QUANTUM HALL FERROMAGNETS AKSHAY KUMAR ADISSERTATION PRESENTED TO THE FACULTY OF PRINCETON UNIVERSITY IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY RECOMMENDED FOR ACCEPTANCE BY THE DEPARTMENT OF PHYSICS ADVISER:SHIVAJI L. SONDHI APRIL 2016 c Copyright by Akshay Kumar, 2016. All rights reserved. Abstract We study several quantum phases that are related to the quantum Hall effect. Our initial focus is on a pair of quantum Hall ferromagnets where the quantum Hall ordering oc- curs simultaneously with a spontaneous breaking of an internal symmetry associated with a semiconductor valley index. In our first example – AlAs heterostructures – we study domain wall structure, role of random-field disorder and dipole moment physics. Then in the second example – Si(111) – we show that symmetry breaking near several integer filling fractions involves a combination of selection by thermal fluctuations known as “order by disorder” and a selection by the energetics of Skyrme lattices induced by moving away from the commensurate fillings, a mechanism we term “order by doping”. We also study ground state of such systems near filling factor one in the absence of valley Zeeman energy. We show that even though the lowest energy charged excitations are charge one skyrmions, the lowest energy skyrmion lattice has charge >1 per unit cell. We then broaden our discussion to include lattice systems having multiple Chern num- ber bands. We find analogs of quantum Hall ferromagnets in the menagerie of fractional Chern insulator phases. Unlike in the AlAs system, here the domain walls come naturally with gapped electronic excitations. We close with a result involving only topology: we show that ABC stacked multilayer graphene placed on boron nitride substrate has flat bands with non-zero local Berry curva- ture but zero Chern number. This allows access to an interaction dominated system with a non-trivial quantum distance metric but without the extra complication of a non-zero Chern number. iii Acknowledgements First of all, I would like to thank my adviser, Shivaji Sondhi, for giving an engineering stu- dent a chance to explore theoretical physics. I am grateful to have had Shivaji as a teacher and counselor. He has always been incredibly patient with me. I have also been fortunate to have closely collaborated with many extremely bright physicists. Sid Parameswaran’s boundless energy and Rahul Nandkishore’s speed of project execution always amazed me. Working with Rahul Roy was simultaneously frustrating and exciting! I also thank my undergraduate adviser Sankalpa Ghosh for motivating me to pursue a PhD. I would also like to thank Ravindra Bhatt for reviewing this dissertation, as well as Simone Giombi and Waseem Bakr for serving on my defense committee. I thank Thomas Gregor for taking me on as an experimental project student. Moreover, I am indebted to the support staff of the Physics Department for greatly simplifying my life as a graduate student. They made it painless for me to attend conferences and summer schools, where I had the chance to meet folks outside of the Princeton bubble; Boulder will always be a favorite of mine. Life at Princeton has not only been about research. I thank all of the wonderful friends I have met here. Dining out at random restaurants with Vedika and Chaney was always fun. I am grateful to them for proofreading parts of my thesis, and I owe it to Vedika for making Dresden tolerable! Discussions with Liangsheng and Bin about life outside of physics were always enlightening. I would like to especially thank my large Indian group of friends for the movies, sports and board games on weekends. Lastly I thank my parents for believing in me at all stages of my life. This dissertation would not have been possible without their unending encouragement. iv Publications associated with this dissertation 1. Akshay Kumar and Rahul Nandkishore. Flat bands with Berry curvature in multi- layer graphene. Phys. Rev. B 87, 241108(R) (2013) 2. Akshay Kumar, S.A.Parameswaran and S. L. Sondhi. Microscopic theory of a quan- tum Hall Ising nematic: Domain walls and disorder. Phys. Rev. B 88, 045133 (2013) 3. Akshay Kumar, Rahul Roy and S. L. Sondhi. Generalizing quantum Hall ferromag- netism to fractional Chern bands. Phys. Rev. B 90, 245106 (2014) 4. Akshay Kumar, S.A.Parameswaran and S. L. Sondhi. Order by disorder and by dop- ing in quantum Hall valley ferromagnets. Phys. Rev. B 93, 014442 (2016) Materials from this dissertation have been presented at the following places: 1. APS March Meeting 2014, Denver, CO 2. Seminar Series, Max Planck Institute for the Physics of Complex Systems (MPIPKS), Dresden, Germany 3. Schlumberger-Doll Research Center, Cambridge, MA v To my parents. vi Contents Abstract . iii Acknowledgements . iv List of Tables . .x List of Figures . xi 1 Introduction 1 1.1 Quantum Hall Effect . .3 1.2 Chern Insulator . .6 1.3 Quantum Hall Ferromagnet . 10 1.4 Quantum Hall Valley Ferromagnet . 12 1.5 Thesis Outline . 15 2 Microscopic Theory of a Quantum Hall Ising Nematic: Domain Walls and Disorder 18 2.1 Introduction . 18 2.2 Overview: Phases, Transitions and Transport . 22 2.3 Microscopic Theory . 25 2.3.1 Hartree-Fock Formalism . 26 2.3.2 Estimates of Tc ............................ 29 2.3.3 Properties of Sharp Domain Walls . 31 2.3.4 Does the Dipole Moment Matter? . 36 vii 2.3.5 Domain Wall Texturing . 38 2.4 Disorder in the Microscopic Theory . 41 2.4.1 Random Fields from Impurity Potential Scattering . 41 2.4.2 Estimating Disorder Strength from Sample Mobility . 43 2.5 Experiments . 45 2.6 Concluding Remarks . 47 3 Order by Disorder and by Doping in Quantum Hall Valley Ferromagnets 49 3.1 Introduction . 49 3.2 Silicon(111) . 51 3.2.1 Effective Hamiltonian . 52 3.2.2 ν = 1 ................................. 53 3.2.3 ν = 2 ................................. 54 3.2.4 ν = 3 ................................. 56 3.3 Experiments . 57 3.4 Silicon(110) . 58 3.5 Group-theoretic analysis of symmetry breaking . 59 3.5.1 Four-Valley Case . 59 3.5.2 Six-Valley Case . 62 3.6 Valley Skyrmion Crystals . 65 3.6.1 Analytics . 65 3.6.2 Numerical Minimization . 68 3.7 Concluding Remarks . 71 4 Generalizing Quantum Hall Ferromagnetism to Fractional Chern Bands 73 4.1 Introduction . 73 4.2 A Special flat C=2 band at 1/2 filling . 75 4.3 Other flat C=2 bands at 1/2 filling . 79 viii 4.4 Generalization to higher Chern bands . 83 4.5 Fractional states . 83 4.6 Concluding Remarks . 84 5 Flat bands with local Berry curvature in multilayer graphene 86 5.1 Introduction . 86 5.2 ABC stacked graphene . 88 5.3 Effect of BN substrate . 90 5.4 Details of Bandstructure Calculation . 95 5.5 Chern number from adatoms . 98 5.6 Concluding Remarks . 98 6 Conclusion 99 A Theta Functions 101 A.1 Basic Theta Function . 101 A.2 Modified Theta Function . 101 Bibliography 102 ix List of Tables 4.1 Analogies between gas and lattice systems. ................ 73 x List of Figures 2.1 (a) Model band structure used in this chapter, appropriate to describing AlAs wide quantum wells. (b) Different phases as determined by com- paring Imry-Ma domain size ξIM to sample dimensions LS. Top: For ξIM LS we find the QHRFPM. Bottom: For ξIM LS the system is dominated by the properties of a single domain, and is better modeled as a QHIN. At intermediate scales, LS ξIM there is a crossover. 20 ∼ 2.2 Phase diagram as function of temperature (T ) and disorder strength (W ), showing behavior of conductivity. The phases and critical points are defined in the introduction. 24 2.3 Valley symmetry-breaking field permits transport to probe the energy scales of the QHIN/QHRFPM. (Inset) Domain structure as function of disorder strength and valley splitting; dashed line shows a representative path in ∆v leading to a transport signature similar to that in the main figure. ∗ ∆v is the valley splitting for which the system is single-domain dominated. 26 2.4 Mean-field and NLσM estimates of Tc. Dashed line shows the anisotropy (λ2 5:5) appropriate to AlAs. 31 ≈ 2.5 (a) Surface tension and (b) dipole moment of a sharp DW as a function of the effective mass anisotropy. Dashed line shows the anisotropy (λ2 5:5) ≈ appropriate to AlAs. 34 xi 2.6 Domain-wall texturing from Hartee-Fock Theory. (Top) Contour plot of the average in-plane valley pseudospin Sx per unit magnetic length along h i the domain wall, as a function of the mass anisotropy λ2 and the valley Zeeman field gradient g, with the latter on a logarithmic scale. The dashed line marks the anisotropy λ2 5:5 relevant to AlAs; note that there is still ≈ some texturing in this limit. (Bottom) Cut along dashed line, with g on a linear scale. 40 3.1 Valley ordering in Si(111) QH states. (Inset) Model Fermi surface. El- lipses denote constant-energy lines in k-space. (Main figure) Schematic 3 global phase diagram, showing how the G = [SU(2)] o D3 symmetry is broken to H0;HT at zero and finite temperature. The order parameter spaces are O = G=HT for T > 0, and O = HT =H0 at T = 0. For ν = 1; 2, D3 symmetry breaks continuously at Tc, but this becomes first-order around ν = 3. Near ν = 2; 3 order by doping yields to thermal order-by-disorder at T T ∗ ................................... 50 ∼ E-S 3.2 Possible valley-ordered states at ν = 1; 2; 3, including representatives of Class I and II states for ν = 2; 3.