CMSA Quantum Matter in Mathematics and Physics: A Riemann sum of quantum field theory: lattice Hamiltonian realization of TQFTs


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February 24, 2021 10:30 am - 12:00 pm
via Zoom Video Conferencing

Zhenghan Wang - Microsoft Station Q

Walker and I wrote down a lattice model schema to realize the (3+1)-Crane-Yetter TQFTs based on unitary pre-modular categories many years ago, and application of the model is found in a variety of places such as quantum cellular automata and fracton physics.  I will start with the conceptual origin of this model as requested by the organizer.  Then I will discuss a general idea for writing down lattice realizations of state-sum TQFTs based on gluing formulas of TQFTs and explain the model for Crane-Yetter TQFTs on general three manifolds.  In the end, I will mention lattice models that generalize the Haah codes in two directions:  general three manifolds and more than two qubits per site.

If the path integral of a quantum field theory is regarded as a generalization of the ordinary definite integral, then a lattice model of a quantum field theory could be regarded as an analogue of a Riemann sum.  New lattice models in fracton physics raise an interesting question:  what kinds of quantum field theories are they approximating if their continuous limits exist?  Their continuous limits would be rather unusual as the local degrees of freedom of such lattice models increase under entanglement renormalization flow.