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arXiv:1510.02520 · Journal of Statistical Mechanics Theory and Experiment

Integrable supersymmetric chain without particle conservation

Jan de Gier, Gyorgy Z. Feher, Bernard Nienhuis, Magdalena Rusaczonek

Published Feb 1, 2016• 14 citations
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Abstract

Abstract We introduce a new integrable supersymmetric lattice chain which violates fermion conservation and exhibits fermion–hole symmetry. The model displays exponential degeneracy in every eigenstate including the ground state. This degeneracy is expressed in the possibility to create any number of zero modes reminiscent of Cooper pairs.

DOI
10.1088/1742-5468/2016/02/023104

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Abstract We introduce a new integrable supersymmetric lattice chain which violates fermion conservation and exhibits fermion–hole symmetry. The model displays exponential degeneracy in every eigenstate including the ground state. This degeneracy is expressed in the possibility to create any number of zero modes reminiscent of Cooper pairs.
We study a 111D supersymmetric (SUSY) hard-core fermion model first proposed by Fendley, Schoutens, and de Boer [Phys. Rev. Lett. 90, 120402 (2003)]. We focus on the full Hilbert space instead of a restricted subspace. Exact diagonalization shows the degeneracy of zero-energy states scales exponentially with size of the system, with a recurrence relation between different system sizes. We solve the degeneracy problem by showing the ground states can be systematically constructed by inserting immobile walls of fermions into the chain. Mapping the counting problem to a combinatorial one and obtaining the exact generating function, we prove the recurrence relation on both open and periodic chains. We also provide an explicit mapping between ground states, giving a combinatorial explanation of the recurrence relation.
A novel Bethe ansatz scheme is proposed to calculate physical properties of quantum integrable systems without U(1)U(1)U(1) symmetry. As an example, the antiperiodic XXZ spin chain, a typical correlated many-body system embedded in a topological manifold, is examined. Conserved ``momentum'' and ``charge'' operators are constructed despite the absence of translational invariance and U(1)U(1)U(1) symmetry. The ground state energy and elementary excitations are derived exactly. It is found that two intrinsic fractional (one half) zero modes accounting for the double degeneracy exist in the eigenstates. The elementary excitations show quite a different picture from that of a periodic chain. This method can be applied to other quantum integrable models either with or without U(1)U(1)U(1) symmetry.
We introduce a supersymmetric lattice fermion model that contains both fermion pairing and the interacting Nicolai model. This model possesses a single control parameter, ggg, introduced through the anticommutator of the supersymmetry generators (supercharge), and it is shown that supersymmetry is broken in both finite and infinite systems as long as ggg is finite. Additionally, the single-mode approximation is employed to establish an upper bound on the dispersion relation of Nambu-Goldstone (NG) fermions, demonstrating their gaplessness. Finally, it is shown, through numerical and analytic calculations, that the anticipated extensive ground-state degeneracy and a zero-energy flat band does not occur for generic values of the parameter ggg.
We study a model for itinerant, strongly interacting fermions where \na judicious tuning of the interactions leads to a supersymmetric Hamiltonian. \nOn the triangular lattice this model is known to exhibit a property called \nsuperfrustration, which is characterized by an extensive ground state entropy. \nUsing a combination of numerical and analytical methods we study various \nladder geometries obtained by imposing doubly periodic boundary conditions \non the triangular lattice. We compare our results to various bounds on the \nground state degeneracy obtained in the literature. For all systems we find that \nthe number of ground states grows exponentially with system size. For two \nof the models that we study we obtain the exact number of ground states by \nsolving the cohomology problem. For one of these, we find that via a sequence \nof mappings the entire spectrum can be understood. It exhibits a gapped phase at \n1/4 filling and a gapless phase at 1/6 filling and phase separation at intermediate \nfillings. The gapless phase separates into an exponential number of sectors, \nwhere the continuum limit of each sector is described by a superconformal field \ntheory.
Using supersymmetry quantum mechanics, we explain the covariance of the eigenvalue equation for the Jaynes-Cummings (JC) Hamiltonian under the particle-hole transformation. The JC Hamiltonians for the original spin-1/2 ``particle'' and the transformed partner (the ``hole'') are shown to be the fermionic and the bosonic parts of a supersymmetric Hamiltonian. The Hamiltonian, unlike ordinary one-dimensional supersymmetric quantum-mechanical models, can have a degenerate zero-energy ground state despite exact supersymmetry. The degeneracy is shown to arise from the hidden particle-hole exchange symmetry as well as supersymmetry in the Hamiltonian.