**Quantum Simulation of a Transverse-Field Ising Model**

**ABSTRACT**

A popular model that have been used to study ferromagnetism is the Ising Model which is an arrangement of spins along a particular direction and with discrete values of +1. 1-D

Ising model doesn’t show a phase transition to the paramagnetic phase as opposed to the 2-D Ising model which shows a transition at a critical temperature. In this work, I have used Monte Carlo simulation method to study the 1-D quantum Ising model in a transverse eld at a nite temperature to obtain the critical eld when a ferromagnetic material becomes paramagnetic.

**TABLE OF CONTENTS**

Dedication ii

Acknowledgement iii

1 INTRODUCTION 1

1.1 Magnetic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A Brief History of Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 LITERATURE REVIEW 6

2.1 Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Ising Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 1-D Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 2-D Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 MONTE CARLO METHODS 14

3.1 Metropolis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2 Simulation of Ising Models Using Metropolis Algorithm . . . . . . . . . . . . 16

3.2.1 1-D Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2.2 2-D Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

vii

4 QUANTUM LATTICE MODEL 21

4.1 Path Integral Formulation of Ising Model . . . . . . . . . . . . . . . . . . . . 21

4.2 Quantum Monte Carlo Simulation and Numerical Results . . . . . . . . . . . 25

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Bibliography 28

A C++ Codes 29

A.1 1-D Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.2 2-D Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.3 1-D Quantum Ising model in a transverse eld . . . . . . . . . . . . . . . . . 33

**CHAPTER ONE**

**1.1 Magnetic Systems**

Magnetism is a common phenomenon observed in some materials in nature. A magnetic material unlike an electric charge cannot exist as a monopole but as a dipole. The smallest

unit of a magnet which is usually referred to as “magnons” exists as a dipole of north pole and south pole. Opposite poles of a magnet attracts. A magnet is a material that produces a magnetic eld.[1]

When magnetic systems enter into a magnetic eld, they get magnetized, on withdrawing the materials from the eld, some loses their magnetism which makes them to be referred to

as temporary magnet and some retains their magnetism which made them to be called permanent magnet. Examples of permanent magnet are iron, nickel, cobalt, loadstone,

etc. While temporary magnets are just any object attracted by a magnet, those materials will lose their magnetism once the permanent magnet is removed, although they may retain a very weak magnetic strength.

In understanding the origin of magnetism, one has to consider the atomic description of the material. Every matter is made up of atoms which are composed of nucleon (neutron+proton) and electrons on the shells around the atom. According to atomic theory, these electrons orbit round the shells of the atom. According to Lenz law, a moving electron create a magnetic eld. As these electrons orbit around the nucleus, they also spin along their orbit which give the electrons a dipole moment. Therefore, generally there are some atoms that have a magnetic moment. We can consider a crystal that contains such atoms arranged in a regular pattern, such a crystal will become magnetic under suitable conditions of spins alignment and of external conditions such as temperature and external magnetic eld.

Depending on the orientation of the spins of the electrons, the material can be categorized into different types among which are:

**Ferromagnets:** These materials have most of its spins aligned and uniformly ordered. These materials are the only one that can retain magnetism and become permanent

magnets.

**Paramagnets:** When there is no uniform alignment of spins, the material can be referred to as paramagnetic. As such, they are weakly attracted to a magnet

**Diamagnets:** These are considered as materials not possessing any form of magnetism. Every other substance like carbon, waste, plastic, etc are diamagnetic.

Different theoretical model have been used to describe the phenomenon of ferromagnetism, the simplest being the Ising model; a model for ferromagnetism formulated as a

problem by Wilhelm Lenz(1920) and gave it as a problem to his student Ernst Ising and was already solved by 1925 as his PhD theses work[2].

**1.2 A Brief History of Ising Model**

There are different conditions that can affect the magnetism or magnetic strength of a material; i.e. there are harsh conditions that make a material loses it magnetism. Essentially, temperature and applied magnetic eld affects the strength of a magnetic system. Increase in temperature within a material increases thermal actuations which dis orient the spins from their aligned positions and make the material to lose its magnetism. Also, since the electrons of an atom respond to magnetic eld, increase in the applied magnetic eld strength decreases the actuations of the spins which tends to increase the magnetization of the material.

In an attempt to investigate how a ferromagnet respond to an external controllable constraints, the problem which led to the Ising model was formulated.

The 1-D Ising model considers N spins on a linear chain. These spins interacts with each other with an interaction energy denoted by J, which by convention can be (J>0)

for ferromagnetism or (J<0) for anti-ferromagnetism. The spins, as was used by Ising were assigned discrete variables of +1 for spin-up and -1 for spin-down. A simple schematic
diagram is shown below.
– 6u
?
u
?
u 6u
6u
–
J ?
u
?
u -u
-J
6 6u
?
u
0 1 2 N 1 N
Figure 1.1: 1-D lattice of N spins with periodic boundary condition on the Nthspin. The interaction energy +J between nearest neighbors pairs is shown for aligned and opposing spins.
The lines connecting neighboring spins are referred to as links. When an external magnetic eld is applied to a magnetic system, the applied magnetic eld orient the spins
along its direction which increases the internal magnetic strength within the system. Ising solved the 1-D model analytically and found no phase transition from the ferromagnetic phase to the paramagnetic phase but that the magnetization just reduces to zero at a particular temperature and remains zero for all higher temperatures.
However, 2-D model which is the square lattice Ising model, a more complicated model, was solved by Lars Onsager in 1944 and there, a phase transition was observed which renders
Ising’s initial assertion of no phase transition for all dimensions false. A schematic diagram for the 2-D Ising model is shown below.
3
–
6
u
u u u
u
–
6
?
Jx
Jy
0
k 1
k
k + 1
l 1
l
l + 1
x
y
j 6
i-
Figure 1.2: 2-D rectangular lattice arrangement of m n spins with periodic boundary conditions on the nth and mthspins with Jx as the interaction energy on the x axis and Jy as the interaction energy on the y axis
As opposed to the 1-D and 2-D Ising models, the 3-D Ising model hasn’t received any exact solution even though some other approaches have been used to investigate phase transitions at higher dimensions. In an attempt to understand higher-D Ising model, approaches like mean-eld theory, quantum eld theory, computer simulations, etc, have been used.
1.3 Statement of the Problem
The Ising models outlined above are statistical models which can be referred to as classical statistical models because of the discrete nature of the spins. However, these models become quantum-mechanical when those spins are considered as spins of the famous Pauli spin matrices.
The energies now are averages of the quantum Hamiltonian operators which describes the system. In this work, we will solve the 1-D quantum Ising model in a transverse eld which should show quantum phase transitions(i.e. quantum effects) when the temperature is kept very low and we vary the eld to obtain the critical eld. However, there are a number
of challenges faced when one tries to simulate a quantum system as opposed to simulating a classical system. A successful approach to simulating quantum system is to nd a path integral formulation which describes the problem in a classical way.