Topological dualities via tensor networks
- URL: http://arxiv.org/abs/2309.13118v2
- Date: Mon, 30 Oct 2023 10:40:37 GMT
- Title: Topological dualities via tensor networks
- Authors: C. Wille, J. Eisert, A. Altland
- Abstract summary: Ground state of the toric code, that of the two-dimensional class D superconductor, and the partition sum of the two-dimensional Ising model are dual to each other.
Connecting fermionic and bosonic systems, the duality construction is intrinsically non-local.
We propose a unified approach to this duality, whose main protagonist is a tensor network (TN) assuming the role of an intermediate translator.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: The ground state of the toric code, that of the two-dimensional class D
superconductor, and the partition sum of the two-dimensional Ising model are
dual to each other. This duality is remarkable inasmuch as it connects systems
commonly associated to different areas of physics -- that of long range
entangled topological order, (topological) band insulators, and classical
statistical mechanics, respectively. Connecting fermionic and bosonic systems,
the duality construction is intrinsically non-local, a complication that has
been addressed in a plethora of different approaches, including dimensional
reduction to one dimension, conformal field theory methods, and operator
algebra. In this work, we propose a unified approach to this duality, whose
main protagonist is a tensor network (TN) assuming the role of an intermediate
translator. Introducing a fourth node into the net of dualities offers several
advantages: the formulation is integrative in that all links of the duality are
treated on an equal footing, (unlike in field theoretical approaches) it is
formulated with lattice precision, a feature that becomes key in the mapping of
correlation functions, and their possible numerical implementation. Finally,
the passage from bosons to fermions is formulated entirely within the
two-dimensional TN framework where it assumes an intuitive and technically
convenient form. We illustrate the predictive potential of the formalism by
exploring the fate of phase transitions, point and line defects, topological
boundary modes, and other structures under the mapping between system classes.
Having condensed matter readerships in mind, we introduce the construction
pedagogically in a manner assuming only minimal familiarity with the concept of
TNs.
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