DISORDER AT FIRST-ORDER CLASSICAL AND QUANTUM PHASE TRANSITIONS by AHMED

149 Pages·2017·2.35 MB·English

by  Ahmed Khalil Ibrahim

DISORDERATFIRST-ORDERCLASSICALANDQUANTUMPHASE
TRANSITIONS
by
AHMEDKHALILIBRAHIM
ADISSERTATION
PresentedtotheGraduateFacultyofthe
MISSOURIUNIVERSITYOFSCIENCEANDTECHNOLOGY
InPartialFulfillmentoftheRequirementsfortheDegree
DOCTOROFPHILOSOPHY
in
PHYSICS
2018
Approvedby
Dr. ThomasVojta,Advisor
Dr. GeraldWilemski
Dr. PaulE.Parris
Dr. AleksandrChernatynskiy
Dr. JoséA.Hoyos
Copyright2018
AHMEDKHALILIBRAHIM
AllRightsReserved
iii
PUBLICATIONDISSERTATIONOPTION
Thisdissertationhasbeenpreparedintheformoffourpapers:
Paper I, Pages 35–64, has been published as Enhanced rare-region effects in the
contact process with long-range correlated disorder, published in Physical Review E 90,
042132(2014)withHatemBarghathiandThomasVojta.
Paper II, Pages 65–81, has been published as Emerging critical behavior at a
first-order phase transition rounded by disorder, published in Fortschritte der Physik 65,
1600018(2017)withThomasVojta.
Paper III, Pages 82–106, has been published as Monte Carlo simulations of the
disordered three-color quantum Ashkin-Teller chain, published in Physical Review B 95,
054403(2017)withThomasVojta.
Paper IV, Pages 107–127, is a manuscript ready to be published as Numerical
investigation of a disordered superconductor-metal quantum phase transition (2018) with
ThomasVojta.
In addition, the dissertation contains an introductory section (Sec. 1) and a brief
summary(Sec. 2).
iv
ABSTRACT
This dissertation studies the effects of quenched disorder on classical, quantum
and nonequilibrium phase transitions. After a short introduction which covers the basic
concepts of phase transitions, finite-size scaling and random disorder, the dissertation
focusesonfourseparatebutrelatedprojects. First,weinvestigatetheinfluenceofquenched
disorder with long-range spatial correlations on the nonequilibrium phase transitions in
the contact process. We show that the long-range correlations increase the probability to
find rare atypical regions in the sample. This leads to enhanced Griffiths singularities and
changestheuniversalityclassofthetransition.
Project 2 and 3 focus on disorder at first-order phase transitions. In project 2, we
analyze the phase transitions of a classical Ashkin-Teller magnet. We demonstrate that the
first-order classical phase transition is destroyed by disorder, and the resulting continuous
transition belongs to the clean two-dimensional Ising universality class with logarithmic
corrections.
Project 3 investigates the fate of the first-order quantum phase transition in the
quantumAshkin-Tellermodelbylarge-scaleMonteCarlosimulations. Wefindthatdisorder
rounds the first-order quantum phase transition just as in the classical case. The resulting
criticalbehaviordependsonthestrengthoftheinter-colorcouplinginthequantumAshkin-
Teller model. This leads to two different regimes, the weak and strong coupling regimes,
both of which feature infinite-randomness critical behavior but in different universality
classes.
Finally, we study the quantum phase transition of a disordered nanowire from
superconductor to metallic behavior. We show that the critical behavior is of infinite-
randomtypeandbelongstotherandomtransverse-fieldIsinguniversalityclassaspredicted
bystrongdisorderrenormalizationgroupresults.
v
ACKNOWLEDGMENTS
First of all, I would like to express my deepest thanks and gratitude to my advisor
Dr. ThomasVojta,forhispatienceanddedicationinsupportingandguidingmethroughout
thePhDstudyperiodandwritingmydissertation. Ibelievethisworkwouldnothavebeen
possiblewithouthisassistanceandsupport. Thankyou,Dr. Vojta.
Iamgreatlyindebtedtoallmyadvisorycommitteemembers,Dr. GeraldWilemski,
Dr. Paul Parris, Dr. Aleksandr Chernatynskiy, and Dr. José A. Hoyos, for their valuable
discussionsandaccessibility.
I would like to thank our former chairman, Dr. George D. Waddill, our graduate
coordinator, Dr. Jerry Peacher, and also the staff in the Physics Department, Pamela J.
Crabtree, Janice Gargus, Russell L. Summers, Ronald Woody, and Andy Stubbs, for their
help. I am very thankful to the Higher Committee for Education Development (HCED) in
IraqforgrantingmeaPhDscholarship.
Iwouldlikeespeciallytogivemythankstomyfriend,Dr. HatemBarghathi. Thank
youforyourhelpandsupport.
A special thanks to my family, my parents, my brothers, and my sister. Thank you
foryourlove,support,andencouragement.
Finally, I would like to express my gratitude to my beloved wife, Raghdaa "Um
Raneem", and my wonderful daughters, Raneem, Rafeef, and Taleen. Thank you, for all
yourpatience,care,andsupport.
vi
TABLEOFCONTENTS
Page
PUBLICATIONDISSERTATIONOPTION...................................... iii
ABSTRACT .................................................................. iv
ACKNOWLEDGMENTS ...................................................... v
LISTOFILLUSTRATIONS.................................................... x
LISTOFTABLES............................................................. xv
SECTION
1. INTRODUCTION.......................................................... 1
1.1. THERMALANDQUANTUMPHASETRANSITIONS ..................... 1
1.1.1. LandauTheoryofPhaseTransitions ................................... 2
1.1.2. TheScalingHypothesisandRenormalizationGroup ................. 6
1.1.3. QuantumPhaseTransitions ............................................. 10
1.1.4. Transverse-FieldIsingModel........................................... 15
1.2. NON-EQUILIBRIUMPHASETRANSITIONS............................... 16
1.3. FINITE-SIZESCALING ........................................................ 19
1.4. DISORDEREDPHASETRANSITIONS ...................................... 22
1.4.1. QuenchedDisorder...................................................... 23
1.4.2. Imry-MaCriterion....................................................... 24
1.4.3. HarrisCriterion.......................................................... 25
1.4.4. Strong-DisorderRenormalizationGroupTheoryandInfiniteRan-
domnessCriticalPoint .................................................. 27
vii
1.4.5. Rare-RegionsandGriffithsEffects ..................................... 32
PAPER
I. ENHANCEDRARE-REGIONEFFECTSINTHECONTACTPROCESSWITH
LONG-RANGECORRELATEDDISORDER................................. 35
ABSTRACT ............................................................................ 35
1. INTRODUCTION ............................................................... 36
2. CONTACTPROCESSWITHCORRELATEDDISORDER.................. 37
3. THEORY ......................................................................... 39
3.1. RARE-REGIONPROBABILITY...................................... 39
3.2. GRIFFITHSPHASE .................................................... 41
3.3. CRITICALPOINT...................................................... 44
4. MONTE-CARLOSIMULATIONS............................................. 47
4.1. OVERVIEW ............................................................. 47
4.2. RESULTS:CRITICALBEHAVIOR................................... 48
4.3. RESULTS:GRIFFITHSPHASE....................................... 54
5. GENERALIZATIONS........................................................... 56
5.1. HIGHERDIMENSIONS ............................................... 56
5.2. OTHERSYSTEM....................................................... 57
6. CONCLUSIONS................................................................. 60
ACKNOWLEDGEMENTS............................................................ 61
REFERENCES ......................................................................... 62
II. EMERGING CRITICAL BEHAVIOR AT A FIRST-ORDER PHASE TRAN-
SITIONROUNDEDBYDISORDER........................................ 65
ABSTRACT ............................................................................ 65
1. INTRODUCTION ............................................................... 66
viii
2. DILUTEDASHKIN-TELLERMODEL........................................ 67
3. MONTECARLOSIMULATIONS ............................................. 70
3.1. METHODANDOVERVIEW.......................................... 70
3.2. RESULTS................................................................ 71
4. SUMMARYANDCONCLUSIONS ........................................... 75
ACKNOWLEDGEMENTS............................................................ 77
REFERENCES ......................................................................... 78
III. MONTECARLOSIMULATIONSOFTHEDISORDEREDTHREE-COLOR
QUANTUMASHKIN-TELLERCHAIN ..................................... 82
ABSTRACT ............................................................................ 82
1. INTRODUCTION ............................................................... 83
2. MODELANDTHEORY........................................................ 85
2.1. QUANTUMASHKIN-TELLERCHAIN.............................. 85
2.2. RENORMALIZATIONGROUPPREDICTIONS .................... 86
2.3. QUANTUM-TO-CLASSICALMAPPING ............................ 88
3. MONTECARLOSIMULATIONS ............................................. 89
3.1. OVERVIEW ............................................................. 89
3.2. WEAKCOUPLINGREGIME ......................................... 91
3.3. STRONGCOUPLINGREGIME....................................... 96
4. CONCLUSIONS................................................................. 99
ACKNOWLEDGEMENTS............................................................ 101
REFERENCES ......................................................................... 102
IV. NUMERICALINVESTIGATIONOFADISORDEREDSUPERCONDUCTOR-
METALQUANTUMPHASETRANSITION................................. 107
ABSTRACT ............................................................................ 107
ix
1. INTRODUCTION ............................................................... 108
2. THEMODEL .................................................................... 109
3. THEORY ......................................................................... 111
3.1. RENORMALIZATIONGROUPPREDICTIONS .................... 111
4. MONTECARLOSIMULATIONS ............................................. 113
5. THERMODYNAMICS.......................................................... 113
5.1. CLEANSYSTEM....................................................... 113
5.2. DISORDEREDSYSTEM............................................... 117
6. CONCLUSIONS................................................................. 122
ACKNOWLEDGEMENTS............................................................ 123
REFERENCES ......................................................................... 124
SECTION
2. CONCLUSIONSANDOUTLOOK..........................................128
REFERENCES................................................................130
VITA.........................................................................134
x
LISTOFILLUSTRATIONS
Figure Page
SECTION
1.1. Schematic phase diagram of water. The solid lines indicate first-order phase
transitions. The gas-liquid phase boundary ends in a critical point at which
the phase transition is continuous. The intersection of three lines is referred
toasthetriplepointatwhichallthreephasescoexist. ............................. 2
1.2. Schematic of magnetization m vs. temperature T at a ferromagnetic phase
transition. If the temperature T is below the critical temperature T , the
c
substancehasafinite(non-zero)valueofthemagnetizationm (ferromagnetic
phase). WhenT > T ,thenthemagnetization m vanishes(paramagneticphase). 3
c
1.3. Schematic of the Landau free energy F as a function of the order parameter
L
m forvariousvalueofr. ............................................................. 4
1.4. Schematic phase diagram in the vicinity of a quantum critical point (QCP)
located at P . The horizontal axis is the non-thermal control parameter P
c
which can changes the system by means of the zero-temperature (quantum)
phasetransition,whiletheverticalaxisisthetemperatureT. Thesolidlineis
the boundary between finite-temperature phases, near to this line the critical
behaviorisclassical. Thedashedlinesarethebordersofthequantumcritical
regionthenaregivenby k T ∼ (cid:126)w ................................................. 12
B c
1.5. Schematic of contact process in one-dimension. Infected (active) sites infect
their neighbors at rate λ/2. Active sites can spontaneously become inactive
(healthy)withhealingrate µ......................................................... 18
1.6. Schematic of Binder cumulant g vs. temperature T for different system size
L in classical Ising model. All curves cross at the same temperature that
correspondstoacriticaltemperatureT ............................................. 21
c
1.7. BehaviorofBindercumulantintherandomquantumIsingmodel. (a)Binder
cumulant as a function of L for several L at the critical temperature. (b)
τ
Scaling plot of the Binder cumulant at T . Main panel: Power-law scaling g
c
vs L /Lmax. Inset: Activatedscaling g vsln(L )/ln(Lmax)........................ 22
τ τ τ τ
1.8. SchematicoftheImry-Macriterian. ................................................ 25
1.9. SchematicofHarriscriterian. ....................................................... 26
1.10. Schematic of the strong-disorder renormalization group step for decimating a
field.................................................................................... 29