Tensor networks, quantum spin chains, and quantum field theory by Yijian Zou A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics Waterloo, Ontario, Canada, 2020 c Yijian Zou 2020 Examining Committee Membership The following served on the Examining Committee for this thesis. The decision of the Examining Committee is by majority vote. External Examiner: German Sierra Professor, Instituto de Fisica Teorica Supervisor(s): Guifre Vidal Adjunct Faculty, Dept. of Physics, University of Waterloo, Senior Faculty, Perimeter Institute for Theoretical Physics Yin-Chen He Adjunct Faculty, Dept. of Physics, University of Waterloo, Junior Faculty, Perimeter Institute for Theoretical Physics Internal Member: Roger Melko Professor, Dept. of Physics, University of Waterloo Internal-External Member: Eduardo Martin-Martinez Assistant Professor, Dept. of Applied Mathematics, University of Waterloo Other Member(s): Chong Wang Adjunct Faculty, Dept. of Physics, University of Waterloo, Junior Faculty, Perimeter Institute for Theoretical Physics ii Author's Declaration I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Statement of Contributions Yijian Zou was the sole author for Chapters 1, 2 and 7 which were written under the supervision of Prof. Guifre Vidal and were not written for publication. Prof. Guifre Vidal was a senior faculty of Perimeter Institute. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. Guifre Vidal received support from Simons Foundation (Many Electron Collaboration) and Compute Canada in the period of working on this thesis. This thesis consists in part of three manuscripts written for publication, as follows. The American Physical Society has the copyright of all three manuscripts. Research presented in Chapters 3 and 4: Dr. Ashley Milsted was a postdoc at Perimeter Institute. This research was conducted by Yijian Zou under the supervision of Guifre Vidal and Ashley Milsted. Yijian Zou contributed to theoretical derivations, part of code writing, part of numerical simulations. Ashley Milsted contributed to part of code writing and part of numerical simulations. Yijian Zou, Ashley Milsted, and Guifre Vidal. Conformal data and renormalization group flow in critical quantum spin chains using periodic uniform matrix product states. Phys. Rev. Lett., 121:230402, Dec 2018. DOI:10.1103/PhysRevLett.121.230402 Research presented in Chapter 5: Dr. Ashley Milsted was a postdoc at Perimeter Institute. This research was conducted by Yijian Zou under the supervision of Guifre Vidal and Ashley Milsted. Yijian Zou contributed to theoretical derivations, all code writing and part of numerical simulations. Ashley Milsted contributed to part of numerical simulations. Yijian Zou, Ashley Milsted, and Guifre Vidal. Conformal fields and operator product expansion in critical quantum spin chains. Phys. Rev. Lett., 124:040604, Jan 2020. DOI:10.1103/PhysRevLett.124.040604 Research presented in Chapter 6: iv This research was conducted solely by Yijian Zou under the supervision of Guifre Vidal. Yijian Zou and Guifre Vidal. Emergence of conformal symmetry in quantum spin chains: Antiperiodic boundary conditions and supersymmetry. Phys. Rev. B, 101:045132, Jan 2020. DOI:10.1103/PhysRevB.101.045132 v Abstract Understanding the universality class of continuous phase transition is of central im- portance in condensed matter physics. In one spatial dimension, the universal properties are encoded in the conformal field theory (CFT), which is in turn specified by the con- formal data. In this thesis, we propose a systematic method to extract complete and accurate conformal data from the critical quantum spin chain based on the operator-state correspondence, thus completing the project initiated by Cardy and others in the 80's. Our method is based on the low-energy eigenstates of the critical quantum spin chain with periodic boundary conditions. First, scaling dimensions and conformal spins are ex- tracted by the energies and momenta. Second, the primary states and conformal towers are identified by using the Koo-Saleur lattice Virasoro generators. Third, we propose a system- atic way of identifying lattice operators with CFT operators, which enables us to compute operator product expansion coefficients from the low-energy eigenstates. Finally, the whole approach is generalized to critical quantum spin chains with antiperiodic boundary condi- tion. In order to reduce finite-size corrections, we put forward the periodic uniform matrix product state (puMPS) algorithm which enables us to compute low-energy eigenstates of a critical quantum spin chain up to several hundreds of spins. Our method also enables us to study nonperturbatively the renormalization group flow between two CFTs as well as explore the emergence of extended symmetries beyond conformal symmetry. We test our method with the Ising model and its generalization due to O'Brien and Fendley. The latter model is featured by a tricritical point described by the tricritical Ising CFT and a line of critical points interpolating between the Ising CFT and the tricritical Ising CFT. We extract complete conformal data from the two models and find excellent agreement with analytical results. Furthermore, we study the spectral renormalization group flow between the two CFTs nonperturbatively. At the tricritical point, the under- lying CFT has an extended symmetry, the superconformal symmetry. We propose lattice operators that correspond to supervirasoro generators and verify their action on low-energy eigenstates. In this way we study the emergence of superconformal symmetry in critical quantum spin chains. vi Acknowledgements First and foremost, I would like to thank Prof. Guifre Vidal, without whom this thesis would be never possible. I have been greatly intrigued and affected by his sharp intuition and deep understanding of physics. I would also like to thank Dr. Ashley Milsted and Ruoshui Wang for nice collaboration and fruitful discussion that is indispensible for the research that we have done. In addition, I would like to thank Martin Ganahl, Qi Hu, Adrian Franco Rubio, Stefan Kuhn, Markus Hauru, Jacob Bridgeman for the Tensor Network Therapy weekly meeting where we made research a pleasure. I am grateful to Yin-Chen He, Chong Wang and Roger Melko who took their time to join my committee. I would like to also thank my colleagues Jingxiang Wu, Shanming Ruan, Weicheng Ye, Yan-yan Li and Zi-wen Liu for help in daily life and emotional support. Finally, I would like to thank my parents who support me throughout my past life. vii Table of Contents List of Figures xiii List of Tables xviii 1 Introduction1 2 Conformal field theory4 2.1 Universality in continuous phase transitions.................4 2.1.1 Phase transitions............................4 2.1.2 Quantum field theory..........................5 2.1.3 Renormalization group.........................6 2.2 Conformal field theory on the plane......................7 2.2.1 Conformal symmetry..........................7 2.2.2 Virasoro algebra............................8 2.2.3 Scaling dimension and conformal spin.................8 2.2.4 Primary operator............................8 2.2.5 Conformal tower............................9 2.2.6 Operator-state correspondence..................... 10 2.2.7 Operator product expansion...................... 11 2.2.8 Conformal data............................. 12 2.3 Conformal field theory on the cylinder.................... 13 viii 2.3.1 Mapping from the plane to the cylinder................ 13 2.3.2 Virasoro generators........................... 14 2.3.3 OPE coefficients............................ 15 2.4 Examples of conformal field theory...................... 15 2.4.1 Ising CFT................................ 16 2.4.2 TCI CFT................................ 17 3 Periodic uniform matrix product states 18 3.1 Quantum spin chains.............................. 18 3.1.1 Hilbert space.............................. 18 3.1.2 Entanglement.............................. 20 3.2 Matrix product states.............................. 21 3.2.1 Notation................................. 21 3.2.2 Uniform infinite matrix product states................ 21 3.2.3 Canonical forms............................. 23 3.2.4 Gradient descent optimiztion..................... 24 3.3 Periodic uniform matrix product states.................... 27 3.3.1 Gradient descent algorithm...................... 27 3.3.2 Preconditioning............................. 30 3.4 Excited states.................................. 30 3.4.1 puMPS Bloch states.......................... 32 3.4.2 The generalized eigenvalue problem.................. 33 3.4.3 Fidelity for the Ising model...................... 35 3.4.4 Validlity for generic critical quantum spin chains........... 36 4 Conformal data and renormalization group flow in critical quantum spin chains 39 4.1 Extraction of conformal data from a critical quantum spin chain...... 39 ix 4.1.1 Scaling dimensions and conformal spins................ 39 4.1.2 Koo-Saleur lattice Virasoro generators................ 40 4.1.3 Finite-size corrections.......................... 41 4.2 Numerical