Exact spectra of conformal supersymmetric nonlinear sigma models in two dimensions

Abstract

We study two-dimensional nonlinear sigma models in which the target spaces are the coset supermanifolds U(n + m|n)/[U(1)×U(n + m − 1|n)] ∼ = CP n+m−1|n (projective superspaces) and OSp(2n + m|2n)/OSp(2n + m − 1|2n) ∼ = S 2n+m−1|2n (superspheres), n, m integers, −2 ≤ m ≤ 2; these quantum field theories live in Hilbert spaces with indefinite inner products. These theories possess non-trivial conformally-invariant renormalization-group fixed points, or in some cases, lines of fixed points. Some of the conformal fixed-point theories can also be obtained within LandauGinzburg theories. We obtain the complete spectra (with multiplicities) of exact conformal weights of states (or corresponding local operators) in the isolated fixed-point conformal field theories, and at one special point on each of the lines of fixed points. Although the conformal weights are rational, the conformal field theories are not, and (with one exception) do not contain the affine versions of their superalgebras in their chiral algebras. The method involves lattice models that represent the strong-coupling region, which can be mapped to loop models, and then to a Coulomb gas with modified boundary conditions. The results apply to percolation, dilute and dense polymers, and other statistical mechanics models, and also to the spin quantum Hall transition in noninteracting fermions with quenched disorder.

ExactS-matrices for supersymmetric sigma models and the Potts model

“…One then argues that the spin chains have transitions, when e.g. 2S is odd and the nearest-neighbor couplings are unstaggered, and for m ≤ 2 these are second order and all in the same universality class for each m, n. Hence the results at S = 1/2 (the V , V * models described above) apply to the critical points in all the spin chain models, and also in the corresponding nonlinear sigma models as θ passes though π [32,29]. Thus the critical theories are closely related to those in the Potts model at Q = m 2 , m integer.…”

“…Notice that the spin-S representations in the supersymmetric representation of TL are multiplets with the degeneracies found in the appendix of Ref. [29], which are larger than those of irreducible representations of sl(m + n|n) for n > 0, except for the cases S = 0, 1, which are the singlet and adjoint of sl(m + n|n). Notice also that the "spin-S" representations here, which are part of the product V ⊗ V * ⊗ V · · · (2S times), [or V * ⊗ V ⊗ V * · · · (2S times), or these alternately if 2S is odd], are different from the "spin-S" representations in Sec.…”

“…There is a transition at the self-dual parameter values, which is first-order for m > 2, and second order for m ≤ 2. The complete exact spectra of the critical theories for −2 ≤ m ≤ 2 have been determined [29]. In particular, the case m = 1 gives a sequence of models that represent percolation, without an ill-defined limit Q → 1 being required.…”

“…We refer to Ref. [29] and references therein for the detailed arguments on all these theories and their transitions.…”

“…This is a consequence of the supersymmetry, which leads to a cancellation of corresponding bosonic and fermionic states in the familiar way. In general, it does not imply that all energy levels of theories that have equal partition functions are the same (if "partition function" means STr e −βH [29] for supersymmetric formulations, and the q-deformed trace for the U q (sl 2 ) formulation), since some may occur in multiplets of vanishing super-(or q-) dimension that do not contribute to the supertrace STr (resp., q-deformed trace). However, for the spin chains constructed here within the TL algebra, all the energy levels for different models do coincide, at least in the m = 2 case, since the spin-1/2 (XXZ or TL) representation of the TL algebra is faithful, as is the supersymmetric one (for n > 0), and this result remains true in the continuum limit.…”