UNIVERSITY OF CINCINNATI Wave Functions of Integrable Models Author: Committee Chair: Zhongtao Mei Prof. Carlos J. Bolech A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the McMicken College of Arts and Sciences Department of Physics M.Sc., University of Cincinnati, Cincinnati, USA B.Sc., Huazhong University of Science and Technology, Wuhan, China June 2018 Abstract This thesis centers on Bethe wave functions for one-dimensional lattice and field theories of quantum systems. The Bethe ansatz is a well established theoretical tool in the cross- cutting research field of integrable systems. However, most of the work in the literature focuses on the eigenenergies, and very limited results are known about the eigenstates. We present here two pieces of work to extend and deepen our understanding about Bethe wave functions. In one work, we recast the Bethe states as exact matrix product states for the Heisenberg 1 XXZ spin- 2 chain and the Lieb-Liniger model with open boundary conditions, and find that the matrices do not depend on the spatial coordinate despite the open boundaries. Based on this result, we suggest generic ways of exploiting translational invariance both for finite size and in the thermodynamic limit. Our work makes the Bethe eigenstates more accessi- ble and informs the choice of ansatz for tensor-network algorithms of both integrable and nonintegrable systems in one dimension. This achievement contributes in this way not only to the basic theory of integrable models but it will also influence the community that works using matrix product states and other tensor networks. In another work, we use the physical interpretation of rapidities in integrable models to calculate the asymptotic expansion velocity of interacting atomic gases. Which is acces- sible in sudden expansion experiments as those done routinely these days using optically- trapped cold atomic gases. Through our research, the calculations of the asymptotic forms of observables of integrable models in quantum quench problems become more clear and theoretically accessible. ii iii Acknowledgments At first, I would like to thank my advisor Prof. Carlos Bolech. He introduced me to the area of integrable models, which I think is an interesting field. Prof. Bolech also provided many useful comments for the writing of this thesis. I have learned much from him. I also want to thank Dr. Nayana Shah. The discussion with her broadens my knowledge of theoretical physics. I would like to thank my committee members Prof. Philip Argyres and Prof. Hans-Peter Wagner for their guidance in the past a few years. Some comments from Prof. Argyres are especially valuable. I am also grateful to Prof. Slava Serota for his help and encouragement during my years at Cincinnati. Next, I want to thank Prof. Michael Ma, Dr. Kuei Sun, and Dr. Sangwoo Chung for interesting discussion about physics and things beyond physics. I also want to thank my father and my mother. Without their constant support, I can not go so far. Last but not least, I want to thank my daughter and my wife. They are my best friends in the world, and they bring much happiness to me. iv Contents Abstract ii Acknowledgments iv List of Symbols vii 1 Introduction1 1.1 Outline......................................... 1 1.2 Bethe Ansatz and Integrability ........................... 2 1.2.1 The Origin of Bethe Ansatz......................... 2 1.2.2 Brief History of Bethe Ansatz........................ 4 1.2.3 Basic Building Blocks of Algebraic Bethe Ansatz............. 4 1.2.4 A Note about Integrability and Exact Solvability............. 8 1.3 Matrix Product States................................. 9 1.4 Cold Atoms and Quantum Quenches........................ 11 1.5 Motivation of Our Research............................. 13 1.5.1 Motivation for Our Research on Matrix Product States ......... 13 1.5.2 Motivation for Our Research on Quantum Quench in Cold Atoms . 14 2 Derivation of MPSs for Integrable Models with OBCs 15 2.1 Background and Motivation............................. 15 2.2 Derivation of MPSs for the Heisenberg Spin Chain with OBCs......... 18 1 2.2.1 ABA for the Heisenberg XXZ Spin- 2 Chain with OBCs......... 18 2.2.2 Comparison between ABA and CBA.................... 21 1 2.2.3 Exact MPSs for the XXZ Spin- 2 Chain with OBCs............ 24 2.2.4 Discussion and Conclusions......................... 28 2.2.5 Comments on the Key Idea of Our Research ............... 30 2.3 MPSs for the Lieb-Liniger Model with OBCs ................... 31 2.3.1 ABA for the Lieb-Liniger Model ...................... 31 2.3.2 Exact MPSs for the Lieb-Liniger Model with OBCs............ 34 3 Expansion Velocity of 1D Hubbard Model 38 3.1 Background ...................................... 38 3.1.1 Introduction.................................. 38 3.1.2 Asymptotic Form of the Quasi-MDF at Half Filling for FHM . 41 3.2 Expansion Velocity of the 1D FHM......................... 43 3.2.1 Eigenvalue Equations of the 1D FHM with OBCs ............ 43 3.2.2 BA Equations of the 1D FHM with OBCs................. 52 v 3.2.3 Expansion Velocity of the 1D FHM..................... 55 3.3 Expansion Velocity of the 1D BHM......................... 61 3.4 Discussion....................................... 63 4 Outlook 65 4.1 Summary........................................ 65 4.2 Possible Future Directions.............................. 65 Bibliography 67 A Proof of Eq. (2.10) for the XXZ Chain with OBCs 76 A.1 An Equation about the R-matrix of the XXZ Spin Chain............. 76 A.2 Proof of Eq. (2.10)................................... 77 B Proof of the Integrability of the XXZ Chain with OBCs 79 C Recover the Hamiltonian of the XXZ Chain with OBCs 82 D Eigenenergy and BAEs of the XXZ Chain with OBCs 85 D.1 Commutation Relations among A, B, C, and D . 85 D.2 Eigenenergy and BAEs................................ 91 E Expansion Velocity of the FHM 97 F Ground State Rapidities for the BHM 98 G List of Publications 101 vi List of Symbols 1ˆ identity matrix h¯ reduced Planck constant kB Boltzmann constant si Pauli matrix jyi wave function ⊗ tensor product dij Kronecker delta D anisotropy parameter for the XXZ spin chain #ijk Levi-Civita symbol XA(a) monodromy matrix for the Hubbard model jwi reference ferromagnetic state for the XXZ spin chain y(x) boson annihilation operator (for field) y†(x) boson creation operator (for field) ai boson annihilation operator (for lattice) [A, B] commutator, defined as AB − BA cis fermion annihilation operator (for lattice) C set of complex numbers E energy H Hamiltonian L number of lattice sites ln x the natural logarithm of x log x the natural logarithm of x R set of real numbers Pij permutation matrix T thermodynamic temperature T0(l) monodromy matrix for the Heisenberg model t(l) transfer matrix Tr taking the trace tr0 trace over the auxiliary space V0 Z partition function Z set of integers vii To my daughter Luchen, and my wife Jingjing. viii Chapter 1 Introduction 1.1 Outline For this thesis, the main focus is on the wave functions of integrable models. We do re- search on two topics related to Bethe wave functions. In one topic, we recast the Bethe wave 1 functions as exact matrix product states (MPSs) for Heisenberg XXZ spin- 2 chain and Lieb- Liniger model with open boundary conditions (OBCs), which was a difficult problem defy- ing resolution for at least 11 years. The matrices do not depend on the spatial coordinate despite the open boundaries. Based on this result, we suggest generic ways of exploiting translational invariance both for finite size and in the thermodynamic limit. The results were published as Ref. [1]. In another topic, we invent a simple approximate method to calculate the asymptotic expansion velocity of interacting atomic gases. The results were published as Ref. [2]. In this chapter, we give the background and motivation of our research. 1 In Chapter2, we derive MPSs for the Heisenberg spin- 2 chain and Lieb-Liniger model with OBCs [1]. For each model, we first give full details about the solution of the model by using algebraic Bethe ansatz (ABA). Then we recast the Bethe eigenstates as exact MPSs, which is our original research. This chapter not only presents our original research, but also it can be used as a detailed introduction to ABA. Chapter3 deals with the sudden expansion of cold atoms in one dimension [2]. Based on a physical interpretation of the rapidities that parameterize the Bethe wave function of integrable models, we invent a simple approximate method to calculate the asymptotic ex- pansion velocity of cold atoms in one dimension. We deal with both Fermi-Hubbard model 1 Chapter 1. Introduction (FHM) and Bose-Hubbard model (BHM). In Chapter4, we discuss the outlook of our work. 1.2 Bethe Ansatz and Integrability 1.2.1 The Origin of Bethe Ansatz Theoretical physics is an exact science to help us to understand the universe. Newtonian mechanics can help us to understand the movements of a particle, a rigid body, a satellite, etc. J. C. Maxwell’s electromagnetic theory can be used to understand the nature of elec- tric and magnetic interactions, and underpins the workings of modern devices like the cell phone, the TV, etc. A. Einstein’s general relativity is important for us to understand the pre- cession of the perihelion of Mercury, and underpins the workings of modern technologies like the global positioning system (GPS), etc. To understand the behavior of a large number of particles, we need statistical physics. Statistical physics was developed by J. C. Maxwell (1831 − 1879), L. Boltzmann (1844 − 1906), J. W. Gibbs (1839 − 1903), etc. An interesting phenomena in nature is ferromagnetism. For a permanent magnet, there is magnetism if the temperature is below the Curie tempera- ture. Above the Curie temperature, there is no magnetism even for permanent magnet.