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arXiv:1703.10079 · SciPost Physics

M$_k$ models: the field theory connection

Thessa Fokkema, Kareljan Schoutens

Published Jul 13, 2017• 9 citations
arXivHTMLPDF

Abstract

The M _k k models for 1D lattice fermions are characterised by {\cal N}=2 supersymmetry and by an order- k k clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of {\cal N}=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 {\cal N}=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 1 , M _2 2 and M _3 3 .

DOI
10.21468/scipostphys.3.1.004

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Mk_kk​ models: the field theory connection
The M _k k models for 1D lattice fermions are characterised by {\cal N}=2 supersymmetry and by an order- k k clustering property. This paper highlights connections with quantum field theories (QFTs) in various regimes. At criticality the QFTs are minimal models of {\cal N}=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 {\cal N}=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 1 , M _2 2 and M _3 3 .
We investigate a family of lattice models with manifest supersymmetry. The models describe fermions on a 1D lattice, subject to the constraint that no more than k consecutive lattice sites may be occupied. We discuss the special properties arising from the supersymmetry, and present Bethe ansatz solutions of the simplest models. We display the connections of the k = 1 model with the spin- antiferromagnetic XXZ chain at Δ = −1/2, and the k = 2 model with both the su (2|1)-symmetric tJ model in the ferromagnetic regime and the integrable spin-1 XXZ chain at . We argue that these models include critical points described by the superconformal minimal models.
We analyse a class of 1D lattice models, known as models, which are characterised by an order- k clustering of spin-less fermions and by lattice supersymmetry. Our main result is the identification of a class of (bulk or edge) defects, that are in one-to-one correspondence with so-called spin fields in a corresponding parafermion CFT. In the gapped regime, injecting such defects leads to ground-state degeneracies that are protected by the supersymmetry. The defects, which are closely analogous to quasi-holes over the fermonic Read-Rezayi quantum Hall states, display characteristic fusion rules, which are of Ising type for k = 2 and of Fibonacci type for k = 3.
We introduce lattice models with explicit N=2 supersymmetry. In these interacting models, the supersymmetry generators Q+/- yield the Hamiltonian H=(Q(+),Q(-)) on any graph. The degrees of freedom can be described as either fermions with hard cores, or as quantum dimers; the Hamiltonian of our simplest model contains a hopping term and a repulsive potential. We analyze these models using conformal field theory, the Bethe ansatz, and cohomology. The simplest model provides a manifestly supersymmetric lattice regulator for the supersymmetric point of the massless (1+1)-dimensional Thirring (Luttinger) model. Generalizations include a quantum monomer-dimer model on a two-leg ladder.
Abstract We present a novel way to organise the finite size spectra of a class of conformal field theories (CFT) with  = 2 or (nonlinear)  = 4 superconformal symmetry. Generalising the spinon basis of the SU ( n ) 1 WZW theories, we introduce supersymmetric spinons ( ϕ − , ϕ + ) , which form a representation of the supersymmetry algebra. In each case, we show how to construct a multi-spinon basis of the chiral CFT spectra. The multi-spinon states are labelled by a collection { n j } of (discrete) momenta. The state-content for given choice of { n j } is determined through a generalised exclusion principle, similar to Haldane's ‘motif’ rules for the SU ( n ) 1 theories. In the simplest case, which is the  = 2 superconformal theory with central charge c = 1, we develop an algebraic framework similar to the Yangian symmetry of the SU ( 2 ) 1 theory. It includes an operator H 2 , akin to a CFT Haldane–Shastry Hamiltonian, which is diagonalised by multi-spinon states. In all cases studied, we obtain finite partition sums by capping the spinon-momenta to some finite value. For the  = 2 superconformal CFTs, this finitisation precisely leads to the so-called M k supersymmetric lattice models with characteristic order- k exclusion rules on the lattice. Finitising the c = 2 CFT with nonlinear  = 4 superconformal symmetry similarly gives lattice model partition sums for spin-full Fermions with on-site and nearest neighbour exclusion.
The Heisenberg XXZ spin-(1/2) chain is considered in the massive antiferromagnetic regime in the presence of a staggered longitudinal magnetic field. The Hamiltonian of the model is characterised by the anisotropy parameter and by the magnetic-field strength h . At zero magnetic field, the model is exactly solvable. In the thermodynamic limit, it has two degenerate vacua and the kinks (which are also called spinons) interpolating between these vacua, as elementary excitations. Application of the staggered magnetic field breaks integrability of the model and induces the long-range attractive potential between two adjacent kinks leading to their confinement into the bound states. The energy spectra of the resulting two-kink bound states are perturbatively calculated in the extreme anisotropic (Ising) limit to the first order in the inverse anisotropy constant , and also for any to the first order in the weak magnetic field h .
We construct fermionic conformal field theories (CFTs) whose spectra are characterized by quantum stabilizer codes. We exploit our construction to search for fermionic CFTs with supersymmetry by focusing on quantum stabilizer codes of the Calderbank-Shor-Steane type, and derive simple criteria for the theories to be supersymmetric. We provide several examples of fermionic CFTs that meet the criteria, and find quantum codes that realize N=4\mathcal{N}=4N=4 supersymmetry. Our work constitutes a new application of quantum codes and paves the way for the methodical search for supersymmetric CFTs.