Population Annealing Monte Carlo Studies of Ising Spin Glasses

147 Pages·2017·4.95 MB·English

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Doctoral Dissertations Dissertations and Theses
November 2015
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wenlong wang
University of Massachusetts Amherst
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POPULATION ANNEALING MONTE CARLO STUDIES OF
ISING SPIN GLASSES
A Dissertation Presented
by
WENLONG WANG
Submitted to the Graduate School of the
University of Massachusetts Amherst in partial fulfillment
of the requirements for the degree of
DOCTOR OF PHILOSOPHY
September 2015
Department of Physics
(cid:13)c Copyright by WENLONG WANG 2015
All Rights Reserved
POPULATION ANNEALING MONTE CARLO STUDIES OF
ISING SPIN GLASSES
A Dissertation Presented
by
WENLONG WANG
Approved as to style and content by:
Jonathan Machta, Chair
Helmut G. Katzgraber, Member
Nikolay Prokofiev, Member
Panayotis G. Kevrekidis, Member
Rory Miskimen, Head
Department of Physics
ACKNOWLEDGMENTS
First and foremost, I want to thank my professor Jonathan Machta for his kind and patient
instructions for many years. It has been an honor to be his student. He has gradually guided
me from a student to a more independent researcher. I learned from him to be precise and clear.
He helped me kindly while I was in perhaps the hardest time of my life here. I appreciate all his
contributions of time, ideas and financial aid to make my PhD experience more interesting and
stimulating. He has also impacted me equally on the attitude of life. Being patient and calm in
all situations has changed my character a lot and is going to be continuously invaluable to me in
the future.
Next, I would like to thank our main collaborator Helmut Katzgraber for helpful discussions
and large-scale computational work. Without him, many of the calculations would perhaps never
finish. I also want to thank all my committee members for critical comments of my work. I am
also in great debt to Nikolay Prokofiev who admitted me into the Physics department and his
instructions on quantum many body Physics. I also want to thank Panayotis Kevrekidis for his
patient instructions on the study of Bose-Einstein condensates of cold atoms. He has opened a
new research area for me on the hot cold atoms.
I want to thank my family and friends who have helped and encouraged me in the past years,
especially for my family’s great understanding in my studying abroad. They have made my life in
America more exciting and meaningful. I also want to thank in particular Dong Yan and Yuping
Zhou who have helped a lot in my early years in America and for being my close friends. I also
would like to thank Jon’s previous student Burcu Yucesoy for helpful discussions and access to her
parallel tempering data, which has played an important role during the research. I thank Zhiyuan
Yao and Qingyou Meng for simulating discussions and computer assistance.
iv
Finally, IwanttothanktheUnitedStatesofAmerica. Ienjoymylivinghereandisexperiencing
a new culture. It has been a great opportunity to stay and study in the US.
v
ABSTRACT
POPULATION ANNEALING MONTE CARLO STUDIES OF
ISING SPIN GLASSES
SEPTEMBER 2015
WENLONG WANG
B.Sc., UNIVERSITY OF SCIENCE AND TECHNOLOGY BEIJING
M.Sc., UNIVERSITY OF MASSACHUSETTS AMHERST
Ph.D., UNIVERSITY OF MASSACHUSETTS AMHERST
Directed by: Professor Jonathan Machta
Spin glasses are spin-lattice models with quenched disorder and frustration. The mean field
long-range Sherrington-Kirkpatrick (SK) model was solved by Parisi and displays replica sym-
metry breaking (RSB), but the more realistic short-range Edwards-Anderson (EA) model is still
not solved. Whether the EA spin glass phase has many pairs of pure states as described by
the RSB scenario or a single pair of pure states as described by two-state scenarios such as the
droplet/scaling picture is not known yet. Rigorous analytical calculations of the EA model are
not available at present and efficient numerical simulations of spin glasses are crucial in making
progresses in the field. In addition to being a prototypical example of a classical disordered sys-
tem with many interesting equilibrium as well as nonequilibrium properties, spin glasses are of
great importance across multiple fields from neural networks, various combinatorial optimization
problems to benchmark tests of quantum annealing machines. Therefore, it is important to gain
a better understanding of the spin glass models.
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In an effort to do so, our work has two main parts, one is to develop an efficient algorithm
called population annealing Monte Carlo and the other is to explore the physics of spin glasses
using thermal boundary conditions. We present a full characterization of the population annealing
algorithm focusing on its equilibration properties and make a systematic comparison of population
annealing with two well established simulation methods, parallel tempering Monte Carlo and
simulated annealing Monte Carlo. We show numerically that population annealing is similar in
performance to parallel tempering, each has its own strengths and weaknesses and both algorithms
outperform simulated annealing in combinatorial optimization problems.
In thermal boundary conditions, all eight combinations of periodic vs antiperiodic boundary
conditions in the three spatial directions appear in the ensemble with their respective Boltzmann
weights, thusminimizingfinite-sizeeffectsduetodomainwalls. Withthermalboundaryconditions
and sample stiffness extrapolation, we show that our data is consistent with a two-state picture,
not the RSB picture for the EA model. Thermal boundary conditions also provides an elegant
way to study the phenomena of temperature chaos and bond chaos, and our results are again in
agreement with the droplet/scaling scenario.
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TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS ........................................................ iv
ABSTRACT..................................................................... vi
LIST OF TABLES ............................................................... xi
LIST OF FIGURES............................................................. xiii
CHAPTER
1. INTRODUCTION .............................................................1
1.1 Spin glasses: a general introduction ............................................ 1
1.2 Spin glass models............................................................ 4
1.2.1 The Edwards-Anderson model .......................................... 5
1.2.2 The Sherrington-Kirkpatrick model ...................................... 7
1.2.3 Replica symmetry breaking vs two-state scenarios ........................ 11
1.3 Numerical methods ......................................................... 14
1.3.1 Simulated annealing Monte Carlo....................................... 15
1.3.2 Population annealing Monte Carlo...................................... 17
1.3.3 Parallel tempering Monte Carlo ........................................ 19
1.3.4 Thermal boundary conditions .......................................... 21
1.3.5 Free energy perturbation method ....................................... 21
1.3.6 The Katzgraber-Young test ............................................ 23
1.4 Overview .................................................................. 24
2. POPULATION ANNEALING MONTE CARLO..............................25
2.1 Weighted averages .......................................................... 25
viii
2.2 Systematic and statistical errors .............................................. 28
2.2.1 Systematic errors and the variance of the free energy...................... 28
2.2.2 Statistical errors ..................................................... 31
2.2.3 Comparison of errors in PA and PT .................................... 33
2.3 Application to the three-dimensional EA model................................. 35
2.3.1 Simulation Details.................................................... 36
2.3.2 Measured Quantities.................................................. 38
2.3.3 Spin overlap measurement ............................................. 39
2.4 Results.................................................................... 40
2.4.1 Spin overlap ......................................................... 40
2.4.2 Characteristic population sizes in PA and correlation times in PT........... 41
2.4.3 Convergence to equilibrium ............................................ 45
2.5 Discussion ................................................................. 51
3. FINDING GROUND STATES OF SPIN GLASSES ...........................53
3.1 Introduction ............................................................... 53
3.2 Measured quantities......................................................... 55
3.3 Comparison between PA and SA.............................................. 57
3.3.1 Finding ground states with population annealing ......................... 57
3.3.2 Detailed comparison for a single sample ................................. 60
3.3.3 Disorder-averaged comparison.......................................... 64
3.4 Comparison between PA and PT ............................................. 65
3.5 Conclusion ................................................................ 66
4. MEASURING FREE ENERGY OF SPIN GLASSES .........................69
4.1 Introduction ............................................................... 69
4.2 The two-stage parallel tempering Monte Carlo.................................. 72
4.3 Results.................................................................... 73
4.3.1 Detailed comparison of a single hard sample ............................. 74
4.3.2 A large scale comparison .............................................. 76
4.4 Conclusions................................................................ 77
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