Quantum orders and symmetric spin liquids
Abstract
A conceptquantum orderis introduced to describe a new kind of orders that generally appear in quantum states at zero temperature. Quantum orders that characterize the universality classes of quantum states (described by complex groundstate wave functions) are much richer than classical orders that characterize the universality classes of finitetemperature classical states (described by positive probability distribution functions). Landau's theory for orders and phase transitions does not apply to quantum orders since they cannot be described by broken symmetries and the associated order parameters. We introduced a mathematical objectprojective symmetry groupto characterize quantum orders. With the help of quantum orders and projective symmetry groups, we construct hundreds of symmetric spin liquids, which have SU(2), U(1), or Z_{2} gauge structures at low energies. We found that various spin liquids can be divided into four classes: (a) Rigid spin liquidspinons (and all other excitations) are fully gapped and may have bosonic, fermionic, or fractional statistics. (b) Fermi spin liquidspinons are gapless and are described by a Fermi liquid theory. (c) Algebraic spin liquidspinons are gapless, but they are not described by free fermionicbosonic quasiparticles. (d) Bose spin liquidlowlying gapless excitations are described by a freeboson theory. The stability of those spin liquids is discussed in detail. We find that stable twodimensional spin liquids exist in the first three classes (a)(c). Those stable spin liquids occupy a finite region in phase space and represent quantum phases. Remarkably, some of the stable quantum phases support gapless excitations even without any spontaneous symmetry breaking. In particular, the gapless excitations in algebraic spin liquids interact down to zero energy and the interaction does not open any energy gap. We propose that it is the quantum orders (instead of symmetries) that protect the gapless excitations and make algebraic spin liquids and Fermi spin liquids stable. Since highT_{c} superconductors are likely to be described by a gapless spin liquid, the quantum orders and their projective symmetry group descriptions lay the foundation for a spin liquid approach to highT_{c} superconductors.
 Publication:

Physical Review B
 Pub Date:
 April 2002
 DOI:
 10.1103/PhysRevB.65.165113
 arXiv:
 arXiv:condmat/0107071
 Bibcode:
 2002PhRvB..65p5113W
 Keywords:

 73.43.Nq;
 74.25.q;
 11.15.Ex;
 Quantum phase transitions;
 Properties of type I and type II superconductors;
 Spontaneous breaking of gauge symmetries;
 Condensed Matter  Strongly Correlated Electrons;
 Condensed Matter  Mesoscale and Nanoscale Physics
 EPrint:
 58 pages, RevTeX4 home page: http://dao.mit.edu/~wen