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arXiv:nucl-th/0306058 · Physical Review C

Dynamical and invariant supersymmetry in the fermion pairing problem

Pavel Cejnar, H. B. Geyer

Published Nov 25, 2003• 3 citations
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Abstract

We argue that fermion-boson mapping techniques represent a natural tool for studying many-body supersymmetry in fermionic systems with pairing. In particular, using the generalized Dyson mapping of a many-level fermion superalgebra with the SU(2) type of pairing we investigate two kinds of supersymmetry connecting excitations in the systems with even and odd particle numbers: dynamical supersymmetry, which ensures a unified classification of states for both even and odd populations, and invariant supersymmetry with actual degeneracies of states within the same supermultiplet. Dynamical supersymmetries based on the dynamical algebra of the type $\mathrm{U}(K∕2\ensuremath{\Omega})$ (where $K$ and $2\ensuremath{\Omega}$ denote the number of fermion-pair and single-fermion states, respectively) naturally arise in the bosonized description of the system. Conditions for invariant supersymmetry are determined in a restricted case of bilinear supercharge operators.

DOI
10.1103/physrevc.68.054324

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We argue that fermion-boson mapping techniques represent a natural tool for studying many-body supersymmetry in fermionic systems with pairing. In particular, using the generalized Dyson mapping of a many-level fermion superalgebra with the SU(2) type of pairing we investigate two kinds of supersymmetry connecting excitations in the systems with even and odd particle numbers: dynamical supersymmetry, which ensures a unified classification of states for both even and odd populations, and invariant supersymmetry with actual degeneracies of states within the same supermultiplet. Dynamical supersymmetries based on the dynamical algebra of the type U(K∕2\ensuremathΩ)\mathrm{U}(K∕2\ensuremath{\Omega})U(K∕2\ensuremathΩ) (where KKK and 2\ensuremathΩ2\ensuremath{\Omega}2\ensuremathΩ denote the number of fermion-pair and single-fermion states, respectively) naturally arise in the bosonized description of the system. Conditions for invariant supersymmetry are determined in a restricted case of bilinear supercharge operators.
Recent experimental results by Metz, Jolie and collaborators have led to renewed interest in the notion of dynamical supersymmetry in fermion many-body systems. By considering the general problem of constructing a boson-fermion realization for Lie algebras with generators expressed in terms of fermion and bifermion operators, also when favored pairs dictate the structure of collective bosons, we demonstrate that dynamical “supersymmetry without bosons” can indeed emerge in a fermion system without any Pauli Violations. Our construction of the general realization utilizes the identification of a physical subspace and allows one to identify the images of single particle transfer operators uniquely, unlike the situation in phenomenological applications of dynamical supersymmetry.
Generalised Dyson boson-fermion mappings are considered. These are techniques used in the analysis of the quantum many-body problem, and are instances of so-called boson expansion methods. A generalised Dyson boson-fermion mapping is a one-to-one linear but non-unitary operator that can be applied to vectors representing the states of a many-fermion system. A vector representing a fermion system maps onto a vector that represents a state of a many-body system that contains both bosons and fermions. The motivation for doing such a mapping is the hope that it will reveal some property of the system that simplifies its analysis and that was hidden in the original form. The aims of this text are to review the theory of generalized Dyson boson-fermion mappings and to find a useful application for a generalized Dyson boson-fermion mapping, by considering a non-trivial model, namely the Richardson model for superconductivity. It is the first time that a boson expansion technique is implemented for a system where the roles of both collective and non-collective fermion pairs are important. The Dyson mapping uncovers non-trivial properties of the system that aid the construction of time-independent as well as time-dependent perturbation expansions. The time-independent expansions agree with results that other authors have obtained through methods other than boson expansions. The time-dependent expansions might in future prove useful in understanding aspects of the dynamics of ultra-cold fermi gases, when time-dependent magnetic fields are used to vary the atom-atom interaction strenght.
Supersymmetry is a powerful concept in quantum many-body physics. It helps to illuminate ground-state properties of complex quantum systems and gives relations between correlation functions. In this Letter, we show that the Sachdev-Ye-Kitaev model, in its simplest form of Majorana fermions with random four-body interactions, is supersymmetric. In contrast to existing explicitly supersymmetric extensions of the model, the supersymmetry we find requires no relations between couplings. The type of supersymmetry and the structure of the supercharges are entirely set by the number of interacting Majorana modes and are thus fundamentally linked to the model's Altland-Zirnbauer classification. The supersymmetry we uncover has a natural interpretation in terms of a one-dimensional topological phase supporting Sachdev-Ye-Kitaev boundary physics and has consequences away from the ground state, including in q-body dynamical correlation functions.
Recently, we have observed that the bosonized supersymmetric quantum mechanics leads naturally to the R-deformed Heisenberg algebra (RDHA) related to parabosons and parafermions. Here we show that RDHA, in turn, gives rise not only to the bosonized supersymmetry associated with Witten's supersymmetric quantum mechanics but also supplies us with a supersymmetry characterized by a nonlinear superalgebra. Such nonlinear supersymmetry may be specified alternatively by a central extension of the usual N=1 superalgebra with grading operator included nontrivially as an even generator. The systems under consideration are described by the Hamiltonians of the simplest form Hn=a+a−H_n=a^+a^-Hn​=a+a− and Ha=a−a+H_a=a^-a^+Ha​=a−a+ and reveal supersymmetries at the special values of the deformation parameter corresponding to parabosons of odd order. The peculiar nature of parabosonic supersymmetries is encoded in essentially nonlinear structure of their supercharges realized in terms of parabosonic creation-annihilation operators $a^\\pm$. At the values of deformation parameter corresponding to the even order parabosons the spectra of the Hamiltonians HnH_nHn​ and HaH_aHa​ have no supersymmetry but reveal `holes' in comparison with bosonic spectra, whose number is correlated with the parastatistics' order. It is shown that RDHA and the associated linear and nonlinear supersymmetries are realizable in the system of two identical fermions. The nonlinear supersymmetry of pure parabosonic systems possesses the structure of a reduced parasupersymmetry and may also be realized by appropriate modification of the classical analog of Witten's supersymmetric quantum mechanics.
Duality relations are explicitly established relating the Hamiltonians and basis classification schemes associated with the number-conserving unitary and number-nonconserving quasispin algebras for the two-level system with pairing interactions. These relations are obtained in a unified formulation for both bosonic and fermionic systems, with arbitrary and, in general, unequal degeneracies for the two levels. Illustrative calculations are carried out comparing the bosonic and fermionic quantum phase transitions.