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Quantum Many-Body States Defined Through Conformal Field Theory

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Quantum Many-Body States Defined Through Conformal Field Theory Technische Universität München Max-Planck-Institut für Quantenoptik Quantum many-body states defined through conformal field theory Benedikt Thomas Herwerth Vollständiger Abdruck der von der Fakultät für Physik der Technischen Universität München zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften (Dr. rer. nat.) genehmigten Dissertation. Vorsitzender: Prof. Dr. Alexander Holleitner Prüfer der Dissertation: 1. Hon.-Prof. J. Ignacio Cirac, Ph.D. 2. Prof. Dr. Frank Pollmann Die Dissertation wurde am 01.08.2018 bei der Technischen Universität München eingereicht und durch die Fakultät für Physik am 05.10.2018 angenommen. Abstract In this thesis, we study quantum many-body systems in one (1D) and in two spatial dimen- sions (2D). We adopt the approach established by Moore and Read, where model states are constructed using conformal field theory (CFT), a scale-invariant quantum field theory. The central themes of this thesis are the definition of states through CFT, their characterization, and the understanding of their properties in terms of the underlying CFT. The first part of this thesis considers a CFT with an additional SU(2) symmetry. We define a 1 map from CFT states to those of spin- 2 systems on lattices. In 1D, the CFT vacuum is mapped to the ground state of a Heisenberg-like spin chain with long-range inverse-square interactions. We show that the excited states of this spin chain can be constructed from excitations of the CFT. Thus, we establish a correspondence between the spectrum of the CFT and that of the spin chain. In a next step, we study these states in 2D, where the CFT ground state corresponds to a fractional quantum Hall (FQH) lattice state. Excited states of the CFT are mapped to wave functions describing edge excitations of the FQH system. Through Monte Carlo simulations, we provide numerical evidence for a central property of these edge modes, namely that their local bulk correlations coincide with those of the corresponding FQH state as the system size becomes large enough. These results confirm the bulk-edge correspondence stating that the CFT associated with the gapless edge can be used to describe wave functions of the bulk. We then consider a larger class of states constructed from CFT. In 1D, these are good 1 descriptions of the ground state of the XXZ spin- 2 chain, and they correspond to FQH lattice states in 2D. Through a path integral representation, we propose an approximation for their spin-spin correlations. The effective action determining this approximation differs from that of the underlying CFT through an additional mass-like term at the lattice positions. This explains the behavior of the correlations: While they decay polynomially in 1D and at the edge of 2D systems, bulk correlations in 2D decrease exponentially as a function of the distance. We test the accuracy of the approximation by comparing it to actual spin-spin correlations obtained through Monte Carlo simulations. Our approximation provides an analytical argument for the screening hypothesis, which states that FQH wave functions have exponentially decaying bulk correlations. The last part of this thesis uses CFT to define nonchiral states with continuous spins on lattices. Opposed to the case of discrete spins, these are Gaussian and thus their properties can be computed efficiently. Through an analysis of entanglement entropies and spectra, we identify signatures of the underlying CFT in 1D. In 2D, we find indications of edge states. Although the topological entanglement entropy vanishes, the entanglement spectrum contains modes that decay exponentially with the distance to the entanglement cut. 3 Zusammenfassung Diese Dissertation untersucht Quanten-Vielteilchensysteme in einer (1D) und in zwei räum- lichen Dimensionen (2D). Wir arbeiten im Rahmen der von Moore und Read etablierten Herangehensweise, wonach Modellzustände durch konforme Feldtheorie (CFT), einer skale- ninvarianten Quantenfeldtheorie, konstruiert werden. Die Schwerpunktthemen dieser Arbeit sind: die Definition von Quantenzuständen vieler Teilchen mittels CFT, deren Charakterisie- rung und das Verständnis ihrer Eigenschaften in Bezug auf die zugrundeliegende CFT. Der erste Teil dieser Dissertation betrachtet eine CFT, welche eine zusätzliche SU(2)-Sym- metrie aufweist. Wir definieren eine Abbildung von Zuständen der CFT zu solchen eines 1 Spin- 2 -Systems auf einem Gitter. In 1D wird das Vakuum der CFT auf den Grundzustand einer Spinkette abgebildet, welche ähnlich einem Heisenberg-Modell ist und langreichweitige Wechselwirkungen hat, die mit dem Quadrat der Distanz abfallen. Wir konstruieren Anre- gungszustände der Spinkette ausgehend von Anregungen der CFT. In einem weiteren Schritt untersuchen wir diese Zustände in 2D, wo der Grundzustand der CFT dem Zustand eines fraktionalen Quanten-Hall-Effekts (FQH) auf einem Gitter entspricht. Anregungen der CFT werden auf Wellenfunktionen abgebildet, welche Randzustände des FQH-Systems beschreiben. Mittels Monte-Carlo-Simulationen finden wir numerische Belege für eine zentrale Eigenschaft solcher Randzustände: Deren lokale Korrelationen im Inneren des Systems stimmen mit denen des zugrundeliegenden FQH-Zustands überein, sobald die Systemgröße groß genug wird. Dieses Ergebnis bestätigt das Konzept der Korrespondenz zwischen dem Inneren und dem Rand. Dieses besagt, dass die CFT des Randes Zustände des Inneren beschreiben kann. Danach betrachten wir eine größere Klasse von Zuständen, welche durch CFT definiert sind. 1 Diese sind gute Näherungen für den Grundzustand der XXZ-Spin- 2 -Kette in 1D und entspre- chen FQH-Zuständen in 2D. Durch eine Pfandintegraldarstellung leiten wir eine Näherung für Spin-Spin-Korrelationen her. Die effektive Wirkung dieser Näherung unterscheidet sich von derjenigen der zugrundeliegenden CFT dadurch, dass sie einen zusätzlichen Term enthält, der einem Masseterm ähnlich ist. Dieser ist an den Gitterplätzen des Systems lokalisiert und erklärt somit das Verhalten der Korrelationen: Diese zerfallen polynomiell in 1D und am Rand eines 2D Systems und exponentiell im Inneren eines 2D Systems. Wir überprüfen die Genauigkeit der Näherung, indem wir sie mit den tatsächlichen Spin-Spin-Korrelationen vergleichen, welche wir mittels Monte-Carlo-Simulationen berechnen. Unsere Näherung stellt einen analytischen Beleg für die Abschirmhypothese dar. Diese besagt, dass Korrelationen von FQH-Zuständen im Inneren des Systems exponentiell zerfallen. Der letzte Teil dieser Dissertation definiert nicht-chirale Zustände mit kontinuierlichem Spin auf Gittern mittels CFT. Im Unterschied zum Fall diskreter Spins sind dies Gaußsche Zu- stände. Daher können deren Eigenschaften effizient berechnet werden. Mittels einer Analyse von Verschränkungsentropien und -spektren identifizieren wir Charakteristika der zugrunde- liegenden CFT in 1D. In 2D finden wir Hinweise auf Randzustände. Obwohl die topologische Verschränkungsentropie verschwindet, enthält das Verschränkungsspektrum Moden, welche exponentiell am Rand lokalisiert sind. 5 Publications Publications related to this thesis 1. B. Herwerth, G. Sierra, H.-H. Tu, and A. E. B. Nielsen, “Excited states in spin chains from conformal blocks”, Phys. Rev. B 91, 235121 (2015), cf. Chapter3. 2. B. Herwerth, G. Sierra, H.-H. Tu, J. I. Cirac, and A. E. B. Nielsen, “Edge states for the Kalmeyer-Laughlin wave function”, Phys. Rev. B 92, 245111 (2015), cf. Chapter3. 3. B. Herwerth, G. Sierra, J. I. Cirac, and A. E. B. Nielsen, “Effective description of correla- tions for states obtained from conformal field theory”, Phys. Rev. B 96, 115139 (2017), cf. Chapter4. 4. B. Herwerth, G. Sierra, J. I. Cirac, and A. E. B. Nielsen, “Bosonic Gaussian states from conformal field theory”, arXiv:1807.01943 (2018), submitted to Phys. Rev. B, cf. Chap- ter5. Further publications 5. J. Erdmenger, B. Herwerth, S. Klug, R. Meyer, and K. Schalm, “S-wave superconductivity in anisotropic holographic insulators”, J. High Energ. Phys. 2015, 94 (2015). 6. B. Herwerth, M. DeKieviet, J. Madroñero, and S. Wimberger, “Quantum reflection from an oscillating surface”, J. Phys. B: At. Mol. Opt. Phys. 46, 141002 (2013) 7 Contents 1 Introduction 13 1.1 Motivation....................................... 13 1.1.1 The quantum Hall effect........................... 13 1.1.2 The quantum Hall effect in lattices and chiral spin liquids . 15 1.1.3 Landau’s theory of phases .......................... 15 1.1.4 Topological phases of matter......................... 15 1.1.5 Conformal field theory for building model systems............ 16 1.1.6 Lattice models from conformal field theory ................ 17 1.2 Purpose and findings of this thesis ......................... 17 1.2.1 Excited and edge states from conformal field theory........... 17 1.2.2 Effective description of correlations..................... 19 1.2.3 States with continuous spins......................... 20 1.3 Structure of this thesis ................................ 20 2 Conformal field theory and model systems 23 2.1 Conformal field theory ................................ 23 2.1.1 The free, massless boson and scale invariance............... 23 2.1.2 Conformal invariance and primary fields.................. 24 2.1.3 Generators of the conformal group and the central charge . 25 2.1.4 Primary fields for the CFT of a free boson................. 26 2.1.5 The free-boson CFT as a low-energy effective theory........... 26 2.2 Quantum Hall model states from conformal field theory............. 27 2.3 Model systems studied in this thesis ........................ 29 2.3.1 Definition of
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