# Quantum Monte-Carlo calculation of Green's function

• Ramil
In summary, the conversation discusses the introduction of a spacetime spherical symmetric lattice and the use of notifications and indices for different coordinates. The individual also shares their algorithm for obtaining r-dependency and mentions having trouble with incorrect results. They ask for advice and suggestions on how to improve their algorithm.
Ramil
Homework Statement
Hi there,

I'm trying to reproduce the results of one article and writing a program that performs calculation of two-point Green's function G(|x'-x|) in 4D using Quantum Monte-Carlo approach.

I'm cosidering the theory of a massive scalar field with a quartic potential (lambda-psi^4).

Here I will denote scalar field as 'psi' just not to mix it up with an azimuthal coordinate 'phi'.

I use spherical coordinates and want to calculate the Green's function along azimuthal coordinate and also to get its dependency on r-coordinate. I consider two points that have similar t, r and theta coordinates but phi-coordinates differ on pi. So that G(|x'-x|) = G(pi).

For different r-coordinates I will get different G(pi).

The problem is that I don't understand how to write down the exact computational formula to obtain this r-dependency.
Relevant Equations
After all the standard steps of Quantum Monte-Carlo simulation I get the configurations of the scalar fields psi(t, r, theta, phi).

Then G(pi) = <psi(t, r, theta, phi)*psi(t, r, theta, phi+pi)>
Introducing the spacetime spherical symmetric lattice, I use the following notifications in my program.
i - index enumerating the nodes along t-coordinate,
j - along the r-coordinate,
k - along the theta-coordinate,
l - along the phi-coordinate.

N_t - the number of nodes along t-coordinate.
N_r - along the r-coordinate,
N_theta - along the theta-coordinate,
N_phi - along the phi-coordinate.

I tried to get the r-dependency as following. I fixed the coordinate r by fixing the j-index, and then calculated G(pi) using the formula:

C++ code

C++:
for(int j=0; j<N_r; j++)
{

G(j) = 0;

for(int i =0: i<N_t; i++)
{
for(int k=0; k<N_theta; k++)
{
for(int l = 0; l<N_phi; l++)
{
G(j)=G(j) + psi(i, j, k, l)*psi(i, j, k, (l+N_phi/2)%N_phi);
}
}
}

G(j) = G(j)/(N_t*N_theta*N_phi) //average along t, theta, phi directions

}
Here (l+N_phi/2)%N_phi ensures that |x'-x| = π.

However, this algorithm seems to be incorrect. Comparing the results with an article, I obtain incorrect graphics for G(pi) dependence on r.

Would be glad if someone gives an advice on it. Thanks in advance!

Hello, thank you for sharing your algorithm with us. Based on a quick review, it seems like there may be a few potential issues that could be causing your incorrect results.

First, it is important to ensure that your indexing is correct. In your code, you have used the index "j" to represent the r-coordinate, but in your explanation, you have stated that "j" represents the index along the r-coordinate. This could potentially lead to confusion and errors in your code.

Second, it seems like you are trying to calculate the function G(pi) for a fixed value of r, by averaging over all values of t, theta, and phi. However, in your code, you are only looping through the indices for t, theta, and phi, without actually fixing their values. This could lead to incorrect results as you are not actually fixing the values of these coordinates.

Additionally, it may be helpful to double check your formula for G(pi). Depending on the specific lattice and coordinate system you are using, the formula may vary. It would also be helpful to compare your results with other sources or publications to see if there are any discrepancies.

I hope this helps and provides some guidance for troubleshooting your algorithm. Good luck!

## 1. What is a Quantum Monte-Carlo calculation of Green's function?

A Quantum Monte-Carlo calculation of Green's function is a computational method used in quantum mechanics to calculate the Green's function, which describes the probability amplitude for a particle to travel from one point to another in a quantum system. It uses Monte-Carlo sampling techniques to approximate the Green's function, making it a powerful tool for studying complex quantum systems.

## 2. How does a Quantum Monte-Carlo calculation of Green's function work?

The method involves randomly sampling points in the quantum system and using them to approximate the Green's function. This is done using a statistical approach, where the more samples are taken, the more accurate the approximation becomes. By using this technique, the Green's function can be calculated for systems that are too complex to solve analytically.

## 3. What are the advantages of using a Quantum Monte-Carlo calculation of Green's function?

One of the main advantages is its ability to handle complex systems that cannot be solved analytically. It also allows for the calculation of the Green's function at any point in the system, providing a detailed understanding of the quantum dynamics. Additionally, it is a non-perturbative method, meaning it can be used for systems with strong interactions between particles.

## 4. What are some applications of Quantum Monte-Carlo calculation of Green's function?

This method has a wide range of applications in quantum mechanics, including studying the properties of materials, such as superconductors and magnets, and understanding the behavior of quantum systems, such as quantum dots and quantum gases. It is also used in the field of nuclear physics to study the properties of atomic nuclei.

## 5. What are the limitations of Quantum Monte-Carlo calculation of Green's function?

One of the main limitations is the statistical error that arises from the Monte-Carlo sampling. This error decreases as more samples are taken, but for complex systems, it may require a large number of samples to achieve an accurate result. Additionally, the method is limited to systems that can be described by a wave function, and it may not be suitable for systems with many interacting particles.

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