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arXiv:hep-th/0210161 · Physical Review Letters

Lattice Models with N=2 Supersymmetry

Paul Fendley, Kareljan Schoutens, Jan De Boer A

Published Mar 25, 2003• 143 citations
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Abstract

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.

DOI
10.1103/physrevlett.90.120402

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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.
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.
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 analyze the quantum ground state structure of a specific model of itinerant, strongly interacting lattice fermions.The interactions are tuned to make the model supersymmetric.Due to this, quantum ground states are in one-to-one correspondence with cohomology classes of the so-called independence complex of the lattice.Our main result is a complete description of the cohomology, and thereby of the quantum ground states, for a two-dimensional square lattice with periodic boundary conditions.Our work builds on results by Jonsson, who determined the Euler characteristic (Witten index) via a correspondence with rhombus tilings of the plane.We prove a theorem, first conjectured by Fendley, which relates dimensions of the cohomology at grade n to the number of rhombus tilings with n rhombi.
A new lattice model is presented for correlated electrons on the unrestricted 4L{4}^{L}4L-dimensional electronic Hilbert space ${\ensuremath{\bigotimes}}_{n\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1}^{L}{\mathbf{C}}^{4}$ (where LLL is the lattice length). It is a supersymmetric generalization of the Hubbard model, but differs from the extended Hubbard model proposed by Essler, Korepin, and Schoutens. The supersymmetry algebra of the new model is superalgebra gl(2 $|$1). The model contains one symmetry-preserving free real parameter which is the Hubbard interaction parameter UUU, and has its origin here in the one-parameter family of inequivalent typical 4-dimensional irreducible representations of gl(2 $|$1). On a one-dimensional lattice, the model is exactly solvable by the Bethe ansatz.
A detailed study of a model for strongly-interacting fermions with exclusion rules and lattice N=2\mathcal N=2N=2 supersymmetry is presented. A submanifold in the space of parameters of the model where it is Bethe-ansatz solvable is identified. The relation between this manifold and the existence of additional, so-called dynamic, supersymmetries is discussed. The ground states are analysed with the help of cohomology techniques, and their exact finite-size Bethe roots are found. Moreover, through analytical and numerical studies it is argued that the model provides a lattice version of the N=1\mathcal N=1N=1 super-sine-Gordon model at a particular coupling where an additional N=(2,2)\mathcal N=(2,2)N=(2,2) supersymmetry is present. The dynamic supersymmetry is shown to allow an exact determination of the gap scaling function of the model.
A one-dimensional electron has both spin and charge degrees of freedom. In the limit of strong on-site repulsion, double occupation of a site can be neglected since it has a large energy cost. If the number N e of electrons is less than the number N of lattice sites, vacant sites appear and the system acquires both spin and charge degrees of freedom. The simplest model to incorporate these degrees of freedom is called the t – J model, which is discussed in this chapter. We begin by reviewing the SU(2,1) supersymmetry (SUSY) as an extension of the SU(2) symmetry in the spin chain. With the supersymmetry, the hopping and exchange in the t – J model can be treated in a unified way as the graded permutation. Then the ground state of the SUSY t – J model is derived in Section 6.2 by the coordinate representation of wave functions by using the fully polarized state as the reference. The static structure factor at zero temperature is completely determined by the wave function at the ground state. The spin and charge components are derived using the determinant and its generalizations in Section 6.3. In Section 6.4, we proceed to derive the spectrum of the SUSY t – J model. The elementary excitations consist of spinons, holons, and their antiparticles. It is proved in Section 6.5 that these excitations span the complete set. The argument relies on a generalization of Young diagrams, which are called ribbon diagrams. At the same time, the degeneracy of energy levels beyond the global SU(2) symmetry is ascribed to the presence of a Yangian supersymmetry Y ( sl 2|1 ).