Talks

I will propose a proof for a monotonicity theorem, or c-theorem, for a three-dimensional Conformal Field Theory (CFT) on a space with a boundary, and for a two-dimensional defect coupled to a higher-dimensional CFT. The proof is applicable only to renormalization group flows that are localized at the boundary or defect, such that the bulk theory remains conformal along the flow, and that preserve locality, reflection positivity, and Euclidean invariance along the defect. The method of proof is a generalization of Komargodski’s proof of Zamolodchikov’s c-theorem. The key ingredient is an external “dilaton” field introduced to match Weyl anomalies between the ultra-violet (UV) and infra-red (IR) fixed points. Reflection positivity in the dilaton’s effective action guarantees that a certain coefficient in the boundary/defect Weyl anomaly must take a value in the UV that is larger than (or equal to) the value in the IR. This boundary/defect c-theorem may have important implications for many theoretical and experimental systems, ranging from graphene to branes in string theory and M-theory.

Identifying the nature of dark  matter is one of the most challenging problems in physics. There is a general consensus that dark matter is a weakly interacting particle and predominantly cold, yet the Cold Dark Matter (CDM) hypothesis remains to be verified.  I will show that next cosmological surveys could play a leading role in understanding the dark matter microphysics.

I'll discuss some recent results, motivated by the black-hole firewall paradox and the AdS/CFT correspondence, about the quantum circuit complexity of preparing certain entangled states and implementing certain unitary transformations.  One result is a strengthening of an argument by Harlow and Hayden: I'll show that, under plausible assumptions, "decoding" useful information from Hawking radiation, as called for by the AMPS "firewall" thought experiment, requires the computational power to invert arbitrary cryptographic one-way functions, something we think not even quantum computers could do in sub-astronomical time.  A second result, joint with Lenny Susskind, considers the circuit complexity of the kinds of states that could arise in AdS/CFT, and shows that, under a reasonable conjecture about complexity classes (PSPACE is not in PP/poly), the complexity indeed becomes superpolynomially large, as predicted by a conjectured relationship between complexity and geometry.  I'll also discuss more general open questions about the complexities of states and unitary transformations (and the connections to quantum money, quantum proofs, and other topics), which I find fascinating even apart from the quantum-gravity motivation.

Parafermions are the simplest generalizations of Majorana fermions that realize topological order. We propose a less restrictive notion of topological order in 1D open chains, which generalizes the seminal work by Fendley [J. Stat. Mech., P11020 (2012)]. The first essential property is that the groundstates are mutually indistinguishable by local, symmetric probes, and the second is a generalized notion of zero edge modes which cyclically permute the groundstates. These two properties are shown to be topologically robust, and applicable to a wider family of topologically-ordered Hamiltonians than has been previously considered. As an application of these edge modes, we formulate a new notion of twisted boundary conditions on a closed chain, which guarantees that the closed-chain groundstate is topological, i.e., it originates from the topological manifold of the open chain. Finally, we generalize these ideas to describe symmetry-breaking phases with a parafermionic order parameter.

These exotic phases are condensates of parafermion multiplets, which generalizes Cooper pairing in superconductors. The stability of these condensates are investigated on both open and closed chains.