M$_k$ models: the field theory connection

Abstract

The M k models for 1D lattice fermions are characterised by = 2 supersymmetry and by an order-k clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of = 2 supersymmetric conformal field theory (CFT) -we analyse finite size spectra on open chains with a variety of supersymmetry preserving boundary conditions. Specific staggering perturbations lead to a gapped regime corresponding to massive = 2 supersymmetric QFT with Chebyshev superpotentials. At 'extreme staggering' we uncover a simple physical picture with degenerate supersymmetric vacua and mobile kinks. We connect this kink-picture to the Chebyshev QFTs and use it to derive novel CFT character formulas. For clarity the focus in this paper is on the simplest models, M 1 , M 2 and M 3 . 5 Off-critical M k models at extreme staggering 15 5.1 M 1 model 15 5.2 M 2 model 15 5.2.1 Kinks and anti-kinks 16 5.2.2 Multiple (anti-)kinks 16 5.3 M 3 model 17 6 Counting of M 2 model kink states and CFT character formulas 17 6.1 Analogy with MR state in thin torus limit 18 6.2 Example 20 6.3 Open/open BC, fusion degeneracies and correspondence to MR quasi-hole state counting 21 6.4 Open/σ BC 22 6.5 σ/σ BC 23 7 M 2 model versus supersymmetric sine-Gordon theory -action of the supercharges 24 7.1 Kinematics 25 7.2 M 2 model supercharges 25 7.3 Supercharges in supersymmetric sine-Gordon theory 26 7.4 M 2 model vs. supersymmetric sine-Gordon theory 26 8 M 2 model versus supersymmetric sine-Gordon theory -finite chains 27 8.1 Mobile M 2 model kinks on open chains 28 8.2 Boundary scattering and kink-spectrum in supersymmetric sine-Gordon theory 29 A Integrable Field Theory 31 A.1 The sine-Gordon theory 31 A.1.1 Sine-Gordon theory with = 2 supersymmetry as a perturbed superconformal field theory 32 A.1.2 Particles in sine-Gordon theory with = 2 supersymmetry 32 A.1.3 Commutation of supercharges with the scattering of solitons 33 A.2 Supersymmetric sine-Gordon theory 33 A.2.1 Particles in supersymmetric sine-Gordon theory 34 A.2.2 = 1 supersymmetry 35 A.2.3 = 3 supersymmetric sine-Gordon as a perturbed conformal field theory 35 A.2.4 Particles in = 3 supersymmetric sine-Gordon theory 36 A.3 Superfields and superpotentials 37 A.3.1 Integrable massive field theories with a Chebyshev superpotential 37 A.3.2 Comparison of k = 1 Chebyshev QFT with sine-Gordon theory 38 References 38The action of the SU(2) currents on the kinks is J + K ±,0 =K ±,0 , J + K 0,± = ±K 0,± , J − K ±,0 = 0, J − K 0,± = 0, J 0 K ±,0 = − 1 2 K ±,0 , J 0 K 0,± = − 1 2 K 0,± , J +K ±,0 = 0, J +K 0,± = 0, J −K ±,0 = K ±,0 , J −K 0,± = ±K 0,± , J 0K ±,0 =

Supersymmetry and Quantum Computation

“…Generalizations of this model, in which up to k consecutive vertices may be occupied, denoted M k , were introduced in [26]. Interestingly, these describe a discretization of N = 2 superconformal minimal models at level k [27]. See [28] for a review of these models.…”

Massless AdS 2 scattering and Bethe ansatz

“…where P, K and C are central bosonic generators, Q and S are fermionic generators. We begin by remarking that, from a purely algebraic viewpoint, these commutation relations already identify N = 1 supersymmetry in 1+1 dimensions [64,67,68] (see also [88] and references therein), which we will take in various massless and relativistic representations in what follows. The (boson,fermion) |φ , |ψ representation we consider is the following:…”

A Curious Mapping Between Supersymmetric Quantum Chains

“…These features were generalized to a class of models called M k , where up to k fermions are allowed on consecutive lattice sites [8]. At critical behaviour of these models is captured by the k-th minimal model of N = 2 superconformal field theory, while massive deformations give rise to integrable massive N = 2 QFT's with superpotentials taking the form of Chebyshev polynomials [9]. This paper is concerned with two other, and seemingly different, incarnations of supersymmetry in one spatial dimension.…”