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Register a new userQuantum-classical correspondence for gl(1|1) supersymmetric Gaudin magnet with boundary
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We extend duality between the quantum integrable Gaudin models with boundary and the classical Calogero-Moser systems associated with root systems of classical Lie algebras $B_N$, $C_N$, $D_N$ to the case of supersymmetric ${rm gl}(m|n)$ Gaudin models with $m+n=2$. Namely, we show that the spectra of quantum Hamiltonians for all such magnets being identified with the classical particles velocities provide the zero level of the classical action variables.
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In this first paper, we start the analysis of correlation functions of quantum spin chains with general integrable boundary conditions. We initiate these computations for the open XXX spin 1/2 quantum chains with some unparallel magnetic fields allowing for a spectrum characterization in terms of homogeneous Baxter like TQ-equations, in the framework of the quantum separation of variables (SoV). Previous SoV analysis leads to the formula for the scalar products of the so-called separate states. Here, we solve the remaining fundamental steps allowing for the computation of correlation functions. In particular, we rederive the ground state density in the thermodynamic limit thanks to SoV approach, we compute the so-called boundary-bulk decomposition of boundary separate states and the action of local operators on these separate states in the case of unparallel boundary magnetic fields. These findings allow us to derive multiple integral formulae for these correlation functions similar to those previously known for the open XXX quantum spin chain with parallel magnetic fields.
We study the $mathfrak{gl}_{1|1}$ supersymmetric XXX spin chains. We give an explicit description of the algebra of Hamiltonians acting on any cyclic tensor products of polynomial evaluation $mathfrak{gl}_{1|1}$ Yangian modules. It follows that there exists a bijection between common eigenvectors (up to proportionality) of the algebra of Hamiltonians and monic divisors of an explicit polynomial written in terms of the Drinfeld polynomials. In particular our result implies that each common eigenspace of the algebra of Hamiltonians has dimension one. We also give dimensions of the generalized eigenspaces. We show that when the tensor product is irreducible, then all eigenvectors can be constructed using Bethe ansatz. We express the transfer matrices associated to symmetrizers and anti-symmetrizers of vector representations in terms of the first transfer matrix and the center of the Yangian.
We review the definition of hypergroups by Sunder, and we associate a hypergroup to a type III subfactor $Nsubset M$ of finite index, whose canonical endomorphism $gammainmathrm{End}(M)$ is multiplicity-free. It is realized by positive maps of $M$ that have $N$ as fixed points. If the depth is $>2$, this hypergroup is different from the hypergroup associated with the fusion algebra of $M$-$M$ bimodules that was Sunders original motivation to introduce hypergroups. We explain how the present hypergroup, associated with a suitable subfactor, controls the composition of transparent boundary conditions between two isomorphic quantum field theories, and that this generalizes to a hypergroupoid of boundary conditions between different quantum field theories sharing a common subtheory.
We establish the most general class of spin-1/2 integrable Richardson-Gaudin models including an arbitrary magnetic field, returning a fully anisotropic (XYZ) model. The restriction to spin-1/2 relaxes the usual integrability constraints, allowing for a general solution where the couplings between spins lack the usual antisymmetric properties of Richardson-Gaudin models. The full set of conserved charges are constructed explicitly and shown to satisfy a set of quadratic equations, allowing for the numerical treatment of a fully anisotropic central spin in an external magnetic field. While this approach does not provide expressions for the exact eigenstates, it allows their eigenvalues to be obtained, and expectation values of local observables can then be calculated from the Hellmann-Feynman theorem.
The most general Dirac Hamiltonians in $(1+1)$ dimensions are revisited under the requirement to exhibit a supersymmetric structure. It is found that supersymmetry allows either for a scalar or a pseudo-scalar potential. Their spectral properties are shown to be represented by those of the associated non-relativistic Witten model. The general discussion is extended to include the corresponding relativistic and non-relativistic resolvents. As example the well-studied relativistic Dirac oscillator is considered and the associated resolved kernel is found in a closed form expression by utilising the energy-dependent Greens function of the non-relativistic harmonic oscillator. The supersymmetric quasi-classical approximation for the Witten model is extended to the associated relativistic model.
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