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/*************************************************************************
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* Copyright (C) 2015 by Andrius Štikonas <andrius@stikonas.eu> * |
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* * |
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* This program is free software; you can redistribute it and/or modify * |
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* it under the terms of the GNU General Public License as published by * |
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* the Free Software Foundation; either version 3 of the License, or * |
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* (at your option) any later version. * |
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* * |
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* This program is distributed in the hope that it will be useful, * |
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* but WITHOUT ANY WARRANTY; without even the implied warranty of * |
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the * |
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* GNU General Public License for more details. * |
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* * |
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* You should have received a copy of the GNU General Public License * |
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* along with this program. If not, see <http://www.gnu.org/licenses/>.*
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*************************************************************************/ |
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// Calculation of mutual information in the setup of http://arxiv.org/abs/1503.08161
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#include <cmath>
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#include <complex>
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#include <iostream>
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#include <iomanip>
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static double alpha, beta; |
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std::complex<double> crossRatio (std::complex<double>, std::complex<double>, std::complex<double>, std::complex<double>); |
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double Fitzpatrick(std::complex<double>, std::complex<double>, double); |
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int main() |
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{ |
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// Parameters that can be changed:
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double tPlus = 0; // time on the left boundary
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double tMinus = 0; // time on the right boundary
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alpha = 0.4; // encodes the conformal dimention of local operator h_Psi
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double y = 1; // endpoint of interval A
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double L = 5; // length of interval A
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double epsilon = 0.01; // smearing parameter
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beta = 10; // inverse temperature
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double tOmega = 5.4418; // thermal state is perturbed by operator inserted at time -tOmega
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double c = 600; // central charge. Must be large in our approximation
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// End of parameters
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// Operator insertion points: Left boundary
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std::complex<double> x1 (0, -epsilon), x4 (0, epsilon), x1bar, x4bar; |
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x1bar = conj(x1); |
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x4bar = conj(x4); |
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double x2 = y - tOmega - tMinus; |
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double x2bar = y + tOmega + tMinus; |
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double x3 = L + x2; |
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double x3bar = L + x2bar; |
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// Operator insertion points: Right boundary
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std::complex<double> x6 (y - tPlus - tOmega, beta/2); |
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std::complex<double> x6bar (y + tPlus + tOmega, -beta/2); |
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std::complex<double> x5 = L + x6; |
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std::complex<double> x5bar = L + x6bar; |
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// Cross-ratios for S_A
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std::complex<double> zA = crossRatio(x1, x2, x3, x4); |
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std::complex<double> zAbar = crossRatio(x1bar, x2bar, x3bar, x4bar); |
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// Cross-ratios for S_B
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std::complex<double> zB = crossRatio(x1, x5, x6, x4); |
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std::complex<double> zBbar = crossRatio(x1bar, x5bar, x6bar, x4bar); |
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// Cross-ratios for S_{A union B}
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std::complex<double> z2 = crossRatio(x1, x2, x6, x4); |
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std::complex<double> z2bar = crossRatio(x1bar, x2bar, x6bar, x4bar); |
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std::complex<double> z5 = crossRatio(x1, x5, x3, x4); |
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std::complex<double> z5bar = crossRatio(x1bar, x5bar, x3bar, x4bar); |
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// Now we calculate entanglement entropies using Fitzpatrick, Kaplan, Walters formula.
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double S_A = c/6 * log(Fitzpatrick(zA, zAbar, 2*M_PI)); |
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double S_B = c/6 * log(Fitzpatrick(zB, zBbar, 0)); |
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double S_union = c/6 * log(Fitzpatrick(z2, z2bar, 2*M_PI) * Fitzpatrick(z5, z5bar, 0)); |
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double S_thermal = 2*c/3 * log(sinh(M_PI*L/beta)/cosh(M_PI/beta*(tMinus-tPlus))); |
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double I = S_A + S_B - S_union + S_thermal; |
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std::cout << I << std::endl; |
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return 0; |
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} |
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std::complex<double> crossRatio (std::complex<double> x1, std::complex<double> x2, std::complex<double> x3, std::complex<double> x4) |
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{ |
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double pb = M_PI/beta; |
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return sinh(pb*(x1-x2))*sinh(pb*(x3-x4))/sinh(pb*(x1-x3))/sinh(pb*(x2-x4)); |
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} |
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double Fitzpatrick(std::complex<double> z, std::complex<double> zbar, double phase) |
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{ |
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// phase is necessary to take into account nontrivial monodromy
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std::complex<double> c1(1, 0); // one as a complex number
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std::complex<double> i(0,1); // imaginary unit
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double alphaExpr = 0.5-alpha/2; |
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std::complex<double> exponent1 = exp(phase*i*alphaExpr); |
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std::complex<double> exponent2 = exp(phase*i*alpha); |
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return real(exponent1 * pow(z, alphaExpr) * pow(zbar, alphaExpr) |
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* (c1 - exponent2 * pow(z, alpha)) * (c1 - pow(zbar, alpha)) |
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/ ( alpha*alpha * (c1-z) * (c1-zbar) ) ); |
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} |