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Journal of Physics Conference Series

Algebraic models of many-body systems and their dynamic symmetries and supersymmetries

FRANCESCO IACHELLO

Published Apr 1, 2019• 6 citations
Publisher

Abstract

An overview of the method of spectrum generating algebras (SGA) and dynamical symmetries (DS) is given and applied to a class of models (the so-called s-b boson models) with SGA≡u(n). Quantum phase transitions (QPT) in these systems are discussed by introducing the coset spaces U(n)/U(n-1)⊗U(1). Finally, spectrum generating superalgebras (SGSA) and dynamical supersymmetries (SUSY) of a class of Bose-Fermi systems with SGA≡u(n/m) are introduced and applied to s-b Bose-Fermi models.

DOI
10.1088/1742-6596/1194/1/012048

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An overview of the method of spectrum generating algebras (SGA) and dynamical symmetries (DS) is given and applied to a class of models (the so-called s-b boson models) with SGA≡u(n). Quantum phase transitions (QPT) in these systems are discussed by introducing the coset spaces U(n)/U(n-1)⊗U(1). Finally, spectrum generating superalgebras (SGSA) and dynamical supersymmetries (SUSY) of a class of Bose-Fermi systems with SGA≡u(n/m) are introduced and applied to s-b Bose-Fermi models.
Supersymmetry (SUSY) postulates that for every fermion there is boson of equal mass (i.e. energy). In quantum mechanics this comes about because for every quantum Hamiltonian there is a partner Hamiltonian that has the same energy spectrum above the ground state of the original system. In particle physics, one “sector” is populated by bosons and the other sector by fermions and SUSY predicts that the lowest lying fermion state is energetically degenerate with the first excited boson state. Evidence for SUSY has proven to be elusive and it is now believed that SUSY is a broken symmetry. In January 2009 at a conference dedicated to Bob Wyatt, we began to look at SUSY as a way to develop new computational methods and approaches. Up until now, SUSY has been more of a mathematical technique that has been used more or less as a way to obtain stationary solutions to the Schrödinger equation for a variety of T&F Cat # K11224, Chapter 4, Page 54, 15-9-2010 one-dimensional potential systems. In this chapter, we will discuss some of the work we have been doing in developing “SUSY” inspired methods for performing quantum many-body calculations and quantum scattering calculations. We begin with a brief overview of the SUSY theory and some of its elementary results. We then discuss how we have used the approach to develop both analytical and numerical solutions of the stationary Schrödinger equation. Finally, we conclude by discussing our recent extension of SUSY to higher dimensions and for scattering theory.
We analyze in detail the quantum phase transitions that arise in models based on the u (2) algebraic description for bosonic systems with two types of scalar bosons. First we discuss the quantum phase transition that occurs in hamiltonians that admix the two dynamical symmetry chains u (2)⊃ u (1) and u (2) ⊃so(2) by diagonalizing the problem exactly in the u(1) basis. Then we apply the coherent state formalism to determine the energy functional. Finally we show that a quantum phase transition of a different nature, but displaying similar characteristics, may arise also within a single chain just by including higher order terms in the hamiltonian.
About 20 years ago almost simultaneously a number of authors had introduced the notion of a dynamic group or dynamic symmetry for quantum systems [ 1 ] (also known as spectrum generating algebra [ 2 ] and non-invariance group [ 3 ]. Introducing this notion required a clearer understanding what is to be called the symmetry of an equation describing a physical system [ 4 ] and to investigate concrete examples of quantum systems. In [ 4–7 ] it had been shown that the dynamic symmetry of hydrogen atom is 0(4,2), as a dynamic group describing Landau levels of an electron in magnetic field can serve either non-homogeneous symplectic group ISp(4,R) or U(2,1) [ 8 , 9 ].
An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.