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Topological Invariants of Gapped Quantum Lattice Systems


Sopenko, Nikita A. (2023) Topological Invariants of Gapped Quantum Lattice Systems. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/715e-q184.


In the first part of the thesis, a systematic way to construct topological invariants of gapped states of quantum lattices systems is proposed. It provides a generalization of the Berry phase and its equivariant analogue to systems with locality in arbitrary dimensions. For a smooth family of gapped ground states in d dimensions, it gives a closed (d + 2)-form on the parameter space which generalizes the curvature of the Berry connection. Its cohomology class is a topological invariant of the family. When the family is equivariant under the action of a compact Lie group G, topological invariants take values in the equivariant cohomology of the parameter space. These invariants unify and generalize the Hall conductance and the Thouless pump. We prove quantization properties of the invariants for low-dimensional invertible systems.

In the second part, we discuss the properties of the invariant associated with the Hall conductance for 2d lattice systems with U(1)-symmetry. We define anyonic states associated with the flux insertions and relate their statistics to this invariant. We also provide the construction of states realizing chiral topological order with a non-trivial value of this invariant. The construction is based on the data of a unitary regular vertex operator algebra.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Topological phases of matter, quantum statistical mechanics.
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Physics
Awards:Robert F. Christy Prize for an Outstanding Doctoral Thesis in Theoretical Physics, 2023. John Stager Stemple Memorial Prize in Physics, 2021.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Kapustin, Anton N.
Thesis Committee:
  • Chen, Xie (chair)
  • Kitaev, Alexei
  • Marcolli, Matilde
  • Kapustin, Anton N.
Defense Date:15 May 2023
Non-Caltech Author Email:niksopenko (AT)
Record Number:CaltechTHESIS:06032023-021309617
Persistent URL:
Related URLs:
URLURL TypeDescription DocumentChiral topologically ordered states on a lattice from vertex operator algebras DocumentLocal Noether theorem for quantum lattice systems and topological invariants of gapped states DocumentA classification of invertible phases of bosonic quantum lattice systems in one dimension DocumentHall conductance and the statistics of flux insertions in gapped interacting lattice systems
Sopenko, Nikita A.0000-0002-8479-1924
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:16072
Deposited By: Nikita Sopenko
Deposited On:08 Jun 2023 15:36
Last Modified:16 Jun 2023 22:39

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