# Simple Quantum Monte Carlo question

Currently I am doing some simple simulation of 1D Transverse field Ising model. I map the quantum mechanical problem into classical 2D classical Ising model with different horizontal interaction and vertical interaction so that the simple Monte Carlo sampling (local update) can be implemented. But I have a problem that isI try to initialised state randomly, I cannot get a converged solution. I saw some people doing this simulation will first polarize the spin (all up or all down) initially. What is the problem of my algorithm ? Sorry for my simple question. Im just a beginner of doing Monte Carlo simulation.

The original Hamiltonain is: $$\hat H = -J \sum_{i = 1}^N \sigma^{z}_{i} \sigma^{z}_{i + 1} - \Gamma \sum_{i = 1}^N \sigma_{i}^{x}$$

After the transform, the effective 2D classical hamiltonain is : $$H = - \sum_{k = 1}^{P} \sum_{i = 1}^{N} (\frac{K_P}{\beta} s_{i}^{k} s_{i}^{k + 1} + \frac{J}{P} s_{i}^{k} s_{i}^{k + 1} - \frac{1}{\beta} \ln C)$$ The second interaction term should be along horizontal direction, there is a typo. The subscript should be $$i+1$$ and the superscript should be $$k$$, where $$C = [\frac{1}{2} \sinh(\frac{2 \Gamma}{k_B TP})]^{PN/2}$$ and $$K_P = \frac{1}{2}log[\coth(\frac{\Gamma}{k_B TP})]$$

• It is hard to say what is happening without knowing some more information. What kind of MC algorithm are you using? (in particular is it a finite temperature algorithm or a zero temperature one?) When you say you do not get a converged solution, what is the state you expect to get - ordered or disordered? Is the acceptance ratio (the probability of accepting an update) very small, or is it reasonable? Commented May 16, 2020 at 9:00
• With a little more detail, you may be able to get a good answer to this question at the new Materials Modeling SE Commented May 16, 2020 at 18:34
• There is also a quantum-monte-carlo tag on Materials Modeling Stack Exchange. Commented May 17, 2020 at 2:06
• I use trotter decomposition to map the 1D transverse field Ising model into 2D classical lattice with different coupling constant in horizontal direction and vertical direction.Then I use very ordinary local update to do the simulation at low finite temperature. I expected to get an ordered phase in spin z direction at low external magnetic field, then the system become disorder with a higher magnetic field like the classical 2D Ising mode with varying the temperaturel. For example, a lattice size L =10, imaginary time slice =10000, it become a classical 10*10000 2-d classical lattice problem. Commented May 17, 2020 at 4:31
• Yes. There is a quantum phase transition with changing the external magnetic field. The spin z change from order phases to disorder phase under the perturbation of external field in x direction. Commented May 17, 2020 at 9:49