[Paper Review] Classical and Quantum sl(1|2) Superalgebras, Casimir Operators and Quantum Chain Hamiltonians

D. Arnaudon, Chryssomalis Chryssomalakos+1 more|arXiv (Cornell University)|Mar 31, 1995

Algebraic structures and combinatorial models12 references15 citations

TL;DR

This paper constructs quantum chain Hamiltonians based on Casimir operators of the quantum superalgebra Uqs(sl(1|2)) using both the Serre presentation and FRT R-matrix formalism. It demonstrates that multiple Casimir operators yield equivalent Hamiltonians up to similarity transformation, leading to two distinct deformations of the supersymmetric t−J model, one corresponding to the fermionic basis and the other to the distinguished basis, with the latter reducing to the Perk–Schultz (1,2) model at s=1.

ABSTRACT

We examine the two parameter deformed superalgebra $U_{qs}(sl(1|2))$ and use the results in the construction of quantum chain Hamiltonians. This study is done both in the framework of the Serre presentation and in the $R$-matrix scheme of Faddeev, Reshetikhin and Takhtajan (FRT). We show that there exists an infinite number of Casimir operators, indexed by integers $p > 1$ in the undeformed case and by $p \in Z$ in the deformed case, which obey quadratic relations. The construction of the dual superalgebra of functions on $SL_{qs}(1|2)$ is also given and higher tensor product representations are discussed. Finally, we construct quantum chain Hamiltonians based on the Casimir operators. In the deformed case we find two Hamiltonians which describe deformed $t-J$ models.

Motivation & Objective

• To explore the existence and structure of Casimir operators in the quantum superalgebra Uqs(sl(1|2)) with two deformation parameters.
• To construct quantum chain Hamiltonians using these Casimir operators in both the Serre presentation and FRT R-matrix framework.
• To investigate whether different Casimir operators or Hopf structures lead to inequivalent Hamiltonians in the quantum deformed case.
• To determine the physical equivalence of the resulting Hamiltonians under similarity transformations, particularly for open boundary conditions.

Proposed method

• Derives an infinite set of Casimir operators Ccl_p for the classical U(sl(1|2)) algebra, indexed by integers p ≥ 2, using the Serre presentation in the fermionic basis.
• Constructs the quantum deformation Uqs(sl(1|2)) with two parameters and derives deformed Casimir operators C_p indexed by p ∈ ℤ, showing they satisfy the same quadratic relations as in the classical case.
• Reformulates Uqs(sl(1|2)) in FRT form using an R-matrix and matrices of generators L±, enabling a dual Hopf algebra construction of functions on SLqs(1|2).
• Introduces a bosonised basis for the algebra and computes the coproduct of the infinite set of Casimirs to facilitate Hamiltonian construction.
• Constructs three-state quantum chain Hamiltonians from Casimir operators, analyzing their invariance under Uqs(sl(1|2)) and their equivalence under similarity transformations.
• Applies a similarity transformation O to eliminate extra deformation parameters (qij), showing that for open chains, only the parameter q remains physically relevant.

Experimental results

Research questions

• RQ1 Are there multiple inequivalent Uqs(sl(1|2))-invariant Hamiltonians constructed from different Casimir operators in the quantum deformed case?
• RQ2 Do different Hopf algebra structures (fermionic vs. distinguished basis) in Uqs(sl(1|2)) lead to physically distinct quantum chain Hamiltonians?
• RQ3 Can the extra deformation parameters (s, qij) in the Hamiltonian be removed via similarity transformation for open boundary conditions?
• RQ4 Is the Hamiltonian constructed from Casimir operators in the fermionic basis equivalent to the one from the distinguished basis?
• RQ5 Does the Perk–Schultz (1,2) Hamiltonian emerge naturally from the Casimir-based construction in the distinguished basis?

Key findings

• The classical universal enveloping superalgebra U(sl(1|2)) contains an infinite set of Casimir operators Ccl_p (p ≥ 2) that satisfy quadratic relations Ccl_p1 Ccl_p2 = Ccl_p3 Ccl_p4 when p1 + p2 = p3 + p4.
• In the quantum case, Uqs(sl(1|2)) has an infinite set of Casimir operators C_p indexed by p ∈ ℤ, which obey the same quadratic relations as in the classical case.
• All Casimir operators in the fermionic basis lead to Hamiltonians proportional to the same two-site Hamiltonian, implying a unique physical model up to normalization.
• For open boundary conditions, a similarity transformation removes all but the q parameter, reducing the two-parameter Hamiltonian to a one-parameter form.
• The Hamiltonian constructed in the distinguished basis is equivalent to the Perk–Schultz (1,2) Hamiltonian at s = 1, and numerical evidence confirms spectral equivalence between the fermionic and distinguished basis Hamiltonians up to seven sites.
• The commutant of the coproduct of Uqs(sl(1|2)) in the tensor square of the fundamental representation is generated by the identity and the R-matrix, confirming the uniqueness of the invariant Hamiltonian up to normalization.