Navigating Frustration

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TECHNICAL SUMMARYThis award supports theoretical research and education on frustrated systems. The PI aims to understand how to handle spins S = 1 and S = 3/2, and how to find emergent local degrees of freedom due to disorder. The research has 4 thrusts:1 Methods - The methods of quasi-degenerate density matrix and/or Renyi entropy will be applied to characterize emergent local degrees of freedom. Locally-dependent variational wavefunctions will be developed to model the ground states of quantum systems lacking translational invariance.2. Highly frustrated antiferromagnets - The spin-1 antiferromagnet on kagome and pyrochlore lattices will be studied, using variational Monte Carlo, also by DMRG using the loop-free cactus lattice as an ersatz. The effective Hamiltonian approach will be applied to situations with random fluxes, both to find the phase diagram of spins coupled to electrons on the kagome lattice, and also to find the correct ground state for spin-1/2 pyrochlore antiferromagnet in the large-N Schwinger boson theory.3. Disorder and frustration - The PI aims to resolve the three major mysteries about the spin-1/2 antiferromagnet at percolation on the Bethe lattice: (1) Why do the emergent spin degrees of freedom decouple? (2) Is the long-range order propagated purely by them? (3) How does the smallest gap scale with system size?4. Interacting fermions - Exact diagonalization and analytic methods will be used to decode the intriguing regularities of the 'coincident amplitude' wavefunctions. The methods the PI will develop will likely be useful for any material with disorder. The PI is completing a graduate text on solid state physics after the introductory texts, but without field theory, a pedagogical middle ground which hopefully will be congenial to experimentalists working on quantum materials. The PI will involve undergraduate students in the research.NONTECHNICAL SUMMARYThis award supports theoretical research and education on frustrated materials. "Frustration" is the technical term for what happens when either electrons which may hop between atoms in a solid, or "spins," which are abstractions of the intrinsic magnetism of electrons, when the electrons don't hop, give conflicting signals to each other. A hallmark of a frustrated system is: rather than having a single collective state which is obviously favored by the interactions, there are numerous collective states which are equally good by technical measures. As a result, small interactions that are usually ignored might tip the balance and favor a particular one of these states. Or, when quantum mechanics is important, a quantum superposition of these states may result, and in some cases such a collective state has exotic properties, such as emergent "particles" having fractional charge. The PI aims to find ways to "navigate" through these states. In many real materials the regular lattice defining the rules for a frustrated model is "diluted" i.e. missing an occasional site. This may have dramatic effects in the case of a frustrated system. The PI will complete an ongoing study that does not include frustration, but combines quantum mechanics with dilution to the percolation threshold, where the network of interacting spins is on the verge of becoming disconnected due to the removed sites. The PI will adapt a computational method study possible non-uniform states in systems where the rules of the game are uniform, but frustrated. Several projects which address dilution will be pursued.The PI aims to adapt a variety of numerical methods. For example, the quantity called "entanglement" which is a measure of how much quantum information a systems has. This can be a powerful way to identify the kind of emergent state the system is in. The PI will evaluate the entanglement of a small region relative the environment that surrounds it and evaluate whether this can be used to identify which kind of emergent local degree of freedom is appearing in that region because of dilution.The methods the PI will develop will likely be useful for any material with disorder. The PI is completing a graduate text on solid state physics after the introductory texts, but without field theory, a pedagogical middle ground which hopefully will be congenial to experimentalists working on quantum materials. The PI will involve undergraduate students in the research.