In this article, we will discuss time crystals produced by topological phases. This is a paper showing the appearance of time crystals in Haldane topological insulators in 1D systems. It uses the same time-varying effective Hamiltonian as usual, but incorporates the periodic changes in the time crystal into the wave function. It is labeled as topological, but unfortunately it does not mean that decoherence does not occur as in a topological quantum computer. For this simulation, we use the following Bose-Hubbard Hamiltonian ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>H</mi><mo>=</mo><msub><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow></msub><msub><mi>t</mi><mi>j</mi></msub><msubsup><mi>a</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow><mo>†</mo></msubsup><msub><mi>a</mi><mi>j</mi></msub><mo>+</mo><msub><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow></msub><mfrac><msub><mi>U</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub><mn>2</mn></mfrac><msub><mi>n</mi><mi>j</mi></msub><mo stretchy="false">(</mo><msub><mi>n</mi><mi>j</mi></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow></msub><msub><mi>V</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>n</mi><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub><msub><mi>n</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">H=\sum_{j=0}t_ja_{j+1}^\dagger a_j+\sum_{j=0}\frac{U_{j,j}}{2}n_{j}(n_j-1)+\sum_{j=0}V_{j,j+1}n_{j+1}n_j</annotation></semantics></math>H=∑j=0​tj​aj+1†​aj​+∑j=0​2Uj,j​​nj​(nj​−1)+∑j=0​Vj,j+1​nj+1​nj​﻿ Of these, ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>t</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">t_{j}</annotation></semantics></math>tj​﻿ and ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>U</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_{j,j}</annotation></semantics></math>Uj,j​﻿ are in a form that includes the integral in the wave function of the localized system. In the Haldane topological insulator, the ratio of the interspin interaction (exchange integral) to ﻿<math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>V</mi><mrow><mi>j</mi><mo separator="true">,</mo><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> V_{j,j+1}</annotation></semantics></math>Vj,j+1​﻿ is V/J=2.5, which is the eigenfunction of the periodically varying Hamiltonian similar to the Floquet lattice. Simulations have confirmed that time crystal oscillations occur with respect to the magnetization, with a period of 0.5 G/ms (0.5 Gauss per millisecond). Research on time crystals has been progressing rapidly in recent years, as simulations on quantum computers have become easier due to the availability of simulation SDKs. Therefore, I am planning to start my own research. <a href="https://arxiv.org/abs/1806.10536" rel="noopener noreferrer" target="_blank">[1806.10536] Topological Time Crystals (arxiv.org)</a>

Paper Introduction: Topological Time Crystals.

Hikaru Wakaura