Quantum Physics Transitions

Resonating valence bonds and odd Z2spin liquids Subir Sachdev Department of Physics, Harvard University, Cambridge MA 02138, USA Abstract This PDF contains the published version of Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000) which is not available online, or in many libraries. Also attached are scanned pages from my notebook #24 written July 30, 1990. The content of these notes were outlined, and final results quoted, in Phys. Rev. B 44, 686 (1991) and Phys. Rev. B45, 12377 (1992). The full contents of these notes were published in Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000). I note that the results in Journal of the Physical Society of Japan 69, Suppl. B, 1 (2000) did not appear elsewhere in the literature, between the 1990 note and the 2000 publication. These publications and notes derive dual models of (what are now called) Z2spin liquids, and show for the first time that Z2spin liquids come in 2 varieties: these are now labeled ‘odd’ and ‘even’ Z2spin liquids. For quantum antiferromagnets with at least one components of the total spin conserved, odd (even) spin liquids appear for half-integer (even-integer) spin per unit cell. The even case corresponds ultimately to the Z2gauge theory studied by Wegner (and later in the toric-code model by Kitaev), and this can have a trivial phase with no broken symmetry. The odd case is new, and was considered here for the first time. In this case, translations along the xand ydirections anti-commute with each other in the m(‘vison’) superselection sector. In the Z2gauge theory description, there is a background gauge charge, and the confining phase is shown here to break lattice symmetry by the appearance of valence bond solid order. This ensures consistency with the Lieb-Schultz-Mattis-Oshikawa-Hastings theorems. It is also shown that vison (or ‘m’ particle) excitations are at least doubly degenerate for the odd Z2spin liquid. Anderson’s resonating valence bond (RVB) theory was for Mott insulators with one electron per site. Such insulators must obey the LSMOH theorem, and cannot have a trivial phase. So Wegner’s Z2gauge theory is not a correct effective theory for the RVB phase. Instead, it is the odd Z2gauge theory which is the correct effective theory for the RVB phase, a claim that appeared first in Phys. Rev. B 44, 686 (1991). The ‘odd’ nature of the effective theory has many important physical consequences, including changes in the universality classes of transitions out of the RVB phase. Subir Sachdev 1