Supersymmetric quantum mechanics on the lattice: I. Loop formulation

Abstract

Simulations of supersymmetric field theories on the lattice with (spontaneously) broken supersymmetry suffer from a fermion sign problem related to the vanishing of the Witten index. We propose a novel approach which solves this problem in low dimensions by formulating the path integral on the lattice in terms of fermion loops. For N = 2 supersymmetric quantum mechanics the loop formulation becomes particularly simple and in this paper - the first in a series of three - we discuss in detail the reformulation of this model in terms of fermionic and bosonic bonds for various lattice discretisations including one which is Q-exact.Comment: 33 pages, 8 figures, 3 tables; typos in text, eq.(107) and table 3 correcte

Supersymmetric quantum mechanics on the lattice: II. Exact results

“…After choosing a suitable discretisation of the derivatives, supersymmetric quantum mechanics can be formulated on the lattice in terms of bosonic and fermionic bond occupation numbers n b i (x) ∈ N 0 and n f (x) = 0, 1, respectively, connecting sites x and x + 1. We refer to our first paper [1] for further details and explanations. In particular, the partition function can be written as a sum over all allowed, possibly constrained, bond configurations…”

“…The occupation of the fermionic bond is represented by the directed full line, cf. the first paper of our series [1] for further explanations on the graphical notation. In fact, since the characterisation of the set of states is independent of the coordinate x, it is sufficient to characterise the states just by |n f , {n b i } and hence the transfer matrix does not depend on x.…”

“…In this paper -the second in a series of three -we present exact results for N = 2 supersymmetric quantum mechanics discretised on the lattice using the bond formulation. This formulation is based on the hopping expansion of the original bosonic and fermionic degrees of freedom and is described in detail in the first paper of our series [1]. For the fermions the bond formulation is more appropriately termed fermion loop formulation since the fermionic bond configurations turn out to be closed fermionic loops.…”

“…This separation into the canonical sectors with fixed fermion number forms the basis for the solution of the fermion sign problem emerging in numerical Monte Carlo simulations of the quantum mechanical system with broken supersymmetry. For a detailed discussion of this issue we refer to the first paper in our series [1].…”

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].…”