@article{Fingerprints:6480,
      recid = {6480},
      author = {Liu, Yuhan},
      title = {Probing Universal Fingerprints of Quantum States: From  Quantum Entanglement to Conformal Field Theory},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2023-06},
      abstract = {Understanding the universal behaviors in quantum phases of  matter lies among the long-term dreams of condensed matter  physicists. However, relating universal behaviors to  physical observables is challenging. Surprisingly,  knowledge from other realms of physics can bring unique  insights. In this thesis, we develop two useful tools:  quantum entanglement and conformal field theory, which are  widely used in quantum information science and string  theory, respectively. We show how they reveal the  previously unknown universal information of three different  systems: topological liquid, random quantum states, and  critical spin chains. These three systems are typical  examples of the area law entanglement scaling phase, volume  law entanglement scaling phase, and logarithmic law  entanglement scaling phase, respectively.

We start by  examing the topological liquids. The universal features of  topological liquids make them useful for topological  quantum computation and quantum error correction. In such  systems, a piece of universal information, the quantum  dimension, is encoded in the constant term of von Neumann  entanglement entropy. However, von Neumann entanglement  entropy only captures the bipartite entanglement structure;  more universal features can be encoded in the multipartite  entanglement structures. Specifically, we investigate the  reflected entropy -- a tripartite entanglement quantity. We  unveil the reflected entropy yields a novel universal form  of two-dimensional chiral topological liquid, capturing its  central charge. To derive this result, we exploit the  bulk-boundary correspondence, approximating the ground  state of chiral topological liquid by vertex state in  boundary conformal field theory. The entanglement  quantities of the vertex state are then computed by  explicit numerical calculation and the conformal interface  method.

We next study the random quantum states. The  entanglement entropy of a typical state in a random  ensemble exhibits the volume law scaling. Understanding the  typical entanglement of random states thus provides  insights into thermalization and quantum chaos.  Specifically, we study von Neumann entanglement entropy of  random free fermion states in the presence of ten  fundamental symmetry classes. We find the fingerprint of  different symmetries lies in the constant term of typical  entanglement entropy and the entanglement variance. Our  results establish the symmetry classification of typical  quantum entanglement and the role of symmetry in quantum  chaos.

The discussion of boundary conformal field theory  brings us to the second important tool, conformal field  theory, a powerful framework for studying the critical  (gapless) phases. Given a critical quantum lattice model,  it is crucial yet difficult to extract the underlying  conformal data. In this subject, we develop the method of  wavefunction overlap, which can extract all conformal data.  Our key insight is that the universal finite-size  correction of the wavefunction overlap is dictated by the  orbifold conformal field theory, obtained by gauging the  symmetry in the replicated theory. Importantly, our method  bridges the gap of operator product expansion coefficient  extraction, where a numerical method was lacking. The power  of our method is demonstrated by unveiling new data of the  newly proposed Haagerup model built from the Haagerup  fusion category.

Finally, we discuss how to generalize  these tools to study more exotic quantum systems, which  might lead to future projects.},
      url = {http://knowledge.uchicago.edu/record/6480},
      doi = {https://doi.org/10.6082/uchicago.6480},
}