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Supersymmetric Models for Fermions on a Lattice

Nevena Ilieva, Heide Narnhofer, Walter E Thirring

Published Jan 1, 2005• 4 citations
OpenAlex

Abstract

We dedicate this article to Julius Wess on the occasion of his 70 th birthday. We investigate the large-N behaviour of simple examples of supersymmetric interactions for fermions on a lattice. Witten’s supersymmetric quantum mechanics and the BCS model appear just as two different aspects of one and the same model. For the BCS model, supersymmetry is only respected in a coherent superposition of Bogoliubov states. In this coherent superposition mesoscopic observables show better stability properties than in a Bogoliubov state.

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arXiv:cond-mat/9403088•Mar 29, 1994•0 citations
We dedicate this article to Julius Wess on the occasion of his 70 th birthday. We investigate the large-N behaviour of simple examples of supersymmetric interactions for fermions on a lattice. Witten’s supersymmetric quantum mechanics and the BCS model appear just as two different aspects of one and the same model. For the BCS model, supersymmetry is only respected in a coherent superposition of Bogoliubov states. In this coherent superposition mesoscopic observables show better stability properties than in a Bogoliubov state.
Supersymmetry is a symmetry between a boson and a fermion. Although there is no apparent supersymmetry in nature, its mathematical consistency and appealing properties have led many people to believe that supersymmetry may exist in nature in the form of a spontaneously broken symmetry. In this paper, we explore an alternative possibility by which supersymmetry is realized in nature, that is, supersymmetry dynamically emerges in the low-energy limit of a nonsupersymmetric condensed matter system. We propose a (2+1)(2+1)(2+1)-dimensional lattice model which exhibits an emergent space-time supersymmetry at a quantum critical point. It is shown that there is only one relevant perturbation at the supersymmetric critical point in the ϵϵϵ expansion and the critical theory is the two copies of the Wess-Zumino theory with four supercharges. Exact critical exponents are predicted.
The interplay between supersymmetry and classical and quantum computation is discussed. First, it is shown that the problem of computing the Witten index of N≤2\mathcal N \leq 2N≤2 quantum mechanical systems is #P\#P#P-complete and therefore intractable. Then, the notions of supersymmetry in the space of qubits and supersymmetric quantum circuits are introduced and some of their properties discussed. In particular, it is shown that these define a nontrivial subclass of quantum algorithms with robustness properties typical of supersymmetric systems. Concrete examples, including the supersymmetric SYK model and fermion hard-core models are discussed. Some applications and open questions are suggested.
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.
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.
Some recent results in supersymmetric quantum mechanics are presented. New semi-classical approximation formulas for Witten's realization of supersymmetric quantum mechanics are discussed. Implications of the supersymmetric structure of Pauli's Hamiltonian are also considered. In particular, the paramagnetisation of a non-interacting electron gas is related to a modified version of Witten's index. We also show that the supersymmetry in this system provides a new counter example for the paramagnetic conjecture.