(sol_2B.t, [imag(data.GL_u.data[idx_1, idx_1, k, k] .+ data.GL_d.data[idx_1, idx_1, k, k]) for k in eachindex(sol_2B.t)], \n", " lw=3, ls = \"--\", c = \"k\", alpha=0.75)\n", "\n", "xlim(0, tmax)\n", "ax.set_xticks(0:tmax / 4:tmax)\n", "ylim(0, 0.3)\n", "xlabel(\"\\$J t\\$\")\n", "ylabel(\"Charge on site 1\", labelpad = 8)\n", "ax.xaxis.set_tick_params(pad = xpad)\n", "ax.yaxis.set_tick_params(pad = ypad)\n", "ticklabel_format(axis = \"y\", style = \"sci\", scilimits = (-0, 0))\n", "ax.legend(loc = \"best\", handlelength = 1, frameon = false, borderpad = 0, labelspacing = 0.25, ncol=2)\n", "\n", "ax = subplot(122)\n", "for (i, (rtol, atol)) in enumerate(reverse(tols))\n", " semilogy(ts[i], [tr(GFs[i][:, :, k, k]) |> imag for k in eachindex(ts[i])] .- sum(N_u .+ N_d) .|> abs, \n", " ls = ls[i], c = colors[i])\n", "end\n", "\n", "xlim(0, tmax)\n", "ylim(1e-10, 2e-4)\n", "ax.set_xticks(0:tmax / 4:tmax)\n", "ylabel(\"\\$Q(t) - Q_0\\$\", labelpad = 16)\n", "xlabel(\"\\$J t\\$\")\n", "ax.xaxis.set_tick_params(pad = xpad)\n", "ax.yaxis.set_tick_params(pad = ypad)\n", "# ticklabel_format(axis = \"y\", style = \"sci\", scilimits = (-0, 0))\n", "ax.yaxis.set_label_position(\"right\")\n", "\n", "tight_layout(pad = 0.1, w_pad = 0.5, h_pad = 0)\n", "# savefig(\"fermi_hubbard_T_matrix_convergence.pdf\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Julia 1.9.4", "language": "julia", "name": "julia-1.9" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.9.4" } }, "nbformat": 4, "nbformat_minor": 2 }