Fig 1 . C oordination 3 Bethe lattice w/ dilution

1. Quantum s=1/2 Heisenberg antiferromagnet on the Bethe lattice at percolation Christopher L. Henley, Cornell University, DMR 1005466 Many atoms in solids carry a spin, a magnetic degree of freedom that (for the "spin 1/2" case we study) has got just two independent quantum states, called "up" and "down". Spin 1/2 systems are as quantum-mechanical as anything can be, hence they are studied by many groups in a search for exotic quantum excitations. In our numerical and theoretical studies, we're specifically interested in the case of "disorder", meaning each site in the system can have different parameters in our model. The simplest form of disorder is "dilution", meaning some of the spin carrying atoms are replaced by spinless atoms. We specialize to a particular case of dilution called a "percolation cluster", which means a cluster formed when the non-diluted atoms are just barely connected to others for an indefinite distance. Percolation clusters are irregular and different from each other, but statistically they are fractals, exhibiting many properties that are seen on all length scales. Rather than work on the usual model of a diluted square lattice,we use the "Bethe lattice" which is simpler since it branches forever without ever re-connecting to the original point; at the percolation point, both are qualitatively similar. Earlier simulations by A. Sandvik studied the antiferromagnet on a percolation cluster, meaning adjacent spins are favored to have opposite directions. He showed that a regular pattern is formed of alternating spin directions (depicted by red/green dots), extending arbitrarily far (long range order). Furthermore, his student L. Wang found that, in "unbalanced" corners of the cluster where the numbers of red and green dots aren't equal, there exist degrees of freedom that are almost decoupled from the long range order. Our work (with Hitesh Changlani, ShivamGhosh, and SumiranPujari) characterized these degrees of freedom by exactly calculating the energy levels and wave function for many different percolation clusters. We discovered that the degree of freedom at each unbalanced place is an emergent spin-1/2. (It is called "emergent" because it does not come from one of the spin-1/2 atoms that the system is made of, but instead is made up from correlated motions of the several nearby spin-1/2's.) We have also characterized the interactions between these emergent spin 1/2's. Fig 1. Coordination 3 Bethe lattice w/ dilution Fig 2. The geometry of the percolation cluster influences the nature and number of low lying energy levels. This can be attributed to the “dangling” degrees of freedom or the “emergent” spin ½ objects we refer to in the text. Their approximate spatial locations are shown with circles above

2. Quantum s=1/2 Heisenberg antiferromagnet on the Bethe lattice at percolation Christopher L. Henley, Cornell University, DMR 1005466 • The broader picture is that the whole field of quantum spin systems and exotic (so-called "frustrated") states has mostly avoided considering disorder, even though it is present in all real solids and is likely to have particularly strong effects on exotic quantum states of interest. This project aims to develop methods for numerical studies of quantum spin systems with disorder. • In particular, Hitesh Changlani implemented a new variation of the Density Matrix Renormalization Group (DMRG). This is a numerical method, a workhorse for computing properties usually of one dimensional quantum systems. However, Changlani has adapted it to the disordered Bethe lattice, taking advantage of its purely branching nature. • This work provides training for graduate students in collaboratively applying to interacting systems a mix of analytic and numerical techniques. The computational physicists that will make breakthroughs in the future are those with a deep understanding of both sides. Fig 3. Renormalization step involved in the DMRG on the Bethe lattice