Abstract
Supersymmetric models with spontaneous supersymmetry breaking suffer from the notorious sign problem in stochastic approaches. By contrast, the tensor network approaches do not have such a problem since they are based on deterministic procedures. In this work, we present a tensor network formulation of the two-dimensional lattice mathcal{N} = 1 Wess-Zumino model while showing that numerical results agree with the exact solutions for the free case.
Highlights
Introduction of auxiliary fieldsThe boson action SB in eq (2.12) is transformed into a nearest-neighbor form using two real auxiliary fields G and H: ZB = DφDGDHe−SB, (3.16) SB SB,naiveG2n + Hn2 − rW ′ + αGn + βHn φn+ˆ1 + φn−ˆ1 − 2φn n∈Γ− rW ′ + αGn − βHn φn+ˆ2 + φn−ˆ2 − 2φn (3.17)√ with SB,√naive given in eq (2.13), α = (1 − 2r2)/2, and√β = 1/ 2
We present a tensor network formulation of the two-dimensional lattice N = 1 Wess-Zumino model while showing that numerical results agree with the exact solutions for the free case
In general, two auxiliary fields are necessary for the next-nearest-neighbor interactions in two directions, it is somewhat √surprising to find that G is decoupled from the other fields for particular values r = ±1/ 2, and the required auxiliary field turns out to be only H
Summary
Two-dimensional N = 1 Wess-Zumino model is a supersymmetric theory that consists of a real scalar field φ (x) and a Majorana fermion field ψ (x). In the Euclidean space-time, the corresponding action is given by. Showing the indices in the spinor space explicitly, γμ and ψ (x) are written as (γμ)αβ and ψα (x) for α, β = 1, 2. The spinor index α and the space-time coordinate x are often suppressed without notice. W (φ) is an arbitrary real function of φ, which is referred to as the superpotential in the superfield formalism, and gives the Yukawa- and φn-type interactions with common coupling constants. For any W (φ), the action in eq (2.1) is invariant under the supersymmetry transformation δφ (x) = ǫψ (x) , δψ (x) = γμ∂μφ (x) − W ′ (φ (x)) ǫ,. Where ǫ is a global Grassmann parameter with two components and ǫsatisfies eq (2.3)
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