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Andrius Štikonas 4 years ago
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      CMakeLists.txt
  2. 99
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      scrambling.cpp

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CMakeLists.txt View File

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cmake_minimum_required(VERSION 2.8)
project(Scrambling)

add_definitions(-std=gnu++11)

add_executable(scrambling scrambling.cpp)

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scrambling.cpp View File

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/*************************************************************************
* Copyright (C) 2015 by Andrius Štikonas <andrius@stikonas.eu> *
* *
* This program is free software; you can redistribute it and/or modify *
* it under the terms of the GNU General Public License as published by *
* the Free Software Foundation; either version 3 of the License, or *
* (at your option) any later version. *
* *
* This program is distributed in the hope that it will be useful, *
* but WITHOUT ANY WARRANTY; without even the implied warranty of *
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the *
* GNU General Public License for more details. *
* *
* You should have received a copy of the GNU General Public License *
* along with this program. If not, see <http://www.gnu.org/licenses/>.*
*************************************************************************/

// Calculation of mutual information in the setup of http://arxiv.org/abs/1503.08161

#include <cmath>
#include <complex>
#include <iostream>
#include <iomanip>

static double alpha, beta;
std::complex<double> crossRatio (std::complex<double>, std::complex<double>, std::complex<double>, std::complex<double>);
double Fitzpatrick(std::complex<double>, std::complex<double>, double);

int main()
{
// Parameters that can be changed:
double tPlus = 0; // time on the left boundary
double tMinus = 0; // time on the right boundary
alpha = 0.4; // encodes the conformal dimention of local operator h_Psi
double y = 1; // endpoint of interval A
double L = 5; // length of interval A
double epsilon = 0.01; // smearing parameter
beta = 10; // inverse temperature
double tOmega = 5.4418; // thermal state is perturbed by operator inserted at time -tOmega
double c = 600; // central charge. Must be large in our approximation
// End of parameters

// Operator insertion points: Left boundary
std::complex<double> x1 (0, -epsilon), x4 (0, epsilon), x1bar, x4bar;
x1bar = conj(x1);
x4bar = conj(x4);
double x2 = y - tOmega - tMinus;
double x2bar = y + tOmega + tMinus;
double x3 = L + x2;
double x3bar = L + x2bar;

// Operator insertion points: Right boundary
std::complex<double> x6 (y - tPlus - tOmega, beta/2);
std::complex<double> x6bar (y + tPlus + tOmega, -beta/2);
std::complex<double> x5 = L + x6;
std::complex<double> x5bar = L + x6bar;

// Cross-ratios for S_A
std::complex<double> zA = crossRatio(x1, x2, x3, x4);
std::complex<double> zAbar = crossRatio(x1bar, x2bar, x3bar, x4bar);

// Cross-ratios for S_B
std::complex<double> zB = crossRatio(x1, x5, x6, x4);
std::complex<double> zBbar = crossRatio(x1bar, x5bar, x6bar, x4bar);

// Cross-ratios for S_{A union B}
std::complex<double> z2 = crossRatio(x1, x2, x6, x4);
std::complex<double> z2bar = crossRatio(x1bar, x2bar, x6bar, x4bar);
std::complex<double> z5 = crossRatio(x1, x5, x3, x4);
std::complex<double> z5bar = crossRatio(x1bar, x5bar, x3bar, x4bar);

// Now we calculate entanglement entropies using Fitzpatrick, Kaplan, Walters formula.
double S_A = c/6 * log(Fitzpatrick(zA, zAbar, 2*M_PI));
double S_B = c/6 * log(Fitzpatrick(zB, zBbar, 0));
double S_union = c/6 * log(Fitzpatrick(z2, z2bar, 2*M_PI) * Fitzpatrick(z5, z5bar, 0));
double S_thermal = 2*c/3 * log(sinh(M_PI*L/beta)/cosh(M_PI/beta*(tMinus-tPlus)));
double I = S_A + S_B - S_union + S_thermal;
std::cout << I << std::endl;
return 0;
}

std::complex<double> crossRatio (std::complex<double> x1, std::complex<double> x2, std::complex<double> x3, std::complex<double> x4)
{
double pb = M_PI/beta;
return sinh(pb*(x1-x2))*sinh(pb*(x3-x4))/sinh(pb*(x1-x3))/sinh(pb*(x2-x4));
}

double Fitzpatrick(std::complex<double> z, std::complex<double> zbar, double phase)
{
// phase is necessary to take into account nontrivial monodromy
std::complex<double> c1(1, 0); // one as a complex number
std::complex<double> i(0,1); // imaginary unit
double alphaExpr = 0.5-alpha/2;
std::complex<double> exponent1 = exp(phase*i*alphaExpr);
std::complex<double> exponent2 = exp(phase*i*alpha);
return real(exponent1 * pow(z, alphaExpr) * pow(zbar, alphaExpr)
* (c1 - exponent2 * pow(z, alpha)) * (c1 - pow(zbar, alpha))
/ ( alpha*alpha * (c1-z) * (c1-zbar) ) );
}

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