Supersymmetric quantum mechanics on the lattice: III. Simulations and algorithms

Abstract

In the fermion loop formulation the contributions to the partition function naturally separate into topological equivalence classes with a definite sign. This separation forms the basis for an efficient fermion simulation algorithm using a fluctuating open fermion string. It guarantees sufficient tunnelling between the topological sectors, and hence provides a solution to the fermion sign problem affecting systems with broken supersymmetry. Moreover, the algorithm shows no critical slowing down even in the massless limit and can hence handle the massless Goldstino mode emerging in the supersymmetry broken phase. In this paper -the third in a series of three -we present the details of the simulation algorithm and demonstrate its efficiency by means of a few examples. arXiv:1505.07397v2 [hep-lat] 3 Sep 2015

Supersymmetric quantum mechanics on the lattice: I. Loop formulation

“…In fact, the explicit separation of the sectors in the fermion loop formulation allows the construction of a simulation algorithm which samples these sectors separately, and more importantly also samples the relative weights between them. We demonstrate in the third paper of this series [45] that in this way, the loop formulation indeed provides a solution to the fermion sign problem. The solution is not restricted to the quantum mechanics case, but it is in fact also applicable in higher dimensions.…”

“…Since this so-called open fermion string algorithm directly samples the Goldstino mode, there is no critical slowing down and the physics of the Goldstino is properly captured. The application of the algorithm to the quantum mechanical system is the topic of our third paper in the series [45].…”

“…In the following subsections, we will derive the representation of these observables in the loop formulation. This will turn out to be very useful also for the exact calculation of two-point functions and other observables using transfer matrices in the second paper of this series [37], where we discuss a plethora of results, and for the discussion of the simulation algorithm in the third paper of this series [45].…”

“…The picture suggests to interpret the fermionic correlation function C f (x − y) as an open fermion string on the background of bosonic bond configurations in sector Z 0 , or as an open antifermion string on the background of bond configurations in sector Z 1 , i.e., as a antifermionic correlation function −C f (y − x). It is this property which forms the basis for an efficient simulation algorithm which will be discussed in detail in the third paper of this series [45]. Finally, the reformulation of the fermionic correlation functions in terms of bond variables can be generalised to include more complicated fermionic source and sink operators such as ψ x φ k x or ψ x φ k x .…”

Canonical simulations of supersymmetric SU(N) Yang-Mills quantum mechanics

“…To define the theory we employ the lattice regularisation in Euclidean time proposed in [3,4]. The bosonisation of the theory on the lattice by means of the fermion loop formulation [5,6,7] decomposes the fermion contributions into fermion sectors with fixed fermion number. Our recent progress in understanding the algebraic structure of the fermion loop formulation allows the explicit construction of transfer matrices [8].…”