#### Presentation Title

Using Tensor Networks to Simulate Topological Phase Transitions

#### Faculty Mentor

Xie Chen

#### Start Date

18-11-2017 10:00 AM

#### End Date

18-11-2017 11:00 AM

#### Location

BSC-Ursa Minor 112

#### Session

Poster 1

#### Type of Presentation

Poster

#### Subject Area

physical_mathematical_sciences

#### Abstract

Typical phase transitions in condensed matter systems involve the breaking of a symmetry, which happens at a critical point. Such transitions can be well understood using Landau's symmetry breaking theory while other types of phase transitions, which we call topological, are not nearly as well understood. So far, physicists have been trying to study topological phase transitions by identifying Landau systems which can be mapped to the topological ones, and drawing parallels between the two. Unfortunately, this is not a general approach and only works occasionally. There is, however, a powerful technique called tensor networks that can be used to simulate topological phase transitions equally well across all locally interacting models. Tensor networks are a way of numerically representing a many-body system's wave functions without causing a conventional computer to run out of memory. Recent work has improved the tensor network approach by identifying which parameters and mechanisms are responsible for driving the phase transitions, ultimately allowing us to begin programming a topological phase transition simulator. Here, we develop an algorithm that can compute the entropy of a quantum mechanical system called the cylindrical toric code. We show how it produces log(2) for the ground state of this system, an analytically verified result. The wide applicability of the tensor network approach will allow future studies to adapt this algorithm to compute the ground state energy across topological phase transitions in more complicated models. The information from these simulations may ultimately lead to a more general theory of topological phase transitions.

#### Summary of research results to be presented

We developed a numerical algorithm implementing tensor networks to compute the entropy of a quantum mechanical system called the cylindrical toric code. We used our program to find that the ground state of our system has topological entanglement entropy equal to log(2), a result which we confirmed with an analytical calculation. One of the most important features of our program is that it can easily be adapted to compute the ground state energy of a system across a phase transition. This will allow us to simulate topological phase transitions in any locally interacting model, perhaps shedding light on many unanswered questions about topological phase transitions.

Using Tensor Networks to Simulate Topological Phase Transitions

BSC-Ursa Minor 112

Typical phase transitions in condensed matter systems involve the breaking of a symmetry, which happens at a critical point. Such transitions can be well understood using Landau's symmetry breaking theory while other types of phase transitions, which we call topological, are not nearly as well understood. So far, physicists have been trying to study topological phase transitions by identifying Landau systems which can be mapped to the topological ones, and drawing parallels between the two. Unfortunately, this is not a general approach and only works occasionally. There is, however, a powerful technique called tensor networks that can be used to simulate topological phase transitions equally well across all locally interacting models. Tensor networks are a way of numerically representing a many-body system's wave functions without causing a conventional computer to run out of memory. Recent work has improved the tensor network approach by identifying which parameters and mechanisms are responsible for driving the phase transitions, ultimately allowing us to begin programming a topological phase transition simulator. Here, we develop an algorithm that can compute the entropy of a quantum mechanical system called the cylindrical toric code. We show how it produces log(2) for the ground state of this system, an analytically verified result. The wide applicability of the tensor network approach will allow future studies to adapt this algorithm to compute the ground state energy across topological phase transitions in more complicated models. The information from these simulations may ultimately lead to a more general theory of topological phase transitions.