MANY-BODY FERMION DENSITY MATRICES

VOLUME I

A Dissertation

Presented to the Faculty of the Graduate School

of Cornell University

in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

by

Siew-Ann Cheong

May 2006

2006 Siew-Ann Cheong

ALL RIGHTS RESERVED

MANY-BODY FERMION DENSITY MATRICES

Siew-Ann Cheong, Ph.D.

Cornell University 2006

This four-part thesis is on the reduced many-body density matrices of systems of

noninteracting and interacting spinless fermions, and the exact solution of ladder mod-

els of interacting spinless fermions. In the first part (Chapters 2 and 3), we derived an

exact formula relating the density matrix and Green function for a cluster of sites within

a system of noninteracting spinless fermions in any dimensions. Based on the thermody-

namic form of the cluster density matrix in this exact formula, we proposed a truncation

scheme in which the new Hilbert space is built from a truncated set of spinless fermion

operators.

In the second part (Chapter 4), we studied various finite size effects in the cluster

density-matrix spectra, and looked at how these can be reduced or eliminated using the

method of twist boundary conditions averaging, for finite two-dimensional systems of

noninteracting and interacting spinless fermions. We also checked the feasibility of the

operator-based truncation scheme for interacting systems.

In the third part (Chapters 5, 6, and 8), we developed a systematic and unbiased

machinery, based on the decomposition of the density matrix of two disjoint clusters

a and b, into a sum of products of an operator on cluster a and an operator on clus-

ter b, to extract the various quantum-mechanical correlations, from a numerical exact-

diagonalization ground-state wave function. This machinery was applied to explore the

ground-state phase diagram of the extended Hubbard ladder of spinless fermions with

correlated hops (which are next-nearest-neighbor hops that occur in the presence of oc-

cupied nearest neighbors).

In the fourth part (Chapter 7), we introduced three analytical maps which allow us to

ultimately write the ground states, of the extended Hubbard ladder of spinless fermions

in three limiting cases, in terms of the one-dimensional Fermi-sea ground state. We

also introduced the technique of intervening-particle expansion, where a ladder corre-

lation function is written as a sum over conditional expectations, each of which can be

ultimately mapped to a one-dimensional Fermi-sea expectation, to calculate various cor-

relation functions in the weak inter-leg, strong inter-leg, and strong correlated hopping

limits.

BIOGRAPHICAL SKETCH

Cheong, Siew-Ann (

) was born on November 4, 1969 to Cheong Ah Sang

and Guh Ah Chu, the eldest of three sons born to them. His father Cheong Ah Sang was a

vocational institute instructor specializing in metal welding until he retired in 2003. His

mother Guh Ah Chu has had so many jobs that nobody in the family remembers every

job that she has been in. She has been a housewife, a seamstress, and a canteen operator.

As of mid-2005, she has worked as a coffeeshop assistant for ten years. Siew-Anns

two younger brothers, Siew Vim and Siew Kuan, are in the engineering and teaching

profession respectively.

Growing up in the rural northwest of Singapore, Siew-Ann has always been fasci-

nated by all manner of plants and animals, though he had never found the neighbors

irritable dogs endearing. On rare occasions, when he was not avoiding, or running for

dear life, from these furry creatures, he would end up eating one of them, when a neigh-

bor develop an irresistible craving for dog meat. At other times, he would unwittingly

sample pythons, monitor lizards, terrapins, and other forms of wild birds and beasts

which inhabit the swampy neighborhood. Gastronomical adventures aside, Siew-Ann

also enjoyed scampering up the hills during the December monsoon seasons to fly his

home-made kites, or make his way down to the streams to trudge in the foul-smelling

mud, always hoping to catch a small fish or two. He would also lose track of time play-

ing hide-and-seek in the vegetable plots with kids from the neighborhood, and come

home after sunset to find his father or mother with a cane in the hand. Sometimes,

when Siew-Ann was not being whipped for his pranks, his father would bring him on

a trek around the neighborhood, or drive to the marshes and nature reserves in Singa-

pore, or take week-long trips to various parts of peninsular Malaysia to collect butterfly

specimens.

iii

Siew-Ann never remembered much of what he learnt in Ama Keng Primary School,

where he went for his elementary school education. Apart from the stern form teacher in

Primary Two, who spanked him for fumbling on his multiplication tables, and thus left

an indelible impression on the young boy, Siew-Ann can only remember playing mar-

bles, tops, trading cards, five-stones, zero-point, Rubik cubes and other rural children

games not designed to bankrupt working-class parents. He distinctly remembers feel-

ing a sense of relief, after squatting down in the school field every Friday after recess,

trying to brush his teeth with no toothpaste and a minimal amount of gargling water,

to move on to the Chinese High School for his middle school education. He cannot

remember very clearly what he learnt there as well, and did not really mind the fact that

he was coming in near the end of the his class after every examination. He remembers

vividly, however, a swarm of young school boys (him included) in white T-shirt and red

shorts, pretending to run four kilometers every Saturday morning. When the head of

the schools Physical Education Department decided that the students were not getting

fitter, Siew-Ann and company trekked up Bukit Timah Nature Reserve every other Sat-

urday morning. He was not quite sure he had gotten any fitter, but the monkeys calling

Bukit Timah Nature Reserve home did not care, so long as the middle school students

continued to generously hand out food.

After a four-year stint in the Chinese High School, Siew-Ann moved on to the Hwa

Chong Junior College for his high school education. He has many strong memories in

this period of his life. First he was infatuated with and spurned by (approximate order

only) a girl in his class. Then he re-connected with badminton, which he picked up

as an elementary school kid in his front yard, and which is the only sports he has ever

played well at. And just as the hectic high-school life was beginning to settle into some

semblance of monotony, the school buildings were declared structurally unsafe. Thence

iv

he began a nomadic life along with the rest of his Class of 1986, attending lectures and

tutorials, first at the Ngee Ann Polytechnic, and then at the newly-completed and yet-to-

be-occupied Riverside Secondary School. He enjoyed the flux and uncertainty so much

that it actually felt fun taking the Cambridge GCE A Levels examination.

However, life was not always kind to him. A few weeks after graduating from Hwa

Chong Junior College, Siew-Ann was enlisted for military service, mandatory for every

able-bodied Singaporean male. The regimental life in the army was a shock to him,

but he managed to complete his basic military training with limbs and sanity intact,

before the powers-that-be decided that he had performed well enough to be sent to the

infantry section leader course. After spending weeks after weeks after weeks in the

jungle, combating a lack of sleep and jungle rot, as well as some fictitious enemies,

Siew-Ann finally lost his marbles and decided that he like Army life. He signed some

forms that he never looked at ever again, and was promptly put behind a desk as a

contract soldier. His jungle rot improved, but his morale did not, and so he started

writing songs to vent the frustrations of being chained to a desk, mindlessly shoving

papers.

Music and songwriting opened a whole new world to Siew-Ann: one in which lifes

most enduring moments are documented in terms of ephemeral and transient notes; one

of fleeting emotions which brush the heart and soul, and yet leave lasting scars; one of

passion and disdain; and one of things we would see with our eyes, but missed with

our hearts. In the quest for kindred spirits, Siew-Ann met a few songwriters who would

become his best friends. Together, they set up a music publishing company, and strived

to produce compilations of songs which they felt have soul and character. In those

dizzying days and nights, Siew-Ann and friends camped in recording studios, became

members of the Composers and Authors Society of Singapore (COMPASS), got swin-

v

dled of money, met with people in the business, whose highest points on their heads are

the tips of their noses, and even appeared on television! Digusted with the lack of sup-

port and resources for budding songwriters, Siew-Ann joined the executive committee

of the Young Songwriters Society (YSS) to work the underground songwriting scene in

Singapore. He organized workshops, help produced concerts, and kept his mind active

and adrenalin rushing, amidst an Army life that had morphed into something utterly

mundane.

Another major transition in Siew-Anns life came in 1993, when he broke his con-

tract with the Army, and went on to tertiary education at the National University of

Singapore (NUS). Siew-Ann found picking up books after laying them off for five years

tough, and it took him quite a while before all the stuff he learned as a high-schooler,

and was now relearning as a 24-year-old, came into focus once again. In this relearning

process, Siew-Ann remembers feeling excited after a classmate showed him how to in-

tegrate

sin2 d with the use of a trigonometric identity. Notwithstanding the fact that

he was juggling the production of a music album with the production of a music concert,

and also trying to make friends with geeks, he topped the Science faculty in his first year

of study. It surprised him very much. He was also very flattered that the professors are

treating him with a great deal of respect, and agreed to an invitation by his chemistry

professor to work with her in the summer, which culminated in the publication [G. K.

Chuah, S. Jaenicke, S. A. Cheong and K. S. Chan, The Influence of Preparation Con-

ditions on Surface Area of Zirconia, Applied Catalysis A: General, 15, 267 (1996)].

He continued to do well in physics and chemistry in his second year, even though he

was now seriously distracted courting one of his classmates. This classmate was to later

become Mrs Cheong in January 2004.

While music and songwriting opened his eyes to the temporal world of love, hope,

vi

and yearning, his stint as a Science student in NUS opened his eyes to yet another brave

new world. This is the world of reason and intellect, where through various arcane

rites of initiation (furiously copying down notes as the professors flashed transparencies

after transparencies, for example) and intense study (mugging real hard to cram one

semesters worth of information into the superficial areas of the brain real fast, so that

these information can be regurgitated on demand within two-hour examinations), one

acquires the ability at least in principle to see what had been, what is, and that

which has yet to be. Weathering an initially rocky relationship, as all couples must, with

his wife-to-be, Siew-Ann continued to do well, in his third and fourth years in NUS,

to win several prestigious academic awards. By the end of his undergraduate studies,

Siew-Ann was convinced a life in academia, a life in which the mind never rests, is right

for him.

After a one-year accelerated M.Sc. program, earning stipend to pay off his tuition

loan, Siew-Ann came to Cornell. Nominally to earn a Ph.D. in Physics, Siew-Anns

real aim was to soak in the excitement of doing science with the best minds in the

world. He was not disappointed. Through tough homeworks and gruelling term papers,

and then on to the never-ending calculations for his thesis projects, there were not many

days in his seven-and-a-half years in Ithaca, where he had failed to learn something new.

Especially memorable were the one- and two-loop calculations in PHYS 651 Relativistic

Quantum Field Theory taught by Prof. Philip Argyres, who is now at the University of

Cincinnati, the Bethe lattice renormalization-group calculations and his first term paper

on cellular automaton models for traffic simulation in PHYS 653 Statistical Physics

taught by his advisor Prof. Chris Henley, and the tight-binding calculations and plotting

of two-dimensional dispersion relations in PHYS 635 Solid State Physics I taught by

Prof. Dan Ralph. Siew-Ann also enjoyed, perhaps more than he should be allowed to,

vii

the laboratory course PHYS 510 required by the department, receiving kind permission

from Prof. Donald Hartill to continue working on the nonlinear oscillator experiment

after the spring semester of 1999 officially ended. Siew-Ann then turned in a 78-page

report for the experiment, far exceeding the usual 10-page length expected of PHYS 510

reports. In retrospect, this was a harbinger of the shape of things to come.

In the summer of 1999, Siew-Ann started working with Prof. James Sethna on the

analytical study of dislocation dynamics. Having labored the first half of summer and

making no progress whatsoever, Siew-Ann was shepherded into doing computational

studies of dislocation dynamics instead. Unfortunately, he found the prospect of work-

ing for months on a huge C++ code base for molecular dynamics simulation, before

it could be ready for actual runs, unappealing. Siew-Ann therefore switched to work-

ing with Prof. Chris Henley in his second summer, developing a numerical real-space

renormalization-group scheme based on the many-body density matrix for interacting

systems. Ironically, he ended up writing a sizeable Octave code base for these density-

matrix computations. After making good progress initially, Siew-Ann found himself

stuck, despite slogging long and hard hours hacking away at the problem. With immense

determination and perseverance, and fighting frequent bouts of disillusion, Siew-Ann fi-

nally managed to push the calculations through, to obtain the exact formula (2.4.30) in

Chapter 2. Steeled by the experience, Siew-Ann then found himself doing projects after

projects, sometimes in parallel, that led further and further away from the original goal

of a numerical renormalization-group scheme for interacting systems.

Before coming to Cornell, Siew-Ann had the naive notion that doing scientific re-

search meant formulating a general outline of a workable research program, with pri-

mary and secondary goals, as well as the methods to use all thought out. He thought that

all that was left to be done was to break the research program down into digestible and

viii

tractable stages, and then work through these stages in sequential fashion, solving piece-

meal problems along the way. Siew-Ann was surprised to learn here in Cornell that no

physicist actually does things this way. Instead, doing research is very much like a man

who finds himself lost in a forest, groping to find a way out. He would first follow what

appears to be the most promising direction out of the woods, but be forced to reconsider

his options when he gets mired down. Frequently, he would have to pick new, and more

promising directions. This was true when Siew-Ann was working with Prof. James

Sethna, but especially true of the research that he did under Prof. Chris Henley, explain-

ing why his 1278-page thesis consists of a seemingly incongruent set of calculations, all

connected in some ways, but not quite advancing an overall program of research. The

forest Siew-Ann found himself in is the forest of Many-Body Fermion Density Ma-

trices, but sometimes he find that he has to make his way across structurally different

vegetations, nested within the forest, of twist boundary conditions averaging, as well

as the exact solution of interacting spinless fermions on ladders.

Through the many thesis research projects that he has carried through to completion,

and many semi-completed mini research projects of his own, Siew-Ann who became

a proud first-time father in May 2005 has learnt, though not yet mastered, the art

of doing scientific research. From finding the right questions to ask, to identifying the

right analytical and numerical tools to use, to scheduling parallel calculations to most

efficiently tap his creative energies, to packaging results to present them in the best

possible light, Siew-Ann is becoming increasingly confident that he would soon be able

to set and pursue research programs of his own. In other words, Siew-Ann thinks he can

make his way out of the woods, should he wake up one day to find himself lost. . .

ix

To Karen and Ernest,

the number-one and number-two persons in my life

x

ACKNOWLEDGEMENTS

This thesis nearly tore our family apart emotionally, especially in the months of October

and November of 2005, when the prospects of finishing seemed most bleak. I thank my

wife, Karen Tan Sue Ling, and my son, Ernest Emmanuel Cheong Haoen, for enduring

my absence and the dire financial situation that we as a family have been in, as well as

for forgiving my missing of all but a few of Ernests milestones in life. Karen and I thank

God for giving us the courage to enter into the Sacrament of Matrimony in January 2004,

the strength to live through two years of poverty, and the inspiration needed to finish up

this thesis. We are grateful to God that Karen and I were able to spend two weeks of

quality time together before May 12, 2005, and we offer up our deepest gratitude to God

for making us a family of three on that fateful day. We thank Mother Mary and Saint

Gerard for their intercessions, constantly thank God for our son Ernest, the greatest gift

of life. We pray that we will continue to learn how to see His Grace in the most trying

of circumstances.

On behalf of Ernest, we thank the doctors and midwives, especially Dr James Fogel,

of Ob/Gyn Associates of Ithaca, and also Dr Lawrence Dolkart, of the Health Center for

Women, and Arnot-Ogden Medical Center in Elmira, New York for their pre-natal care

of Karen, before she return to Singapore in December 2004. We thank Dr Catherine

Yap for her skillful adminstration of the epidural anesthesia for Karen. We are indebted

to Dr Yeoh Swee Choo, Karens gynaecologist and obstetrician, for her pre- and post-

natal care of Karen, and her prompt decision to perform episiotomy while delivering

Ernest, when his heart stopped beating during the pushing stage. We thank Dr Lee Bee

Wah, Ernests pediatrician, for helping clear Ernests lungs of amniotic fluid, so that he

can take his first breath. We thank the midwives and nursing staff of Mount Elizabeth

Hospital. We are especially grateful to lactation specialist Tan Guat Choo, for being

xi

patient with Ernest, as Karen and him both learn the ropes of breast feeding. We thank

Auntie Christina Kwek for her help and encouragement, without which Karen and I

would probably have decided to give up breast feeding.

We thank Karens parents, Tan Heng Gui (Grandpa) and Rosalin Lim Mui Keng

(Grandma), for their care and support of Karen in the months before Ernest was born,

and especially for doting on Ernie boy after he was born. We thank them for giving this

poor little family a roof over our heads, and all the delicious and nutritious food that far

far surpasses mere sustenance. We thank them for all the clothes and toys they bought

for Ernie boy. We thank Grandma for rocking Ernie boy to sleep every night while he

was still a tiny little baby, and for organizing Ernie boys one-month-old party. We thank

Grandpa for giving Ernie boy the car rides he enjoys everyday, and being Ernie boys

favorite play buddy. We thank Auntie Lily for coming almost everyday to cook for the

family, and help look after Ernie boy while Karen was still in confinement. We thank

her for continuing to come help take care of Ernie boy when Karen did a relief teaching

stint from July to November 2005. We thank her for sharing her love with Ernie boy. We

thank my parents, Cheong Ah Sang and Guh Ah Chu (Po Po), for being there for me

when I was young, and for teaching me in their own silent ways, what a family meant.

We thank Po Po for coming to visit Ernie boy and us, even when she is busy ekeing out

a living. We also thank Miss Lim Oon Hua for visiting Karen and Ernest at the hospital,

Mrs Yeo Shih Fong for visiting Karen and Ernest at the hospital and at home, and Mrs

Lek Bee Hoon for making baby-sized pillows and bolsters and their cases, in addition

to visiting Karen and Ernest at home. We thank the Kong family for various gifts and

presents for Karen and Ernest, especially for the hand-sewn patchwork blanket for Ernie

boy. We are very grateful to Joachim and Maria Sagaram for agreeing to be Ernie boys

godparents.

xii

We thank Father Mose Yap, formerly of Saint Mary of the Angels Church in Bukit

Batok, Singapore, and now ministering in Sabah, East Malaysia, for being our spiritual

guide in Singapore before we got married. We thank the Cornell Catholic Community

for being our spiritual pillar, for the year (2003) before we got married, the year (2004)

when both of us were here in Ithaca, and for the year (2005) that we were separated by

20,000 kilometers. We thank Father Michael Mahler, now Pastor of Saint Catherine of

Siena Church in Ithaca, Father Robert Smith and Father Daniel McMullin, for sharing

their wisdom and reflections on the biblical readings, and their constant challenge and

reminder to live an authentic Christian life. We thank the chaplains, Theresa Miller and

Phil Fiadino for being the often overlooked threads that weave the community together.

We especially thank Sister Donna Fannon, for preparing me for the Sacraments of Bap-

tism, First Communion, and Confirmation, and organizing the various Catholic couple

get-togethers. We thank Prof. Piet Brouwer for agreeing to be my godfather, and Piet,

Gerburg and Benedict for inviting us to their home to play with Benedict, and to share

fellowship and meals. We thank the Goettels, Mike, Sara and Hans for their fellowship

at Annabel Taylor and also at their cozy home. We thank the Henrys, Rob and Cassie,

for taking up the unenviable task of organizing the Emmaus Bible discussion group,

from which we have benefitted a great deal. We thank the Smallidges, Pete, Kelly, Na-

talie and Adelaide, for inviting me to their log-cabin home for a late-summer gathering

last year. We thank them all for their comforting presence every Sunday morning, when

Karen and I were both here in Ithaca in 2004, and especially in 2005, when I was here

all alone.

I thank all the past and present members of the Cornell Badminton Club, for the

camaraderie and thrill I enjoyed on the courts of Helen Newman Hall (and occasionally

in Barton Hall). I thank the old guards who are forever here, injuries notwithstanding:

xiii

Prof. Anil Netravali, faculty advisor to the club, Xiaoshi Jin, Cornell alumni and for-

mer president of the club, David Sagan, who has been here and playing badminton for

as far back as anyone can remember, and David Post, for his good humor. I thank all

the members who graciously invited me to join them, when I started playing with the

club in 2000, and who has since left Cornell: George, who was president of the club

for 1999/2000; Panarairat Srichaiyarat, president of the club for 2000/2001 and now an

Assistant Professor at Chiang Mai University, Thailand; Wipas Wimonsate, now at the

Thailand Environment Institute; Jinlin Peng, who used to drive all the way to Ithaca

from Corning to play badminton every Saturday morning; Lidong Zhou, now with Mi-

crosoft Research, and wife; and Zhong Huang. I thank my sparring partners, who has

given me many good games, before going where their careers led them: Martin Roth,

who flew all the way to Singapore for my wedding; Rachel Zhang, former Johnson

School Assistant Professor and now faculty at the Hong Kong University of Science and

Technology, and her husband Paul Zhu; Dewi Novianty; Kimberley Mok, now in New

York City; Jacky Cheung (no, not the Hong Kong Heavenly King, but no less charming

to the ladies!); Betty Ka Bik Luk and Wallace Leung; Lincoln Yeh, the southpaw law

student who transferred from Cornell to Columbia; Xianzheng Zhang, a former post-

doctoral researcher in Cornell, whom I used to have a lot of trouble beating in singles

games; Xiaochun Fan, now at Harvard Medical School as a postdoctoral researcher;

Kiran Kannancheri; Roman Boltnev, who was here for a year as a NATO fellow work-

ing on low-temperature physics with Prof. David Lee, and who was here again for the

summer of 2005; Ishan Sharma, now in the Department of Applied Mathematics and

Theoretical Physics, Cambridge University as a postdoctoral researcher, and plays on

the Cambridge University badminton team. Karen and I thank him for giving us his

microwave for free before he left for Cambridge; Jonas Korlach, now in San Diego; Si-

xiv

mon Maddick, a British student who came to Cornell for a one-year exchange program

last year; Radha Narayan; Arthur Chuan Lee; Jingfeng Zeng, now in Texas, whom I

had many Saturday-afternoon singles games with; Yen Thi Hong Cu, now in Yale as a

doctoral student; Ningxi Zhang; Clement Xiaofei Wang; and Linghao Zhong. I thank

fellow Singaporeans in the club who has graduated from Cornell and returned to fulfill

their promises to the people of Singapore, for conversing with me using the undecipher-

able Singlish: Peichin Lim; James, who diminuitive stature lures you into a false sense

of security, until you get hit by his powerful smashes; and Yiliang Cannonball Ho,

now with the Republic of Singapore Airforce, who makes his statements with the sonic

booms of his smashes. Karen and I are indebted to Feng Zhao, now an Assistant Pro-

fessor of Finance and Economics at Rutgers University, for giving us his 1991 Dodge

Dynasty when he left Cornell in August 2004. Since then Karen and I were spared the

30-minute walk from Maplewood to Helen Newman every morning for Karens aerobics

classes. I thank all the present members who give me fast games: Archit Lal, Shengfeng

Shen, Ram Narayan, Hengky Chandrahalim, Mike Hsu, Chula Watugala, Winson Lin,

Mark Law, Daphne Ng, Zhongtao Fu, Sze Wei Ang, Yuping Zhou, Karen Lin, Seung Jin

Oh, Peeyush Bhargava, Prof. David Strang, Charlie Wang and wife; as well those who

give me slow but fun games: I-ho Chang, Ivan Han, Jeng-Huei Chen, Terence Tan, and

Evelyn Choi. I like them to know that the adrenalin rushes kept me going, through the

frustrating parts of the thesis research, and especially the five agonizing months spent

writing this thesis.

I thank fellow doctoral students Lek Heng Lim, who came to Cornell as a Ph.D.

student in Mathematics in the same year as me, was housemate with me for a year,

and transferred to Stanford University a year after that; Hway Kiong Lim, who came to

Cornell as a Ph.D. student in Mathematics one year after us, and was housemate with me

xv

for three years; and Kelken Chang, who came to Cornell as a Ph.D. student in Physics

in 2002. I thank them for having lunch or dinner with me daily, either at the Ivy Room

and the Trillium on campus; Burger King, Subway and Ling Ling Garden restaurant in

East Hill; the Hong Kong restaurant, the Little Thai House restaurant, the Vietnamese

restaurant, and Sangam Indian restaurant in Collegetown, and understandingly hearing

out my many grouses. I thank Lek Heng and Hway Kiong for frequently chiding me for

the lack of mathematical rigor in my calculations, but yet always remain ready to extend

helpful suggestions. I especially thank Hway Kiong, for his expert help setting up Linux

on my computers, and also for taking so many fantastic badminton action photographs

of me.

I thank Prof. Chris Henley for his guidance throughout my thesis work, writing let-

ters of recommendation to the various positions and fellowships that I have applied to

in the past year, and showing me glimpses of what goes on in a geniuss mind. I thank

Prof. James Sethna and Prof. Paul McEuen for agreeing to sit on my Special Committee,

and Prof. Sethna for writing letters of recommendation for my various job and fellow-

ship applications. While writing up the results in Chapter 3, which became Ref. 198,

I have benefitted from discussions with Prof. James P. Sethna on numerical integration

schemes and their expected dependence on the integration grid size. I thank Prof. Garnet

Chan of the Cornell Chemistry Department, for illuminating me on how to numerically

implement the fast trace-down algorithm used in Chapter 4. I thank Prof. N. David

Mermin for helping clear a confusion of mine, while I was formulating the operator sin-

gular value decomposition of the correlation density matrix in Chapters 5 and 6. I thank

Prof. Piet Brouwer for showing me how to do the contour integration in Figure 5.8. I

also thank Ellen Hertz and Herman Rubin, for pointing out, on sci.math.research,

references to the Lindeberg Theorem used in Chapter 3.

xvi

Karen and I thank Prof. Vinay Ambegaokar, who showed genuine concern for the

long time I have spent on my doctoral studies. Though not on my Special Committee,

he graciously agreed to help me write letters of recommendation, to the postdoctoral po-

sitions I have applied unsuccessfully for, to my application for the Agency for Science,

Technology and Research (A*Star) Postdoctoral Fellowship, and to my application to

join the faculty at the Nanyang Technological University.

Last, and certainly not the least, we thank all those who offer us hopes for a better

tomorrow, in the form of gainful employment! We like to thank Chris Myers and Dave

Schneider, for offering me a postdoctoral position at the Cornell Theory Center, after

my own five-month search for a postdoctoral position has proven fruitless. We also

thank Prof. Lee Soo Ying, Dean of the School of Physical and Mathematical Sciences,

Nanyang Technological University, Singapore, and Prof. Alfred Huan Cheng Hon, Head

of the Physics and Applied Physics Division there, for offering to take me on their

faculty upon my return to Singapore.

The research done for this thesis was first supported by National Science Foundation

grant DMR-9981744, and then by National Science Foundation grant DMR-0240953.

I also made use of the computing facility of the Cornell Center for Materials Research

(CCMR), with support from the National Science Foundation Materials Research Sci-

ence and Engineering Centers (MRSEC) program (DMR-0079992).

xvii

TABLE OF CONTENTS

1 Introduction 11.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Exact Numerical Methods for Finite Lattices . . . . . . . . . . . . . . 4

1.2.1 Exact and Lanczos Diagonalization . . . . . . . . . . . . . . 41.2.2 Quantum Monte Carlo . . . . . . . . . . . . . . . . . . . . . 71.2.3 The ALPS Project . . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Quantum Renormalization Group Methods . . . . . . . . . . . . . . . 191.3.1 Block-Spin Type QRG . . . . . . . . . . . . . . . . . . . . . 201.3.2 QRG With No Blocking Scheme . . . . . . . . . . . . . . . . 21

1.4 Road Map to Chapters and Results . . . . . . . . . . . . . . . . . . . 231.4.1 Overview of the Chapters . . . . . . . . . . . . . . . . . . . 231.4.2 Highlights of Chapter 2 . . . . . . . . . . . . . . . . . . . . . 281.4.3 Highlights of Chapter 3 . . . . . . . . . . . . . . . . . . . . . 311.4.4 Highlights of Chapter 4 . . . . . . . . . . . . . . . . . . . . . 331.4.5 Highlights of Chapter 5 . . . . . . . . . . . . . . . . . . . . . 351.4.6 Highlights of Chapter 6 . . . . . . . . . . . . . . . . . . . . . 361.4.7 Highlights of Chapter 7 . . . . . . . . . . . . . . . . . . . . . 391.4.8 Highlights of Chapter 8 . . . . . . . . . . . . . . . . . . . . . 451.4.9 Highlights of the Appendices . . . . . . . . . . . . . . . . . . 47

2 Exact Formula 502.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2 Density-Matrix Formulation of Quantum Mechanics . . . . . . . . . 53

2.2.1 Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 532.2.2 Reduced Density Matrix of a Subsystem . . . . . . . . . . . . 55

2.3 Density Matrix of a Finite Cluster . . . . . . . . . . . . . . . . . . . 582.3.1 Method of Referencing Operators . . . . . . . . . . . . . . . 592.3.2 Example Calculations for Finite Clusters . . . . . . . . . . . 602.3.3 Index Structure of the Cluster Density-Matrix Elements . . . 732.3.4 Conjecture Based on Index Structure . . . . . . . . . . . . . 82

2.4 Derivation of Closed-Form Formula . . . . . . . . . . . . . . . . . . 832.4.1 Exponential Form for System Density Matrix . . . . . . . . . 852.4.2 Key Formulas Involving Grassmann Variables . . . . . . . . . 862.4.3 Matrix Block Form . . . . . . . . . . . . . . . . . . . . . . . 882.4.4 Tracing Down the System Density Matrix . . . . . . . . . . . 892.4.5 Alternative Regularization . . . . . . . . . . . . . . . . . . . 93

2.5 Structure and Properties of Cluster Density Matrix . . . . . . . . . . 942.5.1 The Single-Particle Pseudo-Energies . . . . . . . . . . . . . . 942.5.2 Particle-Hole Symmetry at Half-Filling . . . . . . . . . . . . 952.5.3 The Many-Particle Cluster Density-Matrix Eigenstates . . . . 100

xviii

3 Operator-Based DM Truncation Scheme 1043.1 A Quick Guide to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 1043.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.3 Operator-Based Density Matrix Truncation . . . . . . . . . . . . . . 109

3.3.1 Structure of Density-Matrix Eigenvalues and Eigenstates . . . 1093.3.2 Pseudo-Energies and Cluster Green-Function Matrix . . . . . 1113.3.3 Recipe for Operator-Based Truncation . . . . . . . . . . . . . 112

3.4 Asymptotics of Eigenvalue Distribution . . . . . . . . . . . . . . . . 1143.5 Scaling Behaviour of Eigenvalue Distribution . . . . . . . . . . . . . 119

3.5.1 Pseudo-Energies and Pseudo-Occupation Numbers . . . . . . 1203.5.2 Normalization Constant . . . . . . . . . . . . . . . . . . . . 124

3.6 Density-Matrix Weights: Implications of Eigenvalue Scaling . . . . . 1263.6.1 The Gapless Chain of Noninteracting Spinless Fermions . . . 1273.6.2 The Gapped Chain of Noninteracting Spinless Fermions . . . 1293.6.3 Largest Density-Matrix Weight . . . . . . . . . . . . . . . . 1333.6.4 Discarded Weight . . . . . . . . . . . . . . . . . . . . . . . . 136

3.7 Entanglement Entropy . . . . . . . . . . . . . . . . . . . . . . . . . 1413.7.1 Relation to Green-Function Matrix Eigenvalues . . . . . . . . 1423.7.2 Numerical Evaluation of the Entanglement Entropy . . . . . . 1443.7.3 Entanglement Entropy for Even- and Odd-Sized Clusters . . . 1473.7.4 Calculating S From Our Scaling Form . . . . . . . . . . . . . 1473.7.5 Improved Scaling Form for Pseudo-Energy . . . . . . . . . . 153

3.8 Single-Particle Density-Matrix Eigenfunctions . . . . . . . . . . . . . 1553.8.1 A Priori Expectations . . . . . . . . . . . . . . . . . . . . . . 1553.8.2 General Features . . . . . . . . . . . . . . . . . . . . . . . . 1573.8.3 Scaling Behaviour . . . . . . . . . . . . . . . . . . . . . . . 158

3.9 Operator-Based Density-Matrix Truncation Calculation of DispersionRelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633.9.1 Energy Gap at Fermi Level . . . . . . . . . . . . . . . . . . . 1633.9.2 Fermi Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 1683.9.3 Real-Space Structure of Eigenfunctions of Truncated

Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.9.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

3.10 Operator-Based Plane-Wave Truncation Scheme . . . . . . . . . . . . 1733.10.1 Exact Dispersion at Zone Center . . . . . . . . . . . . . . . . 1733.10.2 Energy Gap at Zone Boundary . . . . . . . . . . . . . . . . . 1763.10.3 Fermi Velocity . . . . . . . . . . . . . . . . . . . . . . . . . 178

4 Cluster Density Matrices for Two-Dimensional Systems 1834.1 A Quick Guide to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 1834.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.3 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

4.3.1 Occupation Number Basis States . . . . . . . . . . . . . . . . 1874.3.2 Cluster Density-Matrix Elements . . . . . . . . . . . . . . . 189

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4.3.3 Computational Implementation . . . . . . . . . . . . . . . . 2004.4 The Five-Site, Cross-Shaped Cluster . . . . . . . . . . . . . . . . . . 204

4.4.1 Symmetries and Irreducible Representations . . . . . . . . . . 2044.4.2 One-Particle Cluster States . . . . . . . . . . . . . . . . . . . 2044.4.3 Two-Particle Cluster States . . . . . . . . . . . . . . . . . . . 2064.4.4 Symmetry Transformations on the Cluster States . . . . . . . 208

4.5 System Shapes and Sizes . . . . . . . . . . . . . . . . . . . . . . . . 2124.6 Noninteracting Spinless Fermions . . . . . . . . . . . . . . . . . . . 216

4.6.1 Finite Size Effects and the Infinite-System Limit . . . . . . . 2184.6.2 One-Particle Weights at Small Filling Fractions . . . . . . . . 2364.6.3 Twist Boundary Conditions Averaging . . . . . . . . . . . . . 244

4.7 Strongly-Interacting Spinless Fermions . . . . . . . . . . . . . . . . 2544.7.1 Twist Boundary Conditions Averaging . . . . . . . . . . . . . 2544.7.2 Zero-Particle Sector of Cluster Density Matrix . . . . . . . . 2624.7.3 One-Particle Sector of Cluster Density Matrix . . . . . . . . . 2624.7.4 Two-Particle Sector of Cluster Density Matrix . . . . . . . . . 266

5 Correlation Density Matrix 2695.1 A Quick Guide to Chapter 5 . . . . . . . . . . . . . . . . . . . . . . 2695.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2695.3 One-Dimensional Noninteracting Spinless Fermions . . . . . . . . . 271

5.3.1 Two Clusters with One Site Each . . . . . . . . . . . . . . . 2715.3.2 Two Clusters with Two Sites Each . . . . . . . . . . . . . . . 2755.3.3 The (2 + 2) Supercluster Density Matrix . . . . . . . . . . . . 2915.3.4 The (2 + 2) Correlation Density Matrix . . . . . . . . . . . . 297

5.4 One-Dimensional Superconducting Ground State . . . . . . . . . . . 3025.4.1 BCS Ground State . . . . . . . . . . . . . . . . . . . . . . . 3035.4.2 The (1 + 1) Supercluster . . . . . . . . . . . . . . . . . . . . 3255.4.3 The (2 + 2) Supercluster . . . . . . . . . . . . . . . . . . . . 3285.4.4 Particle-Number-Projected BCS Wave Function . . . . . . . . 3345.4.5 The (1 + 1) Supercluster . . . . . . . . . . . . . . . . . . . . 353

Volume II 357

6 Singular Value Decomposition 3586.1 A Quick Guide to Chapter 6 . . . . . . . . . . . . . . . . . . . . . . 3586.2 Matrix Singular Value Decomposition . . . . . . . . . . . . . . . . . 3596.3 Operator Singular Value Decomposition . . . . . . . . . . . . . . . . 362

6.3.1 Frobenius Operator Orthonormalization . . . . . . . . . . . . 3646.3.2 Operator Expansion of Correlation Density Matrix . . . . . . 365

6.4 One-Dimensional Noninteracting Spinless Fermions . . . . . . . . . 3696.4.1 The (1 + 1) Supercluster . . . . . . . . . . . . . . . . . . . . 3696.4.2 The (2 + 2) Supercluster . . . . . . . . . . . . . . . . . . . . 3726.4.3 Approximate Decomposition in the n 0 Limit . . . . . . . 374

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6.4.4 Approximate Decomposition at Half-Filling: Even Cluster Sep-aration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377

6.4.5 Approximate Decomposition at Half-Filling: Odd Cluster Sep-aration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

6.5 One-Dimensional Superconducting Ground State . . . . . . . . . . . 4046.5.1 BCS Ground State . . . . . . . . . . . . . . . . . . . . . . . 404

7 Long-Range Orders in A Luttinger Liquid 4067.1 A Quick Guide to Chapter 7 . . . . . . . . . . . . . . . . . . . . . . 4067.2 The Physics of Tomonaga-Luttinger and Luther-Emery Liquids . . . . 407

7.2.1 The One-Dimensional Hubbard Model . . . . . . . . . . . . 4107.2.2 The One-Dimensional t-J Model . . . . . . . . . . . . . . . . 4137.2.3 The Hubbard and t-J Ladders . . . . . . . . . . . . . . . . . 415

7.3 Analytical Limits of Interacting Spinless Fermions on a Ladder . . . . 4227.3.1 Correlated Hops and Superconductivity . . . . . . . . . . . . 4237.3.2 The Three Limiting Cases: An Overview . . . . . . . . . . . 426

7.4 Analytical Machinery . . . . . . . . . . . . . . . . . . . . . . . . . . 4327.4.1 Jordan-Wigner Transformation . . . . . . . . . . . . . . . . . 4347.4.2 Nearest-Neighbor Exclusion Map . . . . . . . . . . . . . . . 4367.4.3 Configuration-to-Configuration Mapping . . . . . . . . . . . 4377.4.4 Bloch-State-to-Bloch-State Mapping . . . . . . . . . . . . . . 4427.4.5 Wave-Vector-to-Wave-Vector Map . . . . . . . . . . . . . . . 4607.4.6 Relation Between Expectations . . . . . . . . . . . . . . . . 4697.4.7 Intervening-Particle Expansion . . . . . . . . . . . . . . . . . 476

7.5 Infinitely-Strong Correlated Hops . . . . . . . . . . . . . . . . . . . 4807.5.1 Bound Pairs as Effective Degrees of Freedom . . . . . . . . . 4817.5.2 Ground-State Wave Functions For a Single Bound Pair . . . . 4847.5.3 Ground State For Two Bound Pairs . . . . . . . . . . . . . . 4887.5.4 Mapping Bound Pairs to Hardcore Bosons . . . . . . . . . . . 4947.5.5 General Properties of Ladder Correlation Functions . . . . . . 5057.5.6 Superconducting Correlations . . . . . . . . . . . . . . . . . 5077.5.7 Charge-Density-Wave Correlations . . . . . . . . . . . . . . 5387.5.8 Fermi-Liquid Correlations . . . . . . . . . . . . . . . . . . . 554

7.6 Zero Inter-Leg Hopping . . . . . . . . . . . . . . . . . . . . . . . . . 5617.6.1 Structure of Ground State . . . . . . . . . . . . . . . . . . . 5617.6.2 Fermi-Liquid Correlations . . . . . . . . . . . . . . . . . . . 5707.6.3 Charge-Density-Wave Correlations . . . . . . . . . . . . . . 5787.6.4 Superconducting Correlations . . . . . . . . . . . . . . . . . 5937.6.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . 602

7.7 Very Strong Inter-Leg Hopping . . . . . . . . . . . . . . . . . . . . . 6117.7.1 Structure of Ground State . . . . . . . . . . . . . . . . . . . 6117.7.2 Fermi-Liquid Correlation . . . . . . . . . . . . . . . . . . . . 6167.7.3 Charge-Density-Wave Correlation . . . . . . . . . . . . . . . 6307.7.4 Superconducting Correlations . . . . . . . . . . . . . . . . . 639

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8 Numerical Studies of Order Parameters 6538.1 Chapter Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 653

8.1.1 A Quick Guide to Chapter 8 . . . . . . . . . . . . . . . . . . 6538.1.2 Exploring the Ground-State Phase Diagram . . . . . . . . . . 654

8.2 Computational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 6568.2.1 Ground-State Structure Analysis . . . . . . . . . . . . . . . . 6578.2.2 Choice of Clusters and Order Parameters . . . . . . . . . . . 6668.2.3 Singular Value Analysis . . . . . . . . . . . . . . . . . . . . 6758.2.4 Twist Boundary Conditions Averaging . . . . . . . . . . . . . 679

8.3 Analysis of Quarter-Filled ED Ground States . . . . . . . . . . . . . 6818.3.1 Known Limits . . . . . . . . . . . . . . . . . . . . . . . . . 6818.3.2 Phase Boundaries . . . . . . . . . . . . . . . . . . . . . . . 7378.3.3 Tricritical Point . . . . . . . . . . . . . . . . . . . . . . . . 749

8.4 Analysis of Singular Values . . . . . . . . . . . . . . . . . . . . . . . 7588.4.1 The (1 2) + (1 2) Supercluster . . . . . . . . . . . . . . . 7608.4.2 Known Limits . . . . . . . . . . . . . . . . . . . . . . . . . 7668.4.3 Phase Boundaries . . . . . . . . . . . . . . . . . . . . . . . 7898.4.4 Tricritical Point . . . . . . . . . . . . . . . . . . . . . . . . 797

Volume III 800

9 Conclusions 8019.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8019.2 Reduced Density Matrices of Noninteracting Spinless Fermions . . . 802

9.2.1 Method of Referencing Operators . . . . . . . . . . . . . . . 8039.2.2 Exact Formula . . . . . . . . . . . . . . . . . . . . . . . . . 8049.2.3 Structure, Properties and Statistical Mechanics Analogy . . . 8059.2.4 Operator-Based Density-Matrix Truncation Scheme . . . . . . 8089.2.5 Scaling of Density-Matrix Weights and Eigenfunctions . . . . 8129.2.6 Finite Two-Dimensional Systems . . . . . . . . . . . . . . . 8169.2.7 Relevance to Interacting Systems . . . . . . . . . . . . . . . 817

9.3 Reduced Density Matrices of Interacting Spinless Fermions . . . . . . 8199.3.1 The Extended Spinless Hubbard Model . . . . . . . . . . . . 8209.3.2 Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8219.3.3 Computational Implementation . . . . . . . . . . . . . . . . 8229.3.4 Twist Boundary Conditions Averaging . . . . . . . . . . . . . 8239.3.5 Operator-Based Density-Matrix Truncation Scheme . . . . . . 824

9.4 Systematic Extraction of Ground-State Correlations . . . . . . . . . . 8259.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8259.4.2 Correlation Density Matrix . . . . . . . . . . . . . . . . . . . 8269.4.3 Operator Singular Value Decomposition . . . . . . . . . . . . 8279.4.4 Exploration of Phase Diagram . . . . . . . . . . . . . . . . . 8299.4.5 Analysis of Ground-State Wave Function . . . . . . . . . . . 8319.4.6 Analysis of Singular Values . . . . . . . . . . . . . . . . . . 831

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9.5 Critical Study of Twist Boundary Conditions Averaging . . . . . . . . 8339.5.1 Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8339.5.2 Wave Vector Sampling and Ground State Selection . . . . . . 8349.5.3 Twist Boundary Conditions Averaging Correlation Functions . 8369.5.4 Numerical Integration Schemes . . . . . . . . . . . . . . . . 837

9.6 Analytical Limits of Extended Spinless Hubbard Ladder . . . . . . . 8399.6.1 Overview of Methods . . . . . . . . . . . . . . . . . . . . . . 8399.6.2 Weak Inter-Leg Hopping Limit . . . . . . . . . . . . . . . . . 8429.6.3 Strong Inter-Leg Hopping Limit . . . . . . . . . . . . . . . . 8439.6.4 Strong Correlated Hopping Limit . . . . . . . . . . . . . . . 845

A Correlation Functions in a One-Dimensional Fermi Sea 848A.1 Two-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 848A.2 2n-Point Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 849

B Matrix Block Inversion Formula 851

C The Octave Code Base 853C.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853

C.1.1 A Quick Guide to Appendix C . . . . . . . . . . . . . . . . . 853C.1.2 GNU Octave . . . . . . . . . . . . . . . . . . . . . . . . . . 856C.1.3 Model Hamiltonians . . . . . . . . . . . . . . . . . . . . . . 857C.1.4 Design and Organization . . . . . . . . . . . . . . . . . . . . 858

C.2 System Definition and Utilities . . . . . . . . . . . . . . . . . . . . . 861C.2.1 Automatic Site Indexing . . . . . . . . . . . . . . . . . . . . 861C.2.2 List of Nearest Neighbors . . . . . . . . . . . . . . . . . . . 866C.2.3 List of Next-Nearest Neighbors . . . . . . . . . . . . . . . . 869C.2.4 Reciprocal Lattice Vectors in the First Brillouin Zone . . . . . 871C.2.5 Periodic Boundary Conditions . . . . . . . . . . . . . . . . . 873C.2.6 Other Utility Helper Functions . . . . . . . . . . . . . . . . . 877

C.3 Hilbert Space Definition and Utilities . . . . . . . . . . . . . . . . . . 879C.3.1 Occupation Number Representation . . . . . . . . . . . . . . 879C.3.2 Building Up the Hilbert Space Matrix . . . . . . . . . . . . . 880C.3.3 Other Utility Helper Functions . . . . . . . . . . . . . . . . . 895

C.4 Hamiltonian Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . 896C.4.1 Hardcore Bosons . . . . . . . . . . . . . . . . . . . . . . . . 900C.4.2 Spinless Fermions . . . . . . . . . . . . . . . . . . . . . . . 902C.4.3 Faster Algorithm for Spinless Fermions . . . . . . . . . . . . 904C.4.4 Compact Storage of Hamiltonian Matrices . . . . . . . . . . . 909C.4.5 Adding Correlated Hops to the Hamiltonian . . . . . . . . . . 912C.4.6 Anisotropic Spinless Extended Hubbard Hamiltonian . . . . . 919

C.5 Translational Symmetry and Bloch States . . . . . . . . . . . . . . . 922C.5.1 Translating a Many-Particle Configuration . . . . . . . . . . . 922C.5.2 Partitioning the Hilbert Space . . . . . . . . . . . . . . . . . 926

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C.5.3 Constructing the Bloch States . . . . . . . . . . . . . . . . . 933C.5.4 Compact Storage of Bloch States . . . . . . . . . . . . . . . 939

C.6 Bloch-Reduced Hamiltonian . . . . . . . . . . . . . . . . . . . . . . 944C.6.1 Construction from Full Hamiltonian Matrix . . . . . . . . . . 945C.6.2 Construction from Compact Hamiltonian Array . . . . . . . . 947

C.7 Exact Diagonalizing the Bloch-Reduced Hamiltonian Matrix . . . . . 951C.8 Cluster Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . 957

C.8.1 Cluster Occupation Partitioning . . . . . . . . . . . . . . . . 957C.8.2 Fock-Hilbert Space on a Cluster . . . . . . . . . . . . . . . . 960C.8.3 Calculating a Cluster Density Matrix Sector . . . . . . . . . . 963

C.9 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 967C.9.1 Ground-State Degeneracy . . . . . . . . . . . . . . . . . . . 967C.9.2 Alternative Argument for Degeneracy Averaging . . . . . . . 974C.9.3 Orientation of System . . . . . . . . . . . . . . . . . . . . . 975C.9.4 Twist Boundary Conditions . . . . . . . . . . . . . . . . . . 983

C.10 Noninteracting Spinless Fermions . . . . . . . . . . . . . . . . . . . 983C.10.1 Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . 985C.10.2 Cluster Green Function Matrix . . . . . . . . . . . . . . . . . 987C.10.3 Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989

C.11 Correlation Density Matrix . . . . . . . . . . . . . . . . . . . . . . . 990C.11.1 Cluster and Supercluster Fock-Hilbert Spaces . . . . . . . . . 990C.11.2 Compactifying the Cluster Density Matrix . . . . . . . . . . . 993C.11.3 Correlation Density Matrix . . . . . . . . . . . . . . . . . . . 995C.11.4 Correlation-K Matrix . . . . . . . . . . . . . . . . . . . . . . 998

D Twist Boundary Conditions Averaging 1002D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1002D.2 Twist Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 1007

D.2.1 One-Dimensional Noninteracting Spinless Fermions . . . . . 1007D.2.2 Two-Dimensional Noninteracting Spinless Fermions . . . . . 1014D.2.3 Implications for Twist Boundary Conditions Averaging . . . . 1015

D.3 Wave Vector Sampling and Ground-State Selection . . . . . . . . . . 1022D.3.1 One-Dimensional Noninteracting Spinless Fermions . . . . . 1023D.3.2 Two-Dimensional Noninteracting Spinless Fermions . . . . . 1029D.3.3 Restricting Wave Vector Sampling to the FBZ . . . . . . . . . 1063

D.4 Twist Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1067D.4.1 Phase Shifts and Occupations . . . . . . . . . . . . . . . . . 1068

D.5 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 1075D.5.1 Twist Boundary Conditions and FBZ Averaging . . . . . . . . 1075D.5.2 Two-Point Functions in One Dimension . . . . . . . . . . . . 1078D.5.3 2n-Point Functions in One Dimension . . . . . . . . . . . . . 1078D.5.4 2n-Point Functions in Two Dimensions . . . . . . . . . . . . 1081

D.6 Twist Boundary Conditions Averaging and Brillouin Zone Integration 1090D.6.1 Brillouin Zone Integration . . . . . . . . . . . . . . . . . . . 1091

xxiv

D.6.2 Special-Point Integration Scheme . . . . . . . . . . . . . . . 1092D.6.3 Tetrahedron Integration Scheme . . . . . . . . . . . . . . . . 1111D.6.4 Finite-Element Tetrahedron Integration . . . . . . . . . . . . 1121D.6.5 Automatic Coarse Mesh Generation . . . . . . . . . . . . . . 1134D.6.6 Automatic Fine Mesh Generation . . . . . . . . . . . . . . . 1140D.6.7 Improving Upon the FBZ Integration . . . . . . . . . . . . . 1145

D.7 Octave Functions for Twist Boundary Conditions Averaging . . . . . 1148D.7.1 Hamiltonian Matrix Elements . . . . . . . . . . . . . . . . . 1148D.7.2 Bond-to-Boundary Gauge Transformation . . . . . . . . . . . 1152D.7.3 Degeneracy and Shape Averaging . . . . . . . . . . . . . . . 1152D.7.4 Noninteracting Spinless Fermions . . . . . . . . . . . . . . . 1158

E Finite Element Analysis 1163E.1 Standard Quadratic-Order Square Finite Element . . . . . . . . . . . 1163E.2 Standard Triangular Element . . . . . . . . . . . . . . . . . . . . . . 1166E.3 Nonstandard Finite Elements . . . . . . . . . . . . . . . . . . . . . . 1167E.4 Curvilinear Finite Elements . . . . . . . . . . . . . . . . . . . . . . . 1169

Bibliography 1175

xxv

LIST OF TABLES

4.1 List of finite R1R2 systems that can exactly diagonalized on Pentium-4-class machines with up to 1 GB of RAM, within a reasonable amountof time, with twist boundary conditions averaging. The number of sitesN within the system, the size D of the P = 4 particle Hilbert space, theorientation of the system relative to the underlying square lattice, andthe sites of the chosen cluster relative to the system, are also shown. . 213

4.1 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.1 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2154.2 The zero- and one-particle cluster density-matrix weights, and the sine

of the mixing angle , of the (4, 1) (1, 3) system with N = 11 sitesand P = 1, 2, 3, 4, 5 noninteracting spinless fermions subject to periodicboundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.3 The zero- and one-particle cluster density-matrix weights, and the sineof the mixing angle , of the (3,2)(2, 3) system with N = 13 sites andP = 1, 2, 3, 4, 5, 6 noninteracting spinless fermions subject to periodicboundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 219

4.4 The zero- and one-particle cluster density-matrix weights, and the sineof the mixing angle , of the (4, 1) (1, 4) system with N = 15 sitesand P = 1, 2, 3, 4, 5, 6, 7 noninteracting spinless fermions subject toperiodic boundary conditions. . . . . . . . . . . . . . . . . . . . . . 220

4.5 The zero- and one-particle cluster density-matrix weights, and the sineof the mixing angle , of the (4,1) (1, 4) system with N = 17 sitesand P = 1, 2, 3, 4, 5, 6, 7, 8 noninteracting spinless fermions subject toperiodic boundary conditions. . . . . . . . . . . . . . . . . . . . . . 221

4.6 The zero- and one-particle cluster density-matrix weights, and the sineof the mixing angle , of the (4,2)(2, 4) system with N = 20 sites andP = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 noninteracting spinless fermions subject toperiodic boundary conditions. . . . . . . . . . . . . . . . . . . . . . 222

4.7 Power-law fits of the one-particle cluster density-matrix weights to w =A n, for the (64,16) (16, 64) system of 1 P 200 noninteractingspinless fermions subject to periodic boundary conditions. . . . . . . 237

7.1 A summary of the leading correlation exponents of various correlationfunctions that decay as power laws in the (i) t t, t (strong corre-lated hopping) limit; (ii) t t, t = 0 (weak inter-leg hopping) limit;and (iii) t t, t = 0 (strong inter-leg hopping limit. The wave vectork of the oscillatory function modulating the leading terms in the corre-lation functions are reported in terms of kF = N1, where 0 N1 12is the nearest-neighbor excluded chain filling fraction. . . . . . . . . 428

7.1 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4297.2 Ground-state energies of two bound pairs in ladders of even length from

L = 6 to L = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494

xxvi

7.3 Antisymmetrized two-fermion and symmetrized two-hard-core-bosonamplitudes on a chain of length L = 4, for k1 = 4 and k2 = +

4 . . . . 503

7.4 Fitted parameters for a nonlinear curve fit of the infinite-ladder pure-correlated-hopping SC correlation r1/2 j j+r to an asymptotic formA + Br(1/2) cos(kr + ), for nearest-neighbor excluded chain fillingfractions 0.05 N1 0.45, using separations r 4. . . . . . . . . . 524

7.5 Fitted parameters for three series of nonlinear curve fit of the infinite-ladder pure-correlated-hopping SC correlation r1/2 j j+r to an a-symptotic form Ar(1/2)+Br(1/2) cos(2N1r+), for nearest-neigh-bor excluded chain filling fractions 0.05 N1 0.45, using separa-tions r 4. The full five-parameter fit cannot be done for N1 = 0.05and N1 = 0.10 because the iterations would run away. The visual as-sessments of the quality of fits for each fit in the three series are alsoreported, with a qualitative scale going from perfect (cannot discern anydifferences between the data points and the fit), very good (can discernsmall differences between the data points and the fit), good (can discernsmall systematic deviations between data points and fit), not good (sys-tematic deviations between data points and fit become worrying, but fitis generally acceptable), poor (systematic deviations too large for fit tobe acceptable), to very poor (systematic deviations very large). . . . . 530

7.5 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5317.6 Fitted parameters for a nonlinear curve fit of the infinite-ladder pure-

correlated-hopping subtracted CDW- correlation N jN j+rN j N j+rto an asymptotic form Ar cos(kr+)+Br

, for nearest-neighbor ex-

cluded chain filling fractions 0.05 N1 0.45, for r 4. . . . . . . 5447.7 Fitted parameters for a nonlinear curve fit of the infinite-ladder pure-

correlated-hopping subtracted CDW- correlation N jN j+rN j N j+rto an asymptotic form Ar cos(2N1r + ), for nearest-neighbor ex-cluded chain filling fractions 0.05 N1 0.45, for r 4. . . . . . . 549

7.8 Fitted parameters for two series of nonlinear curve fits of the infinite-ladder pure-correlated-hopping subtracted CDW- correlation N jN j+r N j N j+r to an asymptotic form Ar cos(2N1r + ) + Br

, fornearest-neighbor excluded chain filling fractions 0.05 N1 0.45, forr 4. For N1 = 0.40, the full five-parameter fit cannot be done becausethe iterations would run away. . . . . . . . . . . . . . . . . . . . . . 551

7.9 Fitted parameters for a nonlinear curve fit of the infinite-ladder pure-correlated-hopping subtracted CDW- correlation N jN j+rN j N j+rto an asymptotic form Ar(N1) cos(2N1r + ) + Br2, where (N1) =12 +

52 (

12 N1), for nearest-neighbor excluded chain filling fractions

0.05 N1 0.45, for r 4. . . . . . . . . . . . . . . . . . . . . . . 552

xxvii

7.10 Fitted parameters for a nonlinear curve fit of the infinite-ladder pure-correlated-hopping subtracted CDW- correlation N jN j+rN j N j+rto an asymptotic form Ar cos(2N1r + )+ B exp(r/), for nearest-neighbor excluded chain filling fractions 0.05 N1 0.45, for r 4.For N1 = 0.35 and N1 = 0.40, the full five-parameter fit cannot be donebecause the iterations would run away. . . . . . . . . . . . . . . . . 553

7.11 The six Bloch states, their generating configurations, and ground-stateamplitudes for P = 4 particles on a ladder of length L = 6 subject toperiodic boundary conditions. . . . . . . . . . . . . . . . . . . . . . 567

7.12 Fitted parameters for a nonlinear curve fit of the infinite-ladder sub-tracted antisymmetric CDW correlation n, jn, j+r n, j n, j+r, inthe limit t t, t = 0, to an asymptotic form Ar cos(kr + ), forladder filling fractions 0.05 n2 0.45, for r 4. . . . . . . . . . . 584

7.13 Fitted parameters for five series of nonlinear curve fits of the infinite-ladder subtracted antisymmetric CDW correlation n, jn, j+r n, jn, j+r, in the limit t t, t = 0, to the asymptotic form Arcos(2n2r + ) + Br

, for ladder filling fractions 0.05 n2 0.45

and r 4. We start with the constraints = 12 , = , = 2, andthen relax the constraints, one at a time. When the parameter value isquoted as -, that means that the iterations ran away for that parameter. 589

7.13 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5907.13 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5917.14 Fitted parameters for a nonlinear curve fit of the infinite-ladder SC cor-

relation r2 , j, j+r, in the limit t t, t = 0, to an asymptoticform Ar(2) cos(kr + ), for ladder filling fractions 0.05 n2 0.45,for r 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 598

7.15 Fitted parameters for five series of nonlinear curve fits of the infinite-ladder SC correlation r2 , j, j+r, in the limit t t, t = 0, to anasymptotic form Ar(2) cos(2n2r + ) + Br(

2), for ladder fillingfractions 0.05 n2 0.45, for and for separations r 4. When theparameter value is quoted as -, that means that the iterations ran awayfor that parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . 603

7.15 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6047.15 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6057.16 Fitted parameters for a nonlinear curve fit of the rung-ground-state spin-

less fermion FL correlation to CjC j+r = Ar cos(kr + ), for rung-fermion filling fractions over the range 0.02 N1 0.48. . . . . . . 620

7.16 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6217.17 Fitted parameters for a nonlinear curve fit of the rung-ground-state spin-

less fermion FL correlation to Cj C j+r =[

Ar1 + Br14

]

sin(N1r + ),for rung-fermion filling fractions over the range 0.02 N1 0.48. . . 627

7.17 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 628

xxviii

7.18 Fitted parameters for a nonlinear curve fit of the rung-fermion sub-tracted CDW correlation N jN j+r N j N j+r to the asymptotic formAr cos(kr+), for rung-fermion filling fractions over the range 0.04 N1 0.48. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 632

7.19 Fitted parameters for a nonlinear curve fit of the rung-fermion sub-tracted CDW correlation N+, jN+, j+r N+, j N+, j+r to the asymptoticform Ar2 + Br2 cos(2N1r + 1) + Cr

12 cos(2N1r + 2), for rung-

fermion filling fractions over the range 0.04 N1 0.48, and over therange of separations 4 r 17. . . . . . . . . . . . . . . . . . . . . 638

7.20 Convergence of the parameters in the nonlinear curve fit of r2 j j+rto the asymptotic form Ar(2) + Br(2) cos(kr + ) using a iterativerestricted-parameter algorithm, for N1 = 0.20, and 7 r 19. Eachiteration involves restricted-parameter fits to the pair (k, ), followed byfits to the pair (B, 2), and finally by fits to the pair (A, 2). . . . 645

7.21 Fitted parameters for a nonlinear curve fit of the rung-fermion SC cor-relation r2 j j+r to Ar(2) + Br(2) cos(kr + ), for ladder fillingfractions over the range 0.04 N1 0.48, for 7 r 19. . . . . . . 646

7.22 Fitted parameters for a nonlinear curve fit of the rung-fermion SC cor-relation r2 j j+r to Ar

18 +Br

14 cos(2N1r+1)+Cr

32 cos(2N1r+

2)+Dr72 , for ladder filling fractions over the range 0.04 N1 0.48,

and separations 7 r 19. . . . . . . . . . . . . . . . . . . . . . . 650

8.1 Fused indices l1l2 and m1m2 on clusters a and b respectively, and theirassociated products of referencing operators. Here, the label CDW isfor operators which are important in charge-density-wave phases, FL isfor operators which are important in Fermi liquid phases, and SF is foroperators which are important in staggered-flux phases. . . . . . . . 668

8.2 Fused indices l1l2 on cluster a, and their associated CDW-type productsof referencing operators. There is a corresponding set of fused indicesand product of referencing operators for cluster b. . . . . . . . . . . 671

8.3 Fused indices l1l2 on cluster a, and their associated SF-type productsof referencing operators. There is a corresponding set of fused indicesand product of referencing operators for cluster b. . . . . . . . . . . 672

8.4 Fused indices l1l2 on cluster a, and their associated FL-type productsof referencing operators. There is a corresponding set of fused indicesand product of referencing operators for cluster b. . . . . . . . . . . 673

8.4 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6748.5 Fused indices l1l2 on cluster a, and their associated SC-type products

of referencing operators. There is a corresponding set of fused indicesand product of referencing operators for cluster b. . . . . . . . . . . 674

8.6 Convergence of the correlation density-matrix singular values with in-creasing number N of unrestricted twist angles, for P = 2 particles ona ladder of length L = 6, and t = t = t = 1. . . . . . . . . . . . . . 680

xxix

8.7 Large-absolute-amplitude cluster of CDW-type operator components inthe maximum-absolute-singular-value CDW+ left and right eigenvec-tors, XCDW+ and YCDW+, for intercluster separations r = 2, 3, 4, obtainedfrom the operator singular value decomposition of the correlation den-sity matrix at the parameter point (t/t = 100, t/t = 0). . . . . . . . 779

8.8 Large-absolute-amplitude cluster of CDW-type operator components inthe maximum-absolute-singular-value CDW left and right eigenvec-tors, XCDW and YCDW, for intercluster separations r = 2, 3, 4, obtainedfrom the operator singular value decomposition of the correlation den-sity matrix at the parameter point (t/t = 100, t/t = 0). . . . . . . . 780

8.9 Large-absolute-amplitude cluster of FL-type operator components inthe maximum-absolute-singular-value FL+ left and right eigenvectors,XFL+ and YFL+, for intercluster separations r = 2, 3, 4, obtained from theoperator singular value decomposition of the correlation density matrixat the parameter point (t/t = 100, t/t = 0). . . . . . . . . . . . . . 781

8.10 Large-absolute-amplitude cluster of FL-type operator components inthe maximum-absolute-singular-value FL left and right eigenvectors,XFL and YFL, for intercluster separations r = 2, 3, 4, obtained from theoperator singular value decomposition of the correlation density matrixat the parameter point (t/t = 100, t/t = 0). . . . . . . . . . . . . . 782

8.10 (continued) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7838.11 Large-absolute-amplitude cluster of CDW-type operator components in

the maximum-absolute-singular-value CDW+ left eigenvector XCDW+for intercluster separations r = 2, 3, 4, at the parameter point (t/t =0, t/t = 100). The large-absolute-amplitude cluster of CDW operatorcomponents in the right eigenvector YCDW+ are analogs of those in theleft eigenvector XCDW+ for the respective separations. . . . . . . . . . 788

8.12 Large-absolute-amplitude cluster of CDW-type operator components inthe maximum-absolute-singular-value CDW left and right eigenvec-tors, XCDW and YCDW, for intercluster separations r = 2, 3, 4, obtainedfrom the operator singular value decomposition of the correlation den-sity matrix at the parameter point (t/t = 0, t/t = 100). . . . . . . . 790

C.1 The various spinless fermion calculations incurring fermion signs, theFermion code branch helper function designed to handle them, and theFermion code branch core function making the helper function calls. 897

C.2 Amplitude contributions and sums for the high-symmetry configura-tions |8, |32, |45, and |62 for the (6, 0) (0, 2) ladder system, subjectto periodic boundary conditions, in the q = ( 3 , 0) Bloch sector. . . . . 943

C.3 The one-particle eigenvalues and (unnormalized) eigenstates for thedensity matrix of a five-site, cross-shaped cluster in the (4, 1) (1, 3)system with N = 11 sites and P = 4 particles. . . . . . . . . . . . . . 981

xxx

C.4 The one-particle eigenvalues and (unnormalized) eigenstates for thedensity matrix of a five-site, cross-shaped cluster in the (4, 1) (1, 4)system with N = 15 sites and P = 4 particles. . . . . . . . . . . . . . 983

C.5 The two-particle eigenvalues and (unnormalized) eigenstates for thedensity matrix of a five-site, cross-shaped cluster in the (4, 1) (1, 4)system with N = 15 sites and P = 4 particles. . . . . . . . . . . . . . 984

C.6 The three-particle eigenvalues and (unnormalized) eigenstates for thedensity matrix of a five-site, cross-shaped cluster in the (4, 1) (1, 4)system with N = 15 sites and P = 4 particles. . . . . . . . . . . . . . 984

D.1 The set of momentum transfers K that can be obtained from the dif-ference k k of two, k and k, out of five occupied single-particlewave vectors, k2, k4, k5, k6 and k8. When K extends out of the infinite-system FBZ, we have what is known in solid-state physics as an Umk-lapp term. We add appropriate multiples of 2 to each component of Kto determine its equivalent within the FBZ. . . . . . . . . . . . . . . 1084

D.2 The Monkhorst-Pack sampling wave vectors kpr, and their associatedweights wpr in the IBZ, of order q = 1 to q = 5 for a two-dimensionalsquare lattice. Here ur are the fractional indices given by (D.6.8), andN(q) the number of sampling wave vectors within the IBZ. If we hadchose to integrate over the entire FBZ, there would be q2 sampling wavevectors for order q, having equal weights 1/q2. . . . . . . . . . . . . 1096

D.3 The Monkhorst-Pack sampling wave vectors kpr, and their associatedweights wpr in the IBZ, of order q = 6 to q = 7 for a two-dimensionalsquare lattice. Here ur are the fractional indices given by (D.6.8), andN(q) the number of sampling wave vectors within the IBZ. If we hadchose to integrate over the entire FBZ, there would be q2 sampling wavevectors for order q, having equal weights 1/q2. . . . . . . . . . . . . 1097

D.4 The Monkhorst-Pack sampling wave vectors kpr, and their associatedweights wpr in the IBZ, of order q = 8 for a two-dimensional squarelattice. These sampling wave vectors were also derived by Cunninghamusing the Chadi-Cohen method (see reference in text). Here ur are thefractional indices given by (D.6.8), and N(q) the number of samplingwave vectors within the IBZ. If we had chose to integrate over the entireFBZ, there would be q2 sampling wave vectors for order q, having equalweights 1/q2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1098

D.5 Comparison between the infinite-system value of the two-point func-tion |c(0,0)c(1,0)|, and the values coming from twist boundary con-ditions averaging of the (4, 0) (0, 4) system, calculated using unre-stricted full tetrahedron integration (TI), unrestricted coarse-grained TI,and unrestricted q-special-point integration (SPI). . . . . . . . . . . . 1138

xxxi

LIST OF FIGURES

1.1 Schematic diagram showing the main results (boxed) in each chapterof the thesis. The arrows in the diagram point from a latter chapter tospecific result(s) in previous chapter(s), that the latter chapter dependsupon. The columns corresponds to the rough division of the work donein this thesis into studies of (i) reduced density matrices of noninteract-ing spinless fermions (left), (ii) reduced density matrices of interactingspinless fermions (middle), and (iii) use of the correlation density ma-trix as a diagnostic tool (right). . . . . . . . . . . . . . . . . . . . . . 25

3.1 Faked distribution of total pseudo-energy as a function of the or-dinal number L, after shifting by and scaling by the standard de-viation . These faked total pseudo-energies are generated from 6random number uniformly distributed between 0 and 1. Also shown iserf(( )/). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.2 Faked distribution of total pseudo-energy plotted against (log L)2,for < , along with a fit using Okunishi et als asymptotic formula(3.4.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.3 Distribution of l for different filling fractions and cluster sizes. Inorder to compare l for different cluster sizes, the interval l [1,NC] isrescaled such that (l 12 )/NC (0, 1). With this rescaling, =

12 always

occur at [(l 12 )/NC] = n. . . . . . . . . . . . . . . . . . . . . . . . 1213.4 Plot of NC1 log[l/(1 l)] as a function of the scaling variable

x [(l 12 )/NC] 1/2 for various cluster sizes at half-filling, show-ing a scaling collapse onto the scaling function f (x). For NC > 23, thelargest and smallest pseudo-energies are severely affected by numericaltruncation errors in the diagonalization routines, and thus not shown. 123

3.5 Scaling collapses for (from top to bottom) n = 0.1, 0.3, 0.7 and 0.9,plotted against the scaling variable x (l 12 )/NC n, for three differentcluster sizes: NC = 15 (), NC = 23 () and NC = 67 (). . . . . . . . 125

3.6 Schematic diagram showing the three many-particle density-matrix ei-genstates, (a) |F 1, (b) |F and (c) |F + 1, with the largest weights,for a cluster of NC (NC even) sites within a overall system that is half-filled. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

3.7 Plot of NC1 log[l/(1 l)] as a function of the scaling variablex = (l 12 )/NC

12 , for a cluster of size NC = 23, with different hopping

modulations = 0.1, 0.3, 0.5, 0.7, 0.9. Pseudo-energies for |x| > 0.2 for > 0 are not shown because these are severely affected by numericalerrors incurred in the numerical integration and diagonalization rou-tines. The various sets of straight line segments are intended to guidethe eye in visualizing the data. . . . . . . . . . . . . . . . . . . . . . 131

xxxii

3.8 Plot of NC1 log[l/(1 l)] as a function of the scaling variablex = (l 12 )/NC

12 , for hopping modulation = 0.5 and different cluster

sizes B = 9, 13, 17, 23. The pseudo-energies for |x| > 0.3 are not shownas these are severely affected by numerical errors in the numerical in-tegration and diagonalization routines. Also shown, is an approximatedashed curve for the scaling function f (x) for n = 12 and = 0, obtainedfrom the data for NC = 23. The straight line segments are intended toguide the eye in visualizing the data. . . . . . . . . . . . . . . . . . . 132

3.9 Plot of the largest density-matrix weight wF as a function of the clustersize NC for NC even. The solid curve shown is a fit of the form wF =wF,+wF exp(NC/NC,0). The best fit to this small data set is obtainedby neglecting the data points for NC = 2 and NC = 4, for which we getwF, = 0.41, wF = 0.26 and NC,0 = 0.11. . . . . . . . . . . . . . . . 134

3.10 Discarded weight as a function of the number of states kept: weight-ranked (as done in the DMRG), and operator-based. . . . . . . . . . 137

3.11 Plot of the function f () = log (1 ) log(1 ) against theordinal number l of the eigenvalues l of the cluster Green-functionmatrix GC of a cluster of NC = 15 sites within a one-dimensional chainof noninteracting spinless fermions at half-filling. . . . . . . . . . . . 145

3.12 Plot of the cumulative sum

ll f (l), where f () = log (1 ) log(1 ), against the ordinal number l of the eigenvalues l of thecluster Green-function matrix GC, for a cluster of NC = 15 sites withina half-filled one-dimensional chain of noninteracting spinless fermions. 146

3.13 Entanglement entropy S = Tr C log C for clusters of NC sites in aone-dimensional system of noninteracting spinless fermions. The en-tanglement entropy appears to increase without bound following onelogarithmic behaviour for even NC , and another logarithmic behaviourfor odd NC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

3.14 Fits of the entanglement entropy S = Tr C log C of a one-dimensionalsystem of noninteracting spinless fermions to the form S = a log NC+b.For odd cluster sizes, we find that a = 0.33744 and b = 1.4052, whereasfor even cluster sizes, we find that a = 0.3346 and b = 0.72163. . . . 149

3.15 Entanglement entropy S = Tr C log C for clusters of up to NC = 500sites in a one-dimensional system of noninteracting spinless fermions.Also shown is the logarithmic fit S = 0.3346 log NC+0.72163 obtainedfor even NC up to NC = 60. . . . . . . . . . . . . . . . . . . . . . . . 152

3.16 Plot of the reciprocal of the slope of the pseudo-dispersion relation,1/(lF,NC), as a function of the cluster size NC , for a one-dimensionalsystem of noninteracting spinless fermions. Also shown is the linear fit1/(lF,NC) = 0.23209 + 0.10564 log NC . . . . . . . . . . . . . . . . 154

3.17 Linear fit of the form cNC + d applied to the function NC/( + log NC)for cluster sizes up to NC = 70. . . . . . . . . . . . . . . . . . . . . 156

xxxiii

3.18 Probability density of the normalized one-particle eigenfunctions l( j)(plotted against the scaling variable y = ( j 12 )/NC) of C on a clusterof NC = 9 sites at half-filling, showing the particle-hole symmetry ofthe overall system. The subplots are arranged in order of increasingpseudo-energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

3.19 Probability density function |F( j,NC)|2 of the pseudo-Fermi eigen-function for NC = 23, plotted against the scaled variable ( j 12 )/NC . . 159

3.20 Plot of the rescaled envelope function g(y) for various cluster sizesNC = 4p + 1, p = 1, 2, . . . , compared against sin y and 4y(1 y),where y = ( j 12 )/NC is the rescaled coordinate on the cluster. . . . . 161

3.21 Dispersion relation (k) for a cluster of NC = 8 sites, where the effectsof truncating 2, 4 and 6 PELs are shown. For this cluster size, truncating2, 4 and 6 PELs corresponds to fractions = 0.75, 0.50 and 0.25 ofPELs retained. For = 0.75, the energy bands (dashed curves) justbelow and above the Fermi level F = 0 agree with the true dispersionrelation (solid curve) so well that the difference is not discernible at thescales presented in the figure. . . . . . . . . . . . . . . . . . . . . . 164

3.22 Plot of the gap E as a function of cluster size NC for various con-stant fractions = 1/4, 1/3, 1/2, 2/3, 3/4, 4/5 of PELs retained.Also shown are fits to the data points for various to E(,NC) =E0 exp(()NC ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.23 Plot of the attenuation coefficient () with which the spurious gap Edecays, as a function of the fraction of PELs retained. A cubic splinecurve (dashed curve) is superimposed on the data points to aid visual-ization. Also shown (dotted curve) is log(1 ). . . . . . . . . . . 167

3.24 Plot of Fermi velocity deviation (vF 2) calculated from the truncateddispersion relation, as a function of the cluster size NC , for various frac-tions = 1/15, 1/13, 1/11, 1/9, 1/7, 1/5, 3/15, 3/13, 3/11, 3/9, 3/7 ofPELs retained. Fits to average exponential decays are also shown. . . 169

3.25 Plot of the attenuation coefficient () as a function of the fraction of PELs retained. Also shown (dashed line) is the expected behaviour() = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

3.26 Amplitudes of eigenfunctions of the truncated Hamiltonian H(kF) fordifferent numbers of PELs retained, for a cluster of NC = 23 sites athalf-filling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

3.27 Dispersion relation (k) for a cluster of NC = 8 sites, where the effectof truncating 2 and 6 plane waves (PWs) are shown. For NC = 4n, theFermi level is located at the zone center, and (k) is always gapless hereregardless of the number of plane waves truncated. . . . . . . . . . . 174

3.28 Dispersion relation (k) for a cluster of NC = 10 sites, where the effectof truncating 2 and 6 plane waves (PWs) are shown. For NC = 4n + 2,the Fermi level is located at the zone boundary, and the operator-basedplane wave truncation scheme introduces an energy gap E here. . . 177

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3.29 Plot of E as a function of cluster size NC for r = 3, 5 and 9, cor-responding to the fractions = 2/3, 4/5, 6/7 of cluster plane wavestates retained. Also shown are the fits of the data points to the formulaE(NC , ) = E1()/NC . From the fits, we have E1 = 2.58263 for = 2/3, E1 = 1.57991 for = 4/5 and E1 = 1.13532 for = 6/7. 179

3.30 Plot of vF as a function of in the operator-based PW truncation schemefor both the r = 4p 1 series and r = 4p + 1 series. Also shown as thedashed curve is 2[1 (1 )2], which appears to fit the data points wellnear = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

3.31 Plot of c as a function of in the operator-based PW truncation schemefor both the r = 4p1 series (c+()) and r = 4p+1 (c()) series. Bothc+() and c() appears to be converging towards 2. In fact, from thegraph we find that |c() 2| 0.43(1 ). . . . . . . . . . . . . . . 181

4.1 Indexing the sites of the R1 R2 system, where we start from theleftmost, bottommost site, working our way upwards and then right-wards. In the example (5, 1) (1, 4) shown, there are a total of N = 19sites, numbered from the first, at (0, 0), to the nineteenth, at (5, 4). Thesites {3, 5, 6, 7, 10} belonging to our five-site, cross-shaped cluster aremarked as open circles, while the rest of the sites, which constitutes theenvironment of the cluster, are marked as filled circles. . . . . . . . . 187

4.2 The five-site, cross-shaped cluster whose cluster density matrix we areinterested in calculating for both a system of noninteracting, as well asstrongly-interacting spinless fermions. . . . . . . . . . . . . . . . . . 204

4.3 The zero-particle weight of the cluster density matrix of a five-site,cross-shaped cluster for systems of noninteracting spinless fermionssubject to periodic boundary conditions. The data points coming fromthe (3,2)(2, 3) ( ), (4, 1)(1, 3) (#), (4, 1)(1, 4) (), (4,1)(1, 4)() and (4,2) (2, 4) () systems appear to fall onto a smooth curve. 226

4.4 The one-particle weights of the cluster density matrix of a five-site,cross-shaped cluster for the (3,2) (2, 3) ( ), (4, 1) (1, 3) (#),(4, 1) (1, 4) (), (4,1) (1, 4) () and (4,2) (2, 4) () systemsof noninteracting spinless fermions, with periodic boundary conditionsimposed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

4.5 Zero- and one-particle weights of the cluster density matrix of a five-site, cross-shaped cluster for finite systems of noninteracting spinlessfermions with periodic boundary conditions imposed. The systems,(32,8) (8, 32) (solid line), (4,1) (1, 4) ( ), and (8,2) (2, 8)() have the same shape but different sizes. For ws1 , we see the finitedomain effect deviations for the (4,1) (1, 4) system is practicallygone by the time we get to the (8,2) (2, 8) system. For the rest ofthe one-particle weights, the finite domain effect is largely removed inthe (8,2) (2, 8) system, but shell effect persists. In fact, shell effectis still visible in the (32,8) (8, 32) system. . . . . . . . . . . . . . 229

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4.6 Zero- and one-particle weights of the cluster density matrix of a five-site, cross-shaped cluster for the (16,4) (4, 16) (black solid curve)and (32,8) (8, 32) (red solid curve) systems of noninteracting spin-less fermions with periodic boundary conditions imposed. Ignoring thepersistent shell effect remaining, we see that the cluster density-matrixspectra of the two systems agree, and thus the infinite-system limit isfor practical purposes achieved in the (16,4) (4, 16) system. . . . 230

4.7 Sine of the mixing angle , between the one-particle cluster states |sand |s in the one-particle cluster density-matrix eigenstates of s-sym-metry, as a function of the filling fraction n. . . . . . . . . . . . . . . 231

4.8 Shell effect in the one-particle cluster density-matrix weight wd of the(16,4) (4, 16) system of noninteracting spinless fermions. . . . . 232

4.9 Shell effect in the one-particle cluster density-matrix weight wd of the(32,8) (8, 32) system of noninteracting spinless fermions. . . . . 233

4.10 One-particle weights of the cluster density matrix of a five-site, cross-shaped cluster for the (16,4) (4, 16) (black), (16, 4) (4, 16) (red),(16, 4) (4, 12) (green), (16, 3) (2, 13) (blue), and (15,5) (3, 14)(brown) systems of noninteracting spinless fermions subject to periodicboundary conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.11 Power-law fits of the form ws1 = As1 ns1 . Using all 200 data points, we

have ws1 = 4.0318 n0.96329, whereas if the first 20 data points are used,

we have ws1 = 4.8433 n0.99593. The first fit agrees better with the data

for larger n, whereas the latter fit agrees better with the data for smallern. The exponents from the fits are both close to one. . . . . . . . . . 238

4.12 Power-law fits of the form wp = Ap np. A muted shell effect shows upon the log-log plot at very low filling fractions, and we use all but thefirst ten of the 200 data points to fit a power law of wp = 3.1364 n1.8669.If we exclude the first 100 data points, the power-law fit is then wp =2.0853 n1.74876, whereas if we exclude the first ten and the last 50 datapoints, we get a power-law fit of wp = 3.5014 n1.8908. . . . . . . . . . 239

4.13 Power-law fits of the form wd = Ad nd . For this one-particle weight,shell effects are pronounced, and there does not appear to be any lessnoisy region to do the power-law fit. Using all but the first of the200 data points, we obtain a power-law fit of wd = 2.7245 n2.8034. Ifwe exclude the first 99 data points, the power-law fit becomes wd =2.4219 n2.76748. For lower filling fractions, both power laws do not agreewell with the data points. . . . . . . . . . . . . . . . . . . . . . . . . 240

4.14 Power-law fits of the form ws2 = As2 ns2 . For this one-particle weight,

shell effects are less pronouced than in wd, but are still severe at lowfilling fractions. Using all but the first eight of the 200 data points, weobtain a power-law fit ws2 = 0.39149 n

2.89084. If we exclude the first 99data points, the power-law fit becomes ws2 = 0.31948 n

2.8316. Again,both power laws do not agree well with the data points at lower fillingfractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

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4.15 Zero-