Initial commit.

3 files changed, 251 insertions, 0 deletions

diff --git a/.gitignore b/.gitignore
new file mode 100644
index 0000000..0c0a063
--- /dev/null
+++ b/.gitignore
@@ -0,0 +1,3 @@
+*
+!/**/
+!*.*
diff --git a/compile_flags.txt b/compile_flags.txt
new file mode 100644
index 0000000..02d9f07
--- /dev/null
+++ b/compile_flags.txt
@@ -0,0 +1,3 @@
+-std=c++20
+-O3
+-I/usr/include/eigen3
diff --git a/langevin.cpp b/langevin.cpp
new file mode 100644
index 0000000..73cf4c0
--- /dev/null
+++ b/langevin.cpp
@@ -0,0 +1,245 @@
+#include <getopt.h>
+#include <cstdlib>
+#include <random>
+#include <complex>
+#include <array>
+#include <functional>
+
+#include <eigen3/unsupported/Eigen/CXX11/Tensor>
+
+#include "pcg-cpp/include/pcg_random.hpp"
+#include "randutils/randutils.hpp"
+
+#define P_SPIN_P 3
+const unsigned p = P_SPIN_P;
+
+using Tensor = Eigen::Tensor<std::complex<double>, p>;
+using Scalar = Eigen::Tensor<std::complex<double>, 0>;
+using Vector = Eigen::Tensor<std::complex<double>, 1>;
+using Matrix = Eigen::Tensor<std::complex<double>, 2>;
+
+const std::array<Eigen::IndexPair<int>, 1> ip = {Eigen::IndexPair<int>(0,0)};
+
+using Rng = randutils::random_generator<pcg32>;
+
+template <class T = double>
+class complex_normal_distribution {
+  private:
+    std::complex<T> μ;
+    T σ;
+    std::complex<T> κ;
+    std::normal_distribution<T> dist;
+
+  public:
+    complex_normal_distribution(std::complex<T> μ, T σ, std::complex<T> κ) :
+      μ(μ), σ(σ), κ(κ), dist(0, σ / sqrt(2)) {}
+
+    template <class Generator>
+      std::complex<T> operator()(Generator& g) {
+        std::complex<T> x(dist(g) * sqrt(1 + std::abs(κ)), dist(g) * sqrt(1 - std::abs(κ)));
+        return μ + std::polar((T)1, std::arg(κ)) * x;
+      }
+};
+
+long unsigned factorial(unsigned p) {
+  if (p == 0) {
+    return 1;
+  } else {
+    return p * factorial(p - 1);
+  }
+}
+
+Tensor generate_coupling_tensor(unsigned N, std::complex<double> κ, Rng& r) {
+  double σ = sqrt(factorial(p) / (2.0 * pow(N, p - 1)));
+  complex_normal_distribution<> dist(0, σ, κ);
+
+  Tensor J(N, N, N);
+
+  for (unsigned i1 = 0; i1 < N; i1++) {
+    for (unsigned i2 = i1; i2 < N; i2++) {
+      for (unsigned i3 = i2; i3 < N; i3++) {
+        std::complex<double> x = dist(r.engine());
+
+        J(i1, i2, i3) = x;
+        J(i1, i3, i2) = x;
+        J(i2, i1, i3) = x;
+        J(i2, i3, i1) = x;
+        J(i3, i1, i2) = x;
+        J(i3, i2, i1) = x;
+      }
+    }
+  }
+
+  return J;
+}
+
+template <int r>
+Eigen::Tensor<std::complex<double>, r> conj(const Eigen::Tensor<std::complex<double>, r>& t) {
+  return t.unaryExpr(static_cast<std::complex<double> (*)(const std::complex<double>&)>(&std::conj));
+}
+
+template <int r>
+double norm(const Eigen::Tensor<std::complex<double>, r>& t) {
+  Eigen::Tensor<double, 0> t2 = t.unaryExpr(static_cast<double (*)(const std::complex<double>&)>(&std::norm)).sum();
+  return t2(0);
+}
+
+Vector generate_initial_vector(unsigned N) {
+  Vector z(N);
+  z.setConstant(1);
+  return z;
+}
+
+std::complex<double> hamiltonian(const Tensor& J, const Vector& z) {
+  Eigen::Tensor<std::complex<double>, 0> t = z.contract(z.contract(z.contract(J, ip), ip), ip);
+  return t(0) / (double)factorial(p);
+}
+
+Matrix identity(unsigned N) {
+  Matrix I(N, N);
+  for (unsigned i = 0; i < N; i++) {
+    I(i) = 1;
+  }
+  return I;
+}
+
+std::tuple<Vector, Matrix> gradHess(const Tensor& J, const Vector& z, std::complex<double> ε) {
+  Matrix J1 = z.contract(J, ip);
+  Matrix H = ((p - 1) * (double)p / factorial(p)) * J1 - ((double)p * ε) * identity(z.size());
+
+  Vector J2 = z.contract(J1, ip);
+  Vector g = ((double)p / factorial(p)) * J2 - ((double)p * ε) * z;
+
+  return {g, H};
+}
+
+std::tuple<double, Vector, std::complex<double>, Vector> WdW(const Tensor& J, const Vector& z, std::complex<double> ε) {
+  Vector grad;
+  Matrix hess;
+
+  std::tie(grad, hess) = gradHess(J, z, ε);
+
+  double W = norm(grad);
+  Matrix conjHess = conj(hess);
+  Scalar zz = z.pow(2).sum();
+  Vector zc = conj(z);
+
+  Vector dWdz = grad.contract(conjHess, ip) - (pow(p, 2) / 2.0 * ((double)z.size() - zz(0))) * zc;
+  Scalar dWdε = (-(double)p) * grad.contract(zc, ip);
+  Vector dεdz = (1 / (double)z.size()) * ((1 - 1/(double)p) * grad - ((double)p * ε) * z - (1 /(double)p) * z.contract(hess, ip));
+
+  return {W, dWdz, dWdε(0), dεdz};
+
+}
+
+void gradientDescent(const Tensor& J, Vector& z, std::complex<double>& ε, double δ, double γ) {
+  double W;
+  Vector dz;
+  std::complex<double> dε;
+
+  std::tie(W, dz, dε, std::ignore) = WdW(J, z, ε);
+
+  while (W > δ * z.size()) {
+    std::cout << W << std::endl;
+    z = z - (γ / z.size()) * dz;
+    ε = ε - (γ / z.size()) * dε;
+
+    std::tie(W, dz, dε, std::ignore) = WdW(J, z, ε);
+  }
+}
+
+Vector generateKick(const Vector& z, Rng& r) {
+  Vector k(z.size());
+
+  for (unsigned i = 0; i < k.size(); i++) {
+    k(i) = std::complex<double>(r.variate<double>(0, 1), r.variate<double>(0, 1)) / sqrt(2);
+  }
+
+  Scalar kz = z.contract(k, ip);
+  k = k - kz(0) * z;
+
+  return k;
+}
+
+void langevin(const Tensor& J, Vector& z, std::complex<double>& ε, double T, unsigned t, double γ, Rng& r) {
+  double W;
+  Vector dz;
+  Vector dεdz;
+
+  std::tie(W, dz, std::ignore, dεdz) = WdW(J, z, ε);
+
+  for (unsigned i = 0; i < t; i++) {
+    std::cout << W << std::endl;
+
+    Vector η = generateKick(z, r);
+    Vector δz = (γ / z.size()) * (-dz + T * η);
+    Scalar δε = dεdz.contract(δz, ip);
+
+    z = z + δz;
+    ε = ε + δε(0);
+
+    std::tie(W, dz, std::ignore, dεdz) = WdW(J, z, ε);
+  }
+}
+
+int main(int argc, char* argv[]) {
+  // model parameters
+  unsigned N = 10; // number of spins
+  double T = 1; // temperature
+  double Rκ = 1; // real part of distribution parameter
+  double Iκ = 0; // imaginary part of distribution parameter
+
+  // simulation parameters
+  double δ = 1e-2; // threshold for determining saddle
+  double γ = 1e-2; // step size
+
+  int opt;
+
+  while ((opt = getopt(argc, argv, "N:T:E:e:r:i:g:")) != -1) {
+    switch (opt) {
+    case 'N':
+      N = (unsigned)atof(optarg);
+      break;
+    case 'T':
+      T = atof(optarg);
+      break;
+    case 'e':
+      δ = atof(optarg);
+      break;
+    case 'g':
+      γ = atof(optarg);
+    case 'r':
+      Rκ = atof(optarg);
+      break;
+    case 'i':
+      Iκ = atof(optarg);
+      break;
+    default:
+      exit(1);
+    }
+  }
+
+  std::complex<double> κ(Rκ, Iκ);
+
+  Rng r;
+
+  Tensor J = generate_coupling_tensor(N, κ, r);
+  Vector z = generate_initial_vector(N);
+  std::complex<double> ε = hamiltonian(J, z) / (double)N;
+
+  Scalar zz = z.pow(2).sum();
+
+  std::cout << zz(0) << std::endl;
+
+  gradientDescent(J, z, ε, δ, γ);
+  langevin(J, z, ε, T, 1000, γ, r);
+
+  zz = z.pow(2).sum();
+  double a = norm(z);
+
+  std::cout << zz(0) / (double)N << std::endl;
+  std::cout << ε << std::endl;
+  std::cout << hamiltonian(J, z) / (double)N << std::endl;
+  std::cout << a / N << std::endl;
+}
+