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We present a paradigm for effective descriptions of quantum magnets. Typically, a magnet has many classical ground states — configurations of spins (as classical vectors) that have the least energy. The set of all such ground states forms an abstract space. Remarkably, the low energy physics of the quantum magnet maps to that of a single particle moving in this space.

This presents an elegant route to simulate simple quantum mechanical models using molecular magnets. For instance, a dimer coupled by an XY bond maps to a particle moving on a ring. An XY triangular magnet maps to a particle moving on two disjoint rings. We can even simulate Berry phases; when the spin has half-integer values, the particle sees a pi-flux  threaded through the rings.

A particularly interesting example is the XY tetrahedral magnet. Here, the ground state space is a 'non-manifold' due to singularities. These singularities behave like strong impurities to create bound states. The entire low energy physics of the magnet is dominated by these bound states. We call this phenomenon 'order by singularity’. This leads to a preference for certain classical ground states purely due to topology, rather than due to thermal or quantum fluctuations. Unlike order-by-disorder, this effect persists even in the classical limit.

Randomness is a valuable resource in both classical and quantum networks and we wish to generate desired probability distributions as cheaply as possible. If we are allowed to slightly change the distribution under some tolerance level, we can sometimes greatly reduce the cardinality of the randomness or the dimension of the entanglement. By studying statistical inequalities, we show how to upper bound of the amount of randomness required for any given classical network and tolerance level. We also present a problem we encounter when compressing the randomness in a quantum network.

We study remarkable RG flows in 4d QFT where supersymmetry enhances from N=1 to N=2 in the IR. This is triggered by the N=1 preserving deformation of 4d N=2 SCFTs with non-Abelian flavor symmetry by adding a chiral multiplet in the adjoint representation of the flavor symmetry and giving a nilpotent vev to the chiral multiplet. When the original N=2 SCFT and choice of the vev satisfy certain conditions, the resulting RG flows give N=2 Argyres-Douglas theories in the IR. These flows thus enable us to compute partition functions of Argyres-Douglas theories via localization. We also consider the more general and systematic deformation procedure which is applicable to any N=1 SCFTs and thus includes the above nilpotent one. Starting from N=1 SCFTs we allow all possible relevant deformations and/or coupling to a free chiral sector. We apply this to simple N=1 SCFTs and discuss the landscape of the IR fixed points, focusing on the enhancement of supersymmetry.

The Wilson formulation of lattice gauge theories provides a first-principles study of many properties of strongly interacting theories, such as quantum chromodynamics (QCD). Certain other properties, such as real-time dynamics, pose insurmountable challenges in this paradigm. Quantum Link Models are generalized lattice gauge theories, which not only offer novel approaches to study dynamics of gauge theories with quantum simulators, but also connect to
gauge theory descriptions of certain condensed matter systems (such as frustrated magnetism). In this talk, we explore some novel properties of gauge theories using quantum link models on classical computers, and explain how to realize them on quantum simulators.