Loop formulation for the non-linear supersymmetric O(N) sigma-model

Abstract

We derive the fermion loop formulation for the supersymmetric nonlinear O(N) sigma model by performing a hopping expansion using Wilson fermions. In this formulation the fermionic contribution to the partition function becomes a sum over all possible closed non-oriented fermion loop configurations. The interaction between the bosonic and fermionic degrees of freedom is encoded in the constraints arising from the supersymmetry and induces flavour changing fermion loops. For N ≥ 3 this leads to fermion loops which are no longer self-avoiding and hence to a potential sign problem. Since we use Wilson fermions the bare mass needs to be tuned to the chiral point. For N = 2 we determine the critical point and present boson and fermion masses in the critical regime.

Supersymmetric quantum mechanics on the lattice: II. Exact results

“…In contrast, the fermion loop formulation can be used on its own, e.g. [19,20], and in some cases even provides the basis for the solution of the fermion sign problem such as in the N = 1 Wess-Zumino model [7,21].…”

Loop formulation of supersymmetric Yang-Mills quantum mechanics

“…Secondly, we note that the various sectors, in particular the ones with many fermions, can in principle be simulated by open fermion string (fermion worm) algorithms along the lines described in [23,24]. This approach has indeed already been applied successfully in ordinary supersymmetric quantum mechanics [22], in the supersymmetric nonlinear O(N ) sigma model [32] and in the two dimensional N = 1 Wess-Zumino model [25,27] where the transfer matrix techniques discussed here and in [26] are out of reach. Furthermore, for the model discussed in this paper, a discrete bond formulation for the bosonic degrees of freedom is available [33].…”

Supersymmetric quantum mechanics on the lattice: I. Loop formulation

“…Furthermore, for μ > 0 and g ≥ 0, P u (φ) > 0, and Eqs. (35) and (36) then imply that Z p = 0 and hence the Witten index is nonzero, W = 0. Thus, we conclude that for this superpotential supersymmetry is indeed unbroken, in agreement with the generic expectation from supersymmetric quantum mechanics.…”

“…in higher dimensions, or involving gauge fields. In particular it can be applied to supersymmetric Yang-Mills quantum mechanics [30] and certain two-dimensional supersymmetric field theories, such as the N = 1 Wess-Zumino model [31][32][33][34] and the supersymmetric nonlinear O(N ) sigma model [35].…”