{ "cells": [ { "cell_type": "markdown", "id": "ff8465b6", "metadata": {}, "source": [ "| [**Lattice QCD I**](https://homepages.uni-regensburg.de/~lec17310/teaching/wise2122/lqft.html) | - | [**Christoph Lehner**](https://go.ur.de/lehner) |\n", "|:-|--|-:|\n", "\n", "# Chapter 4: scalar field theory on a lattice" ] }, { "cell_type": "markdown", "id": "05018930", "metadata": {}, "source": [ "## Motivation\n", "- So far we have considered one-particle Hamiltonians in one dimension. More interesting systems occur, e.g., when considering possible theories of the fundamental interactions of nature. Experimentally, we observe inelastic processes which change one set of incoming to a different set of outgoing particles. Example: radioactive decay, pion to muon decays in the atmosphere due to cosmic rays, ...\n", "- We will show today that the Hilbert space of a quantum field theory is much better suited to describe such theories.\n", "- In the next chapter, we will then more systematically explore the space of interesting field theories by starting from a thorough discussion of symmetries.\n", "\n", "## Hamiltonian of a real scalar quantum field theory in one spatial dimension" ] }, { "cell_type": "markdown", "id": "9ece1f95", "metadata": {}, "source": [ "- Consider a real scalar quantum field theory with one spatial dimension with periodic boundary conditions on a spatial lattice\n", " with $n_s$ positions defined by the Lagrangian\n", " $$\n", " \\begin{align}\n", " L = \\sum_{x=0}^{n_s-1} \\frac12 \\phi(x,t) \\left[\\overleftarrow{\\partial}_t \\overrightarrow{\\partial}_t + \\Delta^2 - m^2 \\right] \\phi(x,t)\n", " \\end{align}\n", " $$\n", " with $x \\in \\{ 0, \\ldots, n_s - 1 \\}$, $t,m \\in \\mathbb{R}$, and\n", " $$\n", " \\begin{align}\n", " \\Delta^2 \\phi(x,t) \\equiv \\phi(x+1,t) + \\phi(x-1,t) - 2\\phi(x,t)\n", " \\end{align}\n", " $$\n", " and $\\phi(-1,t)=\\phi(n_s-1,t)$, $\\phi(n_s,t)=\\phi(0,t)$." ] }, { "cell_type": "markdown", "id": "724e1562", "metadata": {}, "source": [ "- The canonical momentum for the field is then given by the appropriate functional derivative\n", "$$\n", " \\begin{align}\n", " \\pi(x,t) = \\partial_{\\partial_t \\phi(x,t)} L = \\partial_t \\phi(x,t)\n", " \\end{align}\n", " $$\n", " and the Euler-Lagrange equation\n", " $$\n", " \\begin{align}\n", " \\partial_{\\phi(x,t)} L = \\partial_t \\pi(x,t)\n", " \\end{align}\n", " $$\n", " which is\n", " $$\n", " \\begin{align}\n", " 0 &=\\partial_t^2 \\phi(x,t) - \\partial_{\\phi(x,t)} \\sum_{x=0}^{n_s-1} \\frac12 \\phi(x,t) \\left[\\phi(x+1,t) + \\phi(x-1,t) - (2+m^2)\\phi(x,t) \\right] \\\\\n", " &=\\partial_t^2 \\phi(x,t) - \\left[\\phi(x+1,t) + \\phi(x-1,t) - (2+m^2)\\phi(x,t) \\right] \\\\\n", " &=(\\partial_t^2 - \\Delta^2 + m^2) \\phi(x,t) \\,.\n", " \\end{align}\n", "$$" ] }, { "cell_type": "markdown", "id": "7f672fee", "metadata": {}, "source": [ "- Next, we consider free-field solutions by first performing a discrete Fourier transform\n", " $$\n", " \\begin{align}\n", " \\phi(x,t) = \\frac1{\\sqrt{2\\pi n_s}} \\int dE \\sum_{k=0}^{n_s-1} e^{2\\pi i x k / n_s} e^{i t E} \\tilde\\phi(k,E)\n", " \\end{align}\n", " $$\n", " yielding\n", " $$\n", " \\begin{align}\n", " \\left(-E^2 + \\left(2\\sin\\left(\\frac{\\pi k}{n_s}\\right)\\right)^2 + m^2\\right) \\tilde\\phi(k,E) = 0 \\,.\n", " \\end{align}\n", " $$\n", " " ] }, { "cell_type": "markdown", "id": "f6a077d2", "metadata": {}, "source": [ "- Defining\n", "$$\n", " \\begin{align}\n", " w_k = \\sqrt{m^2 + \\left(2\\sin\\left(\\frac{\\pi k}{n_s}\\right)\\right)^2}\n", " \\end{align}\n", " $$\n", " we can then write the free field solution\n", " $$\n", " \\begin{align}\n", " f_k(x,t) \\equiv \\frac1{\\sqrt{2 w_k n_s}} e^{- i w_k t + i 2\\pi x k / n_s}\n", " \\end{align}\n", " $$\n", " with\n", " $$\n", " \\begin{align}\n", " (\\partial_t^2 - \\Delta^2 + m^2) f_k(x,t) = 0\n", " \\end{align}\n", " $$\n", " and\n", " $$\n", " \\begin{align}\n", " \\sum_{x=0}^{n_s-1} f^*_k(x,t) f_{k'}(x,t) &= \\frac1{2w_k} \\delta_{k,k'} \\,, \\\\\n", " \\sum_{x=0}^{n_s-1} f_k(x,t) f_{k'}(x,t) &= \\frac1{2w_k} \\delta_{k,-k'} e^{-2i w_k t} \\,.\n", " \\end{align}\n", "$$" ] }, { "cell_type": "markdown", "id": "54daf29f", "metadata": {}, "source": [ "- To quantize the theory, we first make the ansatz for the real field\n", "$$\n", " \\begin{align}\n", " \\phi(x,t) = \\sum_{k=0}^{n_s-1} \\left[ f_k(x,t) \\hat{a}_k + f_k^*(x,t) \\hat{a}_k^\\dagger \\right]\n", " \\end{align}\n", " $$\n", " and operators $\\hat{a}_k$ such that\n", " $$\n", " \\begin{align}\n", " \\pi(x,t) = -i\\sum_{k=0}^{n_s-1} w_k \\left[ f_k(x,t) \\hat{a}_k - f_k^*(x,t) \\hat{a}_k^\\dagger \\right] \\,.\n", " \\end{align}\n", " $$\n", " Then\n", " $$\n", " \\begin{align}\n", " \\hat{a}_k &= \\sum_{x=0}^{n_s-1} f^*_k(x,t) (w_k\\phi(x,t) + i \\pi(x,t)) \\,.\n", " \\end{align}\n", " $$\n", " We quantize the fields canonically, i.e., require\n", " $$\n", " \\begin{align}\n", " [ \\phi(x,t), \\pi(y,t) ] &= i \\delta_{x,y} \\,, &\n", " [ \\phi(x,t), \\phi(y,t) ] &= [ \\pi(x,t), \\pi(y,t) ] = 0\n", " \\end{align}\n", " $$\n", " yielding\n", " $$\n", " \\begin{align}\n", " [\\hat{a}_k,\\hat{a}_{k'} ] &= \\sum_{x,y=0}^{n_s-1} f^*_k(x,t) f^*_{k'}(y,t) [w_k\\phi(x,t) + i \\pi(x,t),w_{k'}\\phi(y,t) + i \\pi(y,t)] \\\\\n", " &= \\sum_{x,y=0}^{n_s-1} f^*_k(x,t) f^*_{k'}(y,t) i\\left[ w_{k'} [\\pi(x,t),\\phi(y,t)] + w_k [\\phi(x,t),\\pi(y,t)] \\right] \\\\\n", " &= \\sum_{x=0}^{n_s-1} f^*_k(x,t) f^*_{k'}(x,t) \\left[ w_{k'} - w_{k} \\right] = 0\n", " \\end{align}\n", " $$\n", " and also\n", " $$\n", " \\begin{align}\n", " [\\hat{a}_k,\\hat{a}^\\dagger_{k'} ]\n", " &= \\sum_{x,y=0}^{n_s-1} f^*_k(x,t) f_{k'}(y,t) i\\left[ w_{k'} [\\pi(x,t),\\phi(y,t)] - w_k [\\phi(x,t),\\pi(y,t)] \\right] \\\\\n", " &= \\sum_{x=0}^{n_s-1} f^*_k(x,t) f_{k'}(x,t) \\left[ w_{k'} + w_{k} \\right] = \\delta_{k,k'} \\,.\n", " \\end{align}\n", "$$" ] }, { "cell_type": "markdown", "id": "06701866", "metadata": {}, "source": [ "- Then Hamiltonian is given by a Legendre transformation of the Lagrangian\n", "$$\n", " \\begin{align}\n", " H &= -L + \\sum_{x=0}^{n_s-1} \\pi(x,t) \\partial_{\\pi(x,t)} L \\\\\n", " &= \\sum_{x=0}^{n_s-1} \\frac12 \\phi(x,t)\\left[ \\overleftarrow{\\partial}_t \\overrightarrow{\\partial}_t - \\Delta^2 +m^2 \\right] \\phi(x,t) \\notag\\\\\n", "&=\\sum_{x,k,k'=0}^{n_s-1} \\frac12 \\left[ f_k(x,t) \\hat{a}_k + f_k^*(x,t) \\hat{a}_k^\\dagger \\right]\\left[ \\overleftarrow{\\partial}_t \\overrightarrow{\\partial}_t - \\partial_t^2 \\right]\\left[ f_{k'}(x,t) \\hat{a}_{k'} + f_{k'}^*(x,t) \\hat{a}_{k'}^\\dagger \\right] \\\\\n", "&=\\sum_{x,k,k'=0}^{n_s-1} \\frac12 \\left[ f_k(x,t) \\hat{a}_k + f_k^*(x,t) \\hat{a}_k^\\dagger \\right] w_{k'}^2\\left[ f_{k'}(x,t) \\hat{a}_{k'} + f_{k'}^*(x,t) \\hat{a}_{k'}^\\dagger \\right] \\notag\\\\&\\quad- \\sum_{x,k,k'=0}^{n_s-1} \\frac12\\left[ f_k(x,t) \\hat{a}_k - f_k^*(x,t) \\hat{a}_k^\\dagger \\right] w_{k'} w_{k'}\\left[ f_{k'}(x,t) \\hat{a}_{k'} - f_{k'}^*(x,t) \\hat{a}_{k'}^\\dagger \\right] \\\\\n", "&=\n", "\\frac12 \\sum_{k=0}^{n_s-1} w_{k} \\left[ \\hat{a}_k \\hat{a}_{k}^\\dagger + \\hat{a}_{k}^\\dagger \\hat{a}_k \\right] \\\\\n", "&=\n", " \\sum_{k=0}^{n_s-1} w_{k} \\left[ \\frac12 + \\hat{a}_{k}^\\dagger \\hat{a}_k \\right] \\,.\n", "\\end{align}\n", " $$" ] }, { "cell_type": "markdown", "id": "a78ceccb", "metadata": {}, "source": [ "- This looks like a Harmonic oscillator for each mode number $k$. Note: we have a vacuum energy offset $E_0 = \\frac12 \\sum_{k=0}^{n_s-1} w_k$.\n", "\n", "- Beyond the vacuum energy, there is a finite contribution of energy $\\omega_k$ for each mode $k$\n", "multiplied by the occupation number operator\n", " \\begin{align}\n", " \\hat{n}_k \\equiv \\hat{a}_k^\\dagger \\hat{a}_k \\,.\n", " \\end{align}\n", "\n", "- We therefore have found a theory which can describe an aribtrary number of particles in mode $k$ that each contribute energy $\\omega_k$.\n", "\n", "- We can write for small momenta\n", "$$\n", "p_k = \\frac{2\\pi}{n_s} k\n", "$$\n", "the energies as\n", "$$\n", "\\omega_k^2 \\approx m^2 + p_k^2\n", "$$\n", "which suggests that we found a theory of non-interacting particles that each obey the relativistic energy-momentum relation. In the\n", "next chapter, we illustrate the origin of this phenomenon." ] }, { "cell_type": "markdown", "id": "dbf2c34e", "metadata": {}, "source": [ "## Global and local Metropolis update for scalar fields\n", "\n", "We will next discretize also time in $n_t$ steps with periodic boundary conditions, as we did for the one-dimensional Hamiltonian examples, and study the Euclidean action\n", "$$\n", "\\begin{align}\n", "S_E(\\phi) = \\frac12 \\sum_{x=0}^{n_s-1} \\sum_{t=0}^{n_t-1} \\phi(x,t) ( -\\Delta_t^2 - \\Delta_x^2 + m^2) \\phi(x,t)\n", "\\end{align}\n", "$$\n", "where we used partial integration to identify $\\overleftarrow{\\partial_t}\\overrightarrow{\\partial_t}=-\\partial_t^2$\n", "and\n", "$$\n", "\\begin{align}\n", "\\Delta_x^2 \\phi(x,t) &= \\phi(x+1,t) + \\phi(x-1,t) - 2\\phi(x,t) \\,, \\\\\n", "\\Delta_t^2 \\phi(x,t) &= \\phi(x,t+1) + \\phi(x,t+1) - 2\\phi(x,t) \\,.\n", "\\end{align}\n", "$$\n", "\n", "We demonstrate how to generate a Markov chain with probability distribution $p(\\phi) \\propto e^{-S_E(\\phi)}$\n", "using both global and local updates. For this, we will use for the first time the high-performance library GPT (https://github.com/lehner/gpt)." ] }, { "cell_type": "code", "execution_count": 22, "id": "6ed42a91", "metadata": {}, "outputs": [], "source": [ "import gpt as g\n", "import numpy as np\n", "from matplotlib import pyplot as plt" ] }, { "cell_type": "code", "execution_count": 3, "id": "8cd3acdd", "metadata": {}, "outputs": [], "source": [ "L = [32,64]\n", "grid = g.grid(L, g.double)\n", "grid_eo = g.grid(L, g.double, g.redblack)" ] }, { "cell_type": "code", "execution_count": 4, "id": "6c536119", "metadata": {}, "outputs": [], "source": [ "field = g.real(grid)\n", "field[:] = 0" ] }, { "cell_type": "code", "execution_count": 5, "id": "09061423", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "GPT : 0.806299 s : Initializing gpt.random(seed13,vectorized_ranlux24_389_64) took 0.000421047 s\n" ] } ], "source": [ "rng = g.random(\"seed13\")" ] }, { "cell_type": "code", "execution_count": 6, "id": "86ad1305", "metadata": {}, "outputs": [ { "data": { "text/plain": [ "lattice(ot_real_additive_group,double)" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "rng.element(field)" ] }, { "cell_type": "code", "execution_count": 7, "id": "95149b73", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(327.52809672707235+0j)\n" ] } ], "source": [ "def S(phi, m0):\n", " action = 0.5*g.norm2(phi)*m0**2\n", " for mu in range(2):\n", " action -= 0.5 * g.inner_product(phi, g.cshift(phi,mu,1) + g.cshift(phi,mu,-1) - 2.*phi)\n", " return action\n", "\n", "print(S(field, 0.1))" ] }, { "cell_type": "code", "execution_count": 8, "id": "1ca76a17", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "(327.52809672707235+0j)\n", "(-0.5412363742706248+0j)\n", "(-0.5412363742706198+0j)\n" ] } ], "source": [ "def S_field(phi, m0):\n", " # gather in afld[x] everything that changes if phi[x] changes\n", " afld = g(0.5 * g.adj(phi)*phi*(m0**2 + 4.0))\n", " for mu in range(2):\n", " afld -= g.adj(phi) * (g.cshift(phi,mu,1) + g.cshift(phi,mu,-1))\n", " return afld\n", "\n", "print(S(field,0.1))\n", "\n", "# test change in single point\n", "field[12,17] = 0.5\n", "S0 = S(field,0.1)\n", "Sf0 = S_field(field,0.1)[12,17]\n", "field[12,17] = 0.3\n", "S1 = S(field,0.1)\n", "Sf1 = S_field(field,0.1)[12,17]\n", "print(S1 - S0)\n", "print(Sf1 - Sf0)" ] }, { "cell_type": "code", "execution_count": 9, "id": "85953305", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0 Acceptance rate: 1.0\n", "1000 Acceptance rate: 0.5354645354645354\n", "2000 Acceptance rate: 0.5417291354322838\n", "3000 Acceptance rate: 0.5428190603132289\n", "4000 Acceptance rate: 0.5546113471632091\n", "5000 Acceptance rate: 0.5530893821235753\n", "6000 Acceptance rate: 0.5504082652891185\n", "7000 Acceptance rate: 0.5513498071704043\n", "8000 Acceptance rate: 0.5504311961004874\n", "9000 Acceptance rate: 0.551716475947117\n", "10000 Acceptance rate: 0.5528447155284472\n", "11000 Acceptance rate: 0.5544041450777202\n", "12000 Acceptance rate: 0.5569535872010666\n", "13000 Acceptance rate: 0.5564187370202293\n", "14000 Acceptance rate: 0.556460252839083\n", "15000 Acceptance rate: 0.5568295446970202\n", "16000 Acceptance rate: 0.5552777951378038\n", "17000 Acceptance rate: 0.5561437562496324\n", "18000 Acceptance rate: 0.5571357146825177\n", "19000 Acceptance rate: 0.558549550023683\n", "20000 Acceptance rate: 0.559622018899055\n", "21000 Acceptance rate: 0.5599733346031142\n", "22000 Acceptance rate: 0.5607472387618745\n", "23000 Acceptance rate: 0.5600626059736533\n", "24000 Acceptance rate: 0.5603099870838715\n" ] } ], "source": [ "def metropolis_global(phi, action, step):\n", " global rng\n", " tmp = g.real(grid)\n", " phi_prime = g(phi + rng.element(tmp) * step)\n", " S0 = action(phi)\n", " S1 = action(phi_prime)\n", " r = np.exp(S0 - S1)\n", " l = rng.uniform_real(min=0.0, max=1.0)\n", " if l < r:\n", " # accept\n", " phi @= phi_prime\n", " return True\n", " return False\n", " \n", "field[:] = 0.0\n", "total = 0\n", "accepted = 0\n", "action_list_global = []\n", "for i in range(25000):\n", " if metropolis_global(field, lambda field: S(field, 1.0),0.04):\n", " accepted+=1\n", " total += 1\n", " action_list_global.append(S(field, 1.0).real)\n", " if i % 1000 == 0:\n", " print(i,\"Acceptance rate:\",accepted/total)" ] }, { "cell_type": "code", "execution_count": 10, "id": "886e26c7", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots()\n", "\n", "ax.plot(range(len(action_list_global)), action_list_global, marker='+', ls='', c='blue', label=\"S using global Metropolis update\")\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 11, "id": "826596e8", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Acceptance rate of local update: 0.654296875\n", "Acceptance rate of local update: 0.66748046875\n", "Acceptance rate of local update: 0.66943359375\n", "Acceptance rate of local update: 0.67138671875\n", "Acceptance rate of local update: 0.6787109375\n", "Acceptance rate of local update: 0.67333984375\n", "Acceptance rate of local update: 0.6689453125\n", "Acceptance rate of local update: 0.66162109375\n", "Acceptance rate of local update: 0.6552734375\n", "Acceptance rate of local update: 0.6611328125\n" ] } ], "source": [ "def metropolis_step(phi, action, step):\n", " global rng\n", " # update even fields then odd fiels, in this way we can update\n", " # half of the sites with individual accept/reject at each step\n", " tmp = g.real(grid)\n", " tmp_cb = g.real(grid_eo)\n", " r_cb = g.real(grid_eo)\n", " mask_accept = g.real(grid)\n", " \n", " total_accept = 0\n", " total_step = 0\n", " \n", " for cb in [g.even, g.odd]:\n", " \n", " # p(x) = const * exp(-S(x)) -> p(x') / p(x) = exp(S(x) - S(x'))\n", " S_field_0 = action(phi) # wasteful to compute every time, for now fine\n", " \n", " # create phi_prime which only has shifts on cb sites\n", " tmp[:] = 0\n", " tmp_cb.checkerboard(cb)\n", " rng.element(tmp_cb)\n", " g.set_checkerboard(tmp, tmp_cb)\n", " phi_prime = g(phi + step * tmp)\n", " \n", " # difference of action (per site)\n", " sdiff = g(S_field_0 - action(phi_prime))\n", " g.pick_checkerboard(cb, tmp_cb, sdiff)\n", " pdiff = g(g.component.exp(tmp_cb))\n", " \n", " # on same sites draw independent numbers between 0 and 1\n", " rng.uniform_real(r_cb, min = 0.0, max = 1.0)\n", " \n", " # and locally do an accept/reject\n", " mask_accept_cb = g(pdiff > r_cb)\n", " mask_accept[:] = 0.0\n", " mask_accept_cb.checkerboard(cb) # in future versions of gpt, this will be inferred from >\n", " g.set_checkerboard(mask_accept, mask_accept_cb)\n", " \n", " # statistics\n", " total_accept += g.norm2(mask_accept) \n", " total_step += mask_accept_cb.grid.gsites\n", " \n", " # locally keep accepted changes\n", " phi @= g.where(mask_accept, phi_prime, phi)\n", "\n", " return total_accept / total_step\n", " \n", "field[:] = 0.0\n", "action_list = []\n", "for s in range(100):\n", " acceptance_rate = metropolis_step(field, lambda field: S_field(field, 1.0), 1.6)\n", " action_list.append(S(field, 1.0).real)\n", " if s % 10 == 0:\n", " print(\"Acceptance rate of local update:\",acceptance_rate)\n", "\n", " " ] }, { "cell_type": "code", "execution_count": null, "id": "a2b4cbb9", "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": 12, "id": "41468e7a", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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i5RYRCUM3URERESCqoV9RUegSiIgURDRDX+34IhJRGYe+mfUzszUJjw/M7HtmdrKZPWlmm4LnTsHwZma3mlm1ma01s2HZWwwREUlFxqHv7q+6+xB3HwIMB/YDDwNzgWXu3hdYFrwGuBjoGzxmAXeEKLeIiGQgW807FwKvu/s2YBKwMOi/EJgcdE8C7vG4VUBHMzs1S/MXEZEUZCv0pwH3B91d3X1n0P0W0DXo7gZsTxinJujXgJnNMrMqM6uqra3NUvFERASyEPpmdgzweWBx4/c8/ieAtE6Id/f57h5z91hZWVnY4omISIJs1PQvBl5y97eD12/XNdsEz7uC/juAHgnjdQ/6iYhInmQj9K/gcNMOwFJgetA9HXg0of/XgrN4RgLvJzQDiYhIHrQNM7KZdQDGA99K6H0DsMjMZgLbgKlB/8eBS4Bq4mf6zAgzbxERSV+o0Hf3D4HOjfrtJn42T+NhHZgTZn4iIhJONP+RKyISUQp9EZEIUeiLiESIQl9EJEIU+iIiEaLQFxGJEIW+iEiEKPRFRCJEoS8iEiEKfRGRCFHoi4hEiEJfRCRCohP6lZWFLoGISMFFJ/TnzSt0CURECi46oS8iIiUe+pWVYBZ/wOFuNfWISERZ/N4mxSkWi3lVVVV2JmYGRbysIiLZYmar3T3W1HulXdMXEZEGohP6FRWFLoGISMGFCn0z62hmS8xso5ltMLNRZnaymT1pZpuC507BsGZmt5pZtZmtNbNh2VmEFKkdX0QkdE3/V8Cf3b0/MBjYAMwFlrl7X2BZ8BrgYqBv8JgF3BFy3iIikqaMQ9/MTgLOA+4EcPd/ufseYBKwMBhsITA56J4E3ONxq4COZnZqpvMXEZH0hanp9wFqgbvN7B9m9jsz6wB0dfedwTBvAV2D7m7A9oTxa4J+DZjZLDOrMrOq2traEMUTEZHGwoR+W2AYcIe7DwU+5HBTDgAePx80rfMk3X2+u8fcPVZWVhaieCIi0liY0K8Batz9heD1EuI/Am/XNdsEz7uC93cAPRLG7x70ExGRPMk49N39LWC7mfULel0IrAeWAtODftOBR4PupcDXgrN4RgLvJzQDiYhIHrQNOf53gT+Y2THAZmAG8R+SRWY2E9gGTA2GfRy4BKgG9gfDiohIHoUKfXdfAzT1V98LmxjWgTlh5iciIuFE5x+5IiKi0BcRiRKFvohIhCj0RUQiRKEvIhIhCn0RkQhR6IuIRIhCX0QkQhT6IiIRotAXEYkQhb6ISIQo9EVEIkShLyISIQp9EZEIUeiLiESIQl9EJEIU+iIiEaLQFxGJEIW+iEiEKPRFRCIkVOib2VYz+6eZrTGzqqDfyWb2pJltCp47Bf3NzG41s2ozW2tmw7KxACIikrps1PTPd/ch7h4LXs8Flrl7X2BZ8BrgYqBv8JgF3JGFeYuISBpy0bwzCVgYdC8EJif0v8fjVgEdzezUHMxfRESSCBv6DjxhZqvNbFbQr6u77wy63wK6Bt3dgO0J49YE/Rows1lmVmVmVbW1tSGLJyIiidqGHP9cd99hZv8GPGlmGxPfdHc3M09ngu4+H5gPEIvF0hpXRESaF6qm7+47guddwMPACODtumab4HlXMPgOoEfC6N2DfiIikicZh76ZdTCzE+q6gc8B64ClwPRgsOnAo0H3UuBrwVk8I4H3E5qBREQkD8I073QFHjazuunc5+5/NrMXgUVmNhPYBkwNhn8cuASoBvYDM0LMW0REMpBx6Lv7ZmBwE/13Axc20d+BOZnOT0REwtM/ckVEIkShLyISIQp9EZEIUeiLiESIQl9EJEIU+iIiEaLQFxGJkLDX3hEpuE8++YSamho++uijQhdFJK/atWtH9+7dOfroo1MeR6EvrV5NTQ0nnHACvXv3JviHuEjJc3d2795NTU0Nffr0SXk8Ne9Iq/fRRx/RuXNnBb5EipnRuXPntPdwFfpSEhT4EkWZfO4V+iIiEaLQl+iqrMzapH76058ycOBAysvLGTJkCC+88ELoaX72s5/NQslgxYoVXHrppVmZVp1x48ZRVVXVZP+ePXsSv75i3OTJkzn++OObnd6ePXu4/fbbs1rGZOrK8uabb3L55ZfnZZ6QfJ0luuWWW9i/f39Oy6HQl+iaNy8rk/nb3/7GY489xksvvcTatWt56qmn6NGjR8sjtuCvf/1rFkqXfx07duT5558H4mG+c2fLt81oLvQPHjyY1fLVOe2001iyZElOpp0phb5IK7Bz5066dOnCscceC0CXLl047bTTjhgusab3zjvv0Lt3bwBeeeUVRowYwZAhQygvL2fTpk3A4RrpihUrGDduHJdffjn9+/fnK1/5Sn1N+vHHH6d///4MHz6ca665psUa/bvvvsvkyZMpLy9n5MiRrF27FoB9+/YxY8YMzj77bMrLy/njH/8IwOzZs4nFYgwcOJCKioqU1se0adN44IEHAHjooYeYMmVKg/d//vOfc84551BeXl4/zblz5/L6668zZMgQfvCDH7BixQrGjBnD5z//ec466yw++uij+vINHTqU5cuXA7BgwQImTZrEuHHj6Nu3L/MSfshvvvlmBg0axKBBg7jllluOKOfWrVsZNGhQs9sgUeLeypIlS7jqqqsAuOqqq/j2t79NLBbjzDPP5LHHHgPgwIEDTJs2jQEDBnDZZZdx4MCB+vGbWq+33norb775Jueffz7nn38+AE888QSjRo1i2LBhfPGLX2Tfvn0pbYNmuXvRPoYPH+4iLVm/fn3qA1dUuMORj4qKjOe/d+9eHzx4sPft29dnz57tK1asaHK4sWPH+osvvuju7rW1td6rVy93d7/66qv93nvvdXf3jz/+2Pfv3+/u7h06dHB39+XLl/uJJ57o27dv908//dRHjhzpzz33nB84cMC7d+/umzdvdnf3adOm+cSJE4+Y7/Lly+v7X3311V5ZWenu7suWLfPBgwe7u/sPf/hDv/baa+vHeffdd93dfffu3e7ufvDgQR87dqy//PLLRyxL42VctWqVn3322X7w4EEfP368b9mypX5Z/vKXv/g3v/lNP3TokH/66ac+ceJEf+aZZ3zLli0+cODABmU+7rjj6pftpptu8hkzZri7+4YNG7xHjx5+4MABv/vuu/2UU07xd955x/fv3+8DBw70F1980auqqnzQoEG+b98+37t3r5911ln+0ksvNVivifNMtg0S1Y3n7r548WKfPn26u7tPnz7dL7roIv/000/9tdde827duvmBAwf8F7/4RX2ZX375ZW/Tpk39Oku2Xnv16uW1tbXuHv+MjBkzxvft2+fu7jfccIPPmzfviHI19fkHqjxJrqqmL9FSWXk46uFwd4j2/eOPP57Vq1czf/58ysrK+NKXvsSCBQtSHn/UqFH87Gc/48Ybb2Tbtm20b9/+iGFGjBhB9+7dOeqooxgyZAhbt25l48aNnH766fXnaF9xxRUtzmvlypV89atfBeCCCy5g9+7dfPDBBzz11FPMmXP4HkedOnUCYNGiRQwbNoyhQ4fyyiuvsH79+hbn0aZNG84991weeOABDhw4UL9HA/Ga6xNPPMHQoUMZNmwYGzdubLJWXbfMdcu2cuVKrrzySgD69+9Pr169eO211wAYP348nTt3pn379kyZMoWVK1eycuVKLrvsMjp06MDxxx/PlClTeO6555KWOZVt0JypU6dy1FFH0bdvX04//XQ2btzIs88+W1/m8vJyysvL64dPZb2uWrWK9evXM3r0aIYMGcLChQvZtm1bWuVqSmmHfhYP1Ik0p02bNowbN4558+bx61//ur55JFHbtm05dOgQQINzq7/85S+zdOlS2rdvzyWXXMLTTz99xLh1TUd188pVO3eiLVu2cNNNN7Fs2TLWrl3LxIkTUz4nfNq0aVxzzTVMnTq1QX935/rrr2fNmjWsWbOG6upqZs6c2eQ0OnTokNK8Gp+2mMlpjKlsg8TpNl4P6ZQh1fXq7owfP75+Xa1fv54777wz3UU7QmmHfpYO1EmJSrGNuiWvvvpqg9rqmjVr6NWr1xHD9e7dm9WrVwM0OIC4efNmTj/9dK655homTZpU387ekn79+rF582a2bt0KwIMPPtjiOGPGjOEPf/gDED9W0KVLF0488UTGjx/PbbfdVj/ce++9xwcffECHDh046aSTePvtt/nTn/6UUrnq5nP99dcfsfdx0UUXcdddd9W3Te/YsYNdu3ZxwgknsHfv3pTK/dprr/HGG2/Qr18/AJ588kneffddDhw4wCOPPMLo0aMZM2YMjzzyCPv37+fDDz/k4YcfZsyYMUmnn8o26Nq1Kxs2bODQoUM8/PDDDd5bvHgxhw4d4vXXX2fz5s3069eP8847j/vuuw+AdevW1U+zufWauB5GjhzJ888/T3V1NQAffvhh/d5NGKEvw2BmbYAqYIe7X2pmfYAHgM7AauCr7v4vMzsWuAcYDuwGvuTuW8POXyRjWdoT3LdvH9/97nfZs2cPbdu25YwzzmD+/PlHDHfdddcxdepU5s+fz8SJE+v7L1q0iN///vccffTRnHLKKfzoRz9Kab7t27fn9ttvZ8KECXTo0IFzzjmnxXEqKyv5+te/Tnl5OccddxwLFy4E4Mc//jFz5sxh0KBBtGnThoqKCqZMmcLQoUPp378/PXr0YPTo0SmukXhN97rrrjui/+c+9zk2bNjAqFGjgHjT2L333stnPvMZRo8ezaBBg7j44osbrB+A73znO8yePZuzzz6btm3bsmDBgvq9nxEjRvCFL3yBmpoarrzySmKxGBA/wDpixAgAvvGNbzB06NCk5U1lG9xwww1ceumllJWVEYvFGhxU7dmzJyNGjOCDDz7gN7/5De3atWP27NnMmDGDAQMGMGDAAIYPHw7A4MGDk67XWbNmMWHCBE477TSWL1/OggULuOKKK/j4448B+MlPfsKZZ57Z8gZoTrLG/lQfwPeB+4DHgteLgGlB92+A2UH3d4DfBN3TgAdbmnZGB3JzcKBOiltaB3JLzN69e93d/dChQz579my/+eabC1yi/Lr77rt9zpw5BS3D9OnTffHixQWbf14P5JpZd2Ai8LvgtQEXAHX7rguByUH3pOA1wfsXWi7+O5+DA3Uixeq3v/0tQ4YMYeDAgbz//vt861vfKnSRpMiZJ/xzLu2RzZYA/wWcAFwHXAWscvczgvd7AH9y90Fmtg6Y4O41wXuvA//L3d9pNM1ZwCyAnj17Dg91tNrscPhLydqwYQMDBgwodDFECqKpz7+ZrXb3WFPDZ1zTN7NLgV3uvjrTaTTF3ee7e8zdY2VlZeEmlqUDdVL8wlReRFqrTD73YZp3RgOfN7OtxA/cXgD8CuhoZnUHiLsDO4LuHUAPgOD9k4gf0M0dNelEQrt27di9e7eCXyLFg+vpt2vXLq3xMj57x92vB64HMLNxwHXu/hUzWwxcTvyHYDrwaDDK0uD134L3n3Z9SyULunfvTk1NDbW1tYUuikhe1d05Kx25uHPWfwIPmNlPgH8Adf8muBP4vZlVA+8SP4NHJLSjjz46rTsHiURZVkLf3VcAK4LuzcCIJob5CPhiNuYnIiKZKe1/5IqISAMKfRGRCAl1nn6umVktEOaycl2Ad1ocqrREcZkhmssdxWWGaC53usvcy92bPOe9qEM/LDOrSvYHhVIVxWWGaC53FJcZornc2VxmNe+IiESIQl9EJEJKPfSPvL5t6YviMkM0lzuKywzRXO6sLXNJt+mLiEhDpV7TFxGRBAp9EZEIKcnQN7MJZvaqmVWb2dxClydXzKyHmS03s/Vm9oqZXRv0P9nMnjSzTcFzp0KXNdvMrI2Z/cPMHgte9zGzF4Jt/qCZHVPoMmabmXU0syVmttHMNpjZqFLf1mb2H8Fne52Z3W9m7UpxW5vZXWa2K7jvSF2/Jretxd0aLP9aMxuWzrxKLvSDe/beBlwMnAVcYWZnFbZUOXMQ+N/ufhYwEpgTLOtcYJm79wWWBa9LzbXAhoTXNwK/DG7g8x4wsyClyq1fAX929/7AYOLLX7Lb2sy6AdcAMXcfBLQhfqHGUtzWC4AJjfol27YXA32DxyzgjnRmVHKhT/xib9Xuvtnd/0X8Es+TClymnHD3ne7+UtC9l3gIdKPhrSkTb1lZEtK8TWdJMLOTgPMIrlrr7v9y9z2U+LYmflHI9sE9OI4DdlKC29rdnyV+9eFEybbtJOCe4Ha4q4jfw+TUVOdViqHfDdie8Lom6FfSzKw3MBR4Aejq7juDt94CuhaqXDlyC/BD4FDwujOwx90PBq9LcZv3AWqBu4Nmrd+ZWQdKeFu7+w7gJuAN4mH/PrCa0t/WdZJt21AZV4qhHzlmdjzwR+B77v5B4nvBjWpK5rzcXN2msxVoCwwD7nD3ocCHNGrKKcFt3Yl4rbYPcBrQgSObQCIhm9u2FEO//raMgcRbNpYcMzuaeOD/wd0fCnq/Xbe7FzzvKlT5ciDd23SWihqgxt1fCF4vIf4jUMrb+t+BLe5e6+6fAA8R3/6lvq3rJNu2oTKuFEP/RaBvcIT/GOIHfpYWuEw5EbRl3wlscPebE96quzUlNLxlZavn7te7e3d370182z7t7l8BlhO/DSeU2DIDuPtbwHYz6xf0uhBYTwlva+LNOiPN7Ljgs163zCW9rRMk27ZLga8FZ/GMBN5PaAZqmbuX3AO4BHgNeB34P4UuTw6X81ziu3xrgTXB4xLibdzLgE3AU8DJhS5rjpZ/HPBY0H068HegGlgMHFvo8uVgeYcAVcH2fgToVOrbGpgHbATWAb8Hji3FbQ3cT/y4xSfE9+pmJtu2gBE/Q/F14J/Ez25KeV66DIOISISUYvOOiIgkodAXEYkQhb6ISIQo9EVEIkShLyISIQp9EZEIUeiLiETI/weoHfoYiKo5PAAAAABJRU5ErkJggg==\n", "text/plain": [ "
" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "fig, ax = plt.subplots()\n", "\n", "ax.plot(range(len(action_list_global)), action_list_global, marker='+', ls='', c='blue', label=\"S using global Metropolis update\")\n", "ax.plot(range(len(action_list)), action_list, marker='+', ls='', c='red', label=\"S using local Metropolis update\")\n", "\n", "plt.legend()\n", "plt.show()\n", "\n", "fig, ax = plt.subplots()\n", "\n", "ax.plot(range(len(action_list)), action_list, marker='+', ls='', c='red', label=\"S using local Metropolis update\")\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "803d5da4", "metadata": {}, "source": [ "## Momentum eigenstates and two-point correlators\n", "\n", "Since the Hamiltonian for non-interacting real scalar fields in one spatial dimension is\n", "$$\n", " \\begin{align}\n", " H &= \\sum_{k=0}^{n_s-1} w_{k} \\left[ \\frac12 + \\hat{n}_k \\right]\n", "\\end{align}\n", "$$\n", "with occupation number operator\n", "$$\n", " \\begin{align}\n", " \\hat{n}_k = \\hat{a}_{k}^\\dagger \\hat{a}_k \\,,\n", " \\end{align}\n", "$$\n", "we know that the eigenstates can be classified with quantum numbers $n_1,\\ldots,n_{n_s-1}$\n", "with\n", "$$\n", "\\begin{align}\n", " \\hat{n}_k \\vert n_1,\\ldots,n_{n_s-1} \\rangle = n_k \\vert n_1,\\ldots,n_{n_s-1} \\rangle \\,.\n", "\\end{align}\n", "$$\n", "\n", "We call the special state with lowest energy the vaccum state\n", "$$\n", "\\begin{align}\n", " \\vert 0 \\rangle = \\vert 0, \\ldots, 0 \\rangle\n", "\\end{align}\n", "$$\n", "with vacuum energy $E_0$ already defined above. We can then study the operator\n", "$$\n", "\\begin{align}\n", "\\phi^\\dagger_k &=\\frac1{\\sqrt{2 w_k n_s}} \\sum_{x=0}^{n_s-1} e^{i p_k x} \\phi(x,0) \\\\\n", "&= \\sum_{x=0}^{n_s-1} f_k(x,0) \\phi(x,0) \\\\\n", "&=\\frac1{2\\omega_k} (a^\\dagger_k + a_{-k})\n", "\\end{align}\n", "$$\n", "such that\n", "$$\n", "\\begin{align}\n", "\\phi^\\dagger_k \\vert 0 \\rangle = \\frac1{2\\omega_k} \\vert 0,\\ldots,0,n_k=1,0,\\ldots 0 \\rangle\n", "\\end{align}\n", "$$\n", "where we used $a_k \\vert 0 \\rangle = 0$.\n", "\n", "We can therefore study\n", "$$\n", "\\begin{align}\n", "C(t) &= \\langle \\phi_k(t) \\phi^\\dagger_k(0) \\rangle \\\\\n", "&=\\frac1Z \\sum_{n,m} \\langle n \\vert \\phi_k \\vert m \\rangle\n", "\\langle m \\vert \\phi_k^\\dagger \\vert n \\rangle e^{-t E_m - (T-t)E_n} \\\\\n", "&\\approx \\sum_{m} \\langle 0 \\vert \\phi_k \\vert m \\rangle\n", "\\langle m \\vert \\phi_k^\\dagger \\vert 0 \\rangle e^{-t (E_m - E_0)} \\\\\n", "&\\approx \\sum_{m} \\langle 0 \\vert \\phi_k \\vert m \\rangle\n", "\\langle m \\vert \\phi_k^\\dagger \\vert 0 \\rangle e^{-t (E_m - E_0)} \\\\\n", "&\\approx \\vert \\langle \\phi_k \\vert \\phi_k \\rangle \\vert^2\n", " e^{-t (E_{n_k=1} - E_0)} \\,,\n", "\\end{align}\n", "$$\n", "where we considered the limits $1 \\ll t \\ll T$.\n" ] }, { "cell_type": "code", "execution_count": 13, "id": "a14a8bc4", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "GPT : 62.218938 s : Initializing gpt.random(seed13,vectorized_ranlux24_389_64) took 0.000290871 s\n", "Acceptance rate of local update: 0.60791015625\n", "Acceptance rate of local update: 0.62109375\n", "Acceptance rate of local update: 0.64208984375\n", "Acceptance rate of local update: 0.62255859375\n", "Acceptance rate of local update: 0.64453125\n", "Acceptance rate of local update: 0.6376953125\n", "Acceptance rate of local update: 0.63525390625\n", "Acceptance rate of local update: 0.62744140625\n", "Acceptance rate of local update: 0.6220703125\n", "Acceptance rate of local update: 0.6357421875\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "rng = g.random(\"seed13\")\n", "field[:] = 0.0\n", "action_list = []\n", "m0 = 0.2\n", "\n", "# first thermalize theory for new m0\n", "for s in range(100):\n", " acceptance_rate = metropolis_step(field, lambda field: S_field(field, m0), 2.0)\n", " action_list.append(S(field, m0).real)\n", " if s % 10 == 0:\n", " print(\"Acceptance rate of local update:\",acceptance_rate)\n", " \n", "fig, ax = plt.subplots()\n", "\n", "ax.plot(range(len(action_list)), action_list, marker='+', ls='', c='red', label=\"S using local Metropolis update\")\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "code", "execution_count": 14, "id": "7ad3444e", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "0\n", "1000\n", "2000\n", "3000\n", "4000\n" ] } ], "source": [ "configurations = []\n", "\n", "for s in range(5000):\n", " for sub_steps in range(5):\n", " acceptance_rate = metropolis_step(field, lambda field: S_field(field, m0), 2.0)\n", " configurations.append(g.copy(field))\n", " if s % 1000 == 0:\n", " print(s)" ] }, { "cell_type": "code", "execution_count": 15, "id": "d80e4be9", "metadata": {}, "outputs": [], "source": [ "def correlate(field):\n", " phit = g.slice(field,1)\n", " Nt=len(phit)\n", " return np.array([ sum([ phit[s].conjugate()*phit[(s+t)%Nt] for s in range(Nt) ])/Nt for t in range(Nt) ])\n", "\n", "def two_point_for_momentum(p):\n", " mom_phase = g.exp_ixp(np.array([p,0]))\n", " correlators = [ correlate(g(mom_phase*conf)) for conf in configurations ]\n", " return correlators\n", "\n", "c0 = two_point_for_momentum(0)\n", "c1 = two_point_for_momentum(2.*np.pi/L[0])" ] }, { "cell_type": "code", "execution_count": 16, "id": "468caaac", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def jackknife_variance_estimator(f, x):\n", " N = len(x)\n", " X = sum(x) / N\n", " mean = f(X)\n", " return sum([(f((N*X - x[j])/(N-1)) - mean)**2 for j in range(N)])*(N-1)/N\n", "\n", "def bin_data(res, bin_size):\n", " assert len(res) % bin_size == 0\n", " n = len(res) // bin_size\n", " return [ sum(res[bin_size*i:bin_size*(i+1)])/bin_size for i in range(n) ]\n", "\n", "def time_parity_average(C):\n", " return [0.5*np.roll(x[::-1],1) + 0.5*x for x in C]\n", "\n", "bc0 = time_parity_average(bin_data(c0, 100))\n", "bc1 = time_parity_average(bin_data(c1, 100))\n", "\n", "def emp(C):\n", " return np.array([np.log(C[t].real/C[t+1].real) for t in range(10)])\n", "\n", "c0_mean = np.mean(bc0, axis=0)\n", "emp0_mean = emp(c0_mean)\n", "emp0_err = jackknife_variance_estimator(emp, bc0)**0.5\n", "\n", "c1_mean = np.mean(bc1, axis=0)\n", "emp1_mean = emp(c1_mean)\n", "emp1_err = jackknife_variance_estimator(emp, bc1)**0.5\n", "\n", "fig, ax = plt.subplots()\n", "\n", "plt.ylim([0,0.4])\n", "ax.errorbar(range(len(emp0_mean)), emp0_mean, emp0_err, marker='+', ls='', c='red', label=\"E(p=0) - E_0\")\n", "ax.errorbar(range(len(emp1_mean)), emp1_mean, emp1_err, marker='+', ls='', c='blue', label=\"E(p=2pi/L) - E_0\")\n", "\n", "plt.axhline(y=m0, color='red', linestyle='-')\n", "plt.axhline(y=(0.2**2. + (2.*np.sin(np.pi/L[0]))**2.)**0.5, color='blue', linestyle='-')\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "46914932", "metadata": {}, "source": [ "We can also create two particles, say, in the $k=0$ state by studying\n", "$$\n", "\\begin{align}\n", "C(t) &= \\langle (\\phi_k(t))^2 (\\phi^\\dagger_k(0))^2 \\rangle - \\langle (\\phi_k(t))^2 \\rangle^2 \\,.\n", "\\end{align}\n", "$$\n", "\n", "***Homework***: Show how one can extract a state with occupation number two from this correlator." ] }, { "cell_type": "code", "execution_count": 17, "id": "8582413b", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def correlate2(field):\n", " phit = g.slice(field,1)\n", " Nt=len(phit)\n", " return np.array([ sum([ phit[s].conjugate()**2*phit[(s+t)%Nt]**2 for s in range(Nt) ])/Nt for t in range(Nt) ])\n", "\n", "def two_point_for_momentum2(p):\n", " mom_phase = g.exp_ixp(np.array([p,0]))\n", " correlators = [ correlate2(g(mom_phase*conf)) for conf in configurations ]\n", " return correlators\n", "\n", "c2 = two_point_for_momentum2(0)\n", "\n", "bc2 = time_parity_average(bin_data(c2, 100))\n", "\n", "# variance of this is negligible, so treat it as a constant\n", "c_const = np.ones(len(c0_mean))*c0_mean[0]**2. \n", "\n", "c2_mean = np.mean(bc2, axis=0)\n", "emp2_mean = emp(c2_mean - c_const)\n", "emp2_err = jackknife_variance_estimator(lambda c2: emp(c2 - c_const), bc2)**0.5\n", "\n", "fig, ax = plt.subplots()\n", "\n", "plt.ylim([0,0.5])\n", "ax.errorbar(range(len(emp0_mean)), emp0_mean, emp0_err, marker='+', ls='', c='red', label=\"E(n=1,p=0) - E_0\")\n", "ax.errorbar(range(len(emp1_mean)), emp1_mean, emp1_err, marker='+', ls='', c='blue', label=\"E(n=1,p=2pi/L) - E_0\")\n", "ax.errorbar(range(len(emp2_mean)), emp2_mean, emp2_err, marker='+', ls='', c='green', label=\"E(n=2,p=0) - E_0\")\n", "\n", "plt.axhline(y=m0, color='red', linestyle='-')\n", "plt.axhline(y=(0.2**2. + (2.*np.sin(np.pi/32))**2.)**0.5, color='blue', linestyle='-')\n", "plt.axhline(y=2*0.2, color='green', linestyle='-')\n", "\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "aaafdbe7", "metadata": {}, "source": [ "Next, we can add a small interaction term and observe how the states shift. Note that even the one-particle state can shift and $m_0$ is in general no longer its energy!" ] }, { "cell_type": "code", "execution_count": 19, "id": "1cba0ce4", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "GPT : 442.801170 s : Initializing gpt.random(seed13,vectorized_ranlux24_389_64) took 0.000324965 s\n", "(-0.543908374270643+0j)\n", "(-0.5439083742706198+0j)\n", "Acceptance rate of local update: 0.4384765625\n", "Acceptance rate of local update: 0.48779296875\n", "Acceptance rate of local update: 0.49755859375\n", "Acceptance rate of local update: 0.4951171875\n", "Acceptance rate of local update: 0.49609375\n", "Acceptance rate of local update: 0.5126953125\n", "Acceptance rate of local update: 0.49462890625\n", "Acceptance rate of local update: 0.49365234375\n", "Acceptance rate of local update: 0.494140625\n", "Acceptance rate of local update: 0.4970703125\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" }, { "name": "stdout", "output_type": "stream", "text": [ "0\n", "1000\n", "2000\n", "3000\n", "4000\n" ] } ], "source": [ "rng = g.random(\"seed13\")\n", "action_list = []\n", "m0 = 0.2\n", "lam = 0.005\n", "\n", "def S_ia(phi, m0, lam):\n", " action = 0.5*g.norm2(phi)*m0**2 + lam * g.sum(g.adj(phi)*phi*g.adj(phi)*phi)\n", " for mu in range(2):\n", " action -= 0.5 * g.inner_product(phi, g.cshift(phi,mu,1) + g.cshift(phi,mu,-1) - 2.*phi)\n", " return action\n", "\n", "def S_field_ia(phi, m0, lam):\n", " # gather in afld[x] everything that changes if phi[x] changes\n", " afld = g(0.5 * g.adj(phi)*phi*(m0**2 + 4.0))\n", " afld += lam * g.adj(phi)*phi*g.adj(phi)*phi\n", " for mu in range(2):\n", " afld -= g.adj(phi) * (g.cshift(phi,mu,1) + g.cshift(phi,mu,-1))\n", " return afld\n", "\n", "# test change in single point\n", "rng.element(field)\n", "field[12,17] = 0.5\n", "S0 = S_ia(field,m0,lam)\n", "Sf0 = S_field_ia(field,m0,lam)[12,17]\n", "field[12,17] = 0.3\n", "S1 = S_ia(field,m0,lam)\n", "Sf1 = S_field_ia(field,m0,lam)[12,17]\n", "print(S1 - S0)\n", "print(Sf1 - Sf0)\n", "\n", "\n", "\n", "# first thermalize theory for new m0\n", "field[:] = 0.0\n", "for s in range(100):\n", " acceptance_rate = metropolis_step(field, lambda field: S_field_ia(field, m0, lam), 3.0)\n", " action_list.append(S_ia(field, m0, lam).real)\n", " if s % 10 == 0:\n", " print(\"Acceptance rate of local update:\",acceptance_rate)\n", " \n", "fig, ax = plt.subplots()\n", "\n", "ax.plot(range(len(action_list)), action_list, marker='+', ls='', c='red', label=\"S using local Metropolis update\")\n", "\n", "plt.legend()\n", "plt.show()\n", "\n", "configurations_ia = []\n", "\n", "for s in range(5000):\n", " for sub_steps in range(5):\n", " acceptance_rate = metropolis_step(field, lambda field: S_field_ia(field, m0, lam), 3.0)\n", " configurations_ia.append(g.copy(field))\n", " if s % 1000 == 0:\n", " print(s)" ] }, { "cell_type": "code", "execution_count": 20, "id": "80189574", "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "def two_point_for_momentum_ia(p):\n", " mom_phase = g.exp_ixp(np.array([p,0]))\n", " correlators = [ correlate(g(mom_phase*conf)) for conf in configurations_ia ]\n", " return correlators\n", "\n", "c0_ia = two_point_for_momentum_ia(0)\n", "bc0_ia = time_parity_average(bin_data(c0_ia, 100))\n", "\n", "c0_ia_mean = np.mean(bc0_ia, axis=0)\n", "emp0_ia_mean = emp(c0_ia_mean)\n", "emp0_ia_err = jackknife_variance_estimator(emp, bc0_ia)**0.5\n", "\n", "fig, ax = plt.subplots()\n", "\n", "plt.ylim([0,0.5])\n", "ax.errorbar(range(len(emp0_ia_mean)), emp0_ia_mean, emp0_ia_err, marker='+', ls='', c='red', label=\"E(p=0,IA) - E_0\")\n", "\n", "plt.axhline(y=m0, color='red', linestyle='-')\n", "\n", "plt.legend()\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "cf2fdb43", "metadata": {}, "source": [ "One can then also measure the energy of two particles in the interacting theory. They will no longer be twice as energetic as a single particle due to the interaction energy. By studying these interaction energies at different volumes, one can learn about scattering amplitudes of particles (see the Lüscher formalism https://doi.org/10.1016/0550-3213(91)90366-6 )." ] }, { "cell_type": "markdown", "id": "8f03962c", "metadata": {}, "source": [ "## Field theories in higher dimensions\n", "\n", "It is straightforward to modify the presented code to simulations in higher dimensions. Below for a 3+1d interacting real scalar field theory." ] }, { "cell_type": "code", "execution_count": 21, "id": "6b5b31e7", "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "262144\n", "GPT : 667.002744 s : Initializing gpt.random(seed13,vectorized_ranlux24_389_64) took 0.000406027 s\n", "Acceptance rate of local update: 0.4446144104003906\n", "Acceptance rate of local update: 0.5132102966308594\n", "Acceptance rate of local update: 0.5131492614746094\n", "Acceptance rate of local update: 0.5140342712402344\n", "Acceptance rate of local update: 0.5136299133300781\n", "Acceptance rate of local update: 0.5135154724121094\n", "Acceptance rate of local update: 0.512176513671875\n", "Acceptance rate of local update: 0.5137062072753906\n", "Acceptance rate of local update: 0.5137748718261719\n", "Acceptance rate of local update: 0.5127792358398438\n" ] }, { "data": { "image/png": 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\n", 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" ] }, "metadata": { "needs_background": "light" }, "output_type": "display_data" } ], "source": [ "L = [16,16,16,64]\n", "print(L[0]*L[1]*L[2]*L[3])\n", "grid = g.grid(L, g.double)\n", "grid_eo = g.grid(L, g.double, g.redblack)\n", "\n", "field = g.real(grid)\n", "field[:] = 0\n", "\n", "rng = g.random(\"seed13\")\n", "field[:] = 0.0\n", "action_list = []\n", "m0 = 0.2\n", "lam = 0.005\n", "\n", "def S_4d(phi, m0, lam):\n", " action = 0.5*g.norm2(phi)*m0**2 + lam * g.sum(g.adj(phi)*phi*g.adj(phi)*phi)\n", " for mu in range(4):\n", " action -= 0.5 * g.inner_product(phi, g.cshift(phi,mu,1) + g.cshift(phi,mu,-1) - 2.*phi)\n", " return action\n", "\n", "def S_field_4d(phi, m0, lam):\n", " # gather in afld[x] everything that changes if phi[x] changes\n", " afld = g(0.5 * g.adj(phi)*phi*(m0**2 + 8.0))\n", " afld += lam * g.adj(phi)*phi*g.adj(phi)*phi\n", " for mu in range(4):\n", " afld -= g.adj(phi) * (g.cshift(phi,mu,1) + g.cshift(phi,mu,-1))\n", " return afld\n", "\n", "# first thermalize theory for new m0\n", "for s in range(100):\n", " acceptance_rate = metropolis_step(field, lambda field: S_field_4d(field, m0, lam), 2.0)\n", " action_list.append(S_4d(field, m0, lam).real)\n", " if s % 10 == 0:\n", " print(\"Acceptance rate of local update:\",acceptance_rate)\n", " \n", "fig, ax = plt.subplots()\n", "\n", "ax.plot(range(len(action_list)), action_list, marker='+', ls='', c='red', label=\"S using local Metropolis update\")\n", "\n", "plt.legend()\n", "plt.show()\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.9.6" } }, "nbformat": 4, "nbformat_minor": 5 }