Duality and topology

M. Socolovsky

Howard J. Schnitzer

It is shown, using level-rank duality that a universal topological quantum computer based on Chern-Simons theory for SU(2)$_3$ also implies an analogous universal quantum computer based on SU(3)$_2$. Suggestions are made for the possible role of level-rank duality in entanglement from topology.

K. Birgitta Whaley

Tantivasadakarn, Nathanan

Dualities are central to our understanding of physics, in particular in the study of quantum phases of matter. In this dissertation, we explore exact lattice dualities that relate various topological phases of matter, which elucidates the interrelations between them. Moreover, searching for new dualities can even lead to the theoretical realization of new phases of matter. In the first half, we study local dualities, and how they can be used as quantum circuits to prepare Symmetry-Protected Topological (SPT) phases. We construct explicit quantum circuits that prepare fermionic SPT phases, which gives insight into the strongly interacting nature of these phases in contrast to previous free-fermion realizations. Moreover, we discuss a family of dualities, which not only generates the entanglement necessary to prepare SPT phases, but also generates anomalous symmetries which arise at transitions between such topological phases, giving new insights into topological quantum criticality. In the second half, we explore non-local dualities, exemplified by the famous Kramers-Wannier or Jordan-Wigner dualities in one dimension. we give a systematic method to generalize such non-local dualities to systems with different symmetries and to higher dimensions, which leads to explicit lattice models realizing new phases of matter. In particular, we uncover new families of fracton phases such as (i) Phases where immobile fracton excitations are emergent fermions (ii) Fracton phases that are enriched by spatial and/or time-reversal symmetries (iii) phases that "hybridizes" between conventional topological orders and fracton orders, some of which hosts non-abelian fracton excitations. Finally, we show that the two types of dualities can be related by an extra ingredient: measurement. This gives a practical method to realize non-local dualities in near-term quantum devices, paving a way to efficiently prepare a large class of topological phases of matter in experiments.

Robijn Vanhove, Vibhu Ravindran, David T. Stephen +2 more

Two quantum theories which look different but are secretly describing the same low-energy physics are said to be dual to each other. When realized in the Topological Holography formalism, duality corresponds to changing the gapped boundary condition on the top boundary of a topological field theory, which determines the symmetry of the system, while not affecting the bottom boundary where all the dynamics take place. In this paper, we show that duality in the Topological Holography formalism can be realized with a Sequential Quantum Circuit applied to the top boundary. As a consequence, the Hamiltonians before and after the duality mapping have exactly the same spectrum in the corresponding symmetry sectors, and the entanglement in the corresponding low-energy eigenstates differs by at most an area law term.

David Finkelstein, Ernesto Rodriguez