1. Fermions without Fermi Fields RC Ball Department of Physics University of Warwick Fermions can emerge as the elementary excitations of generalised spin models. Such models can be constructed in any dimension of space. They are built of local ‘spin’ operators with the key locality property that: Operators acting at different locations always commute. For fermion hopping on an arbitrary graph, the corresponding spin model can be “built-to-order”. References: AY Kitaev, http://online.itp.ucsb.edu/online/glasses_03, Ann Phys 303 2 (2003)http://online.itp.ucsb.edu/online/glasses_03 M Levin & X-G Wen, Phys Rev B 67, 245316 (2003) RC Ball, cond-mat / 0409485 (2004)

2. the trouble with fermion operators …

3. Low Dimensional methods

4. Fermions deconstructed -> Majorana operators

5. Fermion Hopping Hamiltonian Require finite coordination number. Unrestricted loops -> arbitrary dimensionality of space.

6. Wen & Levin ‘trick’ A version of this argument was presented by Levin & Wen for square and cubic lattices

8. Local commutation relations on each site

9. Local matrix representation

10. Further developments Interactions and elaborations readily incorporated –any joint function of number operators allowed –physical spin  multiple species  spin dependent interactions. New hamiltonian should have as many conserved quantitities as the number of S operators introduced. These can all be found in terms of the  matrices. For d=1, the familiar anisotropic Heisenberg model is recovered. Simplest d=2 spin model with fermions has one set of Pauli matrices per site of hexagonal lattice.

11. Conservation Laws

12. Example: full fermions on even sites

13. Generalised Anisotropic Heisenberg Model

14. Simpler Majorana Hopping

15. General Majorana Hopping -> Spin Hamiltonian

16. Teasers exact hypothetical 3x3 square lattice Antiferromagnet Strong external field

17. Concluding Remarks Arbitrary fermion spectrum can be mapped into a spin model with no explicit fermion operators Size of local matrix representation set by coordination number  large for simplest lattices in higher d. Downside: separation of non-interacting fermions lost, giving a full many body hamiltonian. Upside: more species, spin, and especially interactions (Coulomb, Hubbard, t-J) are trivial to introduce. Speculation: more unsuspected fermions in other spin models? heavy fermions or other ‘exotic’ fermi liquids? References: AY Kitaev, http://online.itp.ucsb.edu/online/glasses_03, Ann Phys 303 2 (2003)http://online.itp.ucsb.edu/online/glasses_03 M Levin & X-G Wen, Phys Rev B 67, 245316 (2003) RC Ball, cond-mat / 0409485 (2004)