Topological quantization and quantum theory (extended)

The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.

Jul 3, 202312 citations

Quantization, manifolds, topology

J. Tolar

Czechoslovak Journal of PhysicsSep 1, 19870 citations

Interfaces and the extended Hilbert space of Chern-Simons theory

Jackson R. Fliss, Robert G. Leigh

Abstract The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.Abstract The low energy effective field theories of (2 + 1) dimensional topological phases of matter provide powerful avenues for investigating entanglement in their ground states. In [1] the entanglement between distinct Abelian topological phases was investigated through Abelian Chern-Simons theories equipped with a set of topological boundary conditions (TBCs). In the present paper we extend the notion of a TBC to non-Abelian Chern-Simons theories, providing an effective description for a class of gapped interfaces across non-Abelian topological phases. These boundary conditions furnish a defining relation for the extended Hilbert space of the quantum theory and allow the calculation of entanglement directly in the gauge theory. Because we allow for trivial interfaces, this includes a generic construction of the extended Hilbert space in any (compact) Chern-Simons theory quantized on a Riemann surface. Additionally, this provides a constructive and principled definition for the Hilbert space of effective ground states of gapped phases of matter glued along gapped interfaces. Lastly, we describe a generalized notion of surgery, adding a powerful tool from topological field theory to the gapped interface toolbox.

arXiv:2004.05123Journal of High Energy PhysicsJul 1, 202010 citations

Topological gauge theory for mixed Dirac stationary states in all dimensions

Ze-Min Huang, Xiao-Qi Sun, Sebastian Diehl

We derive the universal real-time $\text{U}(1)$ topological gauge field action for mixed Dirac states of fermions, described by a density matrix weighted by a gapped Dirac operator, in all dimensions. The key prerequisites are charge quantization and charge conservation; in fact, charge quantization is the root of the action's topological properties. The gauge action encodes nonquantized linear responses as expected for mixed states, but also quantized nonlinear responses, which are accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to nonequilibrium systems. Finally, we demonstrate the independence of the gauge action of the underlying equilibrium or nonequilibrium nature of dynamics stabilizing the state.We derive the universal real-time $\text{U}(1)$ topological gauge field action for mixed Dirac states of fermions, described by a density matrix weighted by a gapped Dirac operator, in all dimensions. The key prerequisites are charge quantization and charge conservation; in fact, charge quantization is the root of the action's topological properties. The gauge action encodes nonquantized linear responses as expected for mixed states, but also quantized nonlinear responses, which are accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to nonequilibrium systems. Finally, we demonstrate the independence of the gauge action of the underlying equilibrium or nonequilibrium nature of dynamics stabilizing the state.

arXiv:2109.06891Physical review. B./Physical review. BDec 19, 202214 citations

We derive the universal real-time

\text{U}(1)

topological gauge field action for mixed Dirac states of fermions, described by a density matrix weighted by a gapped Dirac operator, in all dimensions. The key prerequisites are charge quantization and charge conservation; in fact, charge quantization is the root of the action's topological properties. The gauge action encodes nonquantized linear responses as expected for mixed states, but also quantized nonlinear responses, which are accessible in experiment. Our construction furthermore demonstrates how the physical pictures of anomaly inflow and bulk-boundary correspondence extend to nonequilibrium systems. Finally, we demonstrate the independence of the gauge action of the underlying equilibrium or nonequilibrium nature of dynamics stabilizing the state.