University of Pennsylvania ScholarlyCommons Publicly Accessible Penn Dissertations 2019 Studies On Fractionalization And Topology In Strongly Correlated Systems From Zero To Two Dimensions Yichen Hu University of Pennsylvania, [email protected] Follow this and additional works at: https://repository.upenn.edu/edissertations Part of the Condensed Matter Physics Commons Recommended Citation Hu, Yichen, "Studies On Fractionalization And Topology In Strongly Correlated Systems From Zero To Two Dimensions" (2019). Publicly Accessible Penn Dissertations. 3282. https://repository.upenn.edu/edissertations/3282 This paper is posted at ScholarlyCommons. https://repository.upenn.edu/edissertations/3282 For more information, please contact [email protected]. Studies On Fractionalization And Topology In Strongly Correlated Systems From Zero To Two Dimensions Abstract The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape. For the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics. For the third part, we argue that a correlated fluid of lee ctrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m>1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed ''fractional excitonic insulator'', within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory. Degree Type Dissertation Degree Name Doctor of Philosophy (PhD) This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/3282 Graduate Group Physics & Astronomy First Advisor Charles L. Kane Subject Categories Condensed Matter Physics This dissertation is available at ScholarlyCommons: https://repository.upenn.edu/edissertations/3282 STUDIES ON FRACTIONALIZATION AND TOPOLOGY IN STRONGLY CORRELATED SYSTEMS FROM ZERO TO TWO DIMENSIONS Yichen Hu A DISSERTATION in Physics and Astronomy Presented to the Faculties of the University of Pennsylvania in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy 2019 Supervisor of Dissertation Charles L. Kane, Christopher H. Browne Distinguished Professor of Physics Graduate Group Chairperson Joshua R. Klein, Edmund J. and Louise W. Kahn Professor Dissertation Committee: Randall D. Kamien, Vicki and William Abrams Professor in the Natural Science Eugene J. Mele, Christopher H. Browne Distinguished Professor of Physics Jonathan J. Heckman, Assistant Professor of Physics Bo Zhen, Assistant Professor of Physics STUDIES ON FRACTIONALIZATION AND TOPOLOGY IN STRONGLY CORRELATED SYSTEMS FROM ZERO TO TWO DIMENSIONS COPYRIGHT c 2019 Yichen Hu Acknowledgments Number is the law of the universe. Pythagoras This is a thesis dedicated to integer numbers and their fractionalizations in condensed matter physics. I would like to begin by thanking my advisor Professor Charles Kane for his support and education during my six years at Penn. My happiest episodes in grad school are the countless afternoon discussions with Charlie about physics. No matter how busy he was, he would nevertheless patiently sat down to chat and share his passion for physics. His good taste in picking the right questions to ask and out-of-the-box ideas to attack them have greatly shaped the way I would approach physics problems. I would like to thank Professor Randall Kamien, Eugene Mele, Jonathan Heck- man and Bo Zhen for providing helping hands whenever I needed them. I have learned a lot from them outside my own research topics. I am also fortunate to have made many friends along this journey. Representing iii the frozen world of Midwest, Yuxuan Wang, Xiaoyu Wang, Jian Kang, Mengxing Ye, Xuzhe Ying, Chun Chen, Yuxiao Wu, Laimei Nie and Fnu Setiawan have all shown me the value of sincerity in doing good research. Friends I met at various workshops, Biao Lian, Juven Wang, Tianci Zhou, Xiao Chen and Shenghan Jiang, I enjoyed many intense discussions with them. Let me not forget my local pals who have always been friendly and supportive, Fan Zhang, Benjamin Wieder, J¨orn Venderbos, Tatyana Gavrilchenko, Eric Horsley, Zachariah Addison, Andrew Li, Arjun Kar, Matthew Decross, Pok Man Tam, Jing Cai. Most importantly, I would not have survived this tough PhD training without my family. My life trajectory could be completely altered without understandings from my parents, encouragements from my grandma and love from my wife. Although my mind is pondering mysteries of physics, my heart is always with you guys. iv ABSTRACT STUDIES ON FRACTIONALIZATION AND TOPOLOGY IN STRONGLY CORRELATED SYSTEMS FROM ZERO TO TWO DIMENSIONS Yichen Hu Charles L. Kane The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, super- conductors and semi-metals. Most examples of the aforementioned topological ma- terials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractional- ized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liq- uid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape. For the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fi- v bonacci topological field theory. Our theory is based on a SO(7)1 = SO(7)1=(G2)1 × (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related \anti-Fibonacci" phase, whose topo- logical order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics. For the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m = 1) exci- tonic pairing. Its ground state wavefunction is then worked out and generalized to m > 1. We give several pieces of evidence that this conjectured wavefunction cor- rectly describes a topological phase, dubbed \fractional excitonic insulator", within the same universality class as the corresponding Laughlin state at filling 1=m. We present physical arguments that gapless states with higher angular momentum pair- ing between energy