Supersymmetric nonlinear O(3) sigma model on the lattice

Abstract

A supersymmetric extension of the nonlinear O(3) sigma model in two spacetime dimensions is investigated by means of Monte Carlo simulations. We argue that it is impossible to construct a lattice action that implements both the O(3) symmetry as well as at least one supersymmetry exactly at finite lattice spacing. It is shown by explicit calculations that previously proposed discretizations fail to reproduce the exact symmetries of the target manifold in the continuum limit. We provide an alternative lattice action with exact O(3) symmetry and compare two approaches based on different derivative operators. Using the nonlocal SLAC derivative for the quenched model on moderately sized lattices we extract the value σ(2, u 0 ) = 1.2604(13) for the step scaling function at u 0 = 1.0595, to be compared with the exact value 1.261210. For the supersymmetric model with SLAC derivative the discrete chiral symmetry is maintained but we encounter strong sign fluctuations, rendering large lattice simulations ineffective. By applying the Wilson prescription, supersymmetry and chiral symmetry are broken explicitly at finite lattice spacing, though there is clear evidence that both are restored in the continuum limit by fine tuning of a single mass parameter.

Finetuning the continuum limit in low-dimensional supersymmetric theories

“…Alas, by simulating the action of [3] we find that the order parameter of the O(3) symmetry does not vanish in the infinite volume limit, as is shown in Fig. 1 (left panel), and the resulting continuum limit does not belong to the supersymmetric O(3) NLSM, therefore rendering the Q-exact discretisation invalid [2]. This failure raises the question whether it is possible at all to find a lattice formulation of this model that preserves both the O(3) symmetry as well as part of supersymmetry.…”

“…In order to simulate the theory on the lattice, it is necessary to rewrite the contraints for the component fields. We choose a stereographic projection of the superfield which resolves the field constraints explicitly [2]. In addition to supersymmetry, the classical action is invariant under SO(3)-rotation of the fields.…”

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

“…As pointed out in [12] there exists an N = 2 extension for supersymmetric nonlinear sigma models which have a Kähler target manifold. This is the case for N = 3 and hence an additional SUSY can be worked out [7]. The constraints in eq.…”

“…In these proceedings we report on our ongoing study to apply such a programme to the supersymmetric nonlinear O(N) sigma model regularised on the lattice. This model has already been investigated numerically using a variety of different discretisations [5,6,7]. Here we concentrate on reformulating the model in terms of fermion loops in order to make use of the efficient simulation algorithms and related methods to control the fermion sign problem accompanying any spontaneous supersymmetry breaking [8].…”

“…When regularising the model on a discrete space-time lattice we follow the strategy in [7] and use the Wilson derivative for both the fermionic and the bosonic action. This strategy is well supported by various theoretical and numerical arguments [3,4,8,13,14,15,16,17,18].…”